Spectral Alignment of Graphs

TL;DR

This paper introduces a generalized spectral graph alignment method that considers both matches and mismatches, improving alignment accuracy especially for graphs with different sizes and densities.

Contribution

It proposes a new formulation of graph alignment as a quadratic assignment problem that accounts for mismatches, and develops spectral methods to solve it effectively.

Findings

Spectral alignment method outperforms existing algorithms on synthetic and real graphs.

Significantly better performance in aligning regular graph structures.

Method is robust across various graph sizes and densities.

Abstract

Graph alignment refers to the problem of finding a bijective mapping across vertices of two graphs such that, if two nodes are connected in the first graph, their images are connected in the second graph. This problem arises in many fields such as computational biology, social sciences, and computer vision and is often cast as a quadratic assignment problem (QAP). Most standard graph alignment methods consider an optimization that maximizes the number of matches between the two graphs, ignoring the effect of mismatches. We propose a generalized graph alignment formulation that considers both matches and mismatches in a standard QAP formulation. This modification can have a major impact in aligning graphs with different sizes and heterogenous edge densities. Moreover, we propose two methods for solving the generalized graph alignment problem based on spectral decomposition of matrices. We compare the performance of proposed methods with some existing graph alignment algorithms including Natalie2, GHOST, IsoRank, NetAlign, Klau's approach as well as a semidefinite programming-based method over various synthetic and real graph models. Our proposed method based on simultaneous alignment of multiple eigenvectors leads to consistently good performance in different graph models. In particular, in the alignment of regular graph structures which is one of the most difficult graph alignment cases, our proposed method significantly outperforms other methods.