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We describe and study a transport-based procedure called network optimal transition coupling (NetOTC) for the comparison and alignment of two networks. The networks of interest may be directed or undirected, weighted or unweighted, and may have distinct vertex sets of different sizes. Given two networks and a cost function relating their vertices, NetOTC finds a transition coupling of their associated random walks having minimum expected cost. The minimizing cost quantifies the difference between the networks, while the optimal transport plan itself provides alignments of both the vertices and the edges of the two networks. Coupling of the full random walks, rather than their marginal distributions, ensures that NetOTC captures local and global information about the networks and preserves edges. NetOTC has no free parameters and does not rely on randomization. We investigate a number of theoretical properties of NetOTC and present experiments establishing its empirical performance. 

Abstract We study optimal transport for stationary stochastic processes taking values in finite spaces. In order to reflect the stationarity of the underlying processes, we restrict attention to stationary couplings, also known as joinings. The resulting optimal joining problem captures differences in the long-run average behavior of the processes of interest. We introduce estimators of both optimal joinings and the optimal joining cost, and establish consistency of the estimators under mild conditions. Furthermore, under stronger mixing assumptions we establish finite-sample error rates for the estimated optimal joining cost that extend the best known results in the iid case. We also extend the consistency and rate analysis to an entropy-penalized version of the optimal joining problem. Finally, we validate our convergence results empirically as well as demonstrate the computational advantage of the entropic problem in a simulation experiment.