Structure properties of a doubly-stochastic process on a network

In this paper, we study how special patterns affect certain dynamic process on networks. The process we analyze is an iteration to generate a doubly-stochastic matrix consistent to the adjacent matrix of a network and the patterns can be described as h non-interconnected vertices only connect other g vertices (h>g). From the perspective of network structure, we prove that the necessary and sufficient condition when the iteration converges is that these patterns do not exist in the network. For BA networks, there is a phase transition. The diverge-converge transition point is that the average degree is about 8, which is theoretically proved. The existence of these patterns depends on two factors: first, higher moments of degree distribution of the network; second, the probability that vertices with degree 1 exist in the network. Simulation results also support our theory.