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A graph homomorphism is a map between two graphs that preserves adjacencyrelations. We consider the problem of sampling a random graph homomorphism froma graph into a large network. We propose two complementary MCMC algorithms forsampling random graph homomorphisms and establish bounds on their mixing timesand the concentration of their time averages. Based on our sampling algorithms,we propose a novel framework for network data analysis that circumvents some ofthe drawbacks in methods based on independent and neighborhood sampling.Various time averages of the MCMC trajectory give us various computableobservables, including well-known ones such as homomorphism density and averageclustering coefficient and their generalizations. Furthermore, we show thatthese network observables are stable with respect to a suitably renormalizedcut distance between networks. We provide various examples and simulationsdemonstrating our framework through synthetic networks. We also\commHL{demonstrate the performance of} our framework on the tasks of networkclustering and subgraph classification on the Facebook100 dataset and on WordAdjacency Networks of a set of classic novels.

Sampling random graph homomorphisms and applications to network data analysis