Spectral Alignment of Networks

Network alignment refers to the problem of finding a bijective mapping across vertices of two or more graphs to maximize the number of overlapping edges and/or to minimize the num-ber of mismatched interactions across networks. This paper introduces a network alignment algorithm inspired by eigenvector analysis which creates a simple relaxation for the underlying quadratic assignment problem. Our method relaxes binary assignment constraints along the leading eigenvector of an alignment matrix which captures the structure of matched and mis-matched interactions across networks. Our proposed algorithm denoted by EigeAlign has two steps. First, it computes the Perron-Frobenius eigenvector of the alignment matrix. Second, it uses this eigenvector in a linear optimization framework of maximum weight bipartite matching to infer bijective mappings across vertices of two graphs. Unlike existing network alignment methods, EigenAlign considers both matched and mismatched interactions in its optimization and therefore, it is effective in aligning networks even with low similarity. We show that, when certain technical conditions hold, the relaxation given by EigenAlign is asymptotically exact over Erdös-Rényi graphs with high probability. Moreover, for modular network structures, w