Spectral Analysis for Billion-Scale Graphs: Discoveries and Implementation

Given a graph with billions of nodes and edges, how can we find patterns and anomalies? Are there nodes that participate in too many or too few triangles? Are there close-knit near-cliques? These questions are expensive to answer unless we have the first several eigenvalues and eigenvectors of the graph adjacency matrix. However, eigensolvers suffer from subtle problems (e.g., convergence) for large sparse matrices, let alone for billion-scale ones.