Maximum Entropy Analysis of Flow Networks with Structural Uncertainty (Graph Ensembles)

Abstract

This study examines MaxEnt methods for probabilistic inference of the state of flow networks, including pipe flow, electrical and transport networks, subject to physical laws and observed moments. While these typically assume networks of invariant graph structure, we here consider higher-level MaxEnt schemes, in which the network structure constitutes part of the uncertainty in the problem specification. In physics, most studies on the statistical mechanics of graphs invoke the Shannon entropy $H_G^{Sh} = - \sum \nolimits _{\Omega _G} P(G) \ln P(G)$, where G is the graph and $\Omega _G$ is the graph ensemble. We argue that these should adopt the relative entropy $H_G = - \sum \nolimits _{\Omega _G} P(G) \ln {P(G)}/{Q(G)}$, where Q(G) is the graph prior associated with the graph macrostate G. By this method, the user is able to employ a simplified accounting over graph macrostates rather than need to count individual graphs. Using combinatorial methods, we here derive a variety of graph priors for different graph ensembles, using different macrostate partitioning schemes based on the node or edge counts. A variety of such priors are listed herein, for ensembles of undirected or directed graphs.