Expected reliability of communication protocols

We consider the problem of sending a message from a sender $s$ to a receiver $r$ through an unreliable network by specifying in a protocol what each vertex is supposed to do if it receives the message from one of its neighbors. A protocol for routing a message in such a graph is finite if it never floods $r$ with an infinite number of copies of the message. The expected reliability of a given protocol is the probability that a message sent from $s$ reaches $r$ when the edges of the network fail independently with probability $1-p$. We discuss, for given networks, the properties of finite protocols with maximum expected reliability in the case when $p$ is close to 0 or 1, and we describe networks for which no one protocol is optimal for all values of $p$. In general, finding an optimal protocol for a given network and fixed probability is challenging and many open problems remain.