ALIBABA 2021 Problem 2

Consider a computer network consisting of servers and bi-directional communication channels among them. Unfortunately, not all channels operate. Each direction of each channel fails with probability $p$ and operates otherwise. (All of these stochastic events are mutually independent, and $0 \le p \le 1$ .) There is a root serve, denoted by $r$ . We call the network operational, if all serves can reach $r$ using only operating channels. Note that we do not require $r$ to be able to reach any servers.

Show that the probability of the network to be operational does not depend on the choice of $r$ . (In other words, for any two distinct root servers $r_1$ and $r_2$ , the operational probability is the same.)