Source code for networkx.algorithms.non_randomness

r"""Computation of graph non-randomness"""

import math

import networkx as nx
from networkx.utils import not_implemented_for

__all__ = ["non_randomness"]




@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatchable(edge_attrs="weight")
def non_randomness(G, k=None, weight="weight"):
    """Compute the non-randomness of graph G.

    The first returned value nr is the sum of non-randomness values of all
    edges within the graph (where the non-randomness of an edge tends to be
    small when the two nodes linked by that edge are from two different
    communities).

    The second computed value nr_rd is a relative measure that indicates
    to what extent graph G is different from random graphs in terms
    of probability. When it is close to 0, the graph tends to be more
    likely generated by an Erdos Renyi model.

    Parameters
    ----------
    G : NetworkX graph
        Graph must be symmetric, connected, and without self-loops.

    k : int
        The number of communities in G.
        If k is not set, the function will use a default community
        detection algorithm to set it.

    weight : string or None, optional (default=None)
        The name of an edge attribute that holds the numerical value used
        as a weight. If None, then each edge has weight 1, i.e., the graph is
        binary.

    Returns
    -------
    non-randomness : (float, float) tuple
        Non-randomness, Relative non-randomness w.r.t.
        Erdos Renyi random graphs.

    Raises
    ------
    NetworkXException
        if the input graph is not connected.
    NetworkXError
        if the input graph contains self-loops or if graph has no edges.

    Examples
    --------
    >>> G = nx.karate_club_graph()
    >>> nr, nr_rd = nx.non_randomness(G, 2)
    >>> nr, nr_rd = nx.non_randomness(G, 2, "weight")

    Notes
    -----
    This computes Eq. (4.4) and (4.5) in Ref. [1]_.

    If a weight field is passed, this algorithm will use the eigenvalues
    of the weighted adjacency matrix to compute Eq. (4.4) and (4.5).

    References
    ----------
    .. [1] Xiaowei Ying and Xintao Wu,
           On Randomness Measures for Social Networks,
           SIAM International Conference on Data Mining. 2009
    """
    import numpy as np

    # corner case: graph has no edges
    if nx.is_empty(G):
        raise nx.NetworkXError("non_randomness not applicable to empty graphs")
    if not nx.is_connected(G):
        raise nx.NetworkXException("Non connected graph.")
    if len(list(nx.selfloop_edges(G))) > 0:
        raise nx.NetworkXError("Graph must not contain self-loops")

    if k is None:
        k = len(tuple(nx.community.label_propagation_communities(G)))

    # eq. 4.4
    eigenvalues = np.linalg.eigvals(nx.to_numpy_array(G, weight=weight))
    nr = float(np.real(np.sum(eigenvalues[:k])))

    n = G.number_of_nodes()
    m = G.number_of_edges()
    p = (2 * k * m) / (n * (n - k))

    # eq. 4.5
    nr_rd = (nr - ((n - 2 * k) * p + k)) / math.sqrt(2 * k * p * (1 - p))

    return nr, nr_rd