Random walks over weighted complex networks: Are the most occupied nodes the nearest ones?

In this paper, we study the relationship between occupation and closeness of nodes for particles moving in a random walk on weighted complex networks, such that the adjacency and transition matrices define the outgoing neighbors of a node and transition probabilities to them, respectively, for packages that pass through the node in question. To answer this question for different network topologies and transition probabilities, we propose two new planes involving occupation, closeness, and transient time, which characterize the transport properties of the networks, as opposed to the more static representations of the network, as previously reported. The first plane provides a local relation between occupation and closeness of nodes, while the second plane relates the average closeness and average transient time to converge to the asymptotic state of the network as a whole. We compare 16 different topologies considering complex real-world and synthetic networks. In all the cases considered, we found an approximate inverse relation between occupation and closeness of nodes, and a direct relation between the global transient time and average closeness of the network. The calculations are done directly from the network topology and transition probabilities, but they can also be estimated by directly simulating the network transport. Hence, these planes provide a complementary view of the transportation dynamics on complex networks.