Maximizing the Mean Number of Communicating Vertex Pairs in Series-Parallel Networks

A communication network can be modelled as a probabilistic graph where each of b edges represents a communication line and each of n vertices represents a communication processor. Each edge e (vertex v) functions with probability p_e (p_v). If edges fail independently with uniform probability p and vertices do not fail, the probability that the network is connected is the probabilistic connectedness and is a standard measure of network reliability. The most reliable maximal series-parallel networks by this measure are those with exactly two vertices of degree two. However, as p becomes small, or n becomes large, the probability that even the most reliable series-parallel network is connected falls very quickly. Therefore, we wish to optimize a network with respect to another reliability measure, mean number of communicating vertex pairs. Experimental results suggest that this measure varies with p, with the diameter of the network, and with the number of minimum edge cutsets. We show that for large p, the most reliable series-parallel network must have the fewest minimum edge cutsets and for small p, the most reliable network must have maximum pairs of adjacent edges. We present a construction which incrementally inproves the communicating vertex pair mean for many networks and demonstrates that a fan maximizes this measure over maximal series parallel networks with exactly two edge cutsets of size two.