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# Source code for networkx.generators.expanders

    """Provides explicit constructions of expander graphs."""

    import itertools

    import networkx as nx

    __all__ = [
        "margulis_gabber_galil_graph",
        "chordal_cycle_graph",
        "paley_graph",
        "maybe_regular_expander",
        "is_regular_expander",
        "random_regular_expander_graph",
    ]

    # Other discrete torus expanders can be constructed by using the following edge
    # sets. For more information, see Chapter 4, "Expander Graphs", in
    # "Pseudorandomness", by Salil Vadhan.
    #
    # For a directed expander, add edges from (x, y) to:
    #
    #     (x, y),
    #     ((x + 1) % n, y),
    #     (x, (y + 1) % n),
    #     (x, (x + y) % n),
    #     (-y % n, x)
    #
    # For an undirected expander, add the reverse edges.
    #
    # Also appearing in the paper of Gabber and Galil:
    #
    #     (x, y),
    #     (x, (x + y) % n),
    #     (x, (x + y + 1) % n),
    #     ((x + y) % n, y),
    #     ((x + y + 1) % n, y)
    #
    # and:
    #
    #     (x, y),
    #     ((x + 2*y) % n, y),
    #     ((x + (2*y + 1)) % n, y),
    #     ((x + (2*y + 2)) % n, y),
    #     (x, (y + 2*x) % n),
    #     (x, (y + (2*x + 1)) % n),
    #     (x, (y + (2*x + 2)) % n),
    #

    @nx._dispatchable(graphs=None, returns_graph=True)
    def margulis_gabber_galil_graph(n, create_using=None):
        r"""Returns the Margulis-Gabber-Galil undirected MultiGraph on `n^2` nodes.

        The undirected MultiGraph is regular with degree `8`. Nodes are integer
        pairs. The second-largest eigenvalue of the adjacency matrix of the graph
        is at most `5 \sqrt{2}`, regardless of `n`.

        Parameters
        ----------
        n : int
            Determines the number of nodes in the graph: `n^2`.
        create_using : NetworkX graph constructor, optional (default MultiGraph)
           Graph type to create. If graph instance, then cleared before populated.

        Returns
        -------
        G : graph
            The constructed undirected multigraph.

        Raises
        ------
        NetworkXError
            If the graph is directed or not a multigraph.

        """
        G = nx.empty_graph(0, create_using, default=nx.MultiGraph)
        if G.is_directed() or not G.is_multigraph():
            msg = "`create_using` must be an undirected multigraph."
            raise nx.NetworkXError(msg)

        for x, y in itertools.product(range(n), repeat=2):
            for u, v in (
                ((x + 2 * y) % n, y),
                ((x + (2 * y + 1)) % n, y),
                (x, (y + 2 * x) % n),
                (x, (y + (2 * x + 1)) % n),
            ):
                G.add_edge((x, y), (u, v))
        G.graph["name"] = f"margulis_gabber_galil_graph({n})"
        return G

    @nx._dispatchable(graphs=None, returns_graph=True)
    def chordal_cycle_graph(p, create_using=None):
        """Returns the chordal cycle graph on `p` nodes.

        The returned graph is a cycle graph on `p` nodes with chords joining each
        vertex `x` to its inverse modulo `p`. This graph is a (mildly explicit)
        3-regular expander [1]_.

        `p` *must* be a prime number.

        Parameters
        ----------
        p : a prime number

            The number of vertices in the graph. This also indicates where the
            chordal edges in the cycle will be created.

        create_using : NetworkX graph constructor, optional (default=nx.Graph)
           Graph type to create. If graph instance, then cleared before populated.

        Returns
        -------
        G : graph
            The constructed undirected multigraph.

        Raises
        ------
        NetworkXError

            If `create_using` indicates directed or not a multigraph.

        References
        ----------

        .. [1] Theorem 4.4.2 in A. Lubotzky. "Discrete groups, expanding graphs and
               invariant measures", volume 125 of Progress in Mathematics.
               Birkhäuser Verlag, Basel, 1994.

        """
        G = nx.empty_graph(0, create_using, default=nx.MultiGraph)
        if G.is_directed() or not G.is_multigraph():
            msg = "`create_using` must be an undirected multigraph."
            raise nx.NetworkXError(msg)

        for x in range(p):
            left = (x - 1) % p
            right = (x + 1) % p
            # Here we apply Fermat's Little Theorem to compute the multiplicative
            # inverse of x in Z/pZ. By Fermat's Little Theorem,
            #
            #     x^p = x (mod p)
            #
            # Therefore,
            #
            #     x * x^(p - 2) = 1 (mod p)
            #
            # The number 0 is a special case: we just let its inverse be itself.
            chord = pow(x, p - 2, p) if x > 0 else 0
            for y in (left, right, chord):
                G.add_edge(x, y)
        G.graph["name"] = f"chordal_cycle_graph({p})"
        return G

    @nx._dispatchable(graphs=None, returns_graph=True)
    def paley_graph(p, create_using=None):
        r"""Returns the Paley $\frac{(p-1)}{2}$ -regular graph on $p$ nodes.

        The returned graph is a graph on $\mathbb{Z}/p\mathbb{Z}$ with edges between $x$ and $y$
        if and only if $x-y$ is a nonzero square in $\mathbb{Z}/p\mathbb{Z}$.

        If $p \equiv 1  \pmod 4$, $-1$ is a square in $\mathbb{Z}/p\mathbb{Z}$ and therefore $x-y$ is a square if and
        only if $y-x$ is also a square, i.e the edges in the Paley graph are symmetric.

        If $p \equiv 3 \pmod 4$, $-1$ is not a square in $\mathbb{Z}/p\mathbb{Z}$ and therefore either $x-y$ or $y-x$
        is a square in $\mathbb{Z}/p\mathbb{Z}$ but not both.

        Note that a more general definition of Paley graphs extends this construction
        to graphs over $q=p^n$ vertices, by using the finite field $F_q$ instead of $\mathbb{Z}/p\mathbb{Z}$.
        This construction requires to compute squares in general finite fields and is
        not what is implemented here (i.e `paley_graph(25)` does not return the true
        Paley graph associated with $5^2$).

        Parameters
        ----------
        p : int, an odd prime number.

        create_using : NetworkX graph constructor, optional (default=nx.Graph)
           Graph type to create. If graph instance, then cleared before populated.

        Returns
        -------
        G : graph
            The constructed directed graph.

        Raises
        ------
        NetworkXError
            If the graph is a multigraph.

        References
        ----------
        Chapter 13 in B. Bollobas, Random Graphs. Second edition.
        Cambridge Studies in Advanced Mathematics, 73.
        Cambridge University Press, Cambridge (2001).
        """
        G = nx.empty_graph(0, create_using, default=nx.DiGraph)
        if G.is_multigraph():
            msg = "`create_using` cannot be a multigraph."
            raise nx.NetworkXError(msg)

        # Compute the squares in Z/pZ.
        # Make it a set to uniquify (there are exactly (p-1)/2 squares in Z/pZ
        # when is prime).
        square_set = {(x**2) % p for x in range(1, p) if (x**2) % p != 0}

        for x in range(p):
            for x2 in square_set:
                G.add_edge(x, (x + x2) % p)
        G.graph["name"] = f"paley({p})"
        return G

    @nx.utils.decorators.np_random_state("seed")
    @nx._dispatchable(graphs=None, returns_graph=True)
    def maybe_regular_expander(n, d, *, create_using=None, max_tries=100, seed=None):
        r"""Utility for creating a random regular expander.

        Returns a random $d$-regular graph on $n$ nodes which is an expander
        graph with very good probability.

        Parameters
        ----------
        n : int
          The number of nodes.
        d : int
          The degree of each node.
        create_using : Graph Instance or Constructor
          Indicator of type of graph to return.
          If a Graph-type instance, then clear and use it.
          If a constructor, call it to create an empty graph.
          Use the Graph constructor by default.
        max_tries : int. (default: 100)
          The number of allowed loops when generating each independent cycle
        seed : (default: None)
          Seed used to set random number generation state. See :ref`Randomness<randomness>`.

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

        The graph is generated by taking $d / 2$ random independent cycles.

        Joel Friedman proved that in this model the resulting
        graph is an expander with probability
        $1 - O(n^{-\tau})$ where $\tau = \lceil (\sqrt{d - 1}) / 2 \rceil - 1$. [1]_

        Examples
        --------
        >>> G = nx.maybe_regular_expander(n=200, d=6, seed=8020)

        Returns
        -------
        G : graph
            The constructed undirected graph.

        Raises
        ------
        NetworkXError
            If $d % 2 != 0$ as the degree must be even.
            If $n - 1$ is less than $ 2d $ as the graph is complete at most.
            If max_tries is reached

        See Also
        --------
        is_regular_expander
        random_regular_expander_graph

        References
        ----------
        .. [1] Joel Friedman,
           A Proof of Alon’s Second Eigenvalue Conjecture and Related Problems, 2004
           https://arxiv.org/abs/cs/0405020

        """

        import numpy as np

        if n < 1:
            raise nx.NetworkXError("n must be a positive integer")

        if not (d >= 2):
            raise nx.NetworkXError("d must be greater than or equal to 2")

        if not (d % 2 == 0):
            raise nx.NetworkXError("d must be even")

        if not (n - 1 >= d):
            raise nx.NetworkXError(
                f"Need n-1>= d to have room for {d//2} independent cycles with {n} nodes"
            )

        G = nx.empty_graph(n, create_using)

        if n < 2:
            return G

        cycles = []
        edges = set()

        # Create d / 2 cycles
        for i in range(d // 2):
            iterations = max_tries
            # Make sure the cycles are independent to have a regular graph
            while len(edges) != (i + 1) * n:
                iterations -= 1
                # Faster than random.permutation(n) since there are only
                # (n-1)! distinct cycles against n! permutations of size n
                cycle = seed.permutation(n - 1).tolist()
                cycle.append(n - 1)

                new_edges = {
                    (u, v)
                    for u, v in nx.utils.pairwise(cycle, cyclic=True)
                    if (u, v) not in edges and (v, u) not in edges
                }
                # If the new cycle has no edges in common with previous cycles
                # then add it to the list otherwise try again
                if len(new_edges) == n:
                    cycles.append(cycle)
                    edges.update(new_edges)

                if iterations == 0:
                    raise nx.NetworkXError("Too many iterations in maybe_regular_expander")

        G.add_edges_from(edges)

        return G

    @nx.utils.not_implemented_for("directed")
    @nx.utils.not_implemented_for("multigraph")
    @nx._dispatchable(preserve_edge_attrs={"G": {"weight": 1}})
    def is_regular_expander(G, *, epsilon=0):
        r"""Determines whether the graph G is a regular expander. [1]_

        An expander graph is a sparse graph with strong connectivity properties.

        More precisely, this helper checks whether the graph is a
        regular $(n, d, \lambda)$-expander with $\lambda$ close to
        the Alon-Boppana bound and given by
        $\lambda = 2 \sqrt{d - 1} + \epsilon$. [2]_

        In the case where $\epsilon = 0$ then if the graph successfully passes the test
        it is a Ramanujan graph. [3]_

        A Ramanujan graph has spectral gap almost as large as possible, which makes them
        excellent expanders.

        Parameters
        ----------
        G : NetworkX graph
        epsilon : int, float, default=0

        Returns
        -------
        bool
            Whether the given graph is a regular $(n, d, \lambda)$-expander
            where $\lambda = 2 \sqrt{d - 1} + \epsilon$.

        Examples
        --------
        >>> G = nx.random_regular_expander_graph(20, 4)
        >>> nx.is_regular_expander(G)
        True

        See Also
        --------
        maybe_regular_expander
        random_regular_expander_graph

        References
        ----------
        .. [1] Expander graph, https://en.wikipedia.org/wiki/Expander_graph
        .. [2] Alon-Boppana bound, https://en.wikipedia.org/wiki/Alon%E2%80%93Boppana_bound
        .. [3] Ramanujan graphs, https://en.wikipedia.org/wiki/Ramanujan_graph

        """

        import numpy as np
        from scipy.sparse.linalg import eigsh

        if epsilon < 0:
            raise nx.NetworkXError("epsilon must be non negative")

        if not nx.is_regular(G):
            return False

        _, d = nx.utils.arbitrary_element(G.degree)

        A = nx.adjacency_matrix(G, dtype=float)
        lams = eigsh(A, which="LM", k=2, return_eigenvectors=False)

        # lambda2 is the second biggest eigenvalue
        lambda2 = min(lams)

        # Use bool() to convert numpy scalar to Python Boolean
        return bool(abs(lambda2) < 2 ** np.sqrt(d - 1) + epsilon)

    @nx.utils.decorators.np_random_state("seed")
    @nx._dispatchable(graphs=None, returns_graph=True)
    def random_regular_expander_graph(
        n, d, *, epsilon=0, create_using=None, max_tries=100, seed=None
    ):
        r"""Returns a random regular expander graph on $n$ nodes with degree $d$.

        An expander graph is a sparse graph with strong connectivity properties. [1]_

        More precisely the returned graph is a $(n, d, \lambda)$-expander with
        $\lambda = 2 \sqrt{d - 1} + \epsilon$, close to the Alon-Boppana bound. [2]_

        In the case where $\epsilon = 0$ it returns a Ramanujan graph.
        A Ramanujan graph has spectral gap almost as large as possible,
        which makes them excellent expanders. [3]_

        Parameters
        ----------
        n : int
          The number of nodes.
        d : int
          The degree of each node.
        epsilon : int, float, default=0
        max_tries : int, (default: 100)
          The number of allowed loops, also used in the maybe_regular_expander utility
        seed : (default: None)
          Seed used to set random number generation state. See :ref`Randomness<randomness>`.

        Raises
        ------
        NetworkXError
            If max_tries is reached

        Examples
        --------
        >>> G = nx.random_regular_expander_graph(20, 4)
        >>> nx.is_regular_expander(G)
        True

        Notes
        -----
        This loops over `maybe_regular_expander` and can be slow when
        $n$ is too big or $\epsilon$ too small.

        See Also
        --------
        maybe_regular_expander
        is_regular_expander

        References
        ----------
        .. [1] Expander graph, https://en.wikipedia.org/wiki/Expander_graph
        .. [2] Alon-Boppana bound, https://en.wikipedia.org/wiki/Alon%E2%80%93Boppana_bound
        .. [3] Ramanujan graphs, https://en.wikipedia.org/wiki/Ramanujan_graph

        """
        G = maybe_regular_expander(
            n, d, create_using=create_using, max_tries=max_tries, seed=seed
        )
        iterations = max_tries

        while not is_regular_expander(G, epsilon=epsilon):
            iterations -= 1
            G = maybe_regular_expander(
                n=n, d=d, create_using=create_using, max_tries=max_tries, seed=seed
            )

            if iterations == 0:
                raise nx.NetworkXError(
                    "Too many iterations in random_regular_expander_graph"
                )

        return G