wg-backend-django/dell-env/lib/python3.11/site-packages/networkx/algorithms/richclub.py

"""Functions for computing rich-club coefficients."""

from itertools import accumulate

import networkx as nx
from networkx.utils import not_implemented_for

__all__ = ["rich_club_coefficient"]


@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatch
def rich_club_coefficient(G, normalized=True, Q=100, seed=None):
    r"""Returns the rich-club coefficient of the graph `G`.

    For each degree *k*, the *rich-club coefficient* is the ratio of the
    number of actual to the number of potential edges for nodes with
    degree greater than *k*:

    .. math::

        \phi(k) = \frac{2 E_k}{N_k (N_k - 1)}

    where `N_k` is the number of nodes with degree larger than *k*, and
    `E_k` is the number of edges among those nodes.

    Parameters
    ----------
    G : NetworkX graph
        Undirected graph with neither parallel edges nor self-loops.
    normalized : bool (optional)
        Normalize using randomized network as in [1]_
    Q : float (optional, default=100)
        If `normalized` is True, perform `Q * m` double-edge
        swaps, where `m` is the number of edges in `G`, to use as a
        null-model for normalization.
    seed : integer, random_state, or None (default)
        Indicator of random number generation state.
        See :ref:`Randomness<randomness>`.

    Returns
    -------
    rc : dictionary
       A dictionary, keyed by degree, with rich-club coefficient values.

    Examples
    --------
    >>> G = nx.Graph([(0, 1), (0, 2), (1, 2), (1, 3), (1, 4), (4, 5)])
    >>> rc = nx.rich_club_coefficient(G, normalized=False, seed=42)
    >>> rc[0]
    0.4

    Notes
    -----
    The rich club definition and algorithm are found in [1]_.  This
    algorithm ignores any edge weights and is not defined for directed
    graphs or graphs with parallel edges or self loops.

    Estimates for appropriate values of `Q` are found in [2]_.

    References
    ----------
    .. [1] Julian J. McAuley, Luciano da Fontoura Costa,
       and Tibério S. Caetano,
       "The rich-club phenomenon across complex network hierarchies",
       Applied Physics Letters Vol 91 Issue 8, August 2007.
       https://arxiv.org/abs/physics/0701290
    .. [2] R. Milo, N. Kashtan, S. Itzkovitz, M. E. J. Newman, U. Alon,
       "Uniform generation of random graphs with arbitrary degree
       sequences", 2006. https://arxiv.org/abs/cond-mat/0312028
    """
    if nx.number_of_selfloops(G) > 0:
        raise Exception(
            "rich_club_coefficient is not implemented for " "graphs with self loops."
        )
    rc = _compute_rc(G)
    if normalized:
        # make R a copy of G, randomize with Q*|E| double edge swaps
        # and use rich_club coefficient of R to normalize
        R = G.copy()
        E = R.number_of_edges()
        nx.double_edge_swap(R, Q * E, max_tries=Q * E * 10, seed=seed)
        rcran = _compute_rc(R)
        rc = {k: v / rcran[k] for k, v in rc.items()}
    return rc


def _compute_rc(G):
    """Returns the rich-club coefficient for each degree in the graph
    `G`.

    `G` is an undirected graph without multiedges.

    Returns a dictionary mapping degree to rich-club coefficient for
    that degree.

    """
    deghist = nx.degree_histogram(G)
    total = sum(deghist)
    # Compute the number of nodes with degree greater than `k`, for each
    # degree `k` (omitting the last entry, which is zero).
    nks = (total - cs for cs in accumulate(deghist) if total - cs > 1)
    # Create a sorted list of pairs of edge endpoint degrees.
    #
    # The list is sorted in reverse order so that we can pop from the
    # right side of the list later, instead of popping from the left
    # side of the list, which would have a linear time cost.
    edge_degrees = sorted((sorted(map(G.degree, e)) for e in G.edges()), reverse=True)
    ek = G.number_of_edges()
    k1, k2 = edge_degrees.pop()
    rc = {}
    for d, nk in enumerate(nks):
        while k1 <= d:
            if len(edge_degrees) == 0:
                ek = 0
                break
            k1, k2 = edge_degrees.pop()
            ek -= 1
        rc[d] = 2 * ek / (nk * (nk - 1))
    return rc