The problem of finding the threshold value in ergodic resource network

Resource network is a graphical model of diffusion proposed earlier in the literature. Every node of the network stores some amount of “resource”. This resource disseminates through networks according to the specified rules. Earlier it was proved that if the total amount of resource in the network does not exceed some threshold value that the diffusion process is equivalent to the random walk in the related Markov chain. The problem of finding the threshold value in ergodic resource network is considered. It is shown that this problem is reduced to solving of system of equations, which describes the principles of redistribution of resource for each node of a network. The $O(n^3)$ complexity algorithm for finding the threshold value in arbitrary ergodic network is developed.