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rl/Lib/site-packages/networkx/algorithms/community/__init__.py

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+"""Functions for computing and measuring community structure.
+
+The ``community`` subpackage can be accessed by using :mod:`networkx.community`, then accessing the
+functions as attributes of ``community``. For example::
+
+    >>> import networkx as nx
+    >>> G = nx.barbell_graph(5, 1)
+    >>> communities_generator = nx.community.girvan_newman(G)
+    >>> top_level_communities = next(communities_generator)
+    >>> next_level_communities = next(communities_generator)
+    >>> sorted(map(sorted, next_level_communities))
+    [[0, 1, 2, 3, 4], [5], [6, 7, 8, 9, 10]]
+
+"""
+
+from networkx.algorithms.community.asyn_fluid import *
+from networkx.algorithms.community.centrality import *
+from networkx.algorithms.community.divisive import *
+from networkx.algorithms.community.kclique import *
+from networkx.algorithms.community.kernighan_lin import *
+from networkx.algorithms.community.label_propagation import *
+from networkx.algorithms.community.lukes import *
+from networkx.algorithms.community.modularity_max import *
+from networkx.algorithms.community.quality import *
+from networkx.algorithms.community.community_utils import *
+from networkx.algorithms.community.louvain import *

rl/Lib/site-packages/networkx/algorithms/community/asyn_fluid.py

@@ -0,0 +1,151 @@
+"""Asynchronous Fluid Communities algorithm for community detection."""
+
+from collections import Counter
+
+import networkx as nx
+from networkx.algorithms.components import is_connected
+from networkx.exception import NetworkXError
+from networkx.utils import groups, not_implemented_for, py_random_state
+
+__all__ = ["asyn_fluidc"]
+
+
+@not_implemented_for("directed")
+@not_implemented_for("multigraph")
+@py_random_state(3)
+@nx._dispatchable
+def asyn_fluidc(G, k, max_iter=100, seed=None):
+    """Returns communities in `G` as detected by Fluid Communities algorithm.
+
+    The asynchronous fluid communities algorithm is described in
+    [1]_. The algorithm is based on the simple idea of fluids interacting
+    in an environment, expanding and pushing each other. Its initialization is
+    random, so found communities may vary on different executions.
+
+    The algorithm proceeds as follows. First each of the initial k communities
+    is initialized in a random vertex in the graph. Then the algorithm iterates
+    over all vertices in a random order, updating the community of each vertex
+    based on its own community and the communities of its neighbors. This
+    process is performed several times until convergence.
+    At all times, each community has a total density of 1, which is equally
+    distributed among the vertices it contains. If a vertex changes of
+    community, vertex densities of affected communities are adjusted
+    immediately. When a complete iteration over all vertices is done, such that
+    no vertex changes the community it belongs to, the algorithm has converged
+    and returns.
+
+    This is the original version of the algorithm described in [1]_.
+    Unfortunately, it does not support weighted graphs yet.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        Graph must be simple and undirected.
+
+    k : integer
+        The number of communities to be found.
+
+    max_iter : integer
+        The number of maximum iterations allowed. By default 100.
+
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    communities : iterable
+        Iterable of communities given as sets of nodes.
+
+    Notes
+    -----
+    k variable is not an optional argument.
+
+    References
+    ----------
+    .. [1] Parés F., Garcia-Gasulla D. et al. "Fluid Communities: A
+       Competitive and Highly Scalable Community Detection Algorithm".
+       [https://arxiv.org/pdf/1703.09307.pdf].
+    """
+    # Initial checks
+    if not isinstance(k, int):
+        raise NetworkXError("k must be an integer.")
+    if not k > 0:
+        raise NetworkXError("k must be greater than 0.")
+    if not is_connected(G):
+        raise NetworkXError("Fluid Communities require connected Graphs.")
+    if len(G) < k:
+        raise NetworkXError("k cannot be bigger than the number of nodes.")
+    # Initialization
+    max_density = 1.0
+    vertices = list(G)
+    seed.shuffle(vertices)
+    communities = {n: i for i, n in enumerate(vertices[:k])}
+    density = {}
+    com_to_numvertices = {}
+    for vertex in communities:
+        com_to_numvertices[communities[vertex]] = 1
+        density[communities[vertex]] = max_density
+    # Set up control variables and start iterating
+    iter_count = 0
+    cont = True
+    while cont:
+        cont = False
+        iter_count += 1
+        # Loop over all vertices in graph in a random order
+        vertices = list(G)
+        seed.shuffle(vertices)
+        for vertex in vertices:
+            # Updating rule
+            com_counter = Counter()
+            # Take into account self vertex community
+            try:
+                com_counter.update({communities[vertex]: density[communities[vertex]]})
+            except KeyError:
+                pass
+            # Gather neighbor vertex communities
+            for v in G[vertex]:
+                try:
+                    com_counter.update({communities[v]: density[communities[v]]})
+                except KeyError:
+                    continue
+            # Check which is the community with highest density
+            new_com = -1
+            if len(com_counter.keys()) > 0:
+                max_freq = max(com_counter.values())
+                best_communities = [
+                    com
+                    for com, freq in com_counter.items()
+                    if (max_freq - freq) < 0.0001
+                ]
+                # If actual vertex com in best communities, it is preserved
+                try:
+                    if communities[vertex] in best_communities:
+                        new_com = communities[vertex]
+                except KeyError:
+                    pass
+                # If vertex community changes...
+                if new_com == -1:
+                    # Set flag of non-convergence
+                    cont = True
+                    # Randomly chose a new community from candidates
+                    new_com = seed.choice(best_communities)
+                    # Update previous community status
+                    try:
+                        com_to_numvertices[communities[vertex]] -= 1
+                        density[communities[vertex]] = (
+                            max_density / com_to_numvertices[communities[vertex]]
+                        )
+                    except KeyError:
+                        pass
+                    # Update new community status
+                    communities[vertex] = new_com
+                    com_to_numvertices[communities[vertex]] += 1
+                    density[communities[vertex]] = (
+                        max_density / com_to_numvertices[communities[vertex]]
+                    )
+        # If maximum iterations reached --> output actual results
+        if iter_count > max_iter:
+            break
+    # Return results by grouping communities as list of vertices
+    return iter(groups(communities).values())

rl/Lib/site-packages/networkx/algorithms/community/centrality.py

@@ -0,0 +1,171 @@
+"""Functions for computing communities based on centrality notions."""
+
+import networkx as nx
+
+__all__ = ["girvan_newman"]
+
+
+@nx._dispatchable(preserve_edge_attrs="most_valuable_edge")
+def girvan_newman(G, most_valuable_edge=None):
+    """Finds communities in a graph using the Girvan–Newman method.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    most_valuable_edge : function
+        Function that takes a graph as input and outputs an edge. The
+        edge returned by this function will be recomputed and removed at
+        each iteration of the algorithm.
+
+        If not specified, the edge with the highest
+        :func:`networkx.edge_betweenness_centrality` will be used.
+
+    Returns
+    -------
+    iterator
+        Iterator over tuples of sets of nodes in `G`. Each set of node
+        is a community, each tuple is a sequence of communities at a
+        particular level of the algorithm.
+
+    Examples
+    --------
+    To get the first pair of communities::
+
+        >>> G = nx.path_graph(10)
+        >>> comp = nx.community.girvan_newman(G)
+        >>> tuple(sorted(c) for c in next(comp))
+        ([0, 1, 2, 3, 4], [5, 6, 7, 8, 9])
+
+    To get only the first *k* tuples of communities, use
+    :func:`itertools.islice`::
+
+        >>> import itertools
+        >>> G = nx.path_graph(8)
+        >>> k = 2
+        >>> comp = nx.community.girvan_newman(G)
+        >>> for communities in itertools.islice(comp, k):
+        ...     print(tuple(sorted(c) for c in communities))
+        ...
+        ([0, 1, 2, 3], [4, 5, 6, 7])
+        ([0, 1], [2, 3], [4, 5, 6, 7])
+
+    To stop getting tuples of communities once the number of communities
+    is greater than *k*, use :func:`itertools.takewhile`::
+
+        >>> import itertools
+        >>> G = nx.path_graph(8)
+        >>> k = 4
+        >>> comp = nx.community.girvan_newman(G)
+        >>> limited = itertools.takewhile(lambda c: len(c) <= k, comp)
+        >>> for communities in limited:
+        ...     print(tuple(sorted(c) for c in communities))
+        ...
+        ([0, 1, 2, 3], [4, 5, 6, 7])
+        ([0, 1], [2, 3], [4, 5, 6, 7])
+        ([0, 1], [2, 3], [4, 5], [6, 7])
+
+    To just choose an edge to remove based on the weight::
+
+        >>> from operator import itemgetter
+        >>> G = nx.path_graph(10)
+        >>> edges = G.edges()
+        >>> nx.set_edge_attributes(G, {(u, v): v for u, v in edges}, "weight")
+        >>> def heaviest(G):
+        ...     u, v, w = max(G.edges(data="weight"), key=itemgetter(2))
+        ...     return (u, v)
+        ...
+        >>> comp = nx.community.girvan_newman(G, most_valuable_edge=heaviest)
+        >>> tuple(sorted(c) for c in next(comp))
+        ([0, 1, 2, 3, 4, 5, 6, 7, 8], [9])
+
+    To utilize edge weights when choosing an edge with, for example, the
+    highest betweenness centrality::
+
+        >>> from networkx import edge_betweenness_centrality as betweenness
+        >>> def most_central_edge(G):
+        ...     centrality = betweenness(G, weight="weight")
+        ...     return max(centrality, key=centrality.get)
+        ...
+        >>> G = nx.path_graph(10)
+        >>> comp = nx.community.girvan_newman(G, most_valuable_edge=most_central_edge)
+        >>> tuple(sorted(c) for c in next(comp))
+        ([0, 1, 2, 3, 4], [5, 6, 7, 8, 9])
+
+    To specify a different ranking algorithm for edges, use the
+    `most_valuable_edge` keyword argument::
+
+        >>> from networkx import edge_betweenness_centrality
+        >>> from random import random
+        >>> def most_central_edge(G):
+        ...     centrality = edge_betweenness_centrality(G)
+        ...     max_cent = max(centrality.values())
+        ...     # Scale the centrality values so they are between 0 and 1,
+        ...     # and add some random noise.
+        ...     centrality = {e: c / max_cent for e, c in centrality.items()}
+        ...     # Add some random noise.
+        ...     centrality = {e: c + random() for e, c in centrality.items()}
+        ...     return max(centrality, key=centrality.get)
+        ...
+        >>> G = nx.path_graph(10)
+        >>> comp = nx.community.girvan_newman(G, most_valuable_edge=most_central_edge)
+
+    Notes
+    -----
+    The Girvan–Newman algorithm detects communities by progressively
+    removing edges from the original graph. The algorithm removes the
+    "most valuable" edge, traditionally the edge with the highest
+    betweenness centrality, at each step. As the graph breaks down into
+    pieces, the tightly knit community structure is exposed and the
+    result can be depicted as a dendrogram.
+
+    """
+    # If the graph is already empty, simply return its connected
+    # components.
+    if G.number_of_edges() == 0:
+        yield tuple(nx.connected_components(G))
+        return
+    # If no function is provided for computing the most valuable edge,
+    # use the edge betweenness centrality.
+    if most_valuable_edge is None:
+
+        def most_valuable_edge(G):
+            """Returns the edge with the highest betweenness centrality
+            in the graph `G`.
+
+            """
+            # We have guaranteed that the graph is non-empty, so this
+            # dictionary will never be empty.
+            betweenness = nx.edge_betweenness_centrality(G)
+            return max(betweenness, key=betweenness.get)
+
+    # The copy of G here must include the edge weight data.
+    g = G.copy().to_undirected()
+    # Self-loops must be removed because their removal has no effect on
+    # the connected components of the graph.
+    g.remove_edges_from(nx.selfloop_edges(g))
+    while g.number_of_edges() > 0:
+        yield _without_most_central_edges(g, most_valuable_edge)
+
+
+def _without_most_central_edges(G, most_valuable_edge):
+    """Returns the connected components of the graph that results from
+    repeatedly removing the most "valuable" edge in the graph.
+
+    `G` must be a non-empty graph. This function modifies the graph `G`
+    in-place; that is, it removes edges on the graph `G`.
+
+    `most_valuable_edge` is a function that takes the graph `G` as input
+    (or a subgraph with one or more edges of `G` removed) and returns an
+    edge. That edge will be removed and this process will be repeated
+    until the number of connected components in the graph increases.
+
+    """
+    original_num_components = nx.number_connected_components(G)
+    num_new_components = original_num_components
+    while num_new_components <= original_num_components:
+        edge = most_valuable_edge(G)
+        G.remove_edge(*edge)
+        new_components = tuple(nx.connected_components(G))
+        num_new_components = len(new_components)
+    return new_components

rl/Lib/site-packages/networkx/algorithms/community/community_utils.py

@@ -0,0 +1,30 @@
+"""Helper functions for community-finding algorithms."""
+
+import networkx as nx
+
+__all__ = ["is_partition"]
+
+
+@nx._dispatchable
+def is_partition(G, communities):
+    """Returns *True* if `communities` is a partition of the nodes of `G`.
+
+    A partition of a universe set is a family of pairwise disjoint sets
+    whose union is the entire universe set.
+
+    Parameters
+    ----------
+    G : NetworkX graph.
+
+    communities : list or iterable of sets of nodes
+        If not a list, the iterable is converted internally to a list.
+        If it is an iterator it is exhausted.
+
+    """
+    # Alternate implementation:
+    # return all(sum(1 if v in c else 0 for c in communities) == 1 for v in G)
+    if not isinstance(communities, list):
+        communities = list(communities)
+    nodes = {n for c in communities for n in c if n in G}
+
+    return len(G) == len(nodes) == sum(len(c) for c in communities)

rl/Lib/site-packages/networkx/algorithms/community/divisive.py

@@ -0,0 +1,216 @@
+import functools
+
+import networkx as nx
+
+__all__ = [
+    "edge_betweenness_partition",
+    "edge_current_flow_betweenness_partition",
+]
+
+
+@nx._dispatchable(edge_attrs="weight")
+def edge_betweenness_partition(G, number_of_sets, *, weight=None):
+    """Partition created by iteratively removing the highest edge betweenness edge.
+
+    This algorithm works by calculating the edge betweenness for all
+    edges and removing the edge with the highest value. It is then
+    determined whether the graph has been broken into at least
+    `number_of_sets` connected components.
+    If not the process is repeated.
+
+    Parameters
+    ----------
+    G : NetworkX Graph, DiGraph or MultiGraph
+      Graph to be partitioned
+
+    number_of_sets : int
+      Number of sets in the desired partition of the graph
+
+    weight : key, optional, default=None
+      The key to use if using weights for edge betweenness calculation
+
+    Returns
+    -------
+    C : list of sets
+      Partition of the nodes of G
+
+    Raises
+    ------
+    NetworkXError
+      If number_of_sets is <= 0 or if number_of_sets > len(G)
+
+    Examples
+    --------
+    >>> G = nx.karate_club_graph()
+    >>> part = nx.community.edge_betweenness_partition(G, 2)
+    >>> {0, 1, 3, 4, 5, 6, 7, 10, 11, 12, 13, 16, 17, 19, 21} in part
+    True
+    >>> {
+    ...     2,
+    ...     8,
+    ...     9,
+    ...     14,
+    ...     15,
+    ...     18,
+    ...     20,
+    ...     22,
+    ...     23,
+    ...     24,
+    ...     25,
+    ...     26,
+    ...     27,
+    ...     28,
+    ...     29,
+    ...     30,
+    ...     31,
+    ...     32,
+    ...     33,
+    ... } in part
+    True
+
+    See Also
+    --------
+    edge_current_flow_betweenness_partition
+
+    Notes
+    -----
+    This algorithm is fairly slow, as both the calculation of connected
+    components and edge betweenness relies on all pairs shortest
+    path algorithms. They could potentially be combined to cut down
+    on overall computation time.
+
+    References
+    ----------
+    .. [1] Santo Fortunato 'Community Detection in Graphs' Physical Reports
+       Volume 486, Issue 3-5 p. 75-174
+       http://arxiv.org/abs/0906.0612
+    """
+    if number_of_sets <= 0:
+        raise nx.NetworkXError("number_of_sets must be >0")
+    if number_of_sets == 1:
+        return [set(G)]
+    if number_of_sets == len(G):
+        return [{n} for n in G]
+    if number_of_sets > len(G):
+        raise nx.NetworkXError("number_of_sets must be <= len(G)")
+
+    H = G.copy()
+    partition = list(nx.connected_components(H))
+    while len(partition) < number_of_sets:
+        ranking = nx.edge_betweenness_centrality(H, weight=weight)
+        edge = max(ranking, key=ranking.get)
+        H.remove_edge(*edge)
+        partition = list(nx.connected_components(H))
+    return partition
+
+
+@nx._dispatchable(edge_attrs="weight")
+def edge_current_flow_betweenness_partition(G, number_of_sets, *, weight=None):
+    """Partition created by removing the highest edge current flow betweenness edge.
+
+    This algorithm works by calculating the edge current flow
+    betweenness for all edges and removing the edge with the
+    highest value. It is then determined whether the graph has
+    been broken into at least `number_of_sets` connected
+    components. If not the process is repeated.
+
+    Parameters
+    ----------
+    G : NetworkX Graph, DiGraph or MultiGraph
+      Graph to be partitioned
+
+    number_of_sets : int
+      Number of sets in the desired partition of the graph
+
+    weight : key, optional (default=None)
+      The edge attribute key to use as weights for
+      edge current flow betweenness calculations
+
+    Returns
+    -------
+    C : list of sets
+      Partition of G
+
+    Raises
+    ------
+    NetworkXError
+      If number_of_sets is <= 0 or number_of_sets > len(G)
+
+    Examples
+    --------
+    >>> G = nx.karate_club_graph()
+    >>> part = nx.community.edge_current_flow_betweenness_partition(G, 2)
+    >>> {0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 16, 17, 19, 21} in part
+    True
+    >>> {8, 14, 15, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} in part
+    True
+
+
+    See Also
+    --------
+    edge_betweenness_partition
+
+    Notes
+    -----
+    This algorithm is extremely slow, as the recalculation of the edge
+    current flow betweenness is extremely slow.
+
+    References
+    ----------
+    .. [1] Santo Fortunato 'Community Detection in Graphs' Physical Reports
+       Volume 486, Issue 3-5 p. 75-174
+       http://arxiv.org/abs/0906.0612
+    """
+    if number_of_sets <= 0:
+        raise nx.NetworkXError("number_of_sets must be >0")
+    elif number_of_sets == 1:
+        return [set(G)]
+    elif number_of_sets == len(G):
+        return [{n} for n in G]
+    elif number_of_sets > len(G):
+        raise nx.NetworkXError("number_of_sets must be <= len(G)")
+
+    rank = functools.partial(
+        nx.edge_current_flow_betweenness_centrality, normalized=False, weight=weight
+    )
+
+    # current flow requires a connected network so we track the components explicitly
+    H = G.copy()
+    partition = list(nx.connected_components(H))
+    if len(partition) > 1:
+        Hcc_subgraphs = [H.subgraph(cc).copy() for cc in partition]
+    else:
+        Hcc_subgraphs = [H]
+
+    ranking = {}
+    for Hcc in Hcc_subgraphs:
+        ranking.update(rank(Hcc))
+
+    while len(partition) < number_of_sets:
+        edge = max(ranking, key=ranking.get)
+        for cc, Hcc in zip(partition, Hcc_subgraphs):
+            if edge[0] in cc:
+                Hcc.remove_edge(*edge)
+                del ranking[edge]
+                splitcc_list = list(nx.connected_components(Hcc))
+                if len(splitcc_list) > 1:
+                    # there are 2 connected components. split off smaller one
+                    cc_new = min(splitcc_list, key=len)
+                    Hcc_new = Hcc.subgraph(cc_new).copy()
+                    # update edge rankings for Hcc_new
+                    newranks = rank(Hcc_new)
+                    for e, r in newranks.items():
+                        ranking[e if e in ranking else e[::-1]] = r
+                    # append new cc and Hcc to their lists.
+                    partition.append(cc_new)
+                    Hcc_subgraphs.append(Hcc_new)
+
+                    # leave existing cc and Hcc in their lists, but shrink them
+                    Hcc.remove_nodes_from(cc_new)
+                    cc.difference_update(cc_new)
+                # update edge rankings for Hcc whether it was split or not
+                newranks = rank(Hcc)
+                for e, r in newranks.items():
+                    ranking[e if e in ranking else e[::-1]] = r
+                break
+    return partition

rl/Lib/site-packages/networkx/algorithms/community/kclique.py

@@ -0,0 +1,79 @@
+from collections import defaultdict
+
+import networkx as nx
+
+__all__ = ["k_clique_communities"]
+
+
+@nx._dispatchable
+def k_clique_communities(G, k, cliques=None):
+    """Find k-clique communities in graph using the percolation method.
+
+    A k-clique community is the union of all cliques of size k that
+    can be reached through adjacent (sharing k-1 nodes) k-cliques.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    k : int
+       Size of smallest clique
+
+    cliques: list or generator
+       Precomputed cliques (use networkx.find_cliques(G))
+
+    Returns
+    -------
+    Yields sets of nodes, one for each k-clique community.
+
+    Examples
+    --------
+    >>> G = nx.complete_graph(5)
+    >>> K5 = nx.convert_node_labels_to_integers(G, first_label=2)
+    >>> G.add_edges_from(K5.edges())
+    >>> c = list(nx.community.k_clique_communities(G, 4))
+    >>> sorted(list(c[0]))
+    [0, 1, 2, 3, 4, 5, 6]
+    >>> list(nx.community.k_clique_communities(G, 6))
+    []
+
+    References
+    ----------
+    .. [1] Gergely Palla, Imre Derényi, Illés Farkas1, and Tamás Vicsek,
+       Uncovering the overlapping community structure of complex networks
+       in nature and society Nature 435, 814-818, 2005,
+       doi:10.1038/nature03607
+    """
+    if k < 2:
+        raise nx.NetworkXError(f"k={k}, k must be greater than 1.")
+    if cliques is None:
+        cliques = nx.find_cliques(G)
+    cliques = [frozenset(c) for c in cliques if len(c) >= k]
+
+    # First index which nodes are in which cliques
+    membership_dict = defaultdict(list)
+    for clique in cliques:
+        for node in clique:
+            membership_dict[node].append(clique)
+
+    # For each clique, see which adjacent cliques percolate
+    perc_graph = nx.Graph()
+    perc_graph.add_nodes_from(cliques)
+    for clique in cliques:
+        for adj_clique in _get_adjacent_cliques(clique, membership_dict):
+            if len(clique.intersection(adj_clique)) >= (k - 1):
+                perc_graph.add_edge(clique, adj_clique)
+
+    # Connected components of clique graph with perc edges
+    # are the percolated cliques
+    for component in nx.connected_components(perc_graph):
+        yield (frozenset.union(*component))
+
+
+def _get_adjacent_cliques(clique, membership_dict):
+    adjacent_cliques = set()
+    for n in clique:
+        for adj_clique in membership_dict[n]:
+            if clique != adj_clique:
+                adjacent_cliques.add(adj_clique)
+    return adjacent_cliques

rl/Lib/site-packages/networkx/algorithms/community/kernighan_lin.py

@@ -0,0 +1,139 @@
+"""Functions for computing the Kernighan–Lin bipartition algorithm."""
+
+from itertools import count
+
+import networkx as nx
+from networkx.algorithms.community.community_utils import is_partition
+from networkx.utils import BinaryHeap, not_implemented_for, py_random_state
+
+__all__ = ["kernighan_lin_bisection"]
+
+
+def _kernighan_lin_sweep(edges, side):
+    """
+    This is a modified form of Kernighan-Lin, which moves single nodes at a
+    time, alternating between sides to keep the bisection balanced.  We keep
+    two min-heaps of swap costs to make optimal-next-move selection fast.
+    """
+    costs0, costs1 = costs = BinaryHeap(), BinaryHeap()
+    for u, side_u, edges_u in zip(count(), side, edges):
+        cost_u = sum(w if side[v] else -w for v, w in edges_u)
+        costs[side_u].insert(u, cost_u if side_u else -cost_u)
+
+    def _update_costs(costs_x, x):
+        for y, w in edges[x]:
+            costs_y = costs[side[y]]
+            cost_y = costs_y.get(y)
+            if cost_y is not None:
+                cost_y += 2 * (-w if costs_x is costs_y else w)
+                costs_y.insert(y, cost_y, True)
+
+    i = 0
+    totcost = 0
+    while costs0 and costs1:
+        u, cost_u = costs0.pop()
+        _update_costs(costs0, u)
+        v, cost_v = costs1.pop()
+        _update_costs(costs1, v)
+        totcost += cost_u + cost_v
+        i += 1
+        yield totcost, i, (u, v)
+
+
+@not_implemented_for("directed")
+@py_random_state(4)
+@nx._dispatchable(edge_attrs="weight")
+def kernighan_lin_bisection(G, partition=None, max_iter=10, weight="weight", seed=None):
+    """Partition a graph into two blocks using the Kernighan–Lin
+    algorithm.
+
+    This algorithm partitions a network into two sets by iteratively
+    swapping pairs of nodes to reduce the edge cut between the two sets.  The
+    pairs are chosen according to a modified form of Kernighan-Lin [1]_, which
+    moves node individually, alternating between sides to keep the bisection
+    balanced.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        Graph must be undirected.
+
+    partition : tuple
+        Pair of iterables containing an initial partition. If not
+        specified, a random balanced partition is used.
+
+    max_iter : int
+        Maximum number of times to attempt swaps to find an
+        improvement before giving up.
+
+    weight : key
+        Edge data key to use as weight. If None, the weights are all
+        set to one.
+
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+        Only used if partition is None
+
+    Returns
+    -------
+    partition : tuple
+        A pair of sets of nodes representing the bipartition.
+
+    Raises
+    ------
+    NetworkXError
+        If partition is not a valid partition of the nodes of the graph.
+
+    References
+    ----------
+    .. [1] Kernighan, B. W.; Lin, Shen (1970).
+       "An efficient heuristic procedure for partitioning graphs."
+       *Bell Systems Technical Journal* 49: 291--307.
+       Oxford University Press 2011.
+
+    """
+    n = len(G)
+    labels = list(G)
+    seed.shuffle(labels)
+    index = {v: i for i, v in enumerate(labels)}
+
+    if partition is None:
+        side = [0] * (n // 2) + [1] * ((n + 1) // 2)
+    else:
+        try:
+            A, B = partition
+        except (TypeError, ValueError) as err:
+            raise nx.NetworkXError("partition must be two sets") from err
+        if not is_partition(G, (A, B)):
+            raise nx.NetworkXError("partition invalid")
+        side = [0] * n
+        for a in A:
+            side[index[a]] = 1
+
+    if G.is_multigraph():
+        edges = [
+            [
+                (index[u], sum(e.get(weight, 1) for e in d.values()))
+                for u, d in G[v].items()
+            ]
+            for v in labels
+        ]
+    else:
+        edges = [
+            [(index[u], e.get(weight, 1)) for u, e in G[v].items()] for v in labels
+        ]
+
+    for i in range(max_iter):
+        costs = list(_kernighan_lin_sweep(edges, side))
+        min_cost, min_i, _ = min(costs)
+        if min_cost >= 0:
+            break
+
+        for _, _, (u, v) in costs[:min_i]:
+            side[u] = 1
+            side[v] = 0
+
+    A = {u for u, s in zip(labels, side) if s == 0}
+    B = {u for u, s in zip(labels, side) if s == 1}
+    return A, B

rl/Lib/site-packages/networkx/algorithms/community/label_propagation.py

@@ -0,0 +1,338 @@
+"""
+Label propagation community detection algorithms.
+"""
+
+from collections import Counter, defaultdict, deque
+
+import networkx as nx
+from networkx.utils import groups, not_implemented_for, py_random_state
+
+__all__ = [
+    "label_propagation_communities",
+    "asyn_lpa_communities",
+    "fast_label_propagation_communities",
+]
+
+
+@py_random_state("seed")
+@nx._dispatchable(edge_attrs="weight")
+def fast_label_propagation_communities(G, *, weight=None, seed=None):
+    """Returns communities in `G` as detected by fast label propagation.
+
+    The fast label propagation algorithm is described in [1]_. The algorithm is
+    probabilistic and the found communities may vary in different executions.
+
+    The algorithm operates as follows. First, the community label of each node is
+    set to a unique label. The algorithm then repeatedly updates the labels of
+    the nodes to the most frequent label in their neighborhood. In case of ties,
+    a random label is chosen from the most frequent labels.
+
+    The algorithm maintains a queue of nodes that still need to be processed.
+    Initially, all nodes are added to the queue in a random order. Then the nodes
+    are removed from the queue one by one and processed. If a node updates its label,
+    all its neighbors that have a different label are added to the queue (if not
+    already in the queue). The algorithm stops when the queue is empty.
+
+    Parameters
+    ----------
+    G : Graph, DiGraph, MultiGraph, or MultiDiGraph
+        Any NetworkX graph.
+
+    weight : string, or None (default)
+        The edge attribute representing a non-negative weight of an edge. If None,
+        each edge is assumed to have weight one. The weight of an edge is used in
+        determining the frequency with which a label appears among the neighbors of
+        a node (edge with weight `w` is equivalent to `w` unweighted edges).
+
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state. See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    communities : iterable
+        Iterable of communities given as sets of nodes.
+
+    Notes
+    -----
+    Edge directions are ignored for directed graphs.
+    Edge weights must be non-negative numbers.
+
+    References
+    ----------
+    .. [1] Vincent A. Traag & Lovro Šubelj. "Large network community detection by
+       fast label propagation." Scientific Reports 13 (2023): 2701.
+       https://doi.org/10.1038/s41598-023-29610-z
+    """
+
+    # Queue of nodes to be processed.
+    nodes_queue = deque(G)
+    seed.shuffle(nodes_queue)
+
+    # Set of nodes in the queue.
+    nodes_set = set(G)
+
+    # Assign unique label to each node.
+    comms = {node: i for i, node in enumerate(G)}
+
+    while nodes_queue:
+        # Remove next node from the queue to process.
+        node = nodes_queue.popleft()
+        nodes_set.remove(node)
+
+        # Isolated nodes retain their initial label.
+        if G.degree(node) > 0:
+            # Compute frequency of labels in node's neighborhood.
+            label_freqs = _fast_label_count(G, comms, node, weight)
+            max_freq = max(label_freqs.values())
+
+            # Always sample new label from most frequent labels.
+            comm = seed.choice(
+                [comm for comm in label_freqs if label_freqs[comm] == max_freq]
+            )
+
+            if comms[node] != comm:
+                comms[node] = comm
+
+                # Add neighbors that have different label to the queue.
+                for nbr in nx.all_neighbors(G, node):
+                    if comms[nbr] != comm and nbr not in nodes_set:
+                        nodes_queue.append(nbr)
+                        nodes_set.add(nbr)
+
+    yield from groups(comms).values()
+
+
+def _fast_label_count(G, comms, node, weight=None):
+    """Computes the frequency of labels in the neighborhood of a node.
+
+    Returns a dictionary keyed by label to the frequency of that label.
+    """
+
+    if weight is None:
+        # Unweighted (un)directed simple graph.
+        if not G.is_multigraph():
+            label_freqs = Counter(map(comms.get, nx.all_neighbors(G, node)))
+
+        # Unweighted (un)directed multigraph.
+        else:
+            label_freqs = defaultdict(int)
+            for nbr in G[node]:
+                label_freqs[comms[nbr]] += len(G[node][nbr])
+
+            if G.is_directed():
+                for nbr in G.pred[node]:
+                    label_freqs[comms[nbr]] += len(G.pred[node][nbr])
+
+    else:
+        # Weighted undirected simple/multigraph.
+        label_freqs = defaultdict(float)
+        for _, nbr, w in G.edges(node, data=weight, default=1):
+            label_freqs[comms[nbr]] += w
+
+        # Weighted directed simple/multigraph.
+        if G.is_directed():
+            for nbr, _, w in G.in_edges(node, data=weight, default=1):
+                label_freqs[comms[nbr]] += w
+
+    return label_freqs
+
+
+@py_random_state(2)
+@nx._dispatchable(edge_attrs="weight")
+def asyn_lpa_communities(G, weight=None, seed=None):
+    """Returns communities in `G` as detected by asynchronous label
+    propagation.
+
+    The asynchronous label propagation algorithm is described in
+    [1]_. The algorithm is probabilistic and the found communities may
+    vary on different executions.
+
+    The algorithm proceeds as follows. After initializing each node with
+    a unique label, the algorithm repeatedly sets the label of a node to
+    be the label that appears most frequently among that nodes
+    neighbors. The algorithm halts when each node has the label that
+    appears most frequently among its neighbors. The algorithm is
+    asynchronous because each node is updated without waiting for
+    updates on the remaining nodes.
+
+    This generalized version of the algorithm in [1]_ accepts edge
+    weights.
+
+    Parameters
+    ----------
+    G : Graph
+
+    weight : string
+        The edge attribute representing the weight of an edge.
+        If None, each edge is assumed to have weight one. In this
+        algorithm, the weight of an edge is used in determining the
+        frequency with which a label appears among the neighbors of a
+        node: a higher weight means the label appears more often.
+
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    communities : iterable
+        Iterable of communities given as sets of nodes.
+
+    Notes
+    -----
+    Edge weight attributes must be numerical.
+
+    References
+    ----------
+    .. [1] Raghavan, Usha Nandini, Réka Albert, and Soundar Kumara. "Near
+           linear time algorithm to detect community structures in large-scale
+           networks." Physical Review E 76.3 (2007): 036106.
+    """
+
+    labels = {n: i for i, n in enumerate(G)}
+    cont = True
+
+    while cont:
+        cont = False
+        nodes = list(G)
+        seed.shuffle(nodes)
+
+        for node in nodes:
+            if not G[node]:
+                continue
+
+            # Get label frequencies among adjacent nodes.
+            # Depending on the order they are processed in,
+            # some nodes will be in iteration t and others in t-1,
+            # making the algorithm asynchronous.
+            if weight is None:
+                # initialising a Counter from an iterator of labels is
+                # faster for getting unweighted label frequencies
+                label_freq = Counter(map(labels.get, G[node]))
+            else:
+                # updating a defaultdict is substantially faster
+                # for getting weighted label frequencies
+                label_freq = defaultdict(float)
+                for _, v, wt in G.edges(node, data=weight, default=1):
+                    label_freq[labels[v]] += wt
+
+            # Get the labels that appear with maximum frequency.
+            max_freq = max(label_freq.values())
+            best_labels = [
+                label for label, freq in label_freq.items() if freq == max_freq
+            ]
+
+            # If the node does not have one of the maximum frequency labels,
+            # randomly choose one of them and update the node's label.
+            # Continue the iteration as long as at least one node
+            # doesn't have a maximum frequency label.
+            if labels[node] not in best_labels:
+                labels[node] = seed.choice(best_labels)
+                cont = True
+
+    yield from groups(labels).values()
+
+
+@not_implemented_for("directed")
+@nx._dispatchable
+def label_propagation_communities(G):
+    """Generates community sets determined by label propagation
+
+    Finds communities in `G` using a semi-synchronous label propagation
+    method [1]_. This method combines the advantages of both the synchronous
+    and asynchronous models. Not implemented for directed graphs.
+
+    Parameters
+    ----------
+    G : graph
+        An undirected NetworkX graph.
+
+    Returns
+    -------
+    communities : iterable
+        A dict_values object that contains a set of nodes for each community.
+
+    Raises
+    ------
+    NetworkXNotImplemented
+       If the graph is directed
+
+    References
+    ----------
+    .. [1] Cordasco, G., & Gargano, L. (2010, December). Community detection
+       via semi-synchronous label propagation algorithms. In Business
+       Applications of Social Network Analysis (BASNA), 2010 IEEE International
+       Workshop on (pp. 1-8). IEEE.
+    """
+    coloring = _color_network(G)
+    # Create a unique label for each node in the graph
+    labeling = {v: k for k, v in enumerate(G)}
+    while not _labeling_complete(labeling, G):
+        # Update the labels of every node with the same color.
+        for color, nodes in coloring.items():
+            for n in nodes:
+                _update_label(n, labeling, G)
+
+    clusters = defaultdict(set)
+    for node, label in labeling.items():
+        clusters[label].add(node)
+    return clusters.values()
+
+
+def _color_network(G):
+    """Colors the network so that neighboring nodes all have distinct colors.
+
+    Returns a dict keyed by color to a set of nodes with that color.
+    """
+    coloring = {}  # color => set(node)
+    colors = nx.coloring.greedy_color(G)
+    for node, color in colors.items():
+        if color in coloring:
+            coloring[color].add(node)
+        else:
+            coloring[color] = {node}
+    return coloring
+
+
+def _labeling_complete(labeling, G):
+    """Determines whether or not LPA is done.
+
+    Label propagation is complete when all nodes have a label that is
+    in the set of highest frequency labels amongst its neighbors.
+
+    Nodes with no neighbors are considered complete.
+    """
+    return all(
+        labeling[v] in _most_frequent_labels(v, labeling, G) for v in G if len(G[v]) > 0
+    )
+
+
+def _most_frequent_labels(node, labeling, G):
+    """Returns a set of all labels with maximum frequency in `labeling`.
+
+    Input `labeling` should be a dict keyed by node to labels.
+    """
+    if not G[node]:
+        # Nodes with no neighbors are themselves a community and are labeled
+        # accordingly, hence the immediate if statement.
+        return {labeling[node]}
+
+    # Compute the frequencies of all neighbors of node
+    freqs = Counter(labeling[q] for q in G[node])
+    max_freq = max(freqs.values())
+    return {label for label, freq in freqs.items() if freq == max_freq}
+
+
+def _update_label(node, labeling, G):
+    """Updates the label of a node using the Prec-Max tie breaking algorithm
+
+    The algorithm is explained in: 'Community Detection via Semi-Synchronous
+    Label Propagation Algorithms' Cordasco and Gargano, 2011
+    """
+    high_labels = _most_frequent_labels(node, labeling, G)
+    if len(high_labels) == 1:
+        labeling[node] = high_labels.pop()
+    elif len(high_labels) > 1:
+        # Prec-Max
+        if labeling[node] not in high_labels:
+            labeling[node] = max(high_labels)

rl/Lib/site-packages/networkx/algorithms/community/louvain.py

@@ -0,0 +1,382 @@
+"""Function for detecting communities based on Louvain Community Detection
+Algorithm"""
+
+import itertools
+from collections import defaultdict, deque
+
+import networkx as nx
+from networkx.algorithms.community import modularity
+from networkx.utils import py_random_state
+
+__all__ = ["louvain_communities", "louvain_partitions"]
+
+
+@py_random_state("seed")
+@nx._dispatchable(edge_attrs="weight")
+def louvain_communities(
+    G, weight="weight", resolution=1, threshold=0.0000001, max_level=None, seed=None
+):
+    r"""Find the best partition of a graph using the Louvain Community Detection
+    Algorithm.
+
+    Louvain Community Detection Algorithm is a simple method to extract the community
+    structure of a network. This is a heuristic method based on modularity optimization. [1]_
+
+    The algorithm works in 2 steps. On the first step it assigns every node to be
+    in its own community and then for each node it tries to find the maximum positive
+    modularity gain by moving each node to all of its neighbor communities. If no positive
+    gain is achieved the node remains in its original community.
+
+    The modularity gain obtained by moving an isolated node $i$ into a community $C$ can
+    easily be calculated by the following formula (combining [1]_ [2]_ and some algebra):
+
+    .. math::
+        \Delta Q = \frac{k_{i,in}}{2m} - \gamma\frac{ \Sigma_{tot} \cdot k_i}{2m^2}
+
+    where $m$ is the size of the graph, $k_{i,in}$ is the sum of the weights of the links
+    from $i$ to nodes in $C$, $k_i$ is the sum of the weights of the links incident to node $i$,
+    $\Sigma_{tot}$ is the sum of the weights of the links incident to nodes in $C$ and $\gamma$
+    is the resolution parameter.
+
+    For the directed case the modularity gain can be computed using this formula according to [3]_
+
+    .. math::
+        \Delta Q = \frac{k_{i,in}}{m}
+        - \gamma\frac{k_i^{out} \cdot\Sigma_{tot}^{in} + k_i^{in} \cdot \Sigma_{tot}^{out}}{m^2}
+
+    where $k_i^{out}$, $k_i^{in}$ are the outer and inner weighted degrees of node $i$ and
+    $\Sigma_{tot}^{in}$, $\Sigma_{tot}^{out}$ are the sum of in-going and out-going links incident
+    to nodes in $C$.
+
+    The first phase continues until no individual move can improve the modularity.
+
+    The second phase consists in building a new network whose nodes are now the communities
+    found in the first phase. To do so, the weights of the links between the new nodes are given by
+    the sum of the weight of the links between nodes in the corresponding two communities. Once this
+    phase is complete it is possible to reapply the first phase creating bigger communities with
+    increased modularity.
+
+    The above two phases are executed until no modularity gain is achieved (or is less than
+    the `threshold`, or until `max_levels` is reached).
+
+    Be careful with self-loops in the input graph. These are treated as
+    previously reduced communities -- as if the process had been started
+    in the middle of the algorithm. Large self-loop edge weights thus
+    represent strong communities and in practice may be hard to add
+    other nodes to.  If your input graph edge weights for self-loops
+    do not represent already reduced communities you may want to remove
+    the self-loops before inputting that graph.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+    weight : string or None, optional (default="weight")
+        The name of an edge attribute that holds the numerical value
+        used as a weight. If None then each edge has weight 1.
+    resolution : float, optional (default=1)
+        If resolution is less than 1, the algorithm favors larger communities.
+        Greater than 1 favors smaller communities
+    threshold : float, optional (default=0.0000001)
+        Modularity gain threshold for each level. If the gain of modularity
+        between 2 levels of the algorithm is less than the given threshold
+        then the algorithm stops and returns the resulting communities.
+    max_level : int or None, optional (default=None)
+        The maximum number of levels (steps of the algorithm) to compute.
+        Must be a positive integer or None. If None, then there is no max
+        level and the threshold parameter determines the stopping condition.
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    list
+        A list of sets (partition of `G`). Each set represents one community and contains
+        all the nodes that constitute it.
+
+    Examples
+    --------
+    >>> import networkx as nx
+    >>> G = nx.petersen_graph()
+    >>> nx.community.louvain_communities(G, seed=123)
+    [{0, 4, 5, 7, 9}, {1, 2, 3, 6, 8}]
+
+    Notes
+    -----
+    The order in which the nodes are considered can affect the final output. In the algorithm
+    the ordering happens using a random shuffle.
+
+    References
+    ----------
+    .. [1] Blondel, V.D. et al. Fast unfolding of communities in
+       large networks. J. Stat. Mech 10008, 1-12(2008). https://doi.org/10.1088/1742-5468/2008/10/P10008
+    .. [2] Traag, V.A., Waltman, L. & van Eck, N.J. From Louvain to Leiden: guaranteeing
+       well-connected communities. Sci Rep 9, 5233 (2019). https://doi.org/10.1038/s41598-019-41695-z
+    .. [3] Nicolas Dugué, Anthony Perez. Directed Louvain : maximizing modularity in directed networks.
+        [Research Report] Université d’Orléans. 2015. hal-01231784. https://hal.archives-ouvertes.fr/hal-01231784
+
+    See Also
+    --------
+    louvain_partitions
+    """
+
+    partitions = louvain_partitions(G, weight, resolution, threshold, seed)
+    if max_level is not None:
+        if max_level <= 0:
+            raise ValueError("max_level argument must be a positive integer or None")
+        partitions = itertools.islice(partitions, max_level)
+    final_partition = deque(partitions, maxlen=1)
+    return final_partition.pop()
+
+
+@py_random_state("seed")
+@nx._dispatchable(edge_attrs="weight")
+def louvain_partitions(
+    G, weight="weight", resolution=1, threshold=0.0000001, seed=None
+):
+    """Yields partitions for each level of the Louvain Community Detection Algorithm
+
+    Louvain Community Detection Algorithm is a simple method to extract the community
+    structure of a network. This is a heuristic method based on modularity optimization. [1]_
+
+    The partitions at each level (step of the algorithm) form a dendrogram of communities.
+    A dendrogram is a diagram representing a tree and each level represents
+    a partition of the G graph. The top level contains the smallest communities
+    and as you traverse to the bottom of the tree the communities get bigger
+    and the overall modularity increases making the partition better.
+
+    Each level is generated by executing the two phases of the Louvain Community
+    Detection Algorithm.
+
+    Be careful with self-loops in the input graph. These are treated as
+    previously reduced communities -- as if the process had been started
+    in the middle of the algorithm. Large self-loop edge weights thus
+    represent strong communities and in practice may be hard to add
+    other nodes to.  If your input graph edge weights for self-loops
+    do not represent already reduced communities you may want to remove
+    the self-loops before inputting that graph.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+    weight : string or None, optional (default="weight")
+     The name of an edge attribute that holds the numerical value
+     used as a weight. If None then each edge has weight 1.
+    resolution : float, optional (default=1)
+        If resolution is less than 1, the algorithm favors larger communities.
+        Greater than 1 favors smaller communities
+    threshold : float, optional (default=0.0000001)
+     Modularity gain threshold for each level. If the gain of modularity
+     between 2 levels of the algorithm is less than the given threshold
+     then the algorithm stops and returns the resulting communities.
+    seed : integer, random_state, or None (default)
+     Indicator of random number generation state.
+     See :ref:`Randomness<randomness>`.
+
+    Yields
+    ------
+    list
+        A list of sets (partition of `G`). Each set represents one community and contains
+        all the nodes that constitute it.
+
+    References
+    ----------
+    .. [1] Blondel, V.D. et al. Fast unfolding of communities in
+       large networks. J. Stat. Mech 10008, 1-12(2008)
+
+    See Also
+    --------
+    louvain_communities
+    """
+
+    partition = [{u} for u in G.nodes()]
+    if nx.is_empty(G):
+        yield partition
+        return
+    mod = modularity(G, partition, resolution=resolution, weight=weight)
+    is_directed = G.is_directed()
+    if G.is_multigraph():
+        graph = _convert_multigraph(G, weight, is_directed)
+    else:
+        graph = G.__class__()
+        graph.add_nodes_from(G)
+        graph.add_weighted_edges_from(G.edges(data=weight, default=1))
+
+    m = graph.size(weight="weight")
+    partition, inner_partition, improvement = _one_level(
+        graph, m, partition, resolution, is_directed, seed
+    )
+    improvement = True
+    while improvement:
+        # gh-5901 protect the sets in the yielded list from further manipulation here
+        yield [s.copy() for s in partition]
+        new_mod = modularity(
+            graph, inner_partition, resolution=resolution, weight="weight"
+        )
+        if new_mod - mod <= threshold:
+            return
+        mod = new_mod
+        graph = _gen_graph(graph, inner_partition)
+        partition, inner_partition, improvement = _one_level(
+            graph, m, partition, resolution, is_directed, seed
+        )
+
+
+def _one_level(G, m, partition, resolution=1, is_directed=False, seed=None):
+    """Calculate one level of the Louvain partitions tree
+
+    Parameters
+    ----------
+    G : NetworkX Graph/DiGraph
+        The graph from which to detect communities
+    m : number
+        The size of the graph `G`.
+    partition : list of sets of nodes
+        A valid partition of the graph `G`
+    resolution : positive number
+        The resolution parameter for computing the modularity of a partition
+    is_directed : bool
+        True if `G` is a directed graph.
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    """
+    node2com = {u: i for i, u in enumerate(G.nodes())}
+    inner_partition = [{u} for u in G.nodes()]
+    if is_directed:
+        in_degrees = dict(G.in_degree(weight="weight"))
+        out_degrees = dict(G.out_degree(weight="weight"))
+        Stot_in = list(in_degrees.values())
+        Stot_out = list(out_degrees.values())
+        # Calculate weights for both in and out neighbors without considering self-loops
+        nbrs = {}
+        for u in G:
+            nbrs[u] = defaultdict(float)
+            for _, n, wt in G.out_edges(u, data="weight"):
+                if u != n:
+                    nbrs[u][n] += wt
+            for n, _, wt in G.in_edges(u, data="weight"):
+                if u != n:
+                    nbrs[u][n] += wt
+    else:
+        degrees = dict(G.degree(weight="weight"))
+        Stot = list(degrees.values())
+        nbrs = {u: {v: data["weight"] for v, data in G[u].items() if v != u} for u in G}
+    rand_nodes = list(G.nodes)
+    seed.shuffle(rand_nodes)
+    nb_moves = 1
+    improvement = False
+    while nb_moves > 0:
+        nb_moves = 0
+        for u in rand_nodes:
+            best_mod = 0
+            best_com = node2com[u]
+            weights2com = _neighbor_weights(nbrs[u], node2com)
+            if is_directed:
+                in_degree = in_degrees[u]
+                out_degree = out_degrees[u]
+                Stot_in[best_com] -= in_degree
+                Stot_out[best_com] -= out_degree
+                remove_cost = (
+                    -weights2com[best_com] / m
+                    + resolution
+                    * (out_degree * Stot_in[best_com] + in_degree * Stot_out[best_com])
+                    / m**2
+                )
+            else:
+                degree = degrees[u]
+                Stot[best_com] -= degree
+                remove_cost = -weights2com[best_com] / m + resolution * (
+                    Stot[best_com] * degree
+                ) / (2 * m**2)
+            for nbr_com, wt in weights2com.items():
+                if is_directed:
+                    gain = (
+                        remove_cost
+                        + wt / m
+                        - resolution
+                        * (
+                            out_degree * Stot_in[nbr_com]
+                            + in_degree * Stot_out[nbr_com]
+                        )
+                        / m**2
+                    )
+                else:
+                    gain = (
+                        remove_cost
+                        + wt / m
+                        - resolution * (Stot[nbr_com] * degree) / (2 * m**2)
+                    )
+                if gain > best_mod:
+                    best_mod = gain
+                    best_com = nbr_com
+            if is_directed:
+                Stot_in[best_com] += in_degree
+                Stot_out[best_com] += out_degree
+            else:
+                Stot[best_com] += degree
+            if best_com != node2com[u]:
+                com = G.nodes[u].get("nodes", {u})
+                partition[node2com[u]].difference_update(com)
+                inner_partition[node2com[u]].remove(u)
+                partition[best_com].update(com)
+                inner_partition[best_com].add(u)
+                improvement = True
+                nb_moves += 1
+                node2com[u] = best_com
+    partition = list(filter(len, partition))
+    inner_partition = list(filter(len, inner_partition))
+    return partition, inner_partition, improvement
+
+
+def _neighbor_weights(nbrs, node2com):
+    """Calculate weights between node and its neighbor communities.
+
+    Parameters
+    ----------
+    nbrs : dictionary
+           Dictionary with nodes' neighbors as keys and their edge weight as value.
+    node2com : dictionary
+           Dictionary with all graph's nodes as keys and their community index as value.
+
+    """
+    weights = defaultdict(float)
+    for nbr, wt in nbrs.items():
+        weights[node2com[nbr]] += wt
+    return weights
+
+
+def _gen_graph(G, partition):
+    """Generate a new graph based on the partitions of a given graph"""
+    H = G.__class__()
+    node2com = {}
+    for i, part in enumerate(partition):
+        nodes = set()
+        for node in part:
+            node2com[node] = i
+            nodes.update(G.nodes[node].get("nodes", {node}))
+        H.add_node(i, nodes=nodes)
+
+    for node1, node2, wt in G.edges(data=True):
+        wt = wt["weight"]
+        com1 = node2com[node1]
+        com2 = node2com[node2]
+        temp = H.get_edge_data(com1, com2, {"weight": 0})["weight"]
+        H.add_edge(com1, com2, weight=wt + temp)
+    return H
+
+
+def _convert_multigraph(G, weight, is_directed):
+    """Convert a Multigraph to normal Graph"""
+    if is_directed:
+        H = nx.DiGraph()
+    else:
+        H = nx.Graph()
+    H.add_nodes_from(G)
+    for u, v, wt in G.edges(data=weight, default=1):
+        if H.has_edge(u, v):
+            H[u][v]["weight"] += wt
+        else:
+            H.add_edge(u, v, weight=wt)
+    return H

rl/Lib/site-packages/networkx/algorithms/community/lukes.py

@@ -0,0 +1,227 @@
+"""Lukes Algorithm for exact optimal weighted tree partitioning."""
+
+from copy import deepcopy
+from functools import lru_cache
+from random import choice
+
+import networkx as nx
+from networkx.utils import not_implemented_for
+
+__all__ = ["lukes_partitioning"]
+
+D_EDGE_W = "weight"
+D_EDGE_VALUE = 1.0
+D_NODE_W = "weight"
+D_NODE_VALUE = 1
+PKEY = "partitions"
+CLUSTER_EVAL_CACHE_SIZE = 2048
+
+
+def _split_n_from(n, min_size_of_first_part):
+    # splits j in two parts of which the first is at least
+    # the second argument
+    assert n >= min_size_of_first_part
+    for p1 in range(min_size_of_first_part, n + 1):
+        yield p1, n - p1
+
+
+@nx._dispatchable(node_attrs="node_weight", edge_attrs="edge_weight")
+def lukes_partitioning(G, max_size, node_weight=None, edge_weight=None):
+    """Optimal partitioning of a weighted tree using the Lukes algorithm.
+
+    This algorithm partitions a connected, acyclic graph featuring integer
+    node weights and float edge weights. The resulting clusters are such
+    that the total weight of the nodes in each cluster does not exceed
+    max_size and that the weight of the edges that are cut by the partition
+    is minimum. The algorithm is based on [1]_.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    max_size : int
+        Maximum weight a partition can have in terms of sum of
+        node_weight for all nodes in the partition
+
+    edge_weight : key
+        Edge data key to use as weight. If None, the weights are all
+        set to one.
+
+    node_weight : key
+        Node data key to use as weight. If None, the weights are all
+        set to one. The data must be int.
+
+    Returns
+    -------
+    partition : list
+        A list of sets of nodes representing the clusters of the
+        partition.
+
+    Raises
+    ------
+    NotATree
+        If G is not a tree.
+    TypeError
+        If any of the values of node_weight is not int.
+
+    References
+    ----------
+    .. [1] Lukes, J. A. (1974).
+       "Efficient Algorithm for the Partitioning of Trees."
+       IBM Journal of Research and Development, 18(3), 217–224.
+
+    """
+    # First sanity check and tree preparation
+    if not nx.is_tree(G):
+        raise nx.NotATree("lukes_partitioning works only on trees")
+    else:
+        if nx.is_directed(G):
+            root = [n for n, d in G.in_degree() if d == 0]
+            assert len(root) == 1
+            root = root[0]
+            t_G = deepcopy(G)
+        else:
+            root = choice(list(G.nodes))
+            # this has the desirable side effect of not inheriting attributes
+            t_G = nx.dfs_tree(G, root)
+
+    # Since we do not want to screw up the original graph,
+    # if we have a blank attribute, we make a deepcopy
+    if edge_weight is None or node_weight is None:
+        safe_G = deepcopy(G)
+        if edge_weight is None:
+            nx.set_edge_attributes(safe_G, D_EDGE_VALUE, D_EDGE_W)
+            edge_weight = D_EDGE_W
+        if node_weight is None:
+            nx.set_node_attributes(safe_G, D_NODE_VALUE, D_NODE_W)
+            node_weight = D_NODE_W
+    else:
+        safe_G = G
+
+    # Second sanity check
+    # The values of node_weight MUST BE int.
+    # I cannot see any room for duck typing without incurring serious
+    # danger of subtle bugs.
+    all_n_attr = nx.get_node_attributes(safe_G, node_weight).values()
+    for x in all_n_attr:
+        if not isinstance(x, int):
+            raise TypeError(
+                "lukes_partitioning needs integer "
+                f"values for node_weight ({node_weight})"
+            )
+
+    # SUBROUTINES -----------------------
+    # these functions are defined here for two reasons:
+    # - brevity: we can leverage global "safe_G"
+    # - caching: signatures are hashable
+
+    @not_implemented_for("undirected")
+    # this is intended to be called only on t_G
+    def _leaves(gr):
+        for x in gr.nodes:
+            if not nx.descendants(gr, x):
+                yield x
+
+    @not_implemented_for("undirected")
+    def _a_parent_of_leaves_only(gr):
+        tleaves = set(_leaves(gr))
+        for n in set(gr.nodes) - tleaves:
+            if all(x in tleaves for x in nx.descendants(gr, n)):
+                return n
+
+    @lru_cache(CLUSTER_EVAL_CACHE_SIZE)
+    def _value_of_cluster(cluster):
+        valid_edges = [e for e in safe_G.edges if e[0] in cluster and e[1] in cluster]
+        return sum(safe_G.edges[e][edge_weight] for e in valid_edges)
+
+    def _value_of_partition(partition):
+        return sum(_value_of_cluster(frozenset(c)) for c in partition)
+
+    @lru_cache(CLUSTER_EVAL_CACHE_SIZE)
+    def _weight_of_cluster(cluster):
+        return sum(safe_G.nodes[n][node_weight] for n in cluster)
+
+    def _pivot(partition, node):
+        ccx = [c for c in partition if node in c]
+        assert len(ccx) == 1
+        return ccx[0]
+
+    def _concatenate_or_merge(partition_1, partition_2, x, i, ref_weight):
+        ccx = _pivot(partition_1, x)
+        cci = _pivot(partition_2, i)
+        merged_xi = ccx.union(cci)
+
+        # We first check if we can do the merge.
+        # If so, we do the actual calculations, otherwise we concatenate
+        if _weight_of_cluster(frozenset(merged_xi)) <= ref_weight:
+            cp1 = list(filter(lambda x: x != ccx, partition_1))
+            cp2 = list(filter(lambda x: x != cci, partition_2))
+
+            option_2 = [merged_xi] + cp1 + cp2
+            return option_2, _value_of_partition(option_2)
+        else:
+            option_1 = partition_1 + partition_2
+            return option_1, _value_of_partition(option_1)
+
+    # INITIALIZATION -----------------------
+    leaves = set(_leaves(t_G))
+    for lv in leaves:
+        t_G.nodes[lv][PKEY] = {}
+        slot = safe_G.nodes[lv][node_weight]
+        t_G.nodes[lv][PKEY][slot] = [{lv}]
+        t_G.nodes[lv][PKEY][0] = [{lv}]
+
+    for inner in [x for x in t_G.nodes if x not in leaves]:
+        t_G.nodes[inner][PKEY] = {}
+        slot = safe_G.nodes[inner][node_weight]
+        t_G.nodes[inner][PKEY][slot] = [{inner}]
+    nx._clear_cache(t_G)
+
+    # CORE ALGORITHM -----------------------
+    while True:
+        x_node = _a_parent_of_leaves_only(t_G)
+        weight_of_x = safe_G.nodes[x_node][node_weight]
+        best_value = 0
+        best_partition = None
+        bp_buffer = {}
+        x_descendants = nx.descendants(t_G, x_node)
+        for i_node in x_descendants:
+            for j in range(weight_of_x, max_size + 1):
+                for a, b in _split_n_from(j, weight_of_x):
+                    if (
+                        a not in t_G.nodes[x_node][PKEY]
+                        or b not in t_G.nodes[i_node][PKEY]
+                    ):
+                        # it's not possible to form this particular weight sum
+                        continue
+
+                    part1 = t_G.nodes[x_node][PKEY][a]
+                    part2 = t_G.nodes[i_node][PKEY][b]
+                    part, value = _concatenate_or_merge(part1, part2, x_node, i_node, j)
+
+                    if j not in bp_buffer or bp_buffer[j][1] < value:
+                        # we annotate in the buffer the best partition for j
+                        bp_buffer[j] = part, value
+
+                    # we also keep track of the overall best partition
+                    if best_value <= value:
+                        best_value = value
+                        best_partition = part
+
+            # as illustrated in Lukes, once we finished a child, we can
+            # discharge the partitions we found into the graph
+            # (the key phrase is make all x == x')
+            # so that they are used by the subsequent children
+            for w, (best_part_for_vl, vl) in bp_buffer.items():
+                t_G.nodes[x_node][PKEY][w] = best_part_for_vl
+            bp_buffer.clear()
+
+        # the absolute best partition for this node
+        # across all weights has to be stored at 0
+        t_G.nodes[x_node][PKEY][0] = best_partition
+        t_G.remove_nodes_from(x_descendants)
+
+        if x_node == root:
+            # the 0-labeled partition of root
+            # is the optimal one for the whole tree
+            return t_G.nodes[root][PKEY][0]

rl/Lib/site-packages/networkx/algorithms/community/modularity_max.py

@@ -0,0 +1,451 @@
+"""Functions for detecting communities based on modularity."""
+
+from collections import defaultdict
+
+import networkx as nx
+from networkx.algorithms.community.quality import modularity
+from networkx.utils import not_implemented_for
+from networkx.utils.mapped_queue import MappedQueue
+
+__all__ = [
+    "greedy_modularity_communities",
+    "naive_greedy_modularity_communities",
+]
+
+
+def _greedy_modularity_communities_generator(G, weight=None, resolution=1):
+    r"""Yield community partitions of G and the modularity change at each step.
+
+    This function performs Clauset-Newman-Moore greedy modularity maximization [2]_
+    At each step of the process it yields the change in modularity that will occur in
+    the next step followed by yielding the new community partition after that step.
+
+    Greedy modularity maximization begins with each node in its own community
+    and repeatedly joins the pair of communities that lead to the largest
+    modularity until one community contains all nodes (the partition has one set).
+
+    This function maximizes the generalized modularity, where `resolution`
+    is the resolution parameter, often expressed as $\gamma$.
+    See :func:`~networkx.algorithms.community.quality.modularity`.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    weight : string or None, optional (default=None)
+        The name of an edge attribute that holds the numerical value used
+        as a weight.  If None, then each edge has weight 1.
+        The degree is the sum of the edge weights adjacent to the node.
+
+    resolution : float (default=1)
+        If resolution is less than 1, modularity favors larger communities.
+        Greater than 1 favors smaller communities.
+
+    Yields
+    ------
+    Alternating yield statements produce the following two objects:
+
+    communities: dict_values
+        A dict_values of frozensets of nodes, one for each community.
+        This represents a partition of the nodes of the graph into communities.
+        The first yield is the partition with each node in its own community.
+
+    dq: float
+        The change in modularity when merging the next two communities
+        that leads to the largest modularity.
+
+    See Also
+    --------
+    modularity
+
+    References
+    ----------
+    .. [1] Newman, M. E. J. "Networks: An Introduction", page 224
+       Oxford University Press 2011.
+    .. [2] Clauset, A., Newman, M. E., & Moore, C.
+       "Finding community structure in very large networks."
+       Physical Review E 70(6), 2004.
+    .. [3] Reichardt and Bornholdt "Statistical Mechanics of Community
+       Detection" Phys. Rev. E74, 2006.
+    .. [4] Newman, M. E. J."Analysis of weighted networks"
+       Physical Review E 70(5 Pt 2):056131, 2004.
+    """
+    directed = G.is_directed()
+    N = G.number_of_nodes()
+
+    # Count edges (or the sum of edge-weights for weighted graphs)
+    m = G.size(weight)
+    q0 = 1 / m
+
+    # Calculate degrees (notation from the papers)
+    # a : the fraction of (weighted) out-degree for each node
+    # b : the fraction of (weighted) in-degree for each node
+    if directed:
+        a = {node: deg_out * q0 for node, deg_out in G.out_degree(weight=weight)}
+        b = {node: deg_in * q0 for node, deg_in in G.in_degree(weight=weight)}
+    else:
+        a = b = {node: deg * q0 * 0.5 for node, deg in G.degree(weight=weight)}
+
+    # this preliminary step collects the edge weights for each node pair
+    # It handles multigraph and digraph and works fine for graph.
+    dq_dict = defaultdict(lambda: defaultdict(float))
+    for u, v, wt in G.edges(data=weight, default=1):
+        if u == v:
+            continue
+        dq_dict[u][v] += wt
+        dq_dict[v][u] += wt
+
+    # now scale and subtract the expected edge-weights term
+    for u, nbrdict in dq_dict.items():
+        for v, wt in nbrdict.items():
+            dq_dict[u][v] = q0 * wt - resolution * (a[u] * b[v] + b[u] * a[v])
+
+    # Use -dq to get a max_heap instead of a min_heap
+    # dq_heap holds a heap for each node's neighbors
+    dq_heap = {u: MappedQueue({(u, v): -dq for v, dq in dq_dict[u].items()}) for u in G}
+    # H -> all_dq_heap holds a heap with the best items for each node
+    H = MappedQueue([dq_heap[n].heap[0] for n in G if len(dq_heap[n]) > 0])
+
+    # Initialize single-node communities
+    communities = {n: frozenset([n]) for n in G}
+    yield communities.values()
+
+    # Merge the two communities that lead to the largest modularity
+    while len(H) > 1:
+        # Find best merge
+        # Remove from heap of row maxes
+        # Ties will be broken by choosing the pair with lowest min community id
+        try:
+            negdq, u, v = H.pop()
+        except IndexError:
+            break
+        dq = -negdq
+        yield dq
+        # Remove best merge from row u heap
+        dq_heap[u].pop()
+        # Push new row max onto H
+        if len(dq_heap[u]) > 0:
+            H.push(dq_heap[u].heap[0])
+        # If this element was also at the root of row v, we need to remove the
+        # duplicate entry from H
+        if dq_heap[v].heap[0] == (v, u):
+            H.remove((v, u))
+            # Remove best merge from row v heap
+            dq_heap[v].remove((v, u))
+            # Push new row max onto H
+            if len(dq_heap[v]) > 0:
+                H.push(dq_heap[v].heap[0])
+        else:
+            # Duplicate wasn't in H, just remove from row v heap
+            dq_heap[v].remove((v, u))
+
+        # Perform merge
+        communities[v] = frozenset(communities[u] | communities[v])
+        del communities[u]
+
+        # Get neighbor communities connected to the merged communities
+        u_nbrs = set(dq_dict[u])
+        v_nbrs = set(dq_dict[v])
+        all_nbrs = (u_nbrs | v_nbrs) - {u, v}
+        both_nbrs = u_nbrs & v_nbrs
+        # Update dq for merge of u into v
+        for w in all_nbrs:
+            # Calculate new dq value
+            if w in both_nbrs:
+                dq_vw = dq_dict[v][w] + dq_dict[u][w]
+            elif w in v_nbrs:
+                dq_vw = dq_dict[v][w] - resolution * (a[u] * b[w] + a[w] * b[u])
+            else:  # w in u_nbrs
+                dq_vw = dq_dict[u][w] - resolution * (a[v] * b[w] + a[w] * b[v])
+            # Update rows v and w
+            for row, col in [(v, w), (w, v)]:
+                dq_heap_row = dq_heap[row]
+                # Update dict for v,w only (u is removed below)
+                dq_dict[row][col] = dq_vw
+                # Save old max of per-row heap
+                if len(dq_heap_row) > 0:
+                    d_oldmax = dq_heap_row.heap[0]
+                else:
+                    d_oldmax = None
+                # Add/update heaps
+                d = (row, col)
+                d_negdq = -dq_vw
+                # Save old value for finding heap index
+                if w in v_nbrs:
+                    # Update existing element in per-row heap
+                    dq_heap_row.update(d, d, priority=d_negdq)
+                else:
+                    # We're creating a new nonzero element, add to heap
+                    dq_heap_row.push(d, priority=d_negdq)
+                # Update heap of row maxes if necessary
+                if d_oldmax is None:
+                    # No entries previously in this row, push new max
+                    H.push(d, priority=d_negdq)
+                else:
+                    # We've updated an entry in this row, has the max changed?
+                    row_max = dq_heap_row.heap[0]
+                    if d_oldmax != row_max or d_oldmax.priority != row_max.priority:
+                        H.update(d_oldmax, row_max)
+
+        # Remove row/col u from dq_dict matrix
+        for w in dq_dict[u]:
+            # Remove from dict
+            dq_old = dq_dict[w][u]
+            del dq_dict[w][u]
+            # Remove from heaps if we haven't already
+            if w != v:
+                # Remove both row and column
+                for row, col in [(w, u), (u, w)]:
+                    dq_heap_row = dq_heap[row]
+                    # Check if replaced dq is row max
+                    d_old = (row, col)
+                    if dq_heap_row.heap[0] == d_old:
+                        # Update per-row heap and heap of row maxes
+                        dq_heap_row.remove(d_old)
+                        H.remove(d_old)
+                        # Update row max
+                        if len(dq_heap_row) > 0:
+                            H.push(dq_heap_row.heap[0])
+                    else:
+                        # Only update per-row heap
+                        dq_heap_row.remove(d_old)
+
+        del dq_dict[u]
+        # Mark row u as deleted, but keep placeholder
+        dq_heap[u] = MappedQueue()
+        # Merge u into v and update a
+        a[v] += a[u]
+        a[u] = 0
+        if directed:
+            b[v] += b[u]
+            b[u] = 0
+
+        yield communities.values()
+
+
+@nx._dispatchable(edge_attrs="weight")
+def greedy_modularity_communities(
+    G,
+    weight=None,
+    resolution=1,
+    cutoff=1,
+    best_n=None,
+):
+    r"""Find communities in G using greedy modularity maximization.
+
+    This function uses Clauset-Newman-Moore greedy modularity maximization [2]_
+    to find the community partition with the largest modularity.
+
+    Greedy modularity maximization begins with each node in its own community
+    and repeatedly joins the pair of communities that lead to the largest
+    modularity until no further increase in modularity is possible (a maximum).
+    Two keyword arguments adjust the stopping condition. `cutoff` is a lower
+    limit on the number of communities so you can stop the process before
+    reaching a maximum (used to save computation time). `best_n` is an upper
+    limit on the number of communities so you can make the process continue
+    until at most n communities remain even if the maximum modularity occurs
+    for more. To obtain exactly n communities, set both `cutoff` and `best_n` to n.
+
+    This function maximizes the generalized modularity, where `resolution`
+    is the resolution parameter, often expressed as $\gamma$.
+    See :func:`~networkx.algorithms.community.quality.modularity`.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    weight : string or None, optional (default=None)
+        The name of an edge attribute that holds the numerical value used
+        as a weight.  If None, then each edge has weight 1.
+        The degree is the sum of the edge weights adjacent to the node.
+
+    resolution : float, optional (default=1)
+        If resolution is less than 1, modularity favors larger communities.
+        Greater than 1 favors smaller communities.
+
+    cutoff : int, optional (default=1)
+        A minimum number of communities below which the merging process stops.
+        The process stops at this number of communities even if modularity
+        is not maximized. The goal is to let the user stop the process early.
+        The process stops before the cutoff if it finds a maximum of modularity.
+
+    best_n : int or None, optional (default=None)
+        A maximum number of communities above which the merging process will
+        not stop. This forces community merging to continue after modularity
+        starts to decrease until `best_n` communities remain.
+        If ``None``, don't force it to continue beyond a maximum.
+
+    Raises
+    ------
+    ValueError : If the `cutoff` or `best_n`  value is not in the range
+        ``[1, G.number_of_nodes()]``, or if `best_n` < `cutoff`.
+
+    Returns
+    -------
+    communities: list
+        A list of frozensets of nodes, one for each community.
+        Sorted by length with largest communities first.
+
+    Examples
+    --------
+    >>> G = nx.karate_club_graph()
+    >>> c = nx.community.greedy_modularity_communities(G)
+    >>> sorted(c[0])
+    [8, 14, 15, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]
+
+    See Also
+    --------
+    modularity
+
+    References
+    ----------
+    .. [1] Newman, M. E. J. "Networks: An Introduction", page 224
+       Oxford University Press 2011.
+    .. [2] Clauset, A., Newman, M. E., & Moore, C.
+       "Finding community structure in very large networks."
+       Physical Review E 70(6), 2004.
+    .. [3] Reichardt and Bornholdt "Statistical Mechanics of Community
+       Detection" Phys. Rev. E74, 2006.
+    .. [4] Newman, M. E. J."Analysis of weighted networks"
+       Physical Review E 70(5 Pt 2):056131, 2004.
+    """
+    if not G.size():
+        return [{n} for n in G]
+
+    if (cutoff < 1) or (cutoff > G.number_of_nodes()):
+        raise ValueError(f"cutoff must be between 1 and {len(G)}. Got {cutoff}.")
+    if best_n is not None:
+        if (best_n < 1) or (best_n > G.number_of_nodes()):
+            raise ValueError(f"best_n must be between 1 and {len(G)}. Got {best_n}.")
+        if best_n < cutoff:
+            raise ValueError(f"Must have best_n >= cutoff. Got {best_n} < {cutoff}")
+        if best_n == 1:
+            return [set(G)]
+    else:
+        best_n = G.number_of_nodes()
+
+    # retrieve generator object to construct output
+    community_gen = _greedy_modularity_communities_generator(
+        G, weight=weight, resolution=resolution
+    )
+
+    # construct the first best community
+    communities = next(community_gen)
+
+    # continue merging communities until one of the breaking criteria is satisfied
+    while len(communities) > cutoff:
+        try:
+            dq = next(community_gen)
+        # StopIteration occurs when communities are the connected components
+        except StopIteration:
+            communities = sorted(communities, key=len, reverse=True)
+            # if best_n requires more merging, merge big sets for highest modularity
+            while len(communities) > best_n:
+                comm1, comm2, *rest = communities
+                communities = [comm1 ^ comm2]
+                communities.extend(rest)
+            return communities
+
+        # keep going unless max_mod is reached or best_n says to merge more
+        if dq < 0 and len(communities) <= best_n:
+            break
+        communities = next(community_gen)
+
+    return sorted(communities, key=len, reverse=True)
+
+
+@not_implemented_for("directed")
+@not_implemented_for("multigraph")
+@nx._dispatchable(edge_attrs="weight")
+def naive_greedy_modularity_communities(G, resolution=1, weight=None):
+    r"""Find communities in G using greedy modularity maximization.
+
+    This implementation is O(n^4), much slower than alternatives, but it is
+    provided as an easy-to-understand reference implementation.
+
+    Greedy modularity maximization begins with each node in its own community
+    and joins the pair of communities that most increases modularity until no
+    such pair exists.
+
+    This function maximizes the generalized modularity, where `resolution`
+    is the resolution parameter, often expressed as $\gamma$.
+    See :func:`~networkx.algorithms.community.quality.modularity`.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        Graph must be simple and undirected.
+
+    resolution : float (default=1)
+        If resolution is less than 1, modularity favors larger communities.
+        Greater than 1 favors smaller communities.
+
+    weight : string or None, optional (default=None)
+        The name of an edge attribute that holds the numerical value used
+        as a weight.  If None, then each edge has weight 1.
+        The degree is the sum of the edge weights adjacent to the node.
+
+    Returns
+    -------
+    list
+        A list of sets of nodes, one for each community.
+        Sorted by length with largest communities first.
+
+    Examples
+    --------
+    >>> G = nx.karate_club_graph()
+    >>> c = nx.community.naive_greedy_modularity_communities(G)
+    >>> sorted(c[0])
+    [8, 14, 15, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]
+
+    See Also
+    --------
+    greedy_modularity_communities
+    modularity
+    """
+    # First create one community for each node
+    communities = [frozenset([u]) for u in G.nodes()]
+    # Track merges
+    merges = []
+    # Greedily merge communities until no improvement is possible
+    old_modularity = None
+    new_modularity = modularity(G, communities, resolution=resolution, weight=weight)
+    while old_modularity is None or new_modularity > old_modularity:
+        # Save modularity for comparison
+        old_modularity = new_modularity
+        # Find best pair to merge
+        trial_communities = list(communities)
+        to_merge = None
+        for i, u in enumerate(communities):
+            for j, v in enumerate(communities):
+                # Skip i==j and empty communities
+                if j <= i or len(u) == 0 or len(v) == 0:
+                    continue
+                # Merge communities u and v
+                trial_communities[j] = u | v
+                trial_communities[i] = frozenset([])
+                trial_modularity = modularity(
+                    G, trial_communities, resolution=resolution, weight=weight
+                )
+                if trial_modularity >= new_modularity:
+                    # Check if strictly better or tie
+                    if trial_modularity > new_modularity:
+                        # Found new best, save modularity and group indexes
+                        new_modularity = trial_modularity
+                        to_merge = (i, j, new_modularity - old_modularity)
+                    elif to_merge and min(i, j) < min(to_merge[0], to_merge[1]):
+                        # Break ties by choosing pair with lowest min id
+                        new_modularity = trial_modularity
+                        to_merge = (i, j, new_modularity - old_modularity)
+                # Un-merge
+                trial_communities[i] = u
+                trial_communities[j] = v
+        if to_merge is not None:
+            # If the best merge improves modularity, use it
+            merges.append(to_merge)
+            i, j, dq = to_merge
+            u, v = communities[i], communities[j]
+            communities[j] = u | v
+            communities[i] = frozenset([])
+    # Remove empty communities and sort
+    return sorted((c for c in communities if len(c) > 0), key=len, reverse=True)

rl/Lib/site-packages/networkx/algorithms/community/quality.py

@@ -0,0 +1,346 @@
+"""Functions for measuring the quality of a partition (into
+communities).
+
+"""
+
+from itertools import combinations
+
+import networkx as nx
+from networkx import NetworkXError
+from networkx.algorithms.community.community_utils import is_partition
+from networkx.utils.decorators import argmap
+
+__all__ = ["modularity", "partition_quality"]
+
+
+class NotAPartition(NetworkXError):
+    """Raised if a given collection is not a partition."""
+
+    def __init__(self, G, collection):
+        msg = f"{collection} is not a valid partition of the graph {G}"
+        super().__init__(msg)
+
+
+def _require_partition(G, partition):
+    """Decorator to check that a valid partition is input to a function
+
+    Raises :exc:`networkx.NetworkXError` if the partition is not valid.
+
+    This decorator should be used on functions whose first two arguments
+    are a graph and a partition of the nodes of that graph (in that
+    order)::
+
+        >>> @require_partition
+        ... def foo(G, partition):
+        ...     print("partition is valid!")
+        ...
+        >>> G = nx.complete_graph(5)
+        >>> partition = [{0, 1}, {2, 3}, {4}]
+        >>> foo(G, partition)
+        partition is valid!
+        >>> partition = [{0}, {2, 3}, {4}]
+        >>> foo(G, partition)
+        Traceback (most recent call last):
+          ...
+        networkx.exception.NetworkXError: `partition` is not a valid partition of the nodes of G
+        >>> partition = [{0, 1}, {1, 2, 3}, {4}]
+        >>> foo(G, partition)
+        Traceback (most recent call last):
+          ...
+        networkx.exception.NetworkXError: `partition` is not a valid partition of the nodes of G
+
+    """
+    if is_partition(G, partition):
+        return G, partition
+    raise nx.NetworkXError("`partition` is not a valid partition of the nodes of G")
+
+
+require_partition = argmap(_require_partition, (0, 1))
+
+
+@nx._dispatchable
+def intra_community_edges(G, partition):
+    """Returns the number of intra-community edges for a partition of `G`.
+
+    Parameters
+    ----------
+    G : NetworkX graph.
+
+    partition : iterable of sets of nodes
+        This must be a partition of the nodes of `G`.
+
+    The "intra-community edges" are those edges joining a pair of nodes
+    in the same block of the partition.
+
+    """
+    return sum(G.subgraph(block).size() for block in partition)
+
+
+@nx._dispatchable
+def inter_community_edges(G, partition):
+    """Returns the number of inter-community edges for a partition of `G`.
+    according to the given
+    partition of the nodes of `G`.
+
+    Parameters
+    ----------
+    G : NetworkX graph.
+
+    partition : iterable of sets of nodes
+        This must be a partition of the nodes of `G`.
+
+    The *inter-community edges* are those edges joining a pair of nodes
+    in different blocks of the partition.
+
+    Implementation note: this function creates an intermediate graph
+    that may require the same amount of memory as that of `G`.
+
+    """
+    # Alternate implementation that does not require constructing a new
+    # graph object (but does require constructing an affiliation
+    # dictionary):
+    #
+    #     aff = dict(chain.from_iterable(((v, block) for v in block)
+    #                                    for block in partition))
+    #     return sum(1 for u, v in G.edges() if aff[u] != aff[v])
+    #
+    MG = nx.MultiDiGraph if G.is_directed() else nx.MultiGraph
+    return nx.quotient_graph(G, partition, create_using=MG).size()
+
+
+@nx._dispatchable
+def inter_community_non_edges(G, partition):
+    """Returns the number of inter-community non-edges according to the
+    given partition of the nodes of `G`.
+
+    Parameters
+    ----------
+    G : NetworkX graph.
+
+    partition : iterable of sets of nodes
+        This must be a partition of the nodes of `G`.
+
+    A *non-edge* is a pair of nodes (undirected if `G` is undirected)
+    that are not adjacent in `G`. The *inter-community non-edges* are
+    those non-edges on a pair of nodes in different blocks of the
+    partition.
+
+    Implementation note: this function creates two intermediate graphs,
+    which may require up to twice the amount of memory as required to
+    store `G`.
+
+    """
+    # Alternate implementation that does not require constructing two
+    # new graph objects (but does require constructing an affiliation
+    # dictionary):
+    #
+    #     aff = dict(chain.from_iterable(((v, block) for v in block)
+    #                                    for block in partition))
+    #     return sum(1 for u, v in nx.non_edges(G) if aff[u] != aff[v])
+    #
+    return inter_community_edges(nx.complement(G), partition)
+
+
+@nx._dispatchable(edge_attrs="weight")
+def modularity(G, communities, weight="weight", resolution=1):
+    r"""Returns the modularity of the given partition of the graph.
+
+    Modularity is defined in [1]_ as
+
+    .. math::
+        Q = \frac{1}{2m} \sum_{ij} \left( A_{ij} - \gamma\frac{k_ik_j}{2m}\right)
+            \delta(c_i,c_j)
+
+    where $m$ is the number of edges (or sum of all edge weights as in [5]_),
+    $A$ is the adjacency matrix of `G`, $k_i$ is the (weighted) degree of $i$,
+    $\gamma$ is the resolution parameter, and $\delta(c_i, c_j)$ is 1 if $i$ and
+    $j$ are in the same community else 0.
+
+    According to [2]_ (and verified by some algebra) this can be reduced to
+
+    .. math::
+       Q = \sum_{c=1}^{n}
+       \left[ \frac{L_c}{m} - \gamma\left( \frac{k_c}{2m} \right) ^2 \right]
+
+    where the sum iterates over all communities $c$, $m$ is the number of edges,
+    $L_c$ is the number of intra-community links for community $c$,
+    $k_c$ is the sum of degrees of the nodes in community $c$,
+    and $\gamma$ is the resolution parameter.
+
+    The resolution parameter sets an arbitrary tradeoff between intra-group
+    edges and inter-group edges. More complex grouping patterns can be
+    discovered by analyzing the same network with multiple values of gamma
+    and then combining the results [3]_. That said, it is very common to
+    simply use gamma=1. More on the choice of gamma is in [4]_.
+
+    The second formula is the one actually used in calculation of the modularity.
+    For directed graphs the second formula replaces $k_c$ with $k^{in}_c k^{out}_c$.
+
+    Parameters
+    ----------
+    G : NetworkX Graph
+
+    communities : list or iterable of set of nodes
+        These node sets must represent a partition of G's nodes.
+
+    weight : string or None, optional (default="weight")
+        The edge attribute that holds the numerical value used
+        as a weight. If None or an edge does not have that attribute,
+        then that edge has weight 1.
+
+    resolution : float (default=1)
+        If resolution is less than 1, modularity favors larger communities.
+        Greater than 1 favors smaller communities.
+
+    Returns
+    -------
+    Q : float
+        The modularity of the partition.
+
+    Raises
+    ------
+    NotAPartition
+        If `communities` is not a partition of the nodes of `G`.
+
+    Examples
+    --------
+    >>> G = nx.barbell_graph(3, 0)
+    >>> nx.community.modularity(G, [{0, 1, 2}, {3, 4, 5}])
+    0.35714285714285715
+    >>> nx.community.modularity(G, nx.community.label_propagation_communities(G))
+    0.35714285714285715
+
+    References
+    ----------
+    .. [1] M. E. J. Newman "Networks: An Introduction", page 224.
+       Oxford University Press, 2011.
+    .. [2] Clauset, Aaron, Mark EJ Newman, and Cristopher Moore.
+       "Finding community structure in very large networks."
+       Phys. Rev. E 70.6 (2004). <https://arxiv.org/abs/cond-mat/0408187>
+    .. [3] Reichardt and Bornholdt "Statistical Mechanics of Community Detection"
+       Phys. Rev. E 74, 016110, 2006. https://doi.org/10.1103/PhysRevE.74.016110
+    .. [4] M. E. J. Newman, "Equivalence between modularity optimization and
+       maximum likelihood methods for community detection"
+       Phys. Rev. E 94, 052315, 2016. https://doi.org/10.1103/PhysRevE.94.052315
+    .. [5] Blondel, V.D. et al. "Fast unfolding of communities in large
+       networks" J. Stat. Mech 10008, 1-12 (2008).
+       https://doi.org/10.1088/1742-5468/2008/10/P10008
+    """
+    if not isinstance(communities, list):
+        communities = list(communities)
+    if not is_partition(G, communities):
+        raise NotAPartition(G, communities)
+
+    directed = G.is_directed()
+    if directed:
+        out_degree = dict(G.out_degree(weight=weight))
+        in_degree = dict(G.in_degree(weight=weight))
+        m = sum(out_degree.values())
+        norm = 1 / m**2
+    else:
+        out_degree = in_degree = dict(G.degree(weight=weight))
+        deg_sum = sum(out_degree.values())
+        m = deg_sum / 2
+        norm = 1 / deg_sum**2
+
+    def community_contribution(community):
+        comm = set(community)
+        L_c = sum(wt for u, v, wt in G.edges(comm, data=weight, default=1) if v in comm)
+
+        out_degree_sum = sum(out_degree[u] for u in comm)
+        in_degree_sum = sum(in_degree[u] for u in comm) if directed else out_degree_sum
+
+        return L_c / m - resolution * out_degree_sum * in_degree_sum * norm
+
+    return sum(map(community_contribution, communities))
+
+
+@require_partition
+@nx._dispatchable
+def partition_quality(G, partition):
+    """Returns the coverage and performance of a partition of G.
+
+    The *coverage* of a partition is the ratio of the number of
+    intra-community edges to the total number of edges in the graph.
+
+    The *performance* of a partition is the number of
+    intra-community edges plus inter-community non-edges divided by the total
+    number of potential edges.
+
+    This algorithm has complexity $O(C^2 + L)$ where C is the number of communities and L is the number of links.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    partition : sequence
+        Partition of the nodes of `G`, represented as a sequence of
+        sets of nodes (blocks). Each block of the partition represents a
+        community.
+
+    Returns
+    -------
+    (float, float)
+        The (coverage, performance) tuple of the partition, as defined above.
+
+    Raises
+    ------
+    NetworkXError
+        If `partition` is not a valid partition of the nodes of `G`.
+
+    Notes
+    -----
+    If `G` is a multigraph;
+        - for coverage, the multiplicity of edges is counted
+        - for performance, the result is -1 (total number of possible edges is not defined)
+
+    References
+    ----------
+    .. [1] Santo Fortunato.
+           "Community Detection in Graphs".
+           *Physical Reports*, Volume 486, Issue 3--5 pp. 75--174
+           <https://arxiv.org/abs/0906.0612>
+    """
+
+    node_community = {}
+    for i, community in enumerate(partition):
+        for node in community:
+            node_community[node] = i
+
+    # `performance` is not defined for multigraphs
+    if not G.is_multigraph():
+        # Iterate over the communities, quadratic, to calculate `possible_inter_community_edges`
+        possible_inter_community_edges = sum(
+            len(p1) * len(p2) for p1, p2 in combinations(partition, 2)
+        )
+
+        if G.is_directed():
+            possible_inter_community_edges *= 2
+    else:
+        possible_inter_community_edges = 0
+
+    # Compute the number of edges in the complete graph -- `n` nodes,
+    # directed or undirected, depending on `G`
+    n = len(G)
+    total_pairs = n * (n - 1)
+    if not G.is_directed():
+        total_pairs //= 2
+
+    intra_community_edges = 0
+    inter_community_non_edges = possible_inter_community_edges
+
+    # Iterate over the links to count `intra_community_edges` and `inter_community_non_edges`
+    for e in G.edges():
+        if node_community[e[0]] == node_community[e[1]]:
+            intra_community_edges += 1
+        else:
+            inter_community_non_edges -= 1
+
+    coverage = intra_community_edges / len(G.edges)
+
+    if G.is_multigraph():
+        performance = -1.0
+    else:
+        performance = (intra_community_edges + inter_community_non_edges) / total_pairs
+
+    return coverage, performance

rl/Lib/site-packages/networkx/algorithms/community/tests/test_asyn_fluid.py

@@ -0,0 +1,136 @@
+import pytest
+
+import networkx as nx
+from networkx import Graph, NetworkXError
+from networkx.algorithms.community import asyn_fluidc
+
+
+@pytest.mark.parametrize("graph_constructor", (nx.DiGraph, nx.MultiGraph))
+def test_raises_on_directed_and_multigraphs(graph_constructor):
+    G = graph_constructor([(0, 1), (1, 2)])
+    with pytest.raises(nx.NetworkXNotImplemented):
+        nx.community.asyn_fluidc(G, 1)
+
+
+def test_exceptions():
+    test = Graph()
+    test.add_node("a")
+    pytest.raises(NetworkXError, asyn_fluidc, test, "hi")
+    pytest.raises(NetworkXError, asyn_fluidc, test, -1)
+    pytest.raises(NetworkXError, asyn_fluidc, test, 3)
+    test.add_node("b")
+    pytest.raises(NetworkXError, asyn_fluidc, test, 1)
+
+
+def test_single_node():
+    test = Graph()
+
+    test.add_node("a")
+
+    # ground truth
+    ground_truth = {frozenset(["a"])}
+
+    communities = asyn_fluidc(test, 1)
+    result = {frozenset(c) for c in communities}
+    assert result == ground_truth
+
+
+def test_two_nodes():
+    test = Graph()
+
+    test.add_edge("a", "b")
+
+    # ground truth
+    ground_truth = {frozenset(["a"]), frozenset(["b"])}
+
+    communities = asyn_fluidc(test, 2)
+    result = {frozenset(c) for c in communities}
+    assert result == ground_truth
+
+
+def test_two_clique_communities():
+    test = Graph()
+
+    # c1
+    test.add_edge("a", "b")
+    test.add_edge("a", "c")
+    test.add_edge("b", "c")
+
+    # connection
+    test.add_edge("c", "d")
+
+    # c2
+    test.add_edge("d", "e")
+    test.add_edge("d", "f")
+    test.add_edge("f", "e")
+
+    # ground truth
+    ground_truth = {frozenset(["a", "c", "b"]), frozenset(["e", "d", "f"])}
+
+    communities = asyn_fluidc(test, 2, seed=7)
+    result = {frozenset(c) for c in communities}
+    assert result == ground_truth
+
+
+def test_five_clique_ring():
+    test = Graph()
+
+    # c1
+    test.add_edge("1a", "1b")
+    test.add_edge("1a", "1c")
+    test.add_edge("1a", "1d")
+    test.add_edge("1b", "1c")
+    test.add_edge("1b", "1d")
+    test.add_edge("1c", "1d")
+
+    # c2
+    test.add_edge("2a", "2b")
+    test.add_edge("2a", "2c")
+    test.add_edge("2a", "2d")
+    test.add_edge("2b", "2c")
+    test.add_edge("2b", "2d")
+    test.add_edge("2c", "2d")
+
+    # c3
+    test.add_edge("3a", "3b")
+    test.add_edge("3a", "3c")
+    test.add_edge("3a", "3d")
+    test.add_edge("3b", "3c")
+    test.add_edge("3b", "3d")
+    test.add_edge("3c", "3d")
+
+    # c4
+    test.add_edge("4a", "4b")
+    test.add_edge("4a", "4c")
+    test.add_edge("4a", "4d")
+    test.add_edge("4b", "4c")
+    test.add_edge("4b", "4d")
+    test.add_edge("4c", "4d")
+
+    # c5
+    test.add_edge("5a", "5b")
+    test.add_edge("5a", "5c")
+    test.add_edge("5a", "5d")
+    test.add_edge("5b", "5c")
+    test.add_edge("5b", "5d")
+    test.add_edge("5c", "5d")
+
+    # connections
+    test.add_edge("1a", "2c")
+    test.add_edge("2a", "3c")
+    test.add_edge("3a", "4c")
+    test.add_edge("4a", "5c")
+    test.add_edge("5a", "1c")
+
+    # ground truth
+    ground_truth = {
+        frozenset(["1a", "1b", "1c", "1d"]),
+        frozenset(["2a", "2b", "2c", "2d"]),
+        frozenset(["3a", "3b", "3c", "3d"]),
+        frozenset(["4a", "4b", "4c", "4d"]),
+        frozenset(["5a", "5b", "5c", "5d"]),
+    }
+
+    communities = asyn_fluidc(test, 5, seed=9)
+    result = {frozenset(c) for c in communities}
+    assert result == ground_truth

rl/Lib/site-packages/networkx/algorithms/community/tests/test_centrality.py

@@ -0,0 +1,85 @@
+"""Unit tests for the :mod:`networkx.algorithms.community.centrality`
+module.
+
+"""
+
+from operator import itemgetter
+
+import networkx as nx
+
+
+def set_of_sets(iterable):
+    return set(map(frozenset, iterable))
+
+
+def validate_communities(result, expected):
+    assert set_of_sets(result) == set_of_sets(expected)
+
+
+def validate_possible_communities(result, *expected):
+    assert any(set_of_sets(result) == set_of_sets(p) for p in expected)
+
+
+class TestGirvanNewman:
+    """Unit tests for the
+    :func:`networkx.algorithms.community.centrality.girvan_newman`
+    function.
+
+    """
+
+    def test_no_edges(self):
+        G = nx.empty_graph(3)
+        communities = list(nx.community.girvan_newman(G))
+        assert len(communities) == 1
+        validate_communities(communities[0], [{0}, {1}, {2}])
+
+    def test_undirected(self):
+        # Start with the graph .-.-.-.
+        G = nx.path_graph(4)
+        communities = list(nx.community.girvan_newman(G))
+        assert len(communities) == 3
+        # After one removal, we get the graph .-. .-.
+        validate_communities(communities[0], [{0, 1}, {2, 3}])
+        # After the next, we get the graph .-. . ., but there are two
+        # symmetric possible versions.
+        validate_possible_communities(
+            communities[1], [{0}, {1}, {2, 3}], [{0, 1}, {2}, {3}]
+        )
+        # After the last removal, we always get the empty graph.
+        validate_communities(communities[2], [{0}, {1}, {2}, {3}])
+
+    def test_directed(self):
+        G = nx.DiGraph(nx.path_graph(4))
+        communities = list(nx.community.girvan_newman(G))
+        assert len(communities) == 3
+        validate_communities(communities[0], [{0, 1}, {2, 3}])
+        validate_possible_communities(
+            communities[1], [{0}, {1}, {2, 3}], [{0, 1}, {2}, {3}]
+        )
+        validate_communities(communities[2], [{0}, {1}, {2}, {3}])
+
+    def test_selfloops(self):
+        G = nx.path_graph(4)
+        G.add_edge(0, 0)
+        G.add_edge(2, 2)
+        communities = list(nx.community.girvan_newman(G))
+        assert len(communities) == 3
+        validate_communities(communities[0], [{0, 1}, {2, 3}])
+        validate_possible_communities(
+            communities[1], [{0}, {1}, {2, 3}], [{0, 1}, {2}, {3}]
+        )
+        validate_communities(communities[2], [{0}, {1}, {2}, {3}])
+
+    def test_most_valuable_edge(self):
+        G = nx.Graph()
+        G.add_weighted_edges_from([(0, 1, 3), (1, 2, 2), (2, 3, 1)])
+        # Let the most valuable edge be the one with the highest weight.
+
+        def heaviest(G):
+            return max(G.edges(data="weight"), key=itemgetter(2))[:2]
+
+        communities = list(nx.community.girvan_newman(G, heaviest))
+        assert len(communities) == 3
+        validate_communities(communities[0], [{0}, {1, 2, 3}])
+        validate_communities(communities[1], [{0}, {1}, {2, 3}])
+        validate_communities(communities[2], [{0}, {1}, {2}, {3}])

rl/Lib/site-packages/networkx/algorithms/community/tests/test_divisive.py

@@ -0,0 +1,106 @@
+import pytest
+
+import networkx as nx
+
+
+def test_edge_betweenness_partition():
+    G = nx.barbell_graph(3, 0)
+    C = nx.community.edge_betweenness_partition(G, 2)
+    answer = [{0, 1, 2}, {3, 4, 5}]
+    assert len(C) == len(answer)
+    for s in answer:
+        assert s in C
+
+    G = nx.barbell_graph(3, 1)
+    C = nx.community.edge_betweenness_partition(G, 3)
+    answer = [{0, 1, 2}, {4, 5, 6}, {3}]
+    assert len(C) == len(answer)
+    for s in answer:
+        assert s in C
+
+    C = nx.community.edge_betweenness_partition(G, 7)
+    answer = [{n} for n in G]
+    assert len(C) == len(answer)
+    for s in answer:
+        assert s in C
+
+    C = nx.community.edge_betweenness_partition(G, 1)
+    assert C == [set(G)]
+
+    C = nx.community.edge_betweenness_partition(G, 1, weight="weight")
+    assert C == [set(G)]
+
+    with pytest.raises(nx.NetworkXError):
+        nx.community.edge_betweenness_partition(G, 0)
+
+    with pytest.raises(nx.NetworkXError):
+        nx.community.edge_betweenness_partition(G, -1)
+
+    with pytest.raises(nx.NetworkXError):
+        nx.community.edge_betweenness_partition(G, 10)
+
+
+def test_edge_current_flow_betweenness_partition():
+    pytest.importorskip("scipy")
+
+    G = nx.barbell_graph(3, 0)
+    C = nx.community.edge_current_flow_betweenness_partition(G, 2)
+    answer = [{0, 1, 2}, {3, 4, 5}]
+    assert len(C) == len(answer)
+    for s in answer:
+        assert s in C
+
+    G = nx.barbell_graph(3, 1)
+    C = nx.community.edge_current_flow_betweenness_partition(G, 2)
+    answers = [[{0, 1, 2, 3}, {4, 5, 6}], [{0, 1, 2}, {3, 4, 5, 6}]]
+    assert len(C) == len(answers[0])
+    assert any(all(s in answer for s in C) for answer in answers)
+
+    C = nx.community.edge_current_flow_betweenness_partition(G, 3)
+    answer = [{0, 1, 2}, {4, 5, 6}, {3}]
+    assert len(C) == len(answer)
+    for s in answer:
+        assert s in C
+
+    C = nx.community.edge_current_flow_betweenness_partition(G, 4)
+    answers = [[{1, 2}, {4, 5, 6}, {3}, {0}], [{0, 1, 2}, {5, 6}, {3}, {4}]]
+    assert len(C) == len(answers[0])
+    assert any(all(s in answer for s in C) for answer in answers)
+
+    C = nx.community.edge_current_flow_betweenness_partition(G, 5)
+    answer = [{1, 2}, {5, 6}, {3}, {0}, {4}]
+    assert len(C) == len(answer)
+    for s in answer:
+        assert s in C
+
+    C = nx.community.edge_current_flow_betweenness_partition(G, 6)
+    answers = [[{2}, {5, 6}, {3}, {0}, {4}, {1}], [{1, 2}, {6}, {3}, {0}, {4}, {5}]]
+    assert len(C) == len(answers[0])
+    assert any(all(s in answer for s in C) for answer in answers)
+
+    C = nx.community.edge_current_flow_betweenness_partition(G, 7)
+    answer = [{n} for n in G]
+    assert len(C) == len(answer)
+    for s in answer:
+        assert s in C
+
+    C = nx.community.edge_current_flow_betweenness_partition(G, 1)
+    assert C == [set(G)]
+
+    C = nx.community.edge_current_flow_betweenness_partition(G, 1, weight="weight")
+    assert C == [set(G)]
+
+    with pytest.raises(nx.NetworkXError):
+        nx.community.edge_current_flow_betweenness_partition(G, 0)
+
+    with pytest.raises(nx.NetworkXError):
+        nx.community.edge_current_flow_betweenness_partition(G, -1)
+
+    with pytest.raises(nx.NetworkXError):
+        nx.community.edge_current_flow_betweenness_partition(G, 10)
+
+    N = 10
+    G = nx.empty_graph(N)
+    for i in range(2, N - 1):
+        C = nx.community.edge_current_flow_betweenness_partition(G, i)
+        assert C == [{n} for n in G]

rl/Lib/site-packages/networkx/algorithms/community/tests/test_kclique.py

@@ -0,0 +1,91 @@
+from itertools import combinations
+
+import pytest
+
+import networkx as nx
+
+
+def test_overlapping_K5():
+    G = nx.Graph()
+    G.add_edges_from(combinations(range(5), 2))  # Add a five clique
+    G.add_edges_from(combinations(range(2, 7), 2))  # Add another five clique
+    c = list(nx.community.k_clique_communities(G, 4))
+    assert c == [frozenset(range(7))]
+    c = set(nx.community.k_clique_communities(G, 5))
+    assert c == {frozenset(range(5)), frozenset(range(2, 7))}
+
+
+def test_isolated_K5():
+    G = nx.Graph()
+    G.add_edges_from(combinations(range(5), 2))  # Add a five clique
+    G.add_edges_from(combinations(range(5, 10), 2))  # Add another five clique
+    c = set(nx.community.k_clique_communities(G, 5))
+    assert c == {frozenset(range(5)), frozenset(range(5, 10))}
+
+
+class TestZacharyKarateClub:
+    def setup_method(self):
+        self.G = nx.karate_club_graph()
+
+    def _check_communities(self, k, expected):
+        communities = set(nx.community.k_clique_communities(self.G, k))
+        assert communities == expected
+
+    def test_k2(self):
+        # clique percolation with k=2 is just connected components
+        expected = {frozenset(self.G)}
+        self._check_communities(2, expected)
+
+    def test_k3(self):
+        comm1 = [
+            0,
+            1,
+            2,
+            3,
+            7,
+            8,
+            12,
+            13,
+            14,
+            15,
+            17,
+            18,
+            19,
+            20,
+            21,
+            22,
+            23,
+            26,
+            27,
+            28,
+            29,
+            30,
+            31,
+            32,
+            33,
+        ]
+        comm2 = [0, 4, 5, 6, 10, 16]
+        comm3 = [24, 25, 31]
+        expected = {frozenset(comm1), frozenset(comm2), frozenset(comm3)}
+        self._check_communities(3, expected)
+
+    def test_k4(self):
+        expected = {
+            frozenset([0, 1, 2, 3, 7, 13]),
+            frozenset([8, 32, 30, 33]),
+            frozenset([32, 33, 29, 23]),
+        }
+        self._check_communities(4, expected)
+
+    def test_k5(self):
+        expected = {frozenset([0, 1, 2, 3, 7, 13])}
+        self._check_communities(5, expected)
+
+    def test_k6(self):
+        expected = set()
+        self._check_communities(6, expected)
+
+
+def test_bad_k():
+    with pytest.raises(nx.NetworkXError):
+        list(nx.community.k_clique_communities(nx.Graph(), 1))

rl/Lib/site-packages/networkx/algorithms/community/tests/test_kernighan_lin.py

@@ -0,0 +1,92 @@
+"""Unit tests for the :mod:`networkx.algorithms.community.kernighan_lin`
+module.
+"""
+
+from itertools import permutations
+
+import pytest
+
+import networkx as nx
+from networkx.algorithms.community import kernighan_lin_bisection
+
+
+def assert_partition_equal(x, y):
+    assert set(map(frozenset, x)) == set(map(frozenset, y))
+
+
+def test_partition():
+    G = nx.barbell_graph(3, 0)
+    C = kernighan_lin_bisection(G)
+    assert_partition_equal(C, [{0, 1, 2}, {3, 4, 5}])
+
+
+def test_partition_argument():
+    G = nx.barbell_graph(3, 0)
+    partition = [{0, 1, 2}, {3, 4, 5}]
+    C = kernighan_lin_bisection(G, partition)
+    assert_partition_equal(C, partition)
+
+
+def test_partition_argument_non_integer_nodes():
+    G = nx.Graph([("A", "B"), ("A", "C"), ("B", "C"), ("C", "D")])
+    partition = ({"A", "B"}, {"C", "D"})
+    C = kernighan_lin_bisection(G, partition)
+    assert_partition_equal(C, partition)
+
+
+def test_seed_argument():
+    G = nx.barbell_graph(3, 0)
+    C = kernighan_lin_bisection(G, seed=1)
+    assert_partition_equal(C, [{0, 1, 2}, {3, 4, 5}])
+
+
+def test_non_disjoint_partition():
+    with pytest.raises(nx.NetworkXError):
+        G = nx.barbell_graph(3, 0)
+        partition = ({0, 1, 2}, {2, 3, 4, 5})
+        kernighan_lin_bisection(G, partition)
+
+
+def test_too_many_blocks():
+    with pytest.raises(nx.NetworkXError):
+        G = nx.barbell_graph(3, 0)
+        partition = ({0, 1}, {2}, {3, 4, 5})
+        kernighan_lin_bisection(G, partition)
+
+
+def test_multigraph():
+    G = nx.cycle_graph(4)
+    M = nx.MultiGraph(G.edges())
+    M.add_edges_from(G.edges())
+    M.remove_edge(1, 2)
+    for labels in permutations(range(4)):
+        mapping = dict(zip(M, labels))
+        A, B = kernighan_lin_bisection(nx.relabel_nodes(M, mapping), seed=0)
+        assert_partition_equal(
+            [A, B], [{mapping[0], mapping[1]}, {mapping[2], mapping[3]}]
+        )
+
+
+def test_max_iter_argument():
+    G = nx.Graph(
+        [
+            ("A", "B", {"weight": 1}),
+            ("A", "C", {"weight": 2}),
+            ("A", "D", {"weight": 3}),
+            ("A", "E", {"weight": 2}),
+            ("A", "F", {"weight": 4}),
+            ("B", "C", {"weight": 1}),
+            ("B", "D", {"weight": 4}),
+            ("B", "E", {"weight": 2}),
+            ("B", "F", {"weight": 1}),
+            ("C", "D", {"weight": 3}),
+            ("C", "E", {"weight": 2}),
+            ("C", "F", {"weight": 1}),
+            ("D", "E", {"weight": 4}),
+            ("D", "F", {"weight": 3}),
+            ("E", "F", {"weight": 2}),
+        ]
+    )
+    partition = ({"A", "B", "C"}, {"D", "E", "F"})
+    C = kernighan_lin_bisection(G, partition, max_iter=1)
+    assert_partition_equal(C, ({"A", "F", "C"}, {"D", "E", "B"}))

rl/Lib/site-packages/networkx/algorithms/community/tests/test_label_propagation.py

@@ -0,0 +1,241 @@
+from itertools import chain, combinations
+
+import pytest
+
+import networkx as nx
+
+
+def test_directed_not_supported():
+    with pytest.raises(nx.NetworkXNotImplemented):
+        # not supported for directed graphs
+        test = nx.DiGraph()
+        test.add_edge("a", "b")
+        test.add_edge("a", "c")
+        test.add_edge("b", "d")
+        result = nx.community.label_propagation_communities(test)
+
+
+def test_iterator_vs_iterable():
+    G = nx.empty_graph("a")
+    assert list(nx.community.label_propagation_communities(G)) == [{"a"}]
+    for community in nx.community.label_propagation_communities(G):
+        assert community == {"a"}
+    pytest.raises(TypeError, next, nx.community.label_propagation_communities(G))
+
+
+def test_one_node():
+    test = nx.Graph()
+    test.add_node("a")
+
+    # The expected communities are:
+    ground_truth = {frozenset(["a"])}
+
+    communities = nx.community.label_propagation_communities(test)
+    result = {frozenset(c) for c in communities}
+    assert result == ground_truth
+
+
+def test_unconnected_communities():
+    test = nx.Graph()
+    # community 1
+    test.add_edge("a", "c")
+    test.add_edge("a", "d")
+    test.add_edge("d", "c")
+    # community 2
+    test.add_edge("b", "e")
+    test.add_edge("e", "f")
+    test.add_edge("f", "b")
+
+    # The expected communities are:
+    ground_truth = {frozenset(["a", "c", "d"]), frozenset(["b", "e", "f"])}
+
+    communities = nx.community.label_propagation_communities(test)
+    result = {frozenset(c) for c in communities}
+    assert result == ground_truth
+
+
+def test_connected_communities():
+    test = nx.Graph()
+    # community 1
+    test.add_edge("a", "b")
+    test.add_edge("c", "a")
+    test.add_edge("c", "b")
+    test.add_edge("d", "a")
+    test.add_edge("d", "b")
+    test.add_edge("d", "c")
+    test.add_edge("e", "a")
+    test.add_edge("e", "b")
+    test.add_edge("e", "c")
+    test.add_edge("e", "d")
+    # community 2
+    test.add_edge("1", "2")
+    test.add_edge("3", "1")
+    test.add_edge("3", "2")
+    test.add_edge("4", "1")
+    test.add_edge("4", "2")
+    test.add_edge("4", "3")
+    test.add_edge("5", "1")
+    test.add_edge("5", "2")
+    test.add_edge("5", "3")
+    test.add_edge("5", "4")
+    # edge between community 1 and 2
+    test.add_edge("a", "1")
+    # community 3
+    test.add_edge("x", "y")
+    # community 4 with only a single node
+    test.add_node("z")
+
+    # The expected communities are:
+    ground_truth1 = {
+        frozenset(["a", "b", "c", "d", "e"]),
+        frozenset(["1", "2", "3", "4", "5"]),
+        frozenset(["x", "y"]),
+        frozenset(["z"]),
+    }
+    ground_truth2 = {
+        frozenset(["a", "b", "c", "d", "e", "1", "2", "3", "4", "5"]),
+        frozenset(["x", "y"]),
+        frozenset(["z"]),
+    }
+    ground_truth = (ground_truth1, ground_truth2)
+
+    communities = nx.community.label_propagation_communities(test)
+    result = {frozenset(c) for c in communities}
+    assert result in ground_truth
+
+
+def test_termination():
+    # ensure termination of asyn_lpa_communities in two cases
+    # that led to an endless loop in a previous version
+    test1 = nx.karate_club_graph()
+    test2 = nx.caveman_graph(2, 10)
+    test2.add_edges_from([(0, 20), (20, 10)])
+    nx.community.asyn_lpa_communities(test1)
+    nx.community.asyn_lpa_communities(test2)
+
+
+class TestAsynLpaCommunities:
+    def _check_communities(self, G, expected):
+        """Checks that the communities computed from the given graph ``G``
+        using the :func:`~networkx.asyn_lpa_communities` function match
+        the set of nodes given in ``expected``.
+
+        ``expected`` must be a :class:`set` of :class:`frozenset`
+        instances, each element of which is a node in the graph.
+
+        """
+        communities = nx.community.asyn_lpa_communities(G)
+        result = {frozenset(c) for c in communities}
+        assert result == expected
+
+    def test_null_graph(self):
+        G = nx.null_graph()
+        ground_truth = set()
+        self._check_communities(G, ground_truth)
+
+    def test_single_node(self):
+        G = nx.empty_graph(1)
+        ground_truth = {frozenset([0])}
+        self._check_communities(G, ground_truth)
+
+    def test_simple_communities(self):
+        # This graph is the disjoint union of two triangles.
+        G = nx.Graph(["ab", "ac", "bc", "de", "df", "fe"])
+        ground_truth = {frozenset("abc"), frozenset("def")}
+        self._check_communities(G, ground_truth)
+
+    def test_seed_argument(self):
+        G = nx.Graph(["ab", "ac", "bc", "de", "df", "fe"])
+        ground_truth = {frozenset("abc"), frozenset("def")}
+        communities = nx.community.asyn_lpa_communities(G, seed=1)
+        result = {frozenset(c) for c in communities}
+        assert result == ground_truth
+
+    def test_several_communities(self):
+        # This graph is the disjoint union of five triangles.
+        ground_truth = {frozenset(range(3 * i, 3 * (i + 1))) for i in range(5)}
+        edges = chain.from_iterable(combinations(c, 2) for c in ground_truth)
+        G = nx.Graph(edges)
+        self._check_communities(G, ground_truth)
+
+
+class TestFastLabelPropagationCommunities:
+    N = 100  # number of nodes
+    K = 15  # average node degree
+
+    def _check_communities(self, G, truth, weight=None, seed=42):
+        C = nx.community.fast_label_propagation_communities(G, weight=weight, seed=seed)
+        assert {frozenset(c) for c in C} == truth
+
+    def test_null_graph(self):
+        G = nx.null_graph()
+        truth = set()
+        self._check_communities(G, truth)
+
+    def test_empty_graph(self):
+        G = nx.empty_graph(self.N)
+        truth = {frozenset([i]) for i in G}
+        self._check_communities(G, truth)
+
+    def test_star_graph(self):
+        G = nx.star_graph(self.N)
+        truth = {frozenset(G)}
+        self._check_communities(G, truth)
+
+    def test_complete_graph(self):
+        G = nx.complete_graph(self.N)
+        truth = {frozenset(G)}
+        self._check_communities(G, truth)
+
+    def test_bipartite_graph(self):
+        G = nx.complete_bipartite_graph(self.N // 2, self.N // 2)
+        truth = {frozenset(G)}
+        self._check_communities(G, truth)
+
+    def test_random_graph(self):
+        G = nx.gnm_random_graph(self.N, self.N * self.K // 2, seed=42)
+        truth = {frozenset(G)}
+        self._check_communities(G, truth)
+
+    def test_disjoin_cliques(self):
+        G = nx.Graph(["ab", "AB", "AC", "BC", "12", "13", "14", "23", "24", "34"])
+        truth = {frozenset("ab"), frozenset("ABC"), frozenset("1234")}
+        self._check_communities(G, truth)
+
+    def test_ring_of_cliques(self):
+        N, K = self.N, self.K
+        G = nx.ring_of_cliques(N, K)
+        truth = {frozenset([K * i + k for k in range(K)]) for i in range(N)}
+        self._check_communities(G, truth)
+
+    def test_larger_graph(self):
+        G = nx.gnm_random_graph(100 * self.N, 50 * self.N * self.K, seed=42)
+        nx.community.fast_label_propagation_communities(G)
+
+    def test_graph_type(self):
+        G1 = nx.complete_graph(self.N, nx.MultiDiGraph())
+        G2 = nx.MultiGraph(G1)
+        G3 = nx.DiGraph(G1)
+        G4 = nx.Graph(G1)
+        truth = {frozenset(G1)}
+        self._check_communities(G1, truth)
+        self._check_communities(G2, truth)
+        self._check_communities(G3, truth)
+        self._check_communities(G4, truth)
+
+    def test_weight_argument(self):
+        G = nx.MultiDiGraph()
+        G.add_edge(1, 2, weight=1.41)
+        G.add_edge(2, 1, weight=1.41)
+        G.add_edge(2, 3)
+        G.add_edge(3, 4, weight=3.14)
+        truth = {frozenset({1, 2}), frozenset({3, 4})}
+        self._check_communities(G, truth, weight="weight")
+
+    def test_seed_argument(self):
+        G = nx.karate_club_graph()
+        C = nx.community.fast_label_propagation_communities(G, seed=2023)
+        truth = {frozenset(c) for c in C}
+        self._check_communities(G, truth, seed=2023)
+        # smoke test that seed=None works
+        C = nx.community.fast_label_propagation_communities(G, seed=None)

rl/Lib/site-packages/networkx/algorithms/community/tests/test_louvain.py

@@ -0,0 +1,264 @@
+import pytest
+
+import networkx as nx
+
+
+def test_modularity_increase():
+    G = nx.LFR_benchmark_graph(
+        250, 3, 1.5, 0.009, average_degree=5, min_community=20, seed=10
+    )
+    partition = [{u} for u in G.nodes()]
+    mod = nx.community.modularity(G, partition)
+    partition = nx.community.louvain_communities(G)
+
+    assert nx.community.modularity(G, partition) > mod
+
+
+def test_valid_partition():
+    G = nx.LFR_benchmark_graph(
+        250, 3, 1.5, 0.009, average_degree=5, min_community=20, seed=10
+    )
+    H = G.to_directed()
+    partition = nx.community.louvain_communities(G)
+    partition2 = nx.community.louvain_communities(H)
+
+    assert nx.community.is_partition(G, partition)
+    assert nx.community.is_partition(H, partition2)
+
+
+def test_karate_club_partition():
+    G = nx.karate_club_graph()
+    part = [
+        {0, 1, 2, 3, 7, 9, 11, 12, 13, 17, 19, 21},
+        {16, 4, 5, 6, 10},
+        {23, 25, 27, 28, 24, 31},
+        {32, 33, 8, 14, 15, 18, 20, 22, 26, 29, 30},
+    ]
+    partition = nx.community.louvain_communities(G, seed=2, weight=None)
+
+    assert part == partition
+
+
+def test_partition_iterator():
+    G = nx.path_graph(15)
+    parts_iter = nx.community.louvain_partitions(G, seed=42)
+    first_part = next(parts_iter)
+    first_copy = [s.copy() for s in first_part]
+
+    # gh-5901 reports sets changing after next partition is yielded
+    assert first_copy[0] == first_part[0]
+    second_part = next(parts_iter)
+    assert first_copy[0] == first_part[0]
+
+
+def test_undirected_selfloops():
+    G = nx.karate_club_graph()
+    expected_partition = nx.community.louvain_communities(G, seed=2, weight=None)
+    part = [
+        {0, 1, 2, 3, 7, 9, 11, 12, 13, 17, 19, 21},
+        {16, 4, 5, 6, 10},
+        {23, 25, 27, 28, 24, 31},
+        {32, 33, 8, 14, 15, 18, 20, 22, 26, 29, 30},
+    ]
+    assert expected_partition == part
+
+    G.add_weighted_edges_from([(i, i, i * 1000) for i in range(9)])
+    # large self-loop weight impacts partition
+    partition = nx.community.louvain_communities(G, seed=2, weight="weight")
+    assert part != partition
+
+    # small self-loop weights aren't enough to impact partition in this graph
+    partition = nx.community.louvain_communities(G, seed=2, weight=None)
+    assert part == partition
+
+
+def test_directed_selfloops():
+    G = nx.DiGraph()
+    G.add_nodes_from(range(11))
+    G_edges = [
+        (0, 2),
+        (0, 1),
+        (1, 0),
+        (2, 1),
+        (2, 0),
+        (3, 4),
+        (4, 3),
+        (7, 8),
+        (8, 7),
+        (9, 10),
+        (10, 9),
+    ]
+    G.add_edges_from(G_edges)
+    G_expected_partition = nx.community.louvain_communities(G, seed=123, weight=None)
+
+    G.add_weighted_edges_from([(i, i, i * 1000) for i in range(3)])
+    # large self-loop weight impacts partition
+    G_partition = nx.community.louvain_communities(G, seed=123, weight="weight")
+    assert G_partition != G_expected_partition
+
+    # small self-loop weights aren't enough to impact partition in this graph
+    G_partition = nx.community.louvain_communities(G, seed=123, weight=None)
+    assert G_partition == G_expected_partition
+
+
+def test_directed_partition():
+    """
+    Test 2 cases that were looping infinitely
+    from issues #5175 and #5704
+    """
+    G = nx.DiGraph()
+    H = nx.DiGraph()
+    G.add_nodes_from(range(10))
+    H.add_nodes_from([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11])
+    G_edges = [
+        (0, 2),
+        (0, 1),
+        (1, 0),
+        (2, 1),
+        (2, 0),
+        (3, 4),
+        (4, 3),
+        (7, 8),
+        (8, 7),
+        (9, 10),
+        (10, 9),
+    ]
+    H_edges = [
+        (1, 2),
+        (1, 6),
+        (1, 9),
+        (2, 3),
+        (2, 4),
+        (2, 5),
+        (3, 4),
+        (4, 3),
+        (4, 5),
+        (5, 4),
+        (6, 7),
+        (6, 8),
+        (9, 10),
+        (9, 11),
+        (10, 11),
+        (11, 10),
+    ]
+    G.add_edges_from(G_edges)
+    H.add_edges_from(H_edges)
+
+    G_expected_partition = [{0, 1, 2}, {3, 4}, {5}, {6}, {8, 7}, {9, 10}]
+    G_partition = nx.community.louvain_communities(G, seed=123, weight=None)
+
+    H_expected_partition = [{2, 3, 4, 5}, {8, 1, 6, 7}, {9, 10, 11}]
+    H_partition = nx.community.louvain_communities(H, seed=123, weight=None)
+
+    assert G_partition == G_expected_partition
+    assert H_partition == H_expected_partition
+
+
+def test_none_weight_param():
+    G = nx.karate_club_graph()
+    nx.set_edge_attributes(
+        G, {edge: i * i for i, edge in enumerate(G.edges)}, name="foo"
+    )
+
+    part = [
+        {0, 1, 2, 3, 7, 9, 11, 12, 13, 17, 19, 21},
+        {16, 4, 5, 6, 10},
+        {23, 25, 27, 28, 24, 31},
+        {32, 33, 8, 14, 15, 18, 20, 22, 26, 29, 30},
+    ]
+    partition1 = nx.community.louvain_communities(G, weight=None, seed=2)
+    partition2 = nx.community.louvain_communities(G, weight="foo", seed=2)
+    partition3 = nx.community.louvain_communities(G, weight="weight", seed=2)
+
+    assert part == partition1
+    assert part != partition2
+    assert part != partition3
+    assert partition2 != partition3
+
+
+def test_quality():
+    G = nx.LFR_benchmark_graph(
+        250, 3, 1.5, 0.009, average_degree=5, min_community=20, seed=10
+    )
+    H = nx.gn_graph(200, seed=1234)
+    I = nx.MultiGraph(G)
+    J = nx.MultiDiGraph(H)
+
+    partition = nx.community.louvain_communities(G)
+    partition2 = nx.community.louvain_communities(H)
+    partition3 = nx.community.louvain_communities(I)
+    partition4 = nx.community.louvain_communities(J)
+
+    quality = nx.community.partition_quality(G, partition)[0]
+    quality2 = nx.community.partition_quality(H, partition2)[0]
+    quality3 = nx.community.partition_quality(I, partition3)[0]
+    quality4 = nx.community.partition_quality(J, partition4)[0]
+
+    assert quality >= 0.65
+    assert quality2 >= 0.65
+    assert quality3 >= 0.65
+    assert quality4 >= 0.65
+
+
+def test_multigraph():
+    G = nx.karate_club_graph()
+    H = nx.MultiGraph(G)
+    G.add_edge(0, 1, weight=10)
+    H.add_edge(0, 1, weight=9)
+    G.add_edge(0, 9, foo=20)
+    H.add_edge(0, 9, foo=20)
+
+    partition1 = nx.community.louvain_communities(G, seed=1234)
+    partition2 = nx.community.louvain_communities(H, seed=1234)
+    partition3 = nx.community.louvain_communities(H, weight="foo", seed=1234)
+
+    assert partition1 == partition2 != partition3
+
+
+def test_resolution():
+    G = nx.LFR_benchmark_graph(
+        250, 3, 1.5, 0.009, average_degree=5, min_community=20, seed=10
+    )
+
+    partition1 = nx.community.louvain_communities(G, resolution=0.5, seed=12)
+    partition2 = nx.community.louvain_communities(G, seed=12)
+    partition3 = nx.community.louvain_communities(G, resolution=2, seed=12)
+
+    assert len(partition1) <= len(partition2) <= len(partition3)
+
+
+def test_threshold():
+    G = nx.LFR_benchmark_graph(
+        250, 3, 1.5, 0.009, average_degree=5, min_community=20, seed=10
+    )
+    partition1 = nx.community.louvain_communities(G, threshold=0.3, seed=2)
+    partition2 = nx.community.louvain_communities(G, seed=2)
+    mod1 = nx.community.modularity(G, partition1)
+    mod2 = nx.community.modularity(G, partition2)
+
+    assert mod1 <= mod2
+
+
+def test_empty_graph():
+    G = nx.Graph()
+    G.add_nodes_from(range(5))
+    expected = [{0}, {1}, {2}, {3}, {4}]
+    assert nx.community.louvain_communities(G) == expected
+
+
+def test_max_level():
+    G = nx.LFR_benchmark_graph(
+        250, 3, 1.5, 0.009, average_degree=5, min_community=20, seed=10
+    )
+    parts_iter = nx.community.louvain_partitions(G, seed=42)
+    for max_level, expected in enumerate(parts_iter, 1):
+        partition = nx.community.louvain_communities(G, max_level=max_level, seed=42)
+        assert partition == expected
+    assert max_level > 1  # Ensure we are actually testing max_level
+    # max_level is an upper limit; it's okay if we stop before it's hit.
+    partition = nx.community.louvain_communities(G, max_level=max_level + 1, seed=42)
+    assert partition == expected
+    with pytest.raises(
+        ValueError, match="max_level argument must be a positive integer"
+    ):
+        nx.community.louvain_communities(G, max_level=0)

rl/Lib/site-packages/networkx/algorithms/community/tests/test_lukes.py

@@ -0,0 +1,152 @@
+from itertools import product
+
+import pytest
+
+import networkx as nx
+
+EWL = "e_weight"
+NWL = "n_weight"
+
+
+# first test from the Lukes original paper
+def paper_1_case(float_edge_wt=False, explicit_node_wt=True, directed=False):
+    # problem-specific constants
+    limit = 3
+
+    # configuration
+    if float_edge_wt:
+        shift = 0.001
+    else:
+        shift = 0
+
+    if directed:
+        example_1 = nx.DiGraph()
+    else:
+        example_1 = nx.Graph()
+
+    # graph creation
+    example_1.add_edge(1, 2, **{EWL: 3 + shift})
+    example_1.add_edge(1, 4, **{EWL: 2 + shift})
+    example_1.add_edge(2, 3, **{EWL: 4 + shift})
+    example_1.add_edge(2, 5, **{EWL: 6 + shift})
+
+    # node weights
+    if explicit_node_wt:
+        nx.set_node_attributes(example_1, 1, NWL)
+        wtu = NWL
+    else:
+        wtu = None
+
+    # partitioning
+    clusters_1 = {
+        frozenset(x)
+        for x in nx.community.lukes_partitioning(
+            example_1, limit, node_weight=wtu, edge_weight=EWL
+        )
+    }
+
+    return clusters_1
+
+
+# second test from the Lukes original paper
+def paper_2_case(explicit_edge_wt=True, directed=False):
+    # problem specific constants
+    byte_block_size = 32
+
+    # configuration
+    if directed:
+        example_2 = nx.DiGraph()
+    else:
+        example_2 = nx.Graph()
+
+    if explicit_edge_wt:
+        edic = {EWL: 1}
+        wtu = EWL
+    else:
+        edic = {}
+        wtu = None
+
+    # graph creation
+    example_2.add_edge("name", "home_address", **edic)
+    example_2.add_edge("name", "education", **edic)
+    example_2.add_edge("education", "bs", **edic)
+    example_2.add_edge("education", "ms", **edic)
+    example_2.add_edge("education", "phd", **edic)
+    example_2.add_edge("name", "telephone", **edic)
+    example_2.add_edge("telephone", "home", **edic)
+    example_2.add_edge("telephone", "office", **edic)
+    example_2.add_edge("office", "no1", **edic)
+    example_2.add_edge("office", "no2", **edic)
+
+    example_2.nodes["name"][NWL] = 20
+    example_2.nodes["education"][NWL] = 10
+    example_2.nodes["bs"][NWL] = 1
+    example_2.nodes["ms"][NWL] = 1
+    example_2.nodes["phd"][NWL] = 1
+    example_2.nodes["home_address"][NWL] = 8
+    example_2.nodes["telephone"][NWL] = 8
+    example_2.nodes["home"][NWL] = 8
+    example_2.nodes["office"][NWL] = 4
+    example_2.nodes["no1"][NWL] = 1
+    example_2.nodes["no2"][NWL] = 1
+
+    # partitioning
+    clusters_2 = {
+        frozenset(x)
+        for x in nx.community.lukes_partitioning(
+            example_2, byte_block_size, node_weight=NWL, edge_weight=wtu
+        )
+    }
+
+    return clusters_2
+
+
+def test_paper_1_case():
+    ground_truth = {frozenset([1, 4]), frozenset([2, 3, 5])}
+
+    tf = (True, False)
+    for flt, nwt, drc in product(tf, tf, tf):
+        part = paper_1_case(flt, nwt, drc)
+        assert part == ground_truth
+
+
+def test_paper_2_case():
+    ground_truth = {
+        frozenset(["education", "bs", "ms", "phd"]),
+        frozenset(["name", "home_address"]),
+        frozenset(["telephone", "home", "office", "no1", "no2"]),
+    }
+
+    tf = (True, False)
+    for ewt, drc in product(tf, tf):
+        part = paper_2_case(ewt, drc)
+        assert part == ground_truth
+
+
+def test_mandatory_tree():
+    not_a_tree = nx.complete_graph(4)
+
+    with pytest.raises(nx.NotATree):
+        nx.community.lukes_partitioning(not_a_tree, 5)
+
+
+def test_mandatory_integrality():
+    byte_block_size = 32
+
+    ex_1_broken = nx.DiGraph()
+
+    ex_1_broken.add_edge(1, 2, **{EWL: 3.2})
+    ex_1_broken.add_edge(1, 4, **{EWL: 2.4})
+    ex_1_broken.add_edge(2, 3, **{EWL: 4.0})
+    ex_1_broken.add_edge(2, 5, **{EWL: 6.3})
+
+    ex_1_broken.nodes[1][NWL] = 1.2  # !
+    ex_1_broken.nodes[2][NWL] = 1
+    ex_1_broken.nodes[3][NWL] = 1
+    ex_1_broken.nodes[4][NWL] = 1
+    ex_1_broken.nodes[5][NWL] = 2
+
+    with pytest.raises(TypeError):
+        nx.community.lukes_partitioning(
+            ex_1_broken, byte_block_size, node_weight=NWL, edge_weight=EWL
+        )

rl/Lib/site-packages/networkx/algorithms/community/tests/test_modularity_max.py

@@ -0,0 +1,340 @@
+import pytest
+
+import networkx as nx
+from networkx.algorithms.community import (
+    greedy_modularity_communities,
+    naive_greedy_modularity_communities,
+)
+
+
+@pytest.mark.parametrize(
+    "func", (greedy_modularity_communities, naive_greedy_modularity_communities)
+)
+def test_modularity_communities(func):
+    G = nx.karate_club_graph()
+    john_a = frozenset(
+        [8, 14, 15, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]
+    )
+    mr_hi = frozenset([0, 4, 5, 6, 10, 11, 16, 19])
+    overlap = frozenset([1, 2, 3, 7, 9, 12, 13, 17, 21])
+    expected = {john_a, overlap, mr_hi}
+    assert set(func(G, weight=None)) == expected
+
+
+@pytest.mark.parametrize(
+    "func", (greedy_modularity_communities, naive_greedy_modularity_communities)
+)
+def test_modularity_communities_categorical_labels(func):
+    # Using other than 0-starting contiguous integers as node-labels.
+    G = nx.Graph(
+        [
+            ("a", "b"),
+            ("a", "c"),
+            ("b", "c"),
+            ("b", "d"),  # inter-community edge
+            ("d", "e"),
+            ("d", "f"),
+            ("d", "g"),
+            ("f", "g"),
+            ("d", "e"),
+            ("f", "e"),
+        ]
+    )
+    expected = {frozenset({"f", "g", "e", "d"}), frozenset({"a", "b", "c"})}
+    assert set(func(G)) == expected
+
+
+def test_greedy_modularity_communities_components():
+    # Test for gh-5530
+    G = nx.Graph([(0, 1), (2, 3), (4, 5), (5, 6)])
+    # usual case with 3 components
+    assert greedy_modularity_communities(G) == [{4, 5, 6}, {0, 1}, {2, 3}]
+    # best_n can make the algorithm continue even when modularity goes down
+    assert greedy_modularity_communities(G, best_n=3) == [{4, 5, 6}, {0, 1}, {2, 3}]
+    assert greedy_modularity_communities(G, best_n=2) == [{0, 1, 4, 5, 6}, {2, 3}]
+    assert greedy_modularity_communities(G, best_n=1) == [{0, 1, 2, 3, 4, 5, 6}]
+
+
+def test_greedy_modularity_communities_relabeled():
+    # Test for gh-4966
+    G = nx.balanced_tree(2, 2)
+    mapping = {0: "a", 1: "b", 2: "c", 3: "d", 4: "e", 5: "f", 6: "g", 7: "h"}
+    G = nx.relabel_nodes(G, mapping)
+    expected = [frozenset({"e", "d", "a", "b"}), frozenset({"c", "f", "g"})]
+    assert greedy_modularity_communities(G) == expected
+
+
+def test_greedy_modularity_communities_directed():
+    G = nx.DiGraph(
+        [
+            ("a", "b"),
+            ("a", "c"),
+            ("b", "c"),
+            ("b", "d"),  # inter-community edge
+            ("d", "e"),
+            ("d", "f"),
+            ("d", "g"),
+            ("f", "g"),
+            ("d", "e"),
+            ("f", "e"),
+        ]
+    )
+    expected = [frozenset({"f", "g", "e", "d"}), frozenset({"a", "b", "c"})]
+    assert greedy_modularity_communities(G) == expected
+
+    # with loops
+    G = nx.DiGraph()
+    G.add_edges_from(
+        [(1, 1), (1, 2), (1, 3), (2, 3), (1, 4), (4, 4), (5, 5), (4, 5), (4, 6), (5, 6)]
+    )
+    expected = [frozenset({1, 2, 3}), frozenset({4, 5, 6})]
+    assert greedy_modularity_communities(G) == expected
+
+
+@pytest.mark.parametrize(
+    "func", (greedy_modularity_communities, naive_greedy_modularity_communities)
+)
+def test_modularity_communities_weighted(func):
+    G = nx.balanced_tree(2, 3)
+    for a, b in G.edges:
+        if ((a == 1) or (a == 2)) and (b != 0):
+            G[a][b]["weight"] = 10.0
+        else:
+            G[a][b]["weight"] = 1.0
+
+    expected = [{0, 1, 3, 4, 7, 8, 9, 10}, {2, 5, 6, 11, 12, 13, 14}]
+
+    assert func(G, weight="weight") == expected
+    assert func(G, weight="weight", resolution=0.9) == expected
+    assert func(G, weight="weight", resolution=0.3) == expected
+    assert func(G, weight="weight", resolution=1.1) != expected
+
+
+def test_modularity_communities_floating_point():
+    # check for floating point error when used as key in the mapped_queue dict.
+    # Test for gh-4992 and gh-5000
+    G = nx.Graph()
+    G.add_weighted_edges_from(
+        [(0, 1, 12), (1, 4, 71), (2, 3, 15), (2, 4, 10), (3, 6, 13)]
+    )
+    expected = [{0, 1, 4}, {2, 3, 6}]
+    assert greedy_modularity_communities(G, weight="weight") == expected
+    assert (
+        greedy_modularity_communities(G, weight="weight", resolution=0.99) == expected
+    )
+
+
+def test_modularity_communities_directed_weighted():
+    G = nx.DiGraph()
+    G.add_weighted_edges_from(
+        [
+            (1, 2, 5),
+            (1, 3, 3),
+            (2, 3, 6),
+            (2, 6, 1),
+            (1, 4, 1),
+            (4, 5, 3),
+            (4, 6, 7),
+            (5, 6, 2),
+            (5, 7, 5),
+            (5, 8, 4),
+            (6, 8, 3),
+        ]
+    )
+    expected = [frozenset({4, 5, 6, 7, 8}), frozenset({1, 2, 3})]
+    assert greedy_modularity_communities(G, weight="weight") == expected
+
+    # A large weight of the edge (2, 6) causes 6 to change group, even if it shares
+    # only one connection with the new group and 3 with the old one.
+    G[2][6]["weight"] = 20
+    expected = [frozenset({1, 2, 3, 6}), frozenset({4, 5, 7, 8})]
+    assert greedy_modularity_communities(G, weight="weight") == expected
+
+
+def test_greedy_modularity_communities_multigraph():
+    G = nx.MultiGraph()
+    G.add_edges_from(
+        [
+            (1, 2),
+            (1, 2),
+            (1, 3),
+            (2, 3),
+            (1, 4),
+            (2, 4),
+            (4, 5),
+            (5, 6),
+            (5, 7),
+            (5, 7),
+            (6, 7),
+            (7, 8),
+            (5, 8),
+        ]
+    )
+    expected = [frozenset({1, 2, 3, 4}), frozenset({5, 6, 7, 8})]
+    assert greedy_modularity_communities(G) == expected
+
+    # Converting (4, 5) into a multi-edge causes node 4 to change group.
+    G.add_edge(4, 5)
+    expected = [frozenset({4, 5, 6, 7, 8}), frozenset({1, 2, 3})]
+    assert greedy_modularity_communities(G) == expected
+
+
+def test_greedy_modularity_communities_multigraph_weighted():
+    G = nx.MultiGraph()
+    G.add_weighted_edges_from(
+        [
+            (1, 2, 5),
+            (1, 2, 3),
+            (1, 3, 6),
+            (1, 3, 6),
+            (2, 3, 4),
+            (1, 4, 1),
+            (1, 4, 1),
+            (2, 4, 3),
+            (2, 4, 3),
+            (4, 5, 1),
+            (5, 6, 3),
+            (5, 6, 7),
+            (5, 6, 4),
+            (5, 7, 9),
+            (5, 7, 9),
+            (6, 7, 8),
+            (7, 8, 2),
+            (7, 8, 2),
+            (5, 8, 6),
+            (5, 8, 6),
+        ]
+    )
+    expected = [frozenset({1, 2, 3, 4}), frozenset({5, 6, 7, 8})]
+    assert greedy_modularity_communities(G, weight="weight") == expected
+
+    # Adding multi-edge (4, 5, 16) causes node 4 to change group.
+    G.add_edge(4, 5, weight=16)
+    expected = [frozenset({4, 5, 6, 7, 8}), frozenset({1, 2, 3})]
+    assert greedy_modularity_communities(G, weight="weight") == expected
+
+    # Increasing the weight of edge (1, 4) causes node 4 to return to the former group.
+    G[1][4][1]["weight"] = 3
+    expected = [frozenset({1, 2, 3, 4}), frozenset({5, 6, 7, 8})]
+    assert greedy_modularity_communities(G, weight="weight") == expected
+
+
+def test_greed_modularity_communities_multidigraph():
+    G = nx.MultiDiGraph()
+    G.add_edges_from(
+        [
+            (1, 2),
+            (1, 2),
+            (3, 1),
+            (2, 3),
+            (2, 3),
+            (3, 2),
+            (1, 4),
+            (2, 4),
+            (4, 2),
+            (4, 5),
+            (5, 6),
+            (5, 6),
+            (6, 5),
+            (5, 7),
+            (6, 7),
+            (7, 8),
+            (5, 8),
+            (8, 4),
+        ]
+    )
+    expected = [frozenset({1, 2, 3, 4}), frozenset({5, 6, 7, 8})]
+    assert greedy_modularity_communities(G, weight="weight") == expected
+
+
+def test_greed_modularity_communities_multidigraph_weighted():
+    G = nx.MultiDiGraph()
+    G.add_weighted_edges_from(
+        [
+            (1, 2, 5),
+            (1, 2, 3),
+            (3, 1, 6),
+            (1, 3, 6),
+            (3, 2, 4),
+            (1, 4, 2),
+            (1, 4, 5),
+            (2, 4, 3),
+            (3, 2, 8),
+            (4, 2, 3),
+            (4, 3, 5),
+            (4, 5, 2),
+            (5, 6, 3),
+            (5, 6, 7),
+            (6, 5, 4),
+            (5, 7, 9),
+            (5, 7, 9),
+            (7, 6, 8),
+            (7, 8, 2),
+            (8, 7, 2),
+            (5, 8, 6),
+            (5, 8, 6),
+        ]
+    )
+    expected = [frozenset({1, 2, 3, 4}), frozenset({5, 6, 7, 8})]
+    assert greedy_modularity_communities(G, weight="weight") == expected
+
+
+def test_resolution_parameter_impact():
+    G = nx.barbell_graph(5, 3)
+
+    gamma = 1
+    expected = [frozenset(range(5)), frozenset(range(8, 13)), frozenset(range(5, 8))]
+    assert greedy_modularity_communities(G, resolution=gamma) == expected
+    assert naive_greedy_modularity_communities(G, resolution=gamma) == expected
+
+    gamma = 2.5
+    expected = [{0, 1, 2, 3}, {9, 10, 11, 12}, {5, 6, 7}, {4}, {8}]
+    assert greedy_modularity_communities(G, resolution=gamma) == expected
+    assert naive_greedy_modularity_communities(G, resolution=gamma) == expected
+
+    gamma = 0.3
+    expected = [frozenset(range(8)), frozenset(range(8, 13))]
+    assert greedy_modularity_communities(G, resolution=gamma) == expected
+    assert naive_greedy_modularity_communities(G, resolution=gamma) == expected
+
+
+def test_cutoff_parameter():
+    G = nx.circular_ladder_graph(4)
+
+    # No aggregation:
+    expected = [{k} for k in range(8)]
+    assert greedy_modularity_communities(G, cutoff=8) == expected
+
+    # Aggregation to half order (number of nodes)
+    expected = [{k, k + 1} for k in range(0, 8, 2)]
+    assert greedy_modularity_communities(G, cutoff=4) == expected
+
+    # Default aggregation case (here, 2 communities emerge)
+    expected = [frozenset(range(4)), frozenset(range(4, 8))]
+    assert greedy_modularity_communities(G, cutoff=1) == expected
+
+
+def test_best_n():
+    G = nx.barbell_graph(5, 3)
+
+    # Same result as without enforcing cutoff:
+    best_n = 3
+    expected = [frozenset(range(5)), frozenset(range(8, 13)), frozenset(range(5, 8))]
+    assert greedy_modularity_communities(G, best_n=best_n) == expected
+
+    # One additional merging step:
+    best_n = 2
+    expected = [frozenset(range(8)), frozenset(range(8, 13))]
+    assert greedy_modularity_communities(G, best_n=best_n) == expected
+
+    # Two additional merging steps:
+    best_n = 1
+    expected = [frozenset(range(13))]
+    assert greedy_modularity_communities(G, best_n=best_n) == expected
+
+
+def test_greedy_modularity_communities_corner_cases():
+    G = nx.empty_graph()
+    assert nx.community.greedy_modularity_communities(G) == []
+    G.add_nodes_from(range(3))
+    assert nx.community.greedy_modularity_communities(G) == [{0}, {1}, {2}]

rl/Lib/site-packages/networkx/algorithms/community/tests/test_quality.py

@@ -0,0 +1,139 @@
+"""Unit tests for the :mod:`networkx.algorithms.community.quality`
+module.
+
+"""
+
+import pytest
+
+import networkx as nx
+from networkx import barbell_graph
+from networkx.algorithms.community import modularity, partition_quality
+from networkx.algorithms.community.quality import inter_community_edges
+
+
+class TestPerformance:
+    """Unit tests for the :func:`performance` function."""
+
+    def test_bad_partition(self):
+        """Tests that a poor partition has a low performance measure."""
+        G = barbell_graph(3, 0)
+        partition = [{0, 1, 4}, {2, 3, 5}]
+        assert 8 / 15 == pytest.approx(partition_quality(G, partition)[1], abs=1e-7)
+
+    def test_good_partition(self):
+        """Tests that a good partition has a high performance measure."""
+        G = barbell_graph(3, 0)
+        partition = [{0, 1, 2}, {3, 4, 5}]
+        assert 14 / 15 == pytest.approx(partition_quality(G, partition)[1], abs=1e-7)
+
+
+class TestCoverage:
+    """Unit tests for the :func:`coverage` function."""
+
+    def test_bad_partition(self):
+        """Tests that a poor partition has a low coverage measure."""
+        G = barbell_graph(3, 0)
+        partition = [{0, 1, 4}, {2, 3, 5}]
+        assert 3 / 7 == pytest.approx(partition_quality(G, partition)[0], abs=1e-7)
+
+    def test_good_partition(self):
+        """Tests that a good partition has a high coverage measure."""
+        G = barbell_graph(3, 0)
+        partition = [{0, 1, 2}, {3, 4, 5}]
+        assert 6 / 7 == pytest.approx(partition_quality(G, partition)[0], abs=1e-7)
+
+
+def test_modularity():
+    G = nx.barbell_graph(3, 0)
+    C = [{0, 1, 4}, {2, 3, 5}]
+    assert (-16 / (14**2)) == pytest.approx(modularity(G, C), abs=1e-7)
+    C = [{0, 1, 2}, {3, 4, 5}]
+    assert (35 * 2) / (14**2) == pytest.approx(modularity(G, C), abs=1e-7)
+
+    n = 1000
+    G = nx.erdos_renyi_graph(n, 0.09, seed=42, directed=True)
+    C = [set(range(n // 2)), set(range(n // 2, n))]
+    assert 0.00017154251389292754 == pytest.approx(modularity(G, C), abs=1e-7)
+
+    G = nx.margulis_gabber_galil_graph(10)
+    mid_value = G.number_of_nodes() // 2
+    nodes = list(G.nodes)
+    C = [set(nodes[:mid_value]), set(nodes[mid_value:])]
+    assert 0.13 == pytest.approx(modularity(G, C), abs=1e-7)
+
+    G = nx.DiGraph()
+    G.add_edges_from([(2, 1), (2, 3), (3, 4)])
+    C = [{1, 2}, {3, 4}]
+    assert 2 / 9 == pytest.approx(modularity(G, C), abs=1e-7)
+
+
+def test_modularity_resolution():
+    G = nx.barbell_graph(3, 0)
+    C = [{0, 1, 4}, {2, 3, 5}]
+    assert modularity(G, C) == pytest.approx(3 / 7 - 100 / 14**2)
+    gamma = 2
+    result = modularity(G, C, resolution=gamma)
+    assert result == pytest.approx(3 / 7 - gamma * 100 / 14**2)
+    gamma = 0.2
+    result = modularity(G, C, resolution=gamma)
+    assert result == pytest.approx(3 / 7 - gamma * 100 / 14**2)
+
+    C = [{0, 1, 2}, {3, 4, 5}]
+    assert modularity(G, C) == pytest.approx(6 / 7 - 98 / 14**2)
+    gamma = 2
+    result = modularity(G, C, resolution=gamma)
+    assert result == pytest.approx(6 / 7 - gamma * 98 / 14**2)
+    gamma = 0.2
+    result = modularity(G, C, resolution=gamma)
+    assert result == pytest.approx(6 / 7 - gamma * 98 / 14**2)
+
+    G = nx.barbell_graph(5, 3)
+    C = [frozenset(range(5)), frozenset(range(8, 13)), frozenset(range(5, 8))]
+    gamma = 1
+    result = modularity(G, C, resolution=gamma)
+    # This C is maximal for gamma=1:  modularity = 0.518229
+    assert result == pytest.approx((22 / 24) - gamma * (918 / (48**2)))
+    gamma = 2
+    result = modularity(G, C, resolution=gamma)
+    assert result == pytest.approx((22 / 24) - gamma * (918 / (48**2)))
+    gamma = 0.2
+    result = modularity(G, C, resolution=gamma)
+    assert result == pytest.approx((22 / 24) - gamma * (918 / (48**2)))
+
+    C = [{0, 1, 2, 3}, {9, 10, 11, 12}, {5, 6, 7}, {4}, {8}]
+    gamma = 1
+    result = modularity(G, C, resolution=gamma)
+    assert result == pytest.approx((14 / 24) - gamma * (598 / (48**2)))
+    gamma = 2.5
+    result = modularity(G, C, resolution=gamma)
+    # This C is maximal for gamma=2.5:  modularity = -0.06553819
+    assert result == pytest.approx((14 / 24) - gamma * (598 / (48**2)))
+    gamma = 0.2
+    result = modularity(G, C, resolution=gamma)
+    assert result == pytest.approx((14 / 24) - gamma * (598 / (48**2)))
+
+    C = [frozenset(range(8)), frozenset(range(8, 13))]
+    gamma = 1
+    result = modularity(G, C, resolution=gamma)
+    assert result == pytest.approx((23 / 24) - gamma * (1170 / (48**2)))
+    gamma = 2
+    result = modularity(G, C, resolution=gamma)
+    assert result == pytest.approx((23 / 24) - gamma * (1170 / (48**2)))
+    gamma = 0.3
+    result = modularity(G, C, resolution=gamma)
+    # This C is maximal for gamma=0.3:  modularity = 0.805990
+    assert result == pytest.approx((23 / 24) - gamma * (1170 / (48**2)))
+
+
+def test_inter_community_edges_with_digraphs():
+    G = nx.complete_graph(2, create_using=nx.DiGraph())
+    partition = [{0}, {1}]
+    assert inter_community_edges(G, partition) == 2
+
+    G = nx.complete_graph(10, create_using=nx.DiGraph())
+    partition = [{0}, {1, 2}, {3, 4, 5}, {6, 7, 8, 9}]
+    assert inter_community_edges(G, partition) == 70
+
+    G = nx.cycle_graph(4, create_using=nx.DiGraph())
+    partition = [{0, 1}, {2, 3}]
+    assert inter_community_edges(G, partition) == 2

rl/Lib/site-packages/networkx/algorithms/community/tests/test_utils.py

@@ -0,0 +1,26 @@
+"""Unit tests for the :mod:`networkx.algorithms.community.utils` module."""
+
+import networkx as nx
+
+
+def test_is_partition():
+    G = nx.empty_graph(3)
+    assert nx.community.is_partition(G, [{0, 1}, {2}])
+    assert nx.community.is_partition(G, ({0, 1}, {2}))
+    assert nx.community.is_partition(G, ([0, 1], [2]))
+    assert nx.community.is_partition(G, [[0, 1], [2]])
+
+
+def test_not_covering():
+    G = nx.empty_graph(3)
+    assert not nx.community.is_partition(G, [{0}, {1}])
+
+
+def test_not_disjoint():
+    G = nx.empty_graph(3)
+    assert not nx.community.is_partition(G, [{0, 1}, {1, 2}])
+
+
+def test_not_node():
+    G = nx.empty_graph(3)
+    assert not nx.community.is_partition(G, [{0, 1}, {3}])