Shared-endpoint correlations and hierarchy in random flows on graphs

We analyze the correlation between randomly chosen edge weights on neighboring edges in a directed graph. This shared-endpoint correlation controls the expected organization of randomly drawn edge flows, assuming each edge’s flow is conditionally independent of others given its endpoints. We model different relationships between endpoint attributes and flow by varying the kernel associated with a Gaussian process evaluated on every vertex. We then relate the expected flow structure to the smoothness of functions generated by the Gaussian process. We investigate the shared-endpoint correlation for the squared exponential, mixture, and Matèrn kernels while exploring asymptotics in smooth and rough limits.