Probabilistic Analysis of Message Forwarding

In this paper, we present a novel algorithm that finds the probability density function for the number of distinct nodes reached, within a specific number of levels of message forwarding, for a fixed size network. The algorithm also finds the expected number of distinct nodes to which a message is forwarded, within a specific number of levels of message forwarding, as the sum of the number of nodes, weighted by the probability of reaching that number of nodes. In addition, the algorithm finds the probability density function for the number of distinct nodes at a given level of message forwarding, and the expected number of distinct nodes at that level. Using the algorithm, we calculate these probability density functions and expected values for various size networks, degrees of message forwarding, probabilities of message forwarding, and levels of message forwarding. The algorithm has application to the Internet, peer-to-peer networks, ad-hoc networks, and social networks, and to multicasting, gossiping, and rumor and epidemic protocols.