Source code for networkx.generators.spectral_graph_forge

"""Generates graphs with a given eigenvector structure"""

import networkx as nx
from networkx.utils import np_random_state

__all__ = ["spectral_graph_forge"]

def _mat_spect_approx(A, level, sorteigs=True, reverse=False, absolute=True):
    """Returns the low-rank approximation of the given matrix A

    A : 2D numpy array
    level : integer
        It represents the fixed rank for the output approximation matrix
    sorteigs : boolean
        Whether eigenvectors should be sorted according to their associated
        eigenvalues before removing the firsts of them
    reverse : boolean
        Whether eigenvectors list should be reversed before removing the firsts
        of them
    absolute : boolean
        Whether eigenvectors should be sorted considering the absolute values
        of the corresponding eigenvalues

    B : 2D numpy array
        low-rank approximation of A

    Low-rank matrix approximation is about finding a fixed rank matrix close
    enough to the input one with respect to a given norm (distance).
    In the case of real symmetric input matrix and euclidean distance, the best
    low-rank approximation is given by the sum of first eigenvector matrices.

    ..  [1] G. Eckart and G. Young, The approximation of one matrix by another
            of lower rank
    ..  [2] L. Mirsky, Symmetric gauge functions and unitarily invariant norms

    import numpy as np

    d, V = np.linalg.eigh(A)
    d = np.ravel(d)
    n = len(d)
    if sorteigs:
        if absolute:
            k = np.argsort(np.abs(d))
            k = np.argsort(d)
        # ordered from the lowest to the highest
        k = range(n)
    if not reverse:
        k = np.flipud(k)

    z = np.zeros(n)
    for i in range(level, n):
        V[:, k[i]] = z

    B = V @ np.diag(d) @ V.T
    return B

[docs]@np_random_state(3) def spectral_graph_forge(G, alpha, transformation="identity", seed=None): """Returns a random simple graph with spectrum resembling that of `G` This algorithm, called Spectral Graph Forge (SGF), computes the eigenvectors of a given graph adjacency matrix, filters them and builds a random graph with a similar eigenstructure. SGF has been proved to be particularly useful for synthesizing realistic social networks and it can also be used to anonymize graph sensitive data. Parameters ---------- G : Graph alpha : float Ratio representing the percentage of eigenvectors of G to consider, values in [0,1]. transformation : string, optional Represents the intended matrix linear transformation, possible values are 'identity' and 'modularity' seed : integer, random_state, or None (default) Indicator of numpy random number generation state. See :ref:`Randomness<randomness>`. Returns ------- H : Graph A graph with a similar eigenvector structure of the input one. Raises ------ NetworkXError If transformation has a value different from 'identity' or 'modularity' Notes ----- Spectral Graph Forge (SGF) generates a random simple graph resembling the global properties of the given one. It leverages the low-rank approximation of the associated adjacency matrix driven by the *alpha* precision parameter. SGF preserves the number of nodes of the input graph and their ordering. This way, nodes of output graphs resemble the properties of the input one and attributes can be directly mapped. It considers the graph adjacency matrices which can optionally be transformed to other symmetric real matrices (currently transformation options include *identity* and *modularity*). The *modularity* transformation, in the sense of Newman's modularity matrix allows the focusing on community structure related properties of the graph. SGF applies a low-rank approximation whose fixed rank is computed from the ratio *alpha* of the input graph adjacency matrix dimension. This step performs a filtering on the input eigenvectors similar to the low pass filtering common in telecommunications. The filtered values (after truncation) are used as input to a Bernoulli sampling for constructing a random adjacency matrix. References ---------- .. [1] L. Baldesi, C. T. Butts, A. Markopoulou, "Spectral Graph Forge: Graph Generation Targeting Modularity", IEEE Infocom, '18. .. [2] M. Newman, "Networks: an introduction", Oxford university press, 2010 Examples -------- >>> G = nx.karate_club_graph() >>> H = nx.spectral_graph_forge(G, 0.3) >>> """ import numpy as np import scipy as sp import scipy.stats # call as sp.stats available_transformations = ["identity", "modularity"] alpha = np.clip(alpha, 0, 1) A = nx.to_numpy_array(G) n = A.shape[1] level = int(round(n * alpha)) if transformation not in available_transformations: msg = f"{transformation!r} is not a valid transformation. " msg += f"Transformations: {available_transformations}" raise nx.NetworkXError(msg) K = np.ones((1, n)) @ A B = A if transformation == "modularity": B -= K.T @ K / K.sum() B = _mat_spect_approx(B, level, sorteigs=True, absolute=True) if transformation == "modularity": B += K.T @ K / K.sum() B = np.clip(B, 0, 1) np.fill_diagonal(B, 0) for i in range(n - 1): B[i, i + 1 :] = sp.stats.bernoulli.rvs(B[i, i + 1 :], random_state=seed) B[i + 1 :, i] = np.transpose(B[i, i + 1 :]) H = nx.from_numpy_array(B) return H