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+"""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