Bounds for All-Terminal Reliability in Planar Networks

摘要

A communication network can be modeled as a graph where the nodes of the graph represent the sites, and the edges represent the links between the sites. The edges of the graph operate with equal probability p. The all-terminal reliability of the network with n nodes and b edges can be written as R(p)= sum over i = 0 to i = b of (F sub i) (p to the (b-i) power) (q to the i power) where F sub i is the number of connected subgraphs with (b-i) edges and q=(1-p). All known subgraph counting bounds for R use exact values for F sub 0,..., F sub c and F sub d where c is the cardinality of a minimum cut and d=b-n + 1. This paper derives upper and lower bounds for R for planar networks by using exact values of F sub d-1, F sub d-2, F sub d-3, F sub c+1, and F sub c+2 where approximations were used before. The effect of using these exact values instead of approximations on Kruskal-Katona bounds and Ball-Proven bounds is studied.