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

TL;DR

This paper introduces a spectral algorithm for inferring edge label distributions in generalized stochastic block models with labeled edges and latent attributes, establishing thresholds for successful inference and connecting to spectral graph theory.

Contribution

It proposes a novel spectral method for edge label inference in complex networks with latent attributes, and characterizes the fundamental limits of inference based on node degree thresholds.

Findings

Spectral algorithm achieves asymptotically correct inference at low degrees

Below a certain degree threshold, inference is impossible better than guessing

Provides a method to construct random graphs with specific spectral properties

Abstract

The classical setting of community detection consists of networks exhibiting a clustered structure. To more accurately model real systems we consider a class of networks (i) whose edges may carry labels and (ii) which may lack a clustered structure. Specifically we assume that nodes possess latent attributes drawn from a general compact space and edges between two nodes are randomly generated and labeled according to some unknown distribution as a function of their latent attributes. Our goal is then to infer the edge label distributions from a partially observed network. We propose a computationally efficient spectral algorithm and show it allows for asymptotically correct inference when the average node degree could be as low as logarithmic in the total number of nodes. Conversely, if the average node degree is below a specific constant threshold, we show that no algorithm can achieve better inference than guessing without using the observations. As a byproduct of our analysis, we show that our model provides a general procedure to construct random graph models with a spectrum asymptotic to a pre-specified eigenvalue distribution such as a power-law distribution.