Concentration of Random Graphs and Application to Community Detection

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.2010 Mathematics Subject Classification. 05C80 (primary) and 05C85, 60B20 (secondary).

Thin-shell theory for rotationally invariant random simplices

“…High-dimensional probability theory is concerned with random objects, their characteristics, and the phenomena that accompany both as the dimension of the ambient space tends to infinity. It is a flourishing area of mathematics not least because of numerous applications in modern statistics and machine learning related to high-dimensional data, for instance, in form of dimensionality reduction [14], clustering [46], principal component regression [54], community detection in networks [22,42], topic discovery [20], or covariance estimation [16,59]. High-dimensional probability bears strong connections to geometric functional analysis and convex geometry and this propinquity is typically reflected both in the flavor of a result and the methods used to obtain it.…”

Sparse and smooth: Improved guarantees for spectral clustering in the dynamic stochastic block model

“…= β r−1 = 1 r and β k = 0 for k r in the uniform case, and β k = λ(1 − λ) k for k < t and β t = (1 − λ) t in the exponential case. As we will see, our results will be valid for any estimator of the form (24), with weights β k 0 that satisfy: there are constants β max , C β , C β > 0 such that:…”

“…In the specific case of independent Bernoulli edges like the SBM and DSBM, this correspond to the mean probability of connection, which will be denoted by α n in this paper, where n is the number of nodes in the graph. The dense case α n ∼ 1 is generally simple to analyze [24]. At the other end of the spectrum, the so-called sparse case α n ∼ 1 n is much more complex, since the graph is not even guaranteed to be connected with high probability [1].…”

“…In the dynamic case, a conjecture on the detectability threshold is given in [15]. In parallel, other works study the sparse case by regularizing the adjacency matrix or normalized Laplacian of the graph before the SC algorithm [23,24,7].…”

“…For an estimator of the form (24), our goal is to bound A smooth t − P t . We define P smooth t def.…”