Spectral clustering under degree heterogeneity: a case for the random walk Laplacian

Abstract

This paper shows that graph spectral embedding using the random walk Laplacian produces vector representations which are completely corrected for node degree. Under a generalised random dot product graph, the embedding provides uniformly consistent estimates of degree-corrected latent positions, with asymptotically Gaussian error. In the special case of a degree-corrected stochastic block model, the embedding concentrates about K distinct points, representing communities. These can be recovered perfectly, asymptotically, through a subsequent clustering step, without spherical projection, as commonly required by algorithms based on the adjacency or normalised, symmetric Laplacian matrices. While the estimand does not depend on degree, the asymptotic variance of its estimate does -higher degree nodes are embedded more accurately than lower degree nodes. Our central limit theorem therefore suggests fitting a weighted Gaussian mixture model as the subsequent clustering step, for which we provide an expectation-maximisation algorithm.

Matrix factorisation and the interpretation of geodesic distance

“…This encompasses the case where A is binary, for example a graph adjacency matrix [33,42]. Similar results are available in the cases where A is a Laplacian [42,36], covariance matrix [16], or the matrix implicitly factorised by node2vec [59]. The methods of this paper are based, in practice, on the distances Xi − Xj , which are invariant to orthogonal transformations and so for purposes of validating Xi − Xj ≈ φ p (Z i ) − φ p (Z j ) the presence of Q in (2) is immaterial.…”

Stochastic networks theory to model single-cell genomic count data

“…Building on existing theory for bootstrapping network data with latent structure [24] will enable estimates of uncertainty to be calculated for estimators of the number of latent clusters, as well as quantifying uncertainty on the estimated Laplacian eigenspace. We also note that there are alternative methods for calculating the graph-Laplacian, such as the random-walk Laplacian [25], which will be interesting to investigate in relation to single-cell genomic data. Finally, we have previously proposed a novel method of estimating a latent space in which different cell-types or clusters can be separated well, based on the Mahalanobis distance [16].…”

Spectral embedding and the latent geometry of multipartite networks

“…Of significant current interest is spectral graph embedding, in which the principal eigenvectors of a matrix representation, such as the adjacency or Laplacian matrix, provide these representations. Drawing connections between observed geometric features in the embedding, and underlying structure in the network is a actively-studied area of research [21,2,24,27,23,3,19]. While the vast majority of the literature considers undirected graphs, many real-world graphs inherently exhibit bipartite [11,10,22] or multipartite structure, meaning that their nodes are organized into groups, called partitions, and connections are only observed between nodes from different partitions.…”