Concentration of random graphs and application to community detection

TL;DR

This paper reviews how random graphs concentrate around their expectations, discusses models and tools relevant to random matrix theory, and explores applications to community detection in networks.

Contribution

It provides a comprehensive review of concentration regimes for random graphs, connecting random matrix theory with network analysis and community detection.

Findings

Dense graphs concentrate around their expectation.

Sparse graphs require regularization to concentrate.

Applications to community detection are extensively discussed.

Abstract

Random matrix theory has played an important role in recent work on statistical network analysis. In this paper, we review recent results on regimes of concentration of random graphs around their expectation, showing that dense graphs concentrate and sparse graphs concentrate after regularization. We also review relevant network models that may be of interest to probabilists considering directions for new random matrix theory developments, and random matrix theory tools that may be of interest to statisticians looking to prove properties of network algorithms. Applications of concentration results to the problem of community detection in networks are discussed in detail.