Exact Distributions of the Number of Pattern Occurrences in Undirected Graphical Models

Abstract

The method of probability generating functions is extended for obtaining exact distributions of the number of occurrences of a discrete pattern in undirected graphical models. General results for deriving the distributions are given with illustrative examples. Further, a device for reducing calculations is proposed. It works effectively when the graphical model is relatively simple. An algorithm for obtaining the distributions including the device is also given. In order to show the feasibility of our method, exact distributions of the number of occurrences of a “1”-run are derived in two undirected graphical models whose vertices are allocated on a sphere and a torus, respectively. As an application of our results, the exact reliabilities of consecutive-k-out-of-n:F systems corresponding to the undirected graphical models are obtained.