Dominant Eigenvalue-Eigenvector Pair Estimation via Graph Infection

TL;DR

This paper introduces a novel graph infection-based method to estimate the dominant eigenvalue and eigenvector of non-negative matrices, outperforming traditional power iteration in complex graph structures.

Contribution

The paper presents the first approach using matrix ODEs and graph infection models to estimate dominant eigenpairs, especially in cases where power iteration fails.

Findings

Our method converges faster than power iteration on complex graphs.

It accurately estimates dominant eigenvalues even with multiple same-magnitude eigenvalues.

The approach is effective on various graph types, including directed and bipartite graphs.

Abstract

We present a novel method to estimate the dominant eigenvalue and eigenvector pair of any non-negative real matrix via graph infection. The key idea in our technique lies in approximating the solution to the first-order matrix ordinary differential equation (ODE) with the Euler method. Graphs, which can be weighted, directed, and with loops, are first converted to its adjacency matrix A. Then by a naive infection model for graphs, we establish the corresponding first-order matrix ODE, through which A's dominant eigenvalue is revealed by the fastest growing term. When there are multiple dominant eigenvalues of the same magnitude, the classical power iteration method can fail. In contrast, our method can converge to the dominant eigenvalue even when same-magnitude counterparts exist, be it complex or opposite in sign. We conduct several experiments comparing the convergence between our method and power iteration. Our results show clear advantages over power iteration for tree graphs, bipartite graphs, directed graphs with periods, and Markov chains with spider-traps. To our knowledge, this is the first work that estimates dominant eigenvalue and eigenvector pair from the perspective of a dynamical system and matrix ODE. We believe our method can be adopted as an alternative to power iteration, especially for graphs.