Return times of random walk on generalized random graphs

Random walks are used for modeling various dynamics in, for example, physical, biological, and social contexts. Furthermore, their characteristics provide us with useful information on the phase transition and critical phenomena of even broader classes of related stochastic models. Abundant results are obtained for random walk on simple graphs such as the regular lattices and the Cayley trees. However, random walks and related processes on more complex networks, which are often more relevant in the real world, are still open issues, possibly yielding different characteristics. In this paper, we investigate the return times of random walks on random graphs with arbitrary vertex degree distributions. We analytically derive the distributions of the return times. The results are applied to some types of networks and compared with numerical data.

Figure 1

Schematic diagram showing random walk on a realization of generalized random graph. Integers denote the time of random walk.

Figure 2

Probability distributions of the first return times of the random walk on the ER random graph with (a) $\lambda =7$ and (b) $\lambda =10$. Numerical and theoretical results are indicated by circles and crosses, respectively. The results for the Cayley trees with the same mean vertex degrees, namely, (a) $d=7$ and (b) $d=10$, are indicated by solid lines.

Figure 3

Probability distributions of the first return times in the case of scale-free networks with $m=4$. Numerical and theoretical results are indicated by circles and crosses, respectively. The results for the Cayley trees with $d=7$ (solid lines) and $d=8$ (dashed lines) are also shown.