Graph powering and spectral robustness

TL;DR

This paper introduces a simple graph powering method that enhances spectral algorithms by regularizing graphs, improving spectral gap, and achieving optimal weak recovery in stochastic block models, with increased robustness to complex structures.

Contribution

Proposes a generic graph powering technique that regularizes spectra, achieves optimal recovery thresholds, and improves robustness over existing spectral methods.

Findings

Graph powering creates a maximal spectral gap in Erdős-Rényi graphs.

Achieves the KS threshold for weak recovery in sparse SBM.

More robust to tangles and cliques than previous spectral algorithms.

Abstract

Spectral algorithms, such as principal component analysis and spectral clustering, typically require careful data transformations to be effective: upon observing a matrix $A$, one may look at the spectrum of $\psi(A)$ for a properly chosen $\psi$. The issue is that the spectrum of $A$ might be contaminated by non-informational top eigenvalues, e.g., due to scale` variations in the data, and the application of $\psi$ aims to remove these. Designing a good functional $\psi$ (and establishing what good means) is often challenging and model dependent. This paper proposes a simple and generic construction for sparse graphs, $$\psi(A) = \1((I+A)^r \ge1),$$ where $A$ denotes the adjacency matrix and $r$ is an integer (less than the graph diameter). This produces a graph connecting vertices from the original graph that are within distance $r$, and is referred to as graph powering. It is shown that graph powering regularizes the graph and decontaminates its spectrum in the following sense: (i) If the graph is drawn from the sparse Erd\H{o}s-R\'enyi ensemble, which has no spectral gap, it is shown that graph powering produces a `maximal' spectral gap, with the latter justified by establishing an Alon-Boppana result for powered graphs; (ii) If the graph is drawn from the sparse SBM, graph powering is shown to achieve the fundamental limit for weak recovery (the KS threshold) similarly to \cite{massoulie-STOC}, settling an open problem therein. Further, graph powering is shown to be significantly more robust to tangles and cliques than previous spectral algorithms based on self-avoiding or nonbacktracking walk counts \cite{massoulie-STOC,Mossel_SBM2,bordenave,colin3}. This is illustrated on a geometric block model that is dense in cliques.