Computing a ranking network with confidence bounds from a graph-based Beta random field

We address two largely overlooked, fundamental issues in computing a ranking hierarchy within a society: which information in the network is relevant, and what effect chance has on the hierarchy. To properly account for uncertainty from limited data, we construct a random field in a matrix form having entry-wise posterior Beta distributions based on a graph of pairwise conflict outcomes. To evaluate relevant network information using information transitivity, another random matrix of synthesized transitive dominance odds is computed collectively along observed dominance paths. These two matrices are coupled together to fuse both direct and indirect dominance information. An ensemble of realizations of this fused random matrix facilitates an ensemble of optimal ranking networks by means of simulated annealing. Conditional statistical inferences regarding network features are derived, manifesting the effect of uncertainty. Our computational approach is suitable for large graphs of pairwise conflict outcomes, and can accommodate tremendous data heterogeneity-a typical feature in such studies. We also demonstrate the infeasibility of the classical maximum-likelihood approach, and expose the mechanistic flaws that stem from completely ignoring relevant information residing in the graph. We analyse two real datasets of decisive conflict outcomes, the first involving college football teams, and the second involving an adult rhesus macaque society in captivity.