  This work considers the distributed computation of the one-to-one vertex
correspondences between two undirected and connected graphs, which is called
\textit{graph matching}, over multi-agent networks. Given two
\textit{isomorphic} and \textit{asymmetric} graphs, there is a unique
permutation matrix that maps the vertices in one graph to the vertices in the
other. Based on a convex relaxation of graph matching in Aflalo et al. (2015),
we propose a distributed computation of graph matching as a distributed convex
optimization problem subject to equality constraints and a global set
constraint, using a network of multiple agents whose interaction graph is
connected. Each agent in the network only knows one column of each of the
adjacency matrices of the two graphs, and all agents collaboratively learn the
graph matching by exchanging information with their neighbors. The proposed
algorithm employs a projected primal-dual gradient method to handle equality
constraints and a set constraint. Under the proposed algorithm, the agents'
estimates of the permutation matrix converge to the optimal permutation
globally and exponentially fast. Finally, simulation results are given to
illustrate the effectiveness of the method.
