Network Probabilistic Connectivity: Optimal Structures II

Abstract

Some special structures of networks with unreliable links are analyzed in the paper. Random undirected graph with independently failed edges and absolutely reliable nodes is used as a model. The tasks optimal placement of new edges, replacement of edges, connection of networks, and choice of best nodes for placement of base stations are analyzed. As goal functions we use: the graph connectivity probability (All Terminal Reliability, ATR), the mathematical expectation of the number of connected pairs of nodes (EDP), which is one-to-one with that of the number of connected pairs of nodes (ECP) and the average pairwise connectivity (APC), and the mathematical expectation of size (number of nodes) of a connected subgraph that contains some special node (c -node). It is shown that, in the general case, different structures are optimal when considering different reliability indicators. Most of the results are obtained for graphs with equally unreliable edges, but some results are also presented for the general case.