Suppose I have an undirected, unweighted network in which some node, $i$, is infected. This (and any) infected node has probability $p$ of infecting adjacent nodes. Is there a closed form expression of the probability that an arbitrary node $j$ becomes infected? Any references would be much appreciated!
My thought is that this should be somehow related to the resistance distance between the nodes.
EDIT: So after some simulations there doesn't appear to be much relationship with resistance distance. I think this is because resistance can be understood as a situation where `voltage' doesn't change as additional paths are added. Here however, adding more connections to the infected source node (in some sense) increases the outward flow of contagion (a voltage analog)
EDIT 2: Let me be more clear about the process I have in mind. Take a connected undirected graph,$G$, as given. Some node, $i$, starts out as infected. In period 1, each node adjacent to $i$ has probability, $p$, of becoming infected. In subsequent periods, for each newly infected node, each of its uninfected neighbors has probability $p$ of becoming newly infected for the next period. Can we characterize the probability that an arbitrary node, $j$, becomes infected after infinitely many periods?