Spectral Alignment of Networks

Network alignment refers to the problem of nding a bijective mapping across vertices of two graphs to maximize the number of overlapping edges and/or to minimize the number of mismatched interactions across networks. This problem arises in many elds such as computational biology, social sciences and computer vision and is often cast as an expensive quadratic assignment problem (QAP). Although spectral methods have received signicant attention in dierent network science problems such as network clustering, the use of spectral techniques in the network alignment problem has been limited partially owing to the lack of principled connections between spectral methods and relaxations of the network alignment optimization. In this paper, we propose a network alignment framework that uses an orthogonal relaxation of the underlying QAP in a maximum weight bipartite matching optimization. Our method takes into account the ellipsoidal level sets of the quadratic objective function by exploiting eigenvalues and eigenvectors of (transformations of) adjacency graphs. Our framework not only can be employed to provide a theoretical justication for existing heuristic spectral network alignment methods, but it also leads to a new scalable network alignment algorithm which outperforms existing ones over various synthetic and real networks. Moreover, we generalize the objective function of the network alignment problem to consider both matched and mismatched interactions in a standard QAP formulation. This can be critical in applications where networks have low similarity and therefore we expect more mismatches than matches. We assess the eectiveness of our proposed method theoretically for certain classes of networks, through simulations over various synthetic network models, and in two real-data applications; in comparative analysis of gene regulatory networks across human, y and worm, and in user de-anonymization over twitter follower subgraphs.