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Graph matching consists of aligning the vertices of two unlabeled graphs inorder to maximize the shared structure across networks; when the graphs areunipartite, this is commonly formulated as minimizing their edge disagreements.In this paper, we address the common setting in which one of the graphs tomatch is a bipartite network and one is unipartite. Commonly, the bipartitenetworks are collapsed or projected into a unipartite graph, and graph matchingproceeds as in the classical setting. This potentially leads to noisy edgeestimates and loss of information. We formulate the graph matching problembetween a bipartite and a unipartite graph using an undirected graphical model,and introduce methods to find the alignment with this model without collapsing.We theoretically demonstrate that our methodology is consistent, and providenon-asymptotic conditions that ensure exact recovery of the matching solution.In simulations and real data examples, we show how our methods can result in amore accurate matching than the naive approach of transforming the bipartitenetworks into unipartite, and we demonstrate the performance gains achieved byour method in simulated and real data networks, including aco-authorship-citation network pair, and brain structural and functional data.

Graph matching between bipartite and unipartite networks: to collapse, or not to collapse, that is the question