Spectral Inference Methods on Sparse Graphs: Theory and Applications

TL;DR

This paper develops spectral inference algorithms for sparse graphs using statistical physics insights, providing a probabilistic framework for large-scale network property inference with applications in community detection, clustering, and matrix completion.

Contribution

It introduces a novel spectral inference theory based on mean-field free energy relaxation, advancing probabilistic methods for graph analysis beyond traditional cost function optimization.

Findings

Effective spectral algorithms for community detection and clustering.

The probabilistic approach outperforms traditional methods in sparse graph scenarios.

Demonstrated success in matrix completion tasks.

Abstract

In an era of unprecedented deluge of (mostly unstructured) data, graphs are proving more and more useful, across the sciences, as a flexible abstraction to capture complex relationships between complex objects. One of the main challenges arising in the study of such networks is the inference of macroscopic, large-scale properties affecting a large number of objects, based solely on the microscopic interactions between their elementary constituents. Statistical physics, precisely created to recover the macroscopic laws of thermodynamics from an idealized model of interacting particles, provides significant insight to tackle such complex networks. In this dissertation, we use methods derived from the statistical physics of disordered systems to design and study new algorithms for inference on graphs. Our focus is on spectral methods, based on certain eigenvectors of carefully chosen matrices, and sparse graphs, containing only a small amount of information. We develop an original theory of spectral inference based on a relaxation of various mean-field free energy optimizations. Our approach is therefore fully probabilistic, and contrasts with more traditional motivations based on the optimization of a cost function. We illustrate the efficiency of our approach on various problems, including community detection, randomized similarity-based clustering, and matrix completion.