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# Source code for networkx.generators.random_graphs

    """
    Generators for random graphs.

    """

    import itertools
    import math
    from collections import defaultdict

    import networkx as nx
    from networkx.utils import py_random_state

    from ..utils.misc import check_create_using
    from .classic import complete_graph, empty_graph, path_graph, star_graph
    from .degree_seq import degree_sequence_tree

    __all__ = [
        "fast_gnp_random_graph",
        "gnp_random_graph",
        "dense_gnm_random_graph",
        "gnm_random_graph",
        "erdos_renyi_graph",
        "binomial_graph",
        "newman_watts_strogatz_graph",
        "watts_strogatz_graph",
        "connected_watts_strogatz_graph",
        "random_regular_graph",
        "barabasi_albert_graph",
        "dual_barabasi_albert_graph",
        "extended_barabasi_albert_graph",
        "powerlaw_cluster_graph",
        "random_lobster",
        "random_shell_graph",
        "random_powerlaw_tree",
        "random_powerlaw_tree_sequence",
        "random_kernel_graph",
    ]

    @py_random_state(2)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def fast_gnp_random_graph(n, p, seed=None, directed=False, *, create_using=None):
        """Returns a $G_{n,p}$ random graph, also known as an Erdős-Rényi graph or
        a binomial graph.

        Parameters
        ----------
        n : int
            The number of nodes.
        p : float
            Probability for edge creation.
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        directed : bool, optional (default=False)
            If True, this function returns a directed graph.
        create_using : Graph constructor, optional (default=nx.Graph or nx.DiGraph)
            Graph type to create. If graph instance, then cleared before populated.
            Multigraph types are not supported and raise a ``NetworkXError``.
            By default NetworkX Graph or DiGraph are used depending on `directed`.

        Notes
        -----
        The $G_{n,p}$ graph algorithm chooses each of the $[n (n - 1)] / 2$
        (undirected) or $n (n - 1)$ (directed) possible edges with probability $p$.

        This algorithm [1]_ runs in $O(n + m)$ time, where `m` is the expected number of
        edges, which equals $p n (n - 1) / 2$. This should be faster than
        :func:`gnp_random_graph` when $p$ is small and the expected number of edges
        is small (that is, the graph is sparse).

        See Also
        --------
        gnp_random_graph

        References
        ----------
        .. [1] Vladimir Batagelj and Ulrik Brandes,
           "Efficient generation of large random networks",
           Phys. Rev. E, 71, 036113, 2005.
        """
        default = nx.DiGraph if directed else nx.Graph
        create_using = check_create_using(
            create_using, directed=directed, multigraph=False, default=default
        )
        if p <= 0 or p >= 1:
            return nx.gnp_random_graph(
                n, p, seed=seed, directed=directed, create_using=create_using
            )

        G = empty_graph(n, create_using=create_using)

        lp = math.log(1.0 - p)

        if directed:
            v = 1
            w = -1
            while v < n:
                lr = math.log(1.0 - seed.random())
                w = w + 1 + int(lr / lp)
                while w >= v and v < n:
                    w = w - v
                    v = v + 1
                if v < n:
                    G.add_edge(w, v)

        # Nodes in graph are from 0,n-1 (start with v as the second node index).
        v = 1
        w = -1
        while v < n:
            lr = math.log(1.0 - seed.random())
            w = w + 1 + int(lr / lp)
            while w >= v and v < n:
                w = w - v
                v = v + 1
            if v < n:
                G.add_edge(v, w)
        return G

    @py_random_state(2)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def gnp_random_graph(n, p, seed=None, directed=False, *, create_using=None):
        """Returns a $G_{n,p}$ random graph, also known as an Erdős-Rényi graph
        or a binomial graph.

        The $G_{n,p}$ model chooses each of the possible edges with probability $p$.

        Parameters
        ----------
        n : int
            The number of nodes.
        p : float
            Probability for edge creation.
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        directed : bool, optional (default=False)
            If True, this function returns a directed graph.
        create_using : Graph constructor, optional (default=nx.Graph or nx.DiGraph)
            Graph type to create. If graph instance, then cleared before populated.
            Multigraph types are not supported and raise a ``NetworkXError``.
            By default NetworkX Graph or DiGraph are used depending on `directed`.

        See Also
        --------
        fast_gnp_random_graph

        Notes
        -----
        This algorithm [2]_ runs in $O(n^2)$ time.  For sparse graphs (that is, for
        small values of $p$), :func:`fast_gnp_random_graph` is a faster algorithm.

        :func:`binomial_graph` and :func:`erdos_renyi_graph` are
        aliases for :func:`gnp_random_graph`.

        >>> nx.binomial_graph is nx.gnp_random_graph
        True
        >>> nx.erdos_renyi_graph is nx.gnp_random_graph
        True

        References
        ----------
        .. [1] P. Erdős and A. Rényi, On Random Graphs, Publ. Math. 6, 290 (1959).
        .. [2] E. N. Gilbert, Random Graphs, Ann. Math. Stat., 30, 1141 (1959).
        """
        default = nx.DiGraph if directed else nx.Graph
        create_using = check_create_using(
            create_using, directed=directed, multigraph=False, default=default
        )
        if p >= 1:
            return complete_graph(n, create_using=create_using)

        G = nx.empty_graph(n, create_using=create_using)
        if p <= 0:
            return G

        edgetool = itertools.permutations if directed else itertools.combinations
        for e in edgetool(range(n), 2):
            if seed.random() < p:
                G.add_edge(*e)
        return G

    # add some aliases to common names
    binomial_graph = gnp_random_graph
    erdos_renyi_graph = gnp_random_graph

    @py_random_state(2)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def dense_gnm_random_graph(n, m, seed=None, *, create_using=None):
        """Returns a $G_{n,m}$ random graph.

        In the $G_{n,m}$ model, a graph is chosen uniformly at random from the set
        of all graphs with $n$ nodes and $m$ edges.

        This algorithm should be faster than :func:`gnm_random_graph` for dense
        graphs.

        Parameters
        ----------
        n : int
            The number of nodes.
        m : int
            The number of edges.
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        create_using : Graph constructor, optional (default=nx.Graph)
            Graph type to create. If graph instance, then cleared before populated.
            Multigraph and directed types are not supported and raise a ``NetworkXError``.

        See Also
        --------
        gnm_random_graph

        Notes
        -----
        Algorithm by Keith M. Briggs Mar 31, 2006.
        Inspired by Knuth's Algorithm S (Selection sampling technique),
        in section 3.4.2 of [1]_.

        References
        ----------
        .. [1] Donald E. Knuth, The Art of Computer Programming,
            Volume 2/Seminumerical algorithms, Third Edition, Addison-Wesley, 1997.
        """
        create_using = check_create_using(create_using, directed=False, multigraph=False)
        mmax = n * (n - 1) // 2
        if m >= mmax:
            return complete_graph(n, create_using)
        G = empty_graph(n, create_using)

        if n == 1:
            return G

        u = 0
        v = 1
        t = 0
        k = 0
        while True:
            if seed.randrange(mmax - t) < m - k:
                G.add_edge(u, v)
                k += 1
                if k == m:
                    return G
            t += 1
            v += 1
            if v == n:  # go to next row of adjacency matrix
                u += 1
                v = u + 1

    @py_random_state(2)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def gnm_random_graph(n, m, seed=None, directed=False, *, create_using=None):
        """Returns a $G_{n,m}$ random graph.

        In the $G_{n,m}$ model, a graph is chosen uniformly at random from the set
        of all graphs with $n$ nodes and $m$ edges.

        This algorithm should be faster than :func:`dense_gnm_random_graph` for
        sparse graphs.

        Parameters
        ----------
        n : int
            The number of nodes.
        m : int
            The number of edges.
        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
        create_using : Graph constructor, optional (default=nx.Graph or nx.DiGraph)
            Graph type to create. If graph instance, then cleared before populated.
            Multigraph types are not supported and raise a ``NetworkXError``.
            By default NetworkX Graph or DiGraph are used depending on `directed`.

        See also
        --------
        dense_gnm_random_graph

        """
        default = nx.DiGraph if directed else nx.Graph
        create_using = check_create_using(
            create_using, directed=directed, multigraph=False, default=default
        )
        if n == 1:
            return nx.empty_graph(n, create_using=create_using)
        max_edges = n * (n - 1) if directed else n * (n - 1) / 2.0
        if m >= max_edges:
            return complete_graph(n, create_using=create_using)

        G = nx.empty_graph(n, create_using=create_using)
        nlist = list(G)
        edge_count = 0
        while edge_count < m:
            # generate random edge,u,v
            u = seed.choice(nlist)
            v = seed.choice(nlist)
            if u == v or G.has_edge(u, v):
                continue
            else:
                G.add_edge(u, v)
                edge_count = edge_count + 1
        return G

    @py_random_state(3)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def newman_watts_strogatz_graph(n, k, p, seed=None, *, create_using=None):
        """Returns a Newman–Watts–Strogatz small-world graph.

        Parameters
        ----------
        n : int
            The number of nodes.
        k : int
            Each node is joined with its `k` nearest neighbors in a ring
            topology.
        p : float
            The probability of adding a new edge for each edge.
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        create_using : Graph constructor, optional (default=nx.Graph)
            Graph type to create. If graph instance, then cleared before populated.
            Multigraph and directed types are not supported and raise a ``NetworkXError``.

        Notes
        -----
        First create a ring over $n$ nodes [1]_.  Then each node in the ring is
        connected with its $k$ nearest neighbors (or $k - 1$ neighbors if $k$
        is odd).  Then shortcuts are created by adding new edges as follows: for
        each edge $(u, v)$ in the underlying "$n$-ring with $k$ nearest
        neighbors" with probability $p$ add a new edge $(u, w)$ with
        randomly-chosen existing node $w$.  In contrast with
        :func:`watts_strogatz_graph`, no edges are removed.

        See Also
        --------
        watts_strogatz_graph

        References
        ----------
        .. [1] M. E. J. Newman and D. J. Watts,
           Renormalization group analysis of the small-world network model,
           Physics Letters A, 263, 341, 1999.
           https://doi.org/10.1016/S0375-9601(99)00757-4
        """
        create_using = check_create_using(create_using, directed=False, multigraph=False)
        if k > n:
            raise nx.NetworkXError("k>=n, choose smaller k or larger n")

        # If k == n the graph return is a complete graph
        if k == n:
            return nx.complete_graph(n, create_using)

        G = empty_graph(n, create_using)
        nlist = list(G.nodes())
        fromv = nlist
        # connect the k/2 neighbors
        for j in range(1, k // 2 + 1):
            tov = fromv[j:] + fromv[0:j]  # the first j are now last
            for i in range(len(fromv)):
                G.add_edge(fromv[i], tov[i])
        # for each edge u-v, with probability p, randomly select existing
        # node w and add new edge u-w
        e = list(G.edges())
        for u, v in e:
            if seed.random() < p:
                w = seed.choice(nlist)
                # no self-loops and reject if edge u-w exists
                # is that the correct NWS model?
                while w == u or G.has_edge(u, w):
                    w = seed.choice(nlist)
                    if G.degree(u) >= n - 1:
                        break  # skip this rewiring
                else:
                    G.add_edge(u, w)
        return G

    @py_random_state(3)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def watts_strogatz_graph(n, k, p, seed=None, *, create_using=None):
        """Returns a Watts–Strogatz small-world graph.

        Parameters
        ----------
        n : int
            The number of nodes
        k : int
            Each node is joined with its `k` nearest neighbors in a ring
            topology.
        p : float
            The probability of rewiring each edge
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        create_using : Graph constructor, optional (default=nx.Graph)
            Graph type to create. If graph instance, then cleared before populated.
            Multigraph and directed types are not supported and raise a ``NetworkXError``.

        See Also
        --------
        newman_watts_strogatz_graph
        connected_watts_strogatz_graph

        Notes
        -----
        First create a ring over $n$ nodes [1]_.  Then each node in the ring is joined
        to its $k$ nearest neighbors (or $k - 1$ neighbors if $k$ is odd).
        Then shortcuts are created by replacing some edges as follows: for each
        edge $(u, v)$ in the underlying "$n$-ring with $k$ nearest neighbors"
        with probability $p$ replace it with a new edge $(u, w)$ with uniformly
        random choice of existing node $w$.

        In contrast with :func:`newman_watts_strogatz_graph`, the random rewiring
        does not increase the number of edges. The rewired graph is not guaranteed
        to be connected as in :func:`connected_watts_strogatz_graph`.

        References
        ----------
        .. [1] Duncan J. Watts and Steven H. Strogatz,
           Collective dynamics of small-world networks,
           Nature, 393, pp. 440--442, 1998.
        """
        create_using = check_create_using(create_using, directed=False, multigraph=False)
        if k > n:
            raise nx.NetworkXError("k>n, choose smaller k or larger n")

        # If k == n, the graph is complete not Watts-Strogatz
        if k == n:
            G = nx.complete_graph(n, create_using)
            return G

        G = nx.empty_graph(n, create_using=create_using)
        nodes = list(range(n))  # nodes are labeled 0 to n-1
        # connect each node to k/2 neighbors
        for j in range(1, k // 2 + 1):
            targets = nodes[j:] + nodes[0:j]  # first j nodes are now last in list
            G.add_edges_from(zip(nodes, targets))
        # rewire edges from each node
        # loop over all nodes in order (label) and neighbors in order (distance)
        # no self loops or multiple edges allowed
        for j in range(1, k // 2 + 1):  # outer loop is neighbors
            targets = nodes[j:] + nodes[0:j]  # first j nodes are now last in list
            # inner loop in node order
            for u, v in zip(nodes, targets):
                if seed.random() < p:
                    w = seed.choice(nodes)
                    # Enforce no self-loops or multiple edges
                    while w == u or G.has_edge(u, w):
                        w = seed.choice(nodes)
                        if G.degree(u) >= n - 1:
                            break  # skip this rewiring
                    else:
                        G.remove_edge(u, v)
                        G.add_edge(u, w)
        return G

    @py_random_state(4)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def connected_watts_strogatz_graph(n, k, p, tries=100, seed=None, *, create_using=None):
        """Returns a connected Watts–Strogatz small-world graph.

        Attempts to generate a connected graph by repeated generation of
        Watts–Strogatz small-world graphs.  An exception is raised if the maximum
        number of tries is exceeded.

        Parameters
        ----------
        n : int
            The number of nodes
        k : int
            Each node is joined with its `k` nearest neighbors in a ring
            topology.
        p : float
            The probability of rewiring each edge
        tries : int
            Number of attempts to generate a connected graph.
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        create_using : Graph constructor, optional (default=nx.Graph)
            Graph type to create. If graph instance, then cleared before populated.
            Multigraph and directed types are not supported and raise a ``NetworkXError``.

        Notes
        -----
        First create a ring over $n$ nodes [1]_.  Then each node in the ring is joined
        to its $k$ nearest neighbors (or $k - 1$ neighbors if $k$ is odd).
        Then shortcuts are created by replacing some edges as follows: for each
        edge $(u, v)$ in the underlying "$n$-ring with $k$ nearest neighbors"
        with probability $p$ replace it with a new edge $(u, w)$ with uniformly
        random choice of existing node $w$.
        The entire process is repeated until a connected graph results.

        See Also
        --------
        newman_watts_strogatz_graph
        watts_strogatz_graph

        References
        ----------
        .. [1] Duncan J. Watts and Steven H. Strogatz,
           Collective dynamics of small-world networks,
           Nature, 393, pp. 440--442, 1998.
        """
        for i in range(tries):
            # seed is an RNG so should change sequence each call
            G = watts_strogatz_graph(n, k, p, seed, create_using=create_using)
            if nx.is_connected(G):
                return G
        raise nx.NetworkXError("Maximum number of tries exceeded")

    @py_random_state(2)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def random_regular_graph(d, n, seed=None, *, create_using=None):
        r"""Returns a random $d$-regular graph on $n$ nodes.

        A regular graph is a graph where each node has the same number of neighbors.

        The resulting graph has no self-loops or parallel edges.

        Parameters
        ----------
        d : int
          The degree of each node.
        n : integer
          The number of nodes. The value of $n \times d$ must be even.
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        create_using : Graph constructor, optional (default=nx.Graph)
            Graph type to create. If graph instance, then cleared before populated.
            Multigraph and directed types are not supported and raise a ``NetworkXError``.

        Notes
        -----
        The nodes are numbered from $0$ to $n - 1$.

        Kim and Vu's paper [2]_ shows that this algorithm samples in an
        asymptotically uniform way from the space of random graphs when
        $d = O(n^{1 / 3 - \epsilon})$.

        Raises
        ------

        NetworkXError
            If $n \times d$ is odd or $d$ is greater than or equal to $n$.

        References
        ----------
        .. [1] A. Steger and N. Wormald,
           Generating random regular graphs quickly,
           Probability and Computing 8 (1999), 377-396, 1999.
           https://doi.org/10.1017/S0963548399003867

        .. [2] Jeong Han Kim and Van H. Vu,
           Generating random regular graphs,
           Proceedings of the thirty-fifth ACM symposium on Theory of computing,
           San Diego, CA, USA, pp 213--222, 2003.
           http://portal.acm.org/citation.cfm?id=780542.780576
        """
        create_using = check_create_using(create_using, directed=False, multigraph=False)
        if (n * d) % 2 != 0:
            raise nx.NetworkXError("n * d must be even")

        if not 0 <= d < n:
            raise nx.NetworkXError("the 0 <= d < n inequality must be satisfied")

        G = nx.empty_graph(n, create_using=create_using)

        if d == 0:
            return G

        def _suitable(edges, potential_edges):
            # Helper subroutine to check if there are suitable edges remaining
            # If False, the generation of the graph has failed
            if not potential_edges:
                return True
            for s1 in potential_edges:
                for s2 in potential_edges:
                    # Two iterators on the same dictionary are guaranteed
                    # to visit it in the same order if there are no
                    # intervening modifications.
                    if s1 == s2:
                        # Only need to consider s1-s2 pair one time
                        break
                    if s1 > s2:
                        s1, s2 = s2, s1
                    if (s1, s2) not in edges:
                        return True
            return False

        def _try_creation():
            # Attempt to create an edge set

            edges = set()
            stubs = list(range(n)) * d

            while stubs:
                potential_edges = defaultdict(lambda: 0)
                seed.shuffle(stubs)
                stubiter = iter(stubs)
                for s1, s2 in zip(stubiter, stubiter):
                    if s1 > s2:
                        s1, s2 = s2, s1
                    if s1 != s2 and ((s1, s2) not in edges):
                        edges.add((s1, s2))
                    else:
                        potential_edges[s1] += 1
                        potential_edges[s2] += 1

                if not _suitable(edges, potential_edges):
                    return None  # failed to find suitable edge set

                stubs = [
                    node
                    for node, potential in potential_edges.items()
                    for _ in range(potential)
                ]
            return edges

        # Even though a suitable edge set exists,
        # the generation of such a set is not guaranteed.
        # Try repeatedly to find one.
        edges = _try_creation()
        while edges is None:
            edges = _try_creation()
        G.add_edges_from(edges)

        return G

    def _random_subset(seq, m, rng):
        """Return m unique elements from seq.

        This differs from random.sample which can return repeated
        elements if seq holds repeated elements.

        Note: rng is a random.Random or numpy.random.RandomState instance.
        """
        targets = set()
        while len(targets) < m:
            x = rng.choice(seq)
            targets.add(x)
        return targets

    @py_random_state(2)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def barabasi_albert_graph(n, m, seed=None, initial_graph=None, *, create_using=None):
        """Returns a random graph using Barabási–Albert preferential attachment

        A graph of $n$ nodes is grown by attaching new nodes each with $m$
        edges that are preferentially attached to existing nodes with high degree.

        Parameters
        ----------
        n : int
            Number of nodes
        m : int
            Number of edges to attach from a new node to existing nodes
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        initial_graph : Graph or None (default)
            Initial network for Barabási–Albert algorithm.
            It should be a connected graph for most use cases.
            A copy of `initial_graph` is used.
            If None, starts from a star graph on (m+1) nodes.
        create_using : Graph constructor, optional (default=nx.Graph)
            Graph type to create. If graph instance, then cleared before populated.
            Multigraph and directed types are not supported and raise a ``NetworkXError``.

        Returns
        -------
        G : Graph

        Raises
        ------
        NetworkXError
            If `m` does not satisfy ``1 <= m < n``, or
            the initial graph number of nodes m0 does not satisfy ``m <= m0 <= n``.

        References
        ----------
        .. [1] A. L. Barabási and R. Albert "Emergence of scaling in
           random networks", Science 286, pp 509-512, 1999.
        """
        create_using = check_create_using(create_using, directed=False, multigraph=False)
        if m < 1 or m >= n:
            raise nx.NetworkXError(
                f"Barabási–Albert network must have m >= 1 and m < n, m = {m}, n = {n}"
            )

        if initial_graph is None:
            # Default initial graph : star graph on (m + 1) nodes
            G = star_graph(m, create_using)
        else:
            if len(initial_graph) < m or len(initial_graph) > n:
                raise nx.NetworkXError(
                    f"Barabási–Albert initial graph needs between m={m} and n={n} nodes"
                )
            G = initial_graph.copy()

        # List of existing nodes, with nodes repeated once for each adjacent edge
        repeated_nodes = [n for n, d in G.degree() for _ in range(d)]
        # Start adding the other n - m0 nodes.
        source = len(G)
        while source < n:
            # Now choose m unique nodes from the existing nodes
            # Pick uniformly from repeated_nodes (preferential attachment)
            targets = _random_subset(repeated_nodes, m, seed)
            # Add edges to m nodes from the source.
            G.add_edges_from(zip([source] * m, targets))
            # Add one node to the list for each new edge just created.
            repeated_nodes.extend(targets)
            # And the new node "source" has m edges to add to the list.
            repeated_nodes.extend([source] * m)

            source += 1
        return G

    @py_random_state(4)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def dual_barabasi_albert_graph(
        n, m1, m2, p, seed=None, initial_graph=None, *, create_using=None
    ):
        """Returns a random graph using dual Barabási–Albert preferential attachment

        A graph of $n$ nodes is grown by attaching new nodes each with either $m_1$
        edges (with probability $p$) or $m_2$ edges (with probability $1-p$) that
        are preferentially attached to existing nodes with high degree.

        Parameters
        ----------
        n : int
            Number of nodes
        m1 : int
            Number of edges to link each new node to existing nodes with probability $p$
        m2 : int
            Number of edges to link each new node to existing nodes with probability $1-p$
        p : float
            The probability of attaching $m_1$ edges (as opposed to $m_2$ edges)
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        initial_graph : Graph or None (default)
            Initial network for Barabási–Albert algorithm.
            A copy of `initial_graph` is used.
            It should be connected for most use cases.
            If None, starts from an star graph on max(m1, m2) + 1 nodes.
        create_using : Graph constructor, optional (default=nx.Graph)
            Graph type to create. If graph instance, then cleared before populated.
            Multigraph and directed types are not supported and raise a ``NetworkXError``.

        Returns
        -------
        G : Graph

        Raises
        ------
        NetworkXError
            If `m1` and `m2` do not satisfy ``1 <= m1,m2 < n``, or
            `p` does not satisfy ``0 <= p <= 1``, or
            the initial graph number of nodes m0 does not satisfy m1, m2 <= m0 <= n.

        References
        ----------
        .. [1] N. Moshiri "The dual-Barabasi-Albert model", arXiv:1810.10538.
        """
        create_using = check_create_using(create_using, directed=False, multigraph=False)
        if m1 < 1 or m1 >= n:
            raise nx.NetworkXError(
                f"Dual Barabási–Albert must have m1 >= 1 and m1 < n, m1 = {m1}, n = {n}"
            )
        if m2 < 1 or m2 >= n:
            raise nx.NetworkXError(
                f"Dual Barabási–Albert must have m2 >= 1 and m2 < n, m2 = {m2}, n = {n}"
            )
        if p < 0 or p > 1:
            raise nx.NetworkXError(
                f"Dual Barabási–Albert network must have 0 <= p <= 1, p = {p}"
            )

        # For simplicity, if p == 0 or 1, just return BA
        if p == 1:
            return barabasi_albert_graph(n, m1, seed, create_using=create_using)
        elif p == 0:
            return barabasi_albert_graph(n, m2, seed, create_using=create_using)

        if initial_graph is None:
            # Default initial graph : star graph on max(m1, m2) nodes
            G = star_graph(max(m1, m2), create_using)
        else:
            if len(initial_graph) < max(m1, m2) or len(initial_graph) > n:
                raise nx.NetworkXError(
                    f"Barabási–Albert initial graph must have between "
                    f"max(m1, m2) = {max(m1, m2)} and n = {n} nodes"
                )
            G = initial_graph.copy()

        # Target nodes for new edges
        targets = list(G)
        # List of existing nodes, with nodes repeated once for each adjacent edge
        repeated_nodes = [n for n, d in G.degree() for _ in range(d)]
        # Start adding the remaining nodes.
        source = len(G)
        while source < n:
            # Pick which m to use (m1 or m2)
            if seed.random() < p:
                m = m1
            else:
                m = m2
            # Now choose m unique nodes from the existing nodes
            # Pick uniformly from repeated_nodes (preferential attachment)
            targets = _random_subset(repeated_nodes, m, seed)
            # Add edges to m nodes from the source.
            G.add_edges_from(zip([source] * m, targets))
            # Add one node to the list for each new edge just created.
            repeated_nodes.extend(targets)
            # And the new node "source" has m edges to add to the list.
            repeated_nodes.extend([source] * m)

            source += 1
        return G

    @py_random_state(4)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def extended_barabasi_albert_graph(n, m, p, q, seed=None, *, create_using=None):
        """Returns an extended Barabási–Albert model graph.

        An extended Barabási–Albert model graph is a random graph constructed
        using preferential attachment. The extended model allows new edges,
        rewired edges or new nodes. Based on the probabilities $p$ and $q$
        with $p + q < 1$, the growing behavior of the graph is determined as:

        1) With $p$ probability, $m$ new edges are added to the graph,
        starting from randomly chosen existing nodes and attached preferentially at the
        other end.

        2) With $q$ probability, $m$ existing edges are rewired
        by randomly choosing an edge and rewiring one end to a preferentially chosen node.

        3) With $(1 - p - q)$ probability, $m$ new nodes are added to the graph
        with edges attached preferentially.

        When $p = q = 0$, the model behaves just like the Barabási–Alber model.

        Parameters
        ----------
        n : int
            Number of nodes
        m : int
            Number of edges with which a new node attaches to existing nodes
        p : float
            Probability value for adding an edge between existing nodes. p + q < 1
        q : float
            Probability value of rewiring of existing edges. p + q < 1
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        create_using : Graph constructor, optional (default=nx.Graph)
            Graph type to create. If graph instance, then cleared before populated.
            Multigraph and directed types are not supported and raise a ``NetworkXError``.

        Returns
        -------
        G : Graph

        Raises
        ------
        NetworkXError
            If `m` does not satisfy ``1 <= m < n`` or ``1 >= p + q``

        References
        ----------
        .. [1] Albert, R., & Barabási, A. L. (2000)
           Topology of evolving networks: local events and universality
           Physical review letters, 85(24), 5234.
        """
        create_using = check_create_using(create_using, directed=False, multigraph=False)
        if m < 1 or m >= n:
            msg = f"Extended Barabasi-Albert network needs m>=1 and m<n, m={m}, n={n}"
            raise nx.NetworkXError(msg)
        if p + q >= 1:
            msg = f"Extended Barabasi-Albert network needs p + q <= 1, p={p}, q={q}"
            raise nx.NetworkXError(msg)

        # Add m initial nodes (m0 in barabasi-speak)
        G = empty_graph(m, create_using)

        # List of nodes to represent the preferential attachment random selection.
        # At the creation of the graph, all nodes are added to the list
        # so that even nodes that are not connected have a chance to get selected,
        # for rewiring and adding of edges.
        # With each new edge, nodes at the ends of the edge are added to the list.
        attachment_preference = []
        attachment_preference.extend(range(m))

        # Start adding the other n-m nodes. The first node is m.
        new_node = m
        while new_node < n:
            a_probability = seed.random()

            # Total number of edges of a Clique of all the nodes
            clique_degree = len(G) - 1
            clique_size = (len(G) * clique_degree) / 2

            # Adding m new edges, if there is room to add them
            if a_probability < p and G.size() <= clique_size - m:
                # Select the nodes where an edge can be added
                eligible_nodes = [nd for nd, deg in G.degree() if deg < clique_degree]
                for i in range(m):
                    # Choosing a random source node from eligible_nodes
                    src_node = seed.choice(eligible_nodes)

                    # Picking a possible node that is not 'src_node' or
                    # neighbor with 'src_node', with preferential attachment
                    prohibited_nodes = list(G[src_node])
                    prohibited_nodes.append(src_node)
                    # This will raise an exception if the sequence is empty
                    dest_node = seed.choice(
                        [nd for nd in attachment_preference if nd not in prohibited_nodes]
                    )
                    # Adding the new edge
                    G.add_edge(src_node, dest_node)

                    # Appending both nodes to add to their preferential attachment
                    attachment_preference.append(src_node)
                    attachment_preference.append(dest_node)

                    # Adjusting the eligible nodes. Degree may be saturated.
                    if G.degree(src_node) == clique_degree:
                        eligible_nodes.remove(src_node)
                    if G.degree(dest_node) == clique_degree and dest_node in eligible_nodes:
                        eligible_nodes.remove(dest_node)

            # Rewiring m edges, if there are enough edges
            elif p <= a_probability < (p + q) and m <= G.size() < clique_size:
                # Selecting nodes that have at least 1 edge but that are not
                # fully connected to ALL other nodes (center of star).
                # These nodes are the pivot nodes of the edges to rewire
                eligible_nodes = [nd for nd, deg in G.degree() if 0 < deg < clique_degree]
                for i in range(m):
                    # Choosing a random source node
                    node = seed.choice(eligible_nodes)

                    # The available nodes do have a neighbor at least.
                    nbr_nodes = list(G[node])

                    # Choosing the other end that will get detached
                    src_node = seed.choice(nbr_nodes)

                    # Picking a target node that is not 'node' or
                    # neighbor with 'node', with preferential attachment
                    nbr_nodes.append(node)
                    dest_node = seed.choice(
                        [nd for nd in attachment_preference if nd not in nbr_nodes]
                    )
                    # Rewire
                    G.remove_edge(node, src_node)
                    G.add_edge(node, dest_node)

                    # Adjusting the preferential attachment list
                    attachment_preference.remove(src_node)
                    attachment_preference.append(dest_node)

                    # Adjusting the eligible nodes.
                    # nodes may be saturated or isolated.
                    if G.degree(src_node) == 0 and src_node in eligible_nodes:
                        eligible_nodes.remove(src_node)
                    if dest_node in eligible_nodes:
                        if G.degree(dest_node) == clique_degree:
                            eligible_nodes.remove(dest_node)
                    else:
                        if G.degree(dest_node) == 1:
                            eligible_nodes.append(dest_node)

            # Adding new node with m edges
            else:
                # Select the edges' nodes by preferential attachment
                targets = _random_subset(attachment_preference, m, seed)
                G.add_edges_from(zip([new_node] * m, targets))

                # Add one node to the list for each new edge just created.
                attachment_preference.extend(targets)
                # The new node has m edges to it, plus itself: m + 1
                attachment_preference.extend([new_node] * (m + 1))
                new_node += 1
        return G

    @py_random_state(3)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def powerlaw_cluster_graph(n, m, p, seed=None, *, create_using=None):
        """Holme and Kim algorithm for growing graphs with powerlaw
        degree distribution and approximate average clustering.

        Parameters
        ----------
        n : int
            the number of nodes
        m : int
            the number of random edges to add for each new node
        p : float,
            Probability of adding a triangle after adding a random edge
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        create_using : Graph constructor, optional (default=nx.Graph)
            Graph type to create. If graph instance, then cleared before populated.
            Multigraph and directed types are not supported and raise a ``NetworkXError``.

        Notes
        -----
        The average clustering has a hard time getting above a certain
        cutoff that depends on `m`.  This cutoff is often quite low.  The
        transitivity (fraction of triangles to possible triangles) seems to
        decrease with network size.

        It is essentially the Barabási–Albert (BA) growth model with an
        extra step that each random edge is followed by a chance of
        making an edge to one of its neighbors too (and thus a triangle).

        This algorithm improves on BA in the sense that it enables a
        higher average clustering to be attained if desired.

        It seems possible to have a disconnected graph with this algorithm
        since the initial `m` nodes may not be all linked to a new node
        on the first iteration like the BA model.

        Raises
        ------
        NetworkXError
            If `m` does not satisfy ``1 <= m <= n`` or `p` does not
            satisfy ``0 <= p <= 1``.

        References
        ----------
        .. [1] P. Holme and B. J. Kim,
           "Growing scale-free networks with tunable clustering",
           Phys. Rev. E, 65, 026107, 2002.
        """
        create_using = check_create_using(create_using, directed=False, multigraph=False)
        if m < 1 or n < m:
            raise nx.NetworkXError(f"NetworkXError must have m>1 and m<n, m={m},n={n}")

        if p > 1 or p < 0:
            raise nx.NetworkXError(f"NetworkXError p must be in [0,1], p={p}")

        G = empty_graph(m, create_using)  # add m initial nodes (m0 in barabasi-speak)
        repeated_nodes = list(G)  # list of existing nodes to sample from
        # with nodes repeated once for each adjacent edge
        source = m  # next node is m
        while source < n:  # Now add the other n-1 nodes
            possible_targets = _random_subset(repeated_nodes, m, seed)
            # do one preferential attachment for new node
            target = possible_targets.pop()
            G.add_edge(source, target)
            repeated_nodes.append(target)  # add one node to list for each new link
            count = 1
            while count < m:  # add m-1 more new links
                if seed.random() < p:  # clustering step: add triangle
                    neighborhood = [
                        nbr
                        for nbr in G.neighbors(target)
                        if not G.has_edge(source, nbr) and nbr != source
                    ]
                    if neighborhood:  # if there is a neighbor without a link
                        nbr = seed.choice(neighborhood)
                        G.add_edge(source, nbr)  # add triangle
                        repeated_nodes.append(nbr)
                        count = count + 1
                        continue  # go to top of while loop
                # else do preferential attachment step if above fails
                target = possible_targets.pop()
                G.add_edge(source, target)
                repeated_nodes.append(target)
                count = count + 1

            repeated_nodes.extend([source] * m)  # add source node to list m times
            source += 1
        return G

    @py_random_state(3)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def random_lobster(n, p1, p2, seed=None, *, create_using=None):
        """Returns a random lobster graph.

        A lobster is a tree that reduces to a caterpillar when pruning all
        leaf nodes. A caterpillar is a tree that reduces to a path graph
        when pruning all leaf nodes; setting `p2` to zero produces a caterpillar.

        This implementation iterates on the probabilities `p1` and `p2` to add
        edges at levels 1 and 2, respectively. Graphs are therefore constructed
        iteratively with uniform randomness at each level rather than being selected
        uniformly at random from the set of all possible lobsters.

        Parameters
        ----------
        n : int
            The expected number of nodes in the backbone
        p1 : float
            Probability of adding an edge to the backbone
        p2 : float
            Probability of adding an edge one level beyond backbone
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        create_using : Graph constructor, optional (default=nx.Grap)
            Graph type to create. If graph instance, then cleared before populated.
            Multigraph and directed types are not supported and raise a ``NetworkXError``.

        Raises
        ------
        NetworkXError
            If `p1` or `p2` parameters are >= 1 because the while loops would never finish.
        """
        create_using = check_create_using(create_using, directed=False, multigraph=False)
        p1, p2 = abs(p1), abs(p2)
        if any(p >= 1 for p in [p1, p2]):
            raise nx.NetworkXError("Probability values for `p1` and `p2` must both be < 1.")

        # a necessary ingredient in any self-respecting graph library
        llen = int(2 * seed.random() * n + 0.5)
        L = path_graph(llen, create_using)
        # build caterpillar: add edges to path graph with probability p1
        current_node = llen - 1
        for n in range(llen):
            while seed.random() < p1:  # add fuzzy caterpillar parts
                current_node += 1
                L.add_edge(n, current_node)
                cat_node = current_node
                while seed.random() < p2:  # add crunchy lobster bits
                    current_node += 1
                    L.add_edge(cat_node, current_node)
        return L  # voila, un lobster!

    @py_random_state(1)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def random_shell_graph(constructor, seed=None, *, create_using=None):
        """Returns a random shell graph for the constructor given.

        Parameters
        ----------
        constructor : list of three-tuples
            Represents the parameters for a shell, starting at the center
            shell.  Each element of the list must be of the form `(n, m,
            d)`, where `n` is the number of nodes in the shell, `m` is
            the number of edges in the shell, and `d` is the ratio of
            inter-shell (next) edges to intra-shell edges. If `d` is zero,
            there will be no intra-shell edges, and if `d` is one there
            will be all possible intra-shell edges.
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        create_using : Graph constructor, optional (default=nx.Graph)
            Graph type to create. Graph instances are not supported.
            Multigraph and directed types are not supported and raise a ``NetworkXError``.

        Examples
        --------
        >>> constructor = [(10, 20, 0.8), (20, 40, 0.8)]
        >>> G = nx.random_shell_graph(constructor)

        """
        create_using = check_create_using(create_using, directed=False, multigraph=False)
        G = empty_graph(0, create_using)

        glist = []
        intra_edges = []
        nnodes = 0
        # create gnm graphs for each shell
        for n, m, d in constructor:
            inter_edges = int(m * d)
            intra_edges.append(m - inter_edges)
            g = nx.convert_node_labels_to_integers(
                gnm_random_graph(n, inter_edges, seed=seed, create_using=G.__class__),
                first_label=nnodes,
            )
            glist.append(g)
            nnodes += n
            G = nx.operators.union(G, g)

        # connect the shells randomly
        for gi in range(len(glist) - 1):
            nlist1 = list(glist[gi])
            nlist2 = list(glist[gi + 1])
            total_edges = intra_edges[gi]
            edge_count = 0
            while edge_count < total_edges:
                u = seed.choice(nlist1)
                v = seed.choice(nlist2)
                if u == v or G.has_edge(u, v):
                    continue
                else:
                    G.add_edge(u, v)
                    edge_count = edge_count + 1
        return G

    @py_random_state(2)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def random_powerlaw_tree(n, gamma=3, seed=None, tries=100, *, create_using=None):
        """Returns a tree with a power law degree distribution.

        Parameters
        ----------
        n : int
            The number of nodes.
        gamma : float
            Exponent of the power law.
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        tries : int
            Number of attempts to adjust the sequence to make it a tree.
        create_using : Graph constructor, optional (default=nx.Graph)
            Graph type to create. If graph instance, then cleared before populated.
            Multigraph and directed types are not supported and raise a ``NetworkXError``.

        Raises
        ------
        NetworkXError
            If no valid sequence is found within the maximum number of
            attempts.

        Notes
        -----
        A trial power law degree sequence is chosen and then elements are
        swapped with new elements from a powerlaw distribution until the
        sequence makes a tree (by checking, for example, that the number of
        edges is one smaller than the number of nodes).

        """
        create_using = check_create_using(create_using, directed=False, multigraph=False)
        # This call may raise a NetworkXError if the number of tries is succeeded.
        seq = random_powerlaw_tree_sequence(n, gamma=gamma, seed=seed, tries=tries)
        G = degree_sequence_tree(seq, create_using)
        return G

    @py_random_state(2)
    @nx._dispatchable(graphs=None)
    def random_powerlaw_tree_sequence(n, gamma=3, seed=None, tries=100):
        """Returns a degree sequence for a tree with a power law distribution.

        Parameters
        ----------
        n : int,
            The number of nodes.
        gamma : float
            Exponent of the power law.
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        tries : int
            Number of attempts to adjust the sequence to make it a tree.

        Raises
        ------
        NetworkXError
            If no valid sequence is found within the maximum number of
            attempts.

        Notes
        -----
        A trial power law degree sequence is chosen and then elements are
        swapped with new elements from a power law distribution until
        the sequence makes a tree (by checking, for example, that the number of
        edges is one smaller than the number of nodes).

        """
        # get trial sequence
        z = nx.utils.powerlaw_sequence(n, exponent=gamma, seed=seed)
        # round to integer values in the range [0,n]
        zseq = [min(n, max(round(s), 0)) for s in z]

        # another sequence to swap values from
        z = nx.utils.powerlaw_sequence(tries, exponent=gamma, seed=seed)
        # round to integer values in the range [0,n]
        swap = [min(n, max(round(s), 0)) for s in z]

        for deg in swap:
            # If this degree sequence can be the degree sequence of a tree, return
            # it. It can be a tree if the number of edges is one fewer than the
            # number of nodes, or in other words, `n - sum(zseq) / 2 == 1`. We
            # use an equivalent condition below that avoids floating point
            # operations.
            if 2 * n - sum(zseq) == 2:
                return zseq
            index = seed.randint(0, n - 1)
            zseq[index] = swap.pop()

        raise nx.NetworkXError(
            f"Exceeded max ({tries}) attempts for a valid tree sequence."
        )

    @py_random_state(3)
    @nx._dispatchable(graphs=None, returns_graph=True)
    def random_kernel_graph(
        n, kernel_integral, kernel_root=None, seed=None, *, create_using=None
    ):
        r"""Returns an random graph based on the specified kernel.

        The algorithm chooses each of the $[n(n-1)]/2$ possible edges with
        probability specified by a kernel $\kappa(x,y)$ [1]_.  The kernel
        $\kappa(x,y)$ must be a symmetric (in $x,y$), non-negative,
        bounded function.

        Parameters
        ----------
        n : int
            The number of nodes
        kernel_integral : function
            Function that returns the definite integral of the kernel $\kappa(x,y)$,
            $F(y,a,b) := \int_a^b \kappa(x,y)dx$
        kernel_root: function (optional)
            Function that returns the root $b$ of the equation $F(y,a,b) = r$.
            If None, the root is found using :func:`scipy.optimize.brentq`
            (this requires SciPy).
        seed : integer, random_state, or None (default)
            Indicator of random number generation state.
            See :ref:`Randomness<randomness>`.
        create_using : Graph constructor, optional (default=nx.Graph)
            Graph type to create. If graph instance, then cleared before populated.
            Multigraph and directed types are not supported and raise a ``NetworkXError``.

        Notes
        -----
        The kernel is specified through its definite integral which must be
        provided as one of the arguments. If the integral and root of the
        kernel integral can be found in $O(1)$ time then this algorithm runs in
        time $O(n+m)$ where m is the expected number of edges [2]_.

        The nodes are set to integers from $0$ to $n-1$.

        Examples
        --------
        Generate an Erdős–Rényi random graph $G(n,c/n)$, with kernel
        $\kappa(x,y)=c$ where $c$ is the mean expected degree.

        >>> def integral(u, w, z):
        ...     return c * (z - w)
        >>> def root(u, w, r):
        ...     return r / c + w
        >>> c = 1
        >>> graph = nx.random_kernel_graph(1000, integral, root)

        See Also
        --------
        gnp_random_graph
        expected_degree_graph

        References
        ----------
        .. [1] Bollobás, Béla,  Janson, S. and Riordan, O.
           "The phase transition in inhomogeneous random graphs",
           *Random Structures Algorithms*, 31, 3--122, 2007.

        .. [2] Hagberg A, Lemons N (2015),
           "Fast Generation of Sparse Random Kernel Graphs".
           PLoS ONE 10(9): e0135177, 2015. doi:10.1371/journal.pone.0135177
        """
        create_using = check_create_using(create_using, directed=False, multigraph=False)
        if kernel_root is None:
            import scipy as sp

            def kernel_root(y, a, r):
                def my_function(b):
                    return kernel_integral(y, a, b) - r

                return sp.optimize.brentq(my_function, a, 1)

        graph = nx.empty_graph(create_using=create_using)
        graph.add_nodes_from(range(n))
        (i, j) = (1, 1)
        while i < n:
            r = -math.log(1 - seed.random())  # (1-seed.random()) in (0, 1]
            if kernel_integral(i / n, j / n, 1) <= r:
                i, j = i + 1, i + 1
            else:
                j = math.ceil(n * kernel_root(i / n, j / n, r))
                graph.add_edge(i - 1, j - 1)
        return graph