Parsimonious module inference in large networks

TL;DR

This paper introduces an MDL-based method for detecting modules in large networks without prior knowledge of the number of modules, providing bounds on detectability and an efficient inference algorithm.

Contribution

It develops a novel MDL-based framework for module detection in large networks, including bounds on detectability and an efficient inference algorithm.

Findings

Maximum detectable blocks scale as √N

Algorithm complexity is O(τ N log N) when the number of blocks is unknown

Method successfully applied to a large bipartite network with over a million edges

Abstract

We investigate the detectability of modules in large networks when the number of modules is not known in advance. We employ the minimum description length (MDL) principle which seeks to minimize the total amount of information required to describe the network, and avoid overfitting. According to this criterion, we obtain general bounds on the detectability of any prescribed block structure, given the number of nodes and edges in the sampled network. We also obtain that the maximum number of detectable blocks scales as $\sqrt{N}$, where $N$ is the number of nodes in the network, for a fixed average degree $<k>$. We also show that the simplicity of the MDL approach yields an efficient multilevel Monte Carlo inference algorithm with a complexity of $O(\tau N\log N)$, if the number of blocks is unknown, and $O(\tau N)$ if it is known, where $\tau$ is the mixing time of the Markov chain. We illustrate the application of the method on a large network of actors and films with over $10^6$ edges, and a dissortative, bipartite block structure.