Concentration and regularization of random graphs

Abstract

Abstract. This paper studies how close random graphs are typically to their expectations. We interpret this question through the concentration of the adjacency and Laplacian matrices in the spectral norm. We study inhomogeneous Erdös-Rényi random graphs on n vertices, where edges form independently and possibly with different probabilities pij. Sparse random graphs whose expected degrees are o(log n) fail to concentrate; the obstruction is caused by vertices with abnormally high and low degrees. We show that concentration can be restored if we regularize the degrees of such vertices, and one can do this in various ways. As an example, let us reweight or remove enough edges to make all degrees bounded above by O(d) where d = max npij. Then we show that the resulting adjacency matrix A concentrates with the optimal rate:Similarly, if we make all degrees bounded below by d by adding weight d/n to all edges, then the resulting Laplacian concentrates with the optimal rate:Our approach is based on GrothendieckPietsch factorization, using which we construct a new decomposition of random graphs. We illustrate the concentration results with an application to the community detection problem in the analysis of networks.

Constructive Regularization of the Random Matrix Norm

“…To address this issue without making regularization procedure more complex, we are going to use an additional structural decomposition for Bernoulli random matrices, first shown in the work of Le, Levina and Vershynin [9]. The next proposition is a direct corollary of [9, Theorem 2.6]: Proposition 2.3 (Decomposition lemma).…”

“…The rest of the paper is structured as follows. The proof of Theorem 1.1 is based on the previously known regularization results developed for Bernoulli random matrices (mainly in the works of Feige and Ofek [6], and Le, Levina and Vershynin [9]). In Section 2 we review some results specific to the Bernoulli matrices and briefly explain how they will be used later in the text.…”

Message‐Passing Algorithms for Synchronization Problems over Compact Groups

“…The problem of partitioning a graph into two well-connected subcommunities can be viewed as synchronization over the group f˙1g Š Z=2: each vertex has a latent group element g u 2 f˙1g, its community identity, and each edge is a noisy measurement of the relative status g u g 1 v [5]. A number of more structured approaches have been shown to improve over PCA, including modified spectral methods [37,38,44,50,59] and semidefinite programming [1,2,32,33,49]. However, this approach breaks down in sparse graphs: localized noise eigenvectors associated to high-degree vertices dominate the spectrum.…”

“…This is essentially a failure of PCA to adequately exploit the problem structure, as these localized eigenvectors lie far from the constraint that the truth is entrywise f˙1g. A number of more structured approaches have been shown to improve over PCA, including modified spectral methods [37,38,44,50,59] and semidefinite programming [1,2,32,33,49]. A major algorithmic challenge in this problem is to obtain an efficient algorithm that optimally exploits this structure, to obtain the minimum possible estimation error.…”

Bounding Extremal Degrees of Edge-Independent Random Graphs Using Relative Entropy

“…The problem of bounding the difference between eigenvalues of A and those of the adjacency matrix of G n (p ij ), together with its Laplacian spectra version, has been studied intensively recently; see, e.g., [3,8,9]. It is revealed in [9] that large deviation from the expected spectrum is caused by vertices with extremal degrees, where abnormally high-degree and low-degree vertices are obstructions to concentration of the adjacency and the Laplacian matrices, respectively. A regularization technique is employed to address this issue.…”