We study here dynamic antagonism in a fixed network, represented as a graph $G$ of $n$ vertices. In particular, we consider the case of $k \leq n$ particles walking randomly independently around the network. Each particle belongs to exactly one of two antagonistic species, none of which can give birth to children. When two particles meet, they are engaged in a (sometimes mortal) local fight. The outcome of the fight depends on the species to which the particles belong. Our problem is \emphto predict (i.e. to compute) the eventual chances of species survival. We prove here that this can indeed be done in \emphexpected polynomial time on the size of the network, provided that the network is \emphundirected.