  Graph matching consists of aligning the vertices of two unlabeled graphs in
order to maximize the shared structure across networks; when the graphs are
unipartite, this is commonly formulated as minimizing their edge disagreements.
In this paper, we address the common setting in which one of the graphs to
match is a bipartite network and one is unipartite. Commonly, the bipartite
networks are collapsed or projected into a unipartite graph, and graph matching
proceeds as in the classical setting. This potentially leads to noisy edge
estimates and loss of information. We formulate the graph matching problem
between 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 provide
non-asymptotic conditions that ensure exact recovery of the matching solution.
In simulations and real data examples, we show how our methods can result in a
more accurate matching than the naive approach of transforming the bipartite
networks into unipartite, and we demonstrate the performance gains achieved by
our method in simulated and real data networks, including a
co-authorship-citation network pair, and brain structural and functional data.
