Low Rank Spectral Network Alignment

Abstract

Network alignment or graph matching is the classic problem of finding matching vertices between two graphs with applications in network de-anonymization and bioinformatics. There exist a wide variety of algorithms for it, but a challenging scenario for all of the algorithms is aligning two networks without any information about which nodes might be good matches. In this case, the vast majority of principled algorithms demand quadratic memory in the size of the graphs. We show that one such method---the recently proposed and theoretically grounded EigenAlign algorithm---admits a novel implementation which requires memory that is linear in the size of the graphs. The key step to this insight is identifying low-rank structure in the node-similarity matrix used by EigenAlign for determining matches. With an exact, closed-form low-rank structure, we then solve a maximum weight bipartite matching problem on that low-rank matrix to produce the matching between the graphs. For this task, we show a new, a-posteriori, approximation bound for a simple algorithm to approximate a maximum weight bipartite matching problem on a low-rank matrix. The combination of our two new methods then enables us to tackle much larger network alignment problems than previously possible and to do so quickly. Problems that take hours with existing methods take only seconds with our new algorithm. We thoroughly validate our low-rank algorithm against the original EigenAlign approach. We also compare a variety of existing algorithms on problems in bioinformatics and social networks. Our approach can also be combined with existing algorithms to improve their performance and speed.