  Multiple network alignment is the problem of identifying similar and related
regions in a given set of networks. While there are a large number of effective
techniques for pairwise problems with two networks that scale in terms of
edges, these cannot be readily extended to align multiple networks as the
computational complexity will tend to grow exponentially with the number of
networks.In this paper we introduce a new multiple network alignment algorithm
and framework that is effective at aligning thousands of networks with
thousands of nodes. The key enabling technique of our algorithm is identifying
an exact and easy to compute low-rank tensor structure inside of a principled
heuristic procedure for pairwise network alignment called IsoRank. This can be
combined with a new algorithm for $k$-dimensional matching problems on low-rank
tensors to produce the alignment. We demonstrate results on synthetic and
real-world problems that show our technique (i) is as good or better in terms
of quality as existing methods, when they work on small problems, while running
considerably faster and (ii) is able to scale to aligning a number of networks
unreachable by current methods. We show in this paper that our method is the
realistic choice for aligning multiple networks when no prior information is
present.
