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# Source code for networkx.generators.community

    """Generators for classes of graphs used in studying social networks."""

    import itertools
    import math

    import networkx as nx
    from networkx.utils import py_random_state

    __all__ = [
        "caveman_graph",
        "connected_caveman_graph",
        "relaxed_caveman_graph",
        "random_partition_graph",
        "planted_partition_graph",
        "gaussian_random_partition_graph",
        "ring_of_cliques",
        "windmill_graph",
        "stochastic_block_model",
        "LFR_benchmark_graph",
    ]

    @nx._dispatchable(graphs=None, returns_graph=True)
    def caveman_graph(l, k):
        """Returns a caveman graph of `l` cliques of size `k`.

        Parameters
        ----------
        l : int
          Number of cliques
        k : int
          Size of cliques

        Returns
        -------
        G : NetworkX Graph
          caveman graph

        Notes
        -----
        This returns an undirected graph, it can be converted to a directed
        graph using :func:`nx.to_directed`, or a multigraph using
        ``nx.MultiGraph(nx.caveman_graph(l, k))``. Only the undirected version is
        described in [1]_ and it is unclear which of the directed
        generalizations is most useful.

        Examples
        --------
        >>> G = nx.caveman_graph(3, 3)

        See also
        --------

        connected_caveman_graph

        References
        ----------
        .. [1] Watts, D. J. 'Networks, Dynamics, and the Small-World Phenomenon.'
           Amer. J. Soc. 105, 493-527, 1999.
        """
        # l disjoint cliques of size k
        G = nx.empty_graph(l * k)
        if k > 1:
            for start in range(0, l * k, k):
                edges = itertools.combinations(range(start, start + k), 2)
                G.add_edges_from(edges)
        return G

    @nx._dispatchable(graphs=None, returns_graph=True)
    def connected_caveman_graph(l, k):
        """Returns a connected caveman graph of `l` cliques of size `k`.

        The connected caveman graph is formed by creating `n` cliques of size
        `k`, then a single edge in each clique is rewired to a node in an
        adjacent clique.

        Parameters
        ----------
        l : int
          number of cliques
        k : int
          size of cliques (k at least 2 or NetworkXError is raised)

        Returns
        -------
        G : NetworkX Graph
          connected caveman graph

        Raises
        ------
        NetworkXError
            If the size of cliques `k` is smaller than 2.

        Notes
        -----
        This returns an undirected graph, it can be converted to a directed
        graph using :func:`nx.to_directed`, or a multigraph using
        ``nx.MultiGraph(nx.caveman_graph(l, k))``. Only the undirected version is
        described in [1]_ and it is unclear which of the directed
        generalizations is most useful.

        Examples
        --------
        >>> G = nx.connected_caveman_graph(3, 3)

        References
        ----------
        .. [1] Watts, D. J. 'Networks, Dynamics, and the Small-World Phenomenon.'
           Amer. J. Soc. 105, 493-527, 1999.
        """
        if k < 2:
            raise nx.NetworkXError(
                "The size of cliques in a connected caveman graph must be at least 2."
            )

        G = nx.caveman_graph(l, k)
        for start in range(0, l * k, k):
            G.remove_edge(start, start + 1)
            G.add_edge(start, (start - 1) % (l * k))
        return G

    @py_random_state(3)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def relaxed_caveman_graph(l, k, p, seed=None):
        """Returns a relaxed caveman graph.

        A relaxed caveman graph starts with `l` cliques of size `k`.  Edges are
        then randomly rewired with probability `p` to link different cliques.

        Parameters
        ----------
        l : int
          Number of groups
        k : int
          Size of cliques
        p : float
          Probability of rewiring each edge.
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.

        Returns
        -------
        G : NetworkX Graph
          Relaxed Caveman Graph

        Raises
        ------
        NetworkXError
         If p is not in [0,1]

        Examples
        --------
        >>> G = nx.relaxed_caveman_graph(2, 3, 0.1, seed=42)

        References
        ----------
        .. [1] Santo Fortunato, Community Detection in Graphs,
           Physics Reports Volume 486, Issues 3-5, February 2010, Pages 75-174.
           https://arxiv.org/abs/0906.0612
        """
        G = nx.caveman_graph(l, k)
        nodes = list(G)
        for u, v in G.edges():
            if seed.random() < p:  # rewire the edge
                x = seed.choice(nodes)
                if G.has_edge(u, x):
                    continue
                G.remove_edge(u, v)
                G.add_edge(u, x)
        return G

    @py_random_state(3)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def random_partition_graph(sizes, p_in, p_out, seed=None, directed=False):
        """Returns the random partition graph with a partition of sizes.

        A partition graph is a graph of communities with sizes defined by
        s in sizes. Nodes in the same group are connected with probability
        p_in and nodes of different groups are connected with probability
        p_out.

        Parameters
        ----------
        sizes : list of ints
          Sizes of groups
        p_in : float
          probability of edges with in groups
        p_out : float
          probability of edges between groups
        directed : boolean optional, default=False
          Whether to create a directed graph
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.

        Returns
        -------
        G : NetworkX Graph or DiGraph
          random partition graph of size sum(gs)

        Raises
        ------
        NetworkXError
          If p_in or p_out is not in [0,1]

        Examples
        --------
        >>> G = nx.random_partition_graph([10, 10, 10], 0.25, 0.01)
        >>> len(G)
        30
        >>> partition = G.graph["partition"]
        >>> len(partition)
        3

        Notes
        -----
        This is a generalization of the planted-l-partition described in
        [1]_.  It allows for the creation of groups of any size.

        The partition is store as a graph attribute 'partition'.

        References
        ----------
        .. [1] Santo Fortunato 'Community Detection in Graphs' Physical Reports
           Volume 486, Issue 3-5 p. 75-174. https://arxiv.org/abs/0906.0612
        """
        # Use geometric method for O(n+m) complexity algorithm
        # partition = nx.community_sets(nx.get_node_attributes(G, 'affiliation'))
        if not 0.0 <= p_in <= 1.0:
            raise nx.NetworkXError("p_in must be in [0,1]")
        if not 0.0 <= p_out <= 1.0:
            raise nx.NetworkXError("p_out must be in [0,1]")

        # create connection matrix
        num_blocks = len(sizes)
        p = [[p_out for s in range(num_blocks)] for r in range(num_blocks)]
        for r in range(num_blocks):
            p[r][r] = p_in

        return stochastic_block_model(
            sizes,
            p,
            nodelist=None,
            seed=seed,
            directed=directed,
            selfloops=False,
            sparse=True,
        )

    @py_random_state(4)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def planted_partition_graph(l, k, p_in, p_out, seed=None, directed=False):
        """Returns the planted l-partition graph.

        This model partitions a graph with n=l*k vertices in
        l groups with k vertices each. Vertices of the same
        group are linked with a probability p_in, and vertices
        of different groups are linked with probability p_out.

        Parameters
        ----------
        l : int
          Number of groups
        k : int
          Number of vertices in each group
        p_in : float
          probability of connecting vertices within a group
        p_out : float
          probability of connected vertices between groups
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        directed : bool,optional (default=False)
          If True return a directed graph

        Returns
        -------
        G : NetworkX Graph or DiGraph
          planted l-partition graph

        Raises
        ------
        NetworkXError
          If `p_in`, `p_out` are not in `[0, 1]`

        Examples
        --------
        >>> G = nx.planted_partition_graph(4, 3, 0.5, 0.1, seed=42)

        See Also
        --------
        random_partition_model

        References
        ----------
        .. [1] A. Condon, R.M. Karp, Algorithms for graph partitioning
            on the planted partition model,
            Random Struct. Algor. 18 (2001) 116-140.

        .. [2] Santo Fortunato 'Community Detection in Graphs' Physical Reports
           Volume 486, Issue 3-5 p. 75-174. https://arxiv.org/abs/0906.0612
        """
        return random_partition_graph([k] * l, p_in, p_out, seed=seed, directed=directed)

    @py_random_state(6)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def gaussian_random_partition_graph(n, s, v, p_in, p_out, directed=False, seed=None):
        """Generate a Gaussian random partition graph.

        A Gaussian random partition graph is created by creating k partitions
        each with a size drawn from a normal distribution with mean s and variance
        s/v. Nodes are connected within clusters with probability p_in and
        between clusters with probability p_out[1]

        Parameters
        ----------
        n : int
          Number of nodes in the graph
        s : float
          Mean cluster size
        v : float
          Shape parameter. The variance of cluster size distribution is s/v.
        p_in : float
          Probability of intra cluster connection.
        p_out : float
          Probability of inter cluster connection.
        directed : boolean, optional default=False
          Whether to create a directed graph or not
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.

        Returns
        -------
        G : NetworkX Graph or DiGraph
          gaussian random partition graph

        Raises
        ------
        NetworkXError
          If s is > n
          If p_in or p_out is not in [0,1]

        Notes
        -----
        Note the number of partitions is dependent on s,v and n, and that the
        last partition may be considerably smaller, as it is sized to simply
        fill out the nodes [1]

        See Also
        --------
        random_partition_graph

        Examples
        --------
        >>> G = nx.gaussian_random_partition_graph(100, 10, 10, 0.25, 0.1)
        >>> len(G)
        100

        References
        ----------
        .. [1] Ulrik Brandes, Marco Gaertler, Dorothea Wagner,
           Experiments on Graph Clustering Algorithms,
           In the proceedings of the 11th Europ. Symp. Algorithms, 2003.
        """
        if s > n:
            raise nx.NetworkXError("s must be <= n")
        assigned = 0
        sizes = []
        while True:
            size = int(seed.gauss(s, s / v + 0.5))
            if size < 1:  # how to handle 0 or negative sizes?
                continue
            if assigned + size >= n:
                sizes.append(n - assigned)
                break
            assigned += size
            sizes.append(size)
        return random_partition_graph(sizes, p_in, p_out, seed=seed, directed=directed)

    @nx._dispatchable(graphs=None, returns_graph=True)
    def ring_of_cliques(num_cliques, clique_size):
        """Defines a "ring of cliques" graph.

        A ring of cliques graph is consisting of cliques, connected through single
        links. Each clique is a complete graph.

        Parameters
        ----------
        num_cliques : int
            Number of cliques
        clique_size : int
            Size of cliques

        Returns
        -------
        G : NetworkX Graph
            ring of cliques graph

        Raises
        ------
        NetworkXError
            If the number of cliques is lower than 2 or
            if the size of cliques is smaller than 2.

        Examples
        --------
        >>> G = nx.ring_of_cliques(8, 4)

        See Also
        --------
        connected_caveman_graph

        Notes
        -----
        The `connected_caveman_graph` graph removes a link from each clique to
        connect it with the next clique. Instead, the `ring_of_cliques` graph
        simply adds the link without removing any link from the cliques.
        """
        if num_cliques < 2:
            raise nx.NetworkXError("A ring of cliques must have at least two cliques")
        if clique_size < 2:
            raise nx.NetworkXError("The cliques must have at least two nodes")

        G = nx.Graph()
        for i in range(num_cliques):
            edges = itertools.combinations(
                range(i * clique_size, i * clique_size + clique_size), 2
            )
            G.add_edges_from(edges)
            G.add_edge(
                i * clique_size + 1, (i + 1) * clique_size % (num_cliques * clique_size)
            )
        return G

    @nx._dispatchable(graphs=None, returns_graph=True)
    def windmill_graph(n, k):
        """Generate a windmill graph.
        A windmill graph is a graph of `n` cliques each of size `k` that are all
        joined at one node.
        It can be thought of as taking a disjoint union of `n` cliques of size `k`,
        selecting one point from each, and contracting all of the selected points.
        Alternatively, one could generate `n` cliques of size `k-1` and one node
        that is connected to all other nodes in the graph.

        Parameters
        ----------
        n : int
            Number of cliques
        k : int
            Size of cliques

        Returns
        -------
        G : NetworkX Graph
            windmill graph with n cliques of size k

        Raises
        ------
        NetworkXError
            If the number of cliques is less than two
            If the size of the cliques are less than two

        Examples
        --------
        >>> G = nx.windmill_graph(4, 5)

        Notes
        -----
        The node labeled `0` will be the node connected to all other nodes.
        Note that windmill graphs are usually denoted `Wd(k,n)`, so the parameters
        are in the opposite order as the parameters of this method.
        """
        if n < 2:
            msg = "A windmill graph must have at least two cliques"
            raise nx.NetworkXError(msg)
        if k < 2:
            raise nx.NetworkXError("The cliques must have at least two nodes")

        G = nx.disjoint_union_all(
            itertools.chain(
                [nx.complete_graph(k)], (nx.complete_graph(k - 1) for _ in range(n - 1))
            )
        )
        G.add_edges_from((0, i) for i in range(k, G.number_of_nodes()))
        return G

    @py_random_state(3)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def stochastic_block_model(
        sizes, p, nodelist=None, seed=None, directed=False, selfloops=False, sparse=True
    ):
        """Returns a stochastic block model graph.

        This model partitions the nodes in blocks of arbitrary sizes, and places
        edges between pairs of nodes independently, with a probability that depends
        on the blocks.

        Parameters
        ----------
        sizes : list of ints
            Sizes of blocks
        p : list of list of floats
            Element (r,s) gives the density of edges going from the nodes
            of group r to nodes of group s.
            p must match the number of groups (len(sizes) == len(p)),
            and it must be symmetric if the graph is undirected.
        nodelist : list, optional
            The block tags are assigned according to the node identifiers
            in nodelist. If nodelist is None, then the ordering is the
            range [0,sum(sizes)-1].
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        directed : boolean optional, default=False
            Whether to create a directed graph or not.
        selfloops : boolean optional, default=False
            Whether to include self-loops or not.
        sparse: boolean optional, default=True
            Use the sparse heuristic to speed up the generator.

        Returns
        -------
        g : NetworkX Graph or DiGraph
            Stochastic block model graph of size sum(sizes)

        Raises
        ------
        NetworkXError
          If probabilities are not in [0,1].
          If the probability matrix is not square (directed case).
          If the probability matrix is not symmetric (undirected case).
          If the sizes list does not match nodelist or the probability matrix.
          If nodelist contains duplicate.

        Examples
        --------
        >>> sizes = [75, 75, 300]
        >>> probs = [[0.25, 0.05, 0.02], [0.05, 0.35, 0.07], [0.02, 0.07, 0.40]]
        >>> g = nx.stochastic_block_model(sizes, probs, seed=0)
        >>> len(g)
        450
        >>> H = nx.quotient_graph(g, g.graph["partition"], relabel=True)
        >>> for v in H.nodes(data=True):
        ...     print(round(v[1]["density"], 3))
        0.245
        0.348
        0.405
        >>> for v in H.edges(data=True):
        ...     print(round(1.0 * v[2]["weight"] / (sizes[v[0]] * sizes[v[1]]), 3))
        0.051
        0.022
        0.07

        See Also
        --------
        random_partition_graph
        planted_partition_graph
        gaussian_random_partition_graph
        gnp_random_graph

        References
        ----------
        .. [1] Holland, P. W., Laskey, K. B., & Leinhardt, S.,
               "Stochastic blockmodels: First steps",
               Social networks, 5(2), 109-137, 1983.
        """
        # Check if dimensions match
        if len(sizes) != len(p):
            raise nx.NetworkXException("'sizes' and 'p' do not match.")
        # Check for probability symmetry (undirected) and shape (directed)
        for row in p:
            if len(p) != len(row):
                raise nx.NetworkXException("'p' must be a square matrix.")
        if not directed:
            p_transpose = [list(i) for i in zip(*p)]
            for i in zip(p, p_transpose):
                for j in zip(i[0], i[1]):
                    if abs(j[0] - j[1]) > 1e-08:
                        raise nx.NetworkXException("'p' must be symmetric.")
        # Check for probability range
        for row in p:
            for prob in row:
                if prob < 0 or prob > 1:
                    raise nx.NetworkXException("Entries of 'p' not in [0,1].")
        # Check for nodelist consistency
        if nodelist is not None:
            if len(nodelist) != sum(sizes):
                raise nx.NetworkXException("'nodelist' and 'sizes' do not match.")
            if len(nodelist) != len(set(nodelist)):
                raise nx.NetworkXException("nodelist contains duplicate.")
        else:
            nodelist = range(sum(sizes))

        # Setup the graph conditionally to the directed switch.
        block_range = range(len(sizes))
        if directed:
            g = nx.DiGraph()
            block_iter = itertools.product(block_range, block_range)
        else:
            g = nx.Graph()
            block_iter = itertools.combinations_with_replacement(block_range, 2)
        # Split nodelist in a partition (list of sets).
        size_cumsum = [sum(sizes[0:x]) for x in range(len(sizes) + 1)]
        g.graph["partition"] = [
            set(nodelist[size_cumsum[x] : size_cumsum[x + 1]])
            for x in range(len(size_cumsum) - 1)
        ]
        # Setup nodes and graph name
        for block_id, nodes in enumerate(g.graph["partition"]):
            for node in nodes:
                g.add_node(node, block=block_id)

        g.name = "stochastic_block_model"

        # Test for edge existence
        parts = g.graph["partition"]
        for i, j in block_iter:
            if i == j:
                if directed:
                    if selfloops:
                        edges = itertools.product(parts[i], parts[i])
                    else:
                        edges = itertools.permutations(parts[i], 2)
                else:
                    edges = itertools.combinations(parts[i], 2)
                    if selfloops:
                        edges = itertools.chain(edges, zip(parts[i], parts[i]))
                for e in edges:
                    if seed.random() < p[i][j]:
                        g.add_edge(*e)
            else:
                edges = itertools.product(parts[i], parts[j])
            if sparse:
                if p[i][j] == 1:  # Test edges cases p_ij = 0 or 1
                    for e in edges:
                        g.add_edge(*e)
                elif p[i][j] > 0:
                    while True:
                        try:
                            logrand = math.log(seed.random())
                            skip = math.floor(logrand / math.log(1 - p[i][j]))
                            # consume "skip" edges
                            next(itertools.islice(edges, skip, skip), None)
                            e = next(edges)
                            g.add_edge(*e)  # __safe
                        except StopIteration:
                            break
            else:
                for e in edges:
                    if seed.random() < p[i][j]:
                        g.add_edge(*e)  # __safe
        return g

    def _zipf_rv_below(gamma, xmin, threshold, seed):
        """Returns a random value chosen from the bounded Zipf distribution.

        Repeatedly draws values from the Zipf distribution until the
        threshold is met, then returns that value.
        """
        result = nx.utils.zipf_rv(gamma, xmin, seed)
        while result > threshold:
            result = nx.utils.zipf_rv(gamma, xmin, seed)
        return result

    def _powerlaw_sequence(gamma, low, high, condition, length, max_iters, seed):
        """Returns a list of numbers obeying a constrained power law distribution.

        ``gamma`` and ``low`` are the parameters for the Zipf distribution.

        ``high`` is the maximum allowed value for values draw from the Zipf
        distribution. For more information, see :func:`_zipf_rv_below`.

        ``condition`` and ``length`` are Boolean-valued functions on
        lists. While generating the list, random values are drawn and
        appended to the list until ``length`` is satisfied by the created
        list. Once ``condition`` is satisfied, the sequence generated in
        this way is returned.

        ``max_iters`` indicates the number of times to generate a list
        satisfying ``length``. If the number of iterations exceeds this
        value, :exc:`~networkx.exception.ExceededMaxIterations` is raised.

        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        """
        for i in range(max_iters):
            seq = []
            while not length(seq):
                seq.append(_zipf_rv_below(gamma, low, high, seed))
            if condition(seq):
                return seq
        raise nx.ExceededMaxIterations("Could not create power law sequence")

    def _hurwitz_zeta(x, q, tolerance):
        """The Hurwitz zeta function, or the Riemann zeta function of two arguments.

        ``x`` must be greater than one and ``q`` must be positive.

        This function repeatedly computes subsequent partial sums until
        convergence, as decided by ``tolerance``.
        """
        z = 0
        z_prev = -float("inf")
        k = 0
        while abs(z - z_prev) > tolerance:
            z_prev = z
            z += 1 / ((k + q) ** x)
            k += 1
        return z

    def _generate_min_degree(gamma, average_degree, max_degree, tolerance, max_iters):
        """Returns a minimum degree from the given average degree."""
        # Defines zeta function whether or not Scipy is available
        try:
            from scipy.special import zeta
        except ImportError:

            def zeta(x, q):
                return _hurwitz_zeta(x, q, tolerance)

        min_deg_top = max_degree
        min_deg_bot = 1
        min_deg_mid = (min_deg_top - min_deg_bot) / 2 + min_deg_bot
        itrs = 0
        mid_avg_deg = 0
        while abs(mid_avg_deg - average_degree) > tolerance:
            if itrs > max_iters:
                raise nx.ExceededMaxIterations("Could not match average_degree")
            mid_avg_deg = 0
            for x in range(int(min_deg_mid), max_degree + 1):
                mid_avg_deg += (x ** (-gamma + 1)) / zeta(gamma, min_deg_mid)
            if mid_avg_deg > average_degree:
                min_deg_top = min_deg_mid
                min_deg_mid = (min_deg_top - min_deg_bot) / 2 + min_deg_bot
            else:
                min_deg_bot = min_deg_mid
                min_deg_mid = (min_deg_top - min_deg_bot) / 2 + min_deg_bot
            itrs += 1
        # return int(min_deg_mid + 0.5)
        return round(min_deg_mid)

    def _generate_communities(degree_seq, community_sizes, mu, max_iters, seed):
        """Returns a list of sets, each of which represents a community.

        ``degree_seq`` is the degree sequence that must be met by the
        graph.

        ``community_sizes`` is the community size distribution that must be
        met by the generated list of sets.

        ``mu`` is a float in the interval [0, 1] indicating the fraction of
        intra-community edges incident to each node.

        ``max_iters`` is the number of times to try to add a node to a
        community. This must be greater than the length of
        ``degree_seq``, otherwise this function will always fail. If
        the number of iterations exceeds this value,
        :exc:`~networkx.exception.ExceededMaxIterations` is raised.

        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.

        The communities returned by this are sets of integers in the set {0,
        ..., *n* - 1}, where *n* is the length of ``degree_seq``.

        """
        # This assumes the nodes in the graph will be natural numbers.
        result = [set() for _ in community_sizes]
        n = len(degree_seq)
        free = list(range(n))
        for i in range(max_iters):
            v = free.pop()
            c = seed.choice(range(len(community_sizes)))
            # s = int(degree_seq[v] * (1 - mu) + 0.5)
            s = round(degree_seq[v] * (1 - mu))
            # If the community is large enough, add the node to the chosen
            # community. Otherwise, return it to the list of unaffiliated
            # nodes.
            if s < community_sizes[c]:
                result[c].add(v)
            else:
                free.append(v)
            # If the community is too big, remove a node from it.
            if len(result[c]) > community_sizes[c]:
                free.append(result[c].pop())
            if not free:
                return result
        msg = "Could not assign communities; try increasing min_community"
        raise nx.ExceededMaxIterations(msg)

    @py_random_state(11)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def LFR_benchmark_graph(
        n,
        tau1,
        tau2,
        mu,
        average_degree=None,
        min_degree=None,
        max_degree=None,
        min_community=None,
        max_community=None,
        tol=1.0e-7,
        max_iters=500,
        seed=None,
    ):
        r"""Returns the LFR benchmark graph.

        This algorithm proceeds as follows:

        1) Find a degree sequence with a power law distribution, and minimum
           value ``min_degree``, which has approximate average degree
           ``average_degree``. This is accomplished by either

           a) specifying ``min_degree`` and not ``average_degree``,
           b) specifying ``average_degree`` and not ``min_degree``, in which
              case a suitable minimum degree will be found.

           ``max_degree`` can also be specified, otherwise it will be set to
           ``n``. Each node *u* will have $\mu \mathrm{deg}(u)$ edges
           joining it to nodes in communities other than its own and $(1 -
           \mu) \mathrm{deg}(u)$ edges joining it to nodes in its own
           community.
        2) Generate community sizes according to a power law distribution
           with exponent ``tau2``. If ``min_community`` and
           ``max_community`` are not specified they will be selected to be
           ``min_degree`` and ``max_degree``, respectively.  Community sizes
           are generated until the sum of their sizes equals ``n``.
        3) Each node will be randomly assigned a community with the
           condition that the community is large enough for the node's
           intra-community degree, $(1 - \mu) \mathrm{deg}(u)$ as
           described in step 2. If a community grows too large, a random node
           will be selected for reassignment to a new community, until all
           nodes have been assigned a community.
        4) Each node *u* then adds $(1 - \mu) \mathrm{deg}(u)$
           intra-community edges and $\mu \mathrm{deg}(u)$ inter-community
           edges.

        Parameters
        ----------
        n : int
            Number of nodes in the created graph.

        tau1 : float
            Power law exponent for the degree distribution of the created
            graph. This value must be strictly greater than one.

        tau2 : float
            Power law exponent for the community size distribution in the
            created graph. This value must be strictly greater than one.

        mu : float
            Fraction of inter-community edges incident to each node. This
            value must be in the interval [0, 1].

        average_degree : float
            Desired average degree of nodes in the created graph. This value
            must be in the interval [0, *n*]. Exactly one of this and
            ``min_degree`` must be specified, otherwise a
            :exc:`NetworkXError` is raised.

        min_degree : int
            Minimum degree of nodes in the created graph. This value must be
            in the interval [0, *n*]. Exactly one of this and
            ``average_degree`` must be specified, otherwise a
            :exc:`NetworkXError` is raised.

        max_degree : int
            Maximum degree of nodes in the created graph. If not specified,
            this is set to ``n``, the total number of nodes in the graph.

        min_community : int
            Minimum size of communities in the graph. If not specified, this
            is set to ``min_degree``.

        max_community : int
            Maximum size of communities in the graph. If not specified, this
            is set to ``n``, the total number of nodes in the graph.

        tol : float
            Tolerance when comparing floats, specifically when comparing
            average degree values.

        max_iters : int
            Maximum number of iterations to try to create the community sizes,
            degree distribution, and community affiliations.

        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.

        Returns
        -------
        G : NetworkX graph
            The LFR benchmark graph generated according to the specified
            parameters.

            Each node in the graph has a node attribute ``'community'`` that
            stores the community (that is, the set of nodes) that includes
            it.

        Raises
        ------
        NetworkXError
            If any of the parameters do not meet their upper and lower bounds:

            - ``tau1`` and ``tau2`` must be strictly greater than 1.
            - ``mu`` must be in [0, 1].
            - ``max_degree`` must be in {1, ..., *n*}.
            - ``min_community`` and ``max_community`` must be in {0, ...,
              *n*}.

            If not exactly one of ``average_degree`` and ``min_degree`` is
            specified.

            If ``min_degree`` is not specified and a suitable ``min_degree``
            cannot be found.

        ExceededMaxIterations
            If a valid degree sequence cannot be created within
            ``max_iters`` number of iterations.

            If a valid set of community sizes cannot be created within
            ``max_iters`` number of iterations.

            If a valid community assignment cannot be created within ``10 *
            n * max_iters`` number of iterations.

        Examples
        --------
        Basic usage::

            >>> from networkx.generators.community import LFR_benchmark_graph
            >>> n = 250
            >>> tau1 = 3
            >>> tau2 = 1.5
            >>> mu = 0.1
            >>> G = LFR_benchmark_graph(
            ...     n, tau1, tau2, mu, average_degree=5, min_community=20, seed=10
            ... )

        Continuing the example above, you can get the communities from the
        node attributes of the graph::

            >>> communities = {frozenset(G.nodes[v]["community"]) for v in G}

        Notes
        -----
        This algorithm differs slightly from the original way it was
        presented in [1].

        1) Rather than connecting the graph via a configuration model then
           rewiring to match the intra-community and inter-community
           degrees, we do this wiring explicitly at the end, which should be
           equivalent.
        2) The code posted on the author's website [2] calculates the random
           power law distributed variables and their average using
           continuous approximations, whereas we use the discrete
           distributions here as both degree and community size are
           discrete.

        Though the authors describe the algorithm as quite robust, testing
        during development indicates that a somewhat narrower parameter set
        is likely to successfully produce a graph. Some suggestions have
        been provided in the event of exceptions.

        References
        ----------
        .. [1] "Benchmark graphs for testing community detection algorithms",
               Andrea Lancichinetti, Santo Fortunato, and Filippo Radicchi,
               Phys. Rev. E 78, 046110 2008
        .. [2] https://www.santofortunato.net/resources

        """
        # Perform some basic parameter validation.
        if not tau1 > 1:
            raise nx.NetworkXError("tau1 must be greater than one")
        if not tau2 > 1:
            raise nx.NetworkXError("tau2 must be greater than one")
        if not 0 <= mu <= 1:
            raise nx.NetworkXError("mu must be in the interval [0, 1]")

        # Validate parameters for generating the degree sequence.
        if max_degree is None:
            max_degree = n
        elif not 0 < max_degree <= n:
            raise nx.NetworkXError("max_degree must be in the interval (0, n]")
        if not ((min_degree is None) ^ (average_degree is None)):
            raise nx.NetworkXError(
                "Must assign exactly one of min_degree and average_degree"
            )
        if min_degree is None:
            min_degree = _generate_min_degree(
                tau1, average_degree, max_degree, tol, max_iters
            )

        # Generate a degree sequence with a power law distribution.
        low, high = min_degree, max_degree

        def condition(seq):
            return sum(seq) % 2 == 0

        def length(seq):
            return len(seq) >= n

        deg_seq = _powerlaw_sequence(tau1, low, high, condition, length, max_iters, seed)

        # Validate parameters for generating the community size sequence.
        if min_community is None:
            min_community = min(deg_seq)
        if max_community is None:
            max_community = max(deg_seq)

        # Generate a community size sequence with a power law distribution.
        #
        # TODO The original code incremented the number of iterations each
        # time a new Zipf random value was drawn from the distribution. This
        # differed from the way the number of iterations was incremented in
        # `_powerlaw_degree_sequence`, so this code was changed to match
        # that one. As a result, this code is allowed many more chances to
        # generate a valid community size sequence.
        low, high = min_community, max_community

        def condition(seq):
            return sum(seq) == n

        def length(seq):
            return sum(seq) >= n

        comms = _powerlaw_sequence(tau2, low, high, condition, length, max_iters, seed)

        # Generate the communities based on the given degree sequence and
        # community sizes.
        max_iters *= 10 * n
        communities = _generate_communities(deg_seq, comms, mu, max_iters, seed)

        # Finally, generate the benchmark graph based on the given
        # communities, joining nodes according to the intra- and
        # inter-community degrees.
        G = nx.Graph()
        G.add_nodes_from(range(n))
        for c in communities:
            for u in c:
                while G.degree(u) < round(deg_seq[u] * (1 - mu)):
                    v = seed.choice(list(c))
                    G.add_edge(u, v)
                while G.degree(u) < deg_seq[u]:
                    v = seed.choice(range(n))
                    if v not in c:
                        G.add_edge(u, v)
                G.nodes[u]["community"] = c
        return G