Competing bootstrap processes on the random graph $G(n,p)$

We consider a generalization of classic bootstrap percolation in which two competing processes concurrently evolve on the same graph $G(n,p)$. Nodes can be in one of three states, conveniently represented by different colors: red, black and white. Initially, a given number aR of active red nodes (red seeds) are selected uniformly at random among the n nodes. Similarly, a given number aB of active black nodes (black seeds) are selected uniformly at random among the other n-aR nodes. All remaining nodes are initially white (inactive). White nodes wake up at times dictated by independent Poisson clocks of rate 1. When a white node wakes up, it checks the state of its neighbors: if the number of red (black) neighbors exceeds the number of black (red) neighbors by a fixed amount r ≥ 2, the node becomes an active red (black) node, and remains so forever. The parameters of the model are, besides r (fixed) and n (tending to ∞), the numbers aR (aB) of initial red (black) seeds, and the edge existence probability $p=p(n)$. We study the size A^*R (A^*B) of the final set of active red (black) nodes, identifying different regimes which are analyzed under suitable time-scales, allowing us to obtain detailed (asymptotic) temporal dynamics of the two concurrent activation processes.