Beyond the adjacency matrix: random line graphs and inference for networks with edge attributes

by   Zachary Lubberts, et al.

Any modern network inference paradigm must incorporate multiple aspects of network structure, including information that is often encoded both in vertices and in edges. Methodology for handling vertex attributes has been developed for a number of network models, but comparable techniques for edge-related attributes remain largely unavailable. We address this gap in the literature by extending 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, and then 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 also demonstrate that the line graph of a stochastic block model exhibits underlying block structure, and we synthesize and test our methods in simulations for cluster recovery and edge covariate inference in stochastic block model graphs.


page 1

page 2

page 3

page 4


A Polynomial Kernel for Line Graph Deletion

The line graph of a graph G is the graph L(G) whose vertex set is the ed...

On spectral embedding performance and elucidating network structure in stochastic block model graphs

Statistical inference on graphs often proceeds via spectral methods invo...

The Four Point Permutation Test for Latent Block Structure in Incidence Matrices

Transactional data may be represented as a bipartite graph G:=(L ∪ R, E)...

Nonsingular (Vertex-Weighted) Block Graphs

A graph G is nonsingular (singular) if its adjacency matrix A(G) is nons...

Universally Consistent Latent Position Estimation and Vertex Classification for Random Dot Product Graphs

In this work we show that, using the eigen-decomposition of the adjacenc...

Edge Label Inference in Generalized Stochastic Block Models: from Spectral Theory to Impossibility Results

The classical setting of community detection consists of networks exhibi...

On identifying unobserved heterogeneity in stochastic blockmodel graphs with vertex covariates

Both observed and unobserved vertex heterogeneity can influence block st...