We present a geometrically enhanced Markov chain Monte Carlo sampler for networks based on a discrete
curvature measure defined on graphs. Specifically, we incorporate the concept of graph Forman curvature
into sampling procedures on both the nodes and edges of a network explicitly, via the transition probability
of the Markov chain, as well as implicitly, via the target stationary distribution, which gives a novel, curved
Markov chain Monte Carlo approach to learning networks. We show that integrating curvature into the
sampler results in faster convergence to a wide range of network statistics demonstrated on deterministic
networks drawn from real-world data.