Random line graphs and edge-attributed network inference

TL;DR

This paper extends the latent position random graph model to line graphs, providing spectral analysis, methods for edge clustering, and demonstrating improved network inference by leveraging both vertex and edge information.

Contribution

It introduces spectral concentration results and novel methods for edge clustering in line graphs, enhancing network inference capabilities.

Findings

Spectral concentration inequalities for line graphs.

Existence of signal-preserving subspaces for edge clustering.

Effective estimation of edge latent positions in complex line graphs.

Abstract

We extend the latent position random graph model to the line graph of a random graph, which is formed by creating a vertex for each edge in the original random graph, and connecting each pair of edges incident to a common vertex in the original graph. We prove concentration inequalities for the spectrum of a line graph, as well as limiting distribution results for the largest eigenvalue and the empirical spectral distribution in certain settings. For the stochastic blockmodel, we establish that although naive spectral decompositions can fail to extract necessary signal for edge clustering, there exist signal-preserving singular subspaces of the line graph that can be recovered through a carefully-chosen projection. Moreover, we can consistently estimate edge latent positions in a random line graph, even though such graphs are of a random size, typically have high rank, and possess no spectral gap. Our results demonstrate that the line graph of a stochastic block model exhibits underlying block structure, and in simulations, we synthesize and test our methods against several commonly-used techniques, including tensor decompositions, for cluster recovery and edge covariate inference. By naturally incorporating information encoded in both vertices and edges, the random line graph improves network inference.