Graph Powering And Spectral Robustness

Spectral algorithms, such as principal component analysis and spectral clustering, rely on the extremal eigenpairs of a matrix A. However, these may be uninformative without preprocessing A with a proper transformation. The reason is that the spectrum of A may be contaminated by top eigenvalues resulting from scale variations in the data, such as high-degree nodes. Designing a good psi and establishing what good means is often challenging and model dependent. This paper proposes a simple and generic construction for sparse graphs, psi(A) = 1((I + A)(r) >= 1), where A denotes the adjacency matrix, r is an integer, and the indicator function is applied entrywise. We support this "graph powering" construction with the following regularization properties: (i) if the graph is drawn from the sparse Erdos-Renyi ensemble, which has no spectral gap, then graph powering produces a "maximal" spectral gap, comparable to that obtained when powering a random regular graph; (ii) if the graph is drawn from the sparse stochastic block model, graph powering achieves the fundamental limit for weak recovery (the Kesten-Stigum threshold), settling at the same time a related conjecture by Massoulie in 2013; (iii) we also demonstrate that graph powering is significantly more robust to tangles and cliques than previous spectral algorithms based on self-avoiding or nonbacktracking walk counts, using a geometric block model as our benchmark and introducing new conjectures for this model.