Markov Random Fields, Homomorphism Counting, and Sidorenko’s Conjecture

Graph covers and the Bethe free energy (BFE) have been useful theoretical tools for producing lower bounds on a variety of counting problems in graphical models, including the permanent and the ferromagnetic Ising model. Here, we investigate weighted homomorphism counting problems over bipartite graphs that are related to a conjecture of Sidorenko. We show that the BFE does yield a lower bound in a variety of natural settings, and when it does yield a lower bound, it necessarily improves upon the lower bound conjectured by Sidorenko. Conversely, we show that there exist bipartite graphs for which the BFE does not yield a lower bound on the homomorphism number. Finally, we use the characterizations developed as part of this work to provide a simple proof of Sidorenko’s conjecture in a number of special cases.