Message-Passing Algorithms for Sparse Network Alignment

Abstract

Network alignment generalizes and unifies several approaches for forming a matching or alignment between the vertices of two graphs. We study a mathematical programming framework for network alignment problem and a sparse variation of it where only a small number of matches between the vertices of the two graphs are possible. We propose a new message passing algorithm that allows us to compute, very efficiently, approximate solutions to the sparse network alignment problems with graph sizes as large as hundreds of thousands of vertices. We also provide extensive simulations comparing our algorithms with two of the best solvers for network alignment problems on two synthetic matching problems, two bioinformatics problems, and three large ontology alignment problems including a multilingual problem with a known labeled alignment.

Multimodal Network Alignment

“…IsoRank was one of the first methods in this class [22]. Subsequent techniques include belief propagation methods [2], Lagrangian relaxations [11], spectral methods [7], and tensor eigenvectors for motif-alignment [17]. Essentially, all of these methods store a dense, real-valued heuristic matrix Y of size |V A | × |V B |.…”

“…In a small surprise, this is a well-motivated idea. For more details about this, we refer the reader to [2]. We can express this PageRank problem in terms of the final matrix Y where P is the degree-normalized matrix for network A, P ij = A ij /d j , Q is the degree-normalized matrix for B, and S(v, v ) is the apriori similarity of node v in A and v in B:…”

“…A standard technique here is to run a max-weight bipartite matching on the matrix Y for the network alignment case [22,2]. There are two issues with this step.…”

“…Many good ideas result in a polynomial explosion in problem size. This causes methods that have been developed for the pairwise case to be nearly impossible to use for the multimodal setup we propose [2,11]. More specifically, this is a memory bottleneck: they would need terabytes of memory to handle problems that start off as a megabyte of data.…”

Triangular Alignment (TAME). A Tensor-based Approach for Higher-order Network Alignment

“…This problem is also equivalent to a binary linear program through a standard linearizing transformation [35,50]. Different global alignment methods can be viewed as algorithms that either implicitly or explicitly optimize this BQP formulation.…”