My problem is the following. I have a set of vertices $N$ and a set of vertices $H$. Each vertex $n \in N$ is connected by means of an edge to each vertex $h \in H$. So the two sets of vertices and the set of edges form a complete bipartite graph. The graph is also undirected. Let us say that two vertices $n_1,n_2 \in N$ can communicate with each other if there exists a path $n_1h_1n_2$ of length 2, where $h_1 \in H$. Since we start out with a complete bipartite graph, initially all vertices in $N$ can communicate with each other. However, now there is a probability $p$ assigned to each edge. This probability is the same for all edges and it models the probability that a vertex $n \in N$ is disconnected from a vertex $h \in H$ after some fixed amount of time $t$. Moreover, the probability of two vertices being disconnected is independent of the probability of two other vertices being disconnected. Thus, given that vertices can be disconnected, what is the probability that there does not exist a path $n_1h_1n_2$, for some $n_1, n_2 \in N$ and some $h_1 \in H$ when the time $t$ has elapsed? In other words, what is the probability that two vertices from the set $N$ cannot communicate after $t$ has elapsed?
Edit: To give some background, what I am trying to model is a computer network comprised of computing nodes, represented by the set of vertices $N$, and hubs, represented by the set of vertices $H$. Computing nodes cannot communicate directly with each other, but only by means of an intermediate hub. The edges basically represent cables connecting nodes and hubs. The probability $p$ represents the probability that a cable suffers a failure, e.g., breaks, in an interval of time $[0, t]$.