Random walk with memory on complex networks

We study random walks on complex networks with transition probabilities which depend on the current and previously visited nodes. By using an absorbing Markov chain we derive an exact expression for the mean first passage time between pairs of nodes, for a random walk with a memory of one step. We have analyzed one particular model of random walk, where the transition probabilities depend on the number of paths to the second neighbors. The numerical experiments on paradigmatic complex networks verify the validity of the theoretical expressions, and also indicate that the flattening of the stationary occupation probability accompanies a nearly optimal random search.

Figure 1

GrMFPT in (a) BA, (b) ER, and (c) WS networks of $N=100$ nodes with different average node degree $\langle k\rangle$ for the three cases: uniform (red line and circles), inverse degree (blue line and squares) and one-step memory (green line and triangles). The lines are theoretical values (T) and the markers numerical estimates (N).

Figure 2

Kullback-Leibler divergence of the stationary occupation probability of uniform (red), inverse degree (blue), and one-step memory (green) random walks from the uniform density in (a) BA, (b) ER, and (c) WS networks with $N=100$ nodes for different average node degrees.

Figure 3

Random walks in directed ER networks with different average degree $\langle k\rangle$: (a) Comparison of the GrMFPT for uniform (red circles), inverse indegree (blue squares), and one-stop memory (green triangles). (b) Kullback-Leibler divergence of the invariant density from a uniform density for three approaches: uniform (red circles), inverse-indegree biased (blue squares) and one-step memory (green triangles).