Distinct scalings for mean first-passage time of random walks on scale-free networks with the same degree sequence

TL;DR

This paper demonstrates that the mean first-passage time for random walks on scale-free networks varies significantly with network structure, even when degree distributions are identical, challenging the sufficiency of degree distribution alone for characterizing dynamical processes.

Contribution

It introduces a family of scale-free networks with identical degree sequences but different structures, analyzing how their MFPT scaling behaviors differ for the trapping problem.

Findings

MFPT scales as a power-law with network size for all network types.

Scaling exponents vary with network parameter q, indicating structural influence.

Degree distribution alone does not determine MFPT behavior.

Abstract

In general, the power-law degree distribution has profound influence on various dynamical processes defined on scale-free networks. In this paper, we will show that power-law degree distribution alone does not suffice to characterize the behavior of trapping problem on scale-free networks, which is an integral major theme of interest for random walks in the presence of an immobile perfect absorber. In order to achieve this goal, we study random walks on a family of one-parameter (denoted by $q$) scale-free networks with identical degree sequence for the full range of parameter $q$, in which a trap is located at a fixed site. We obtain analytically or numerically the mean first-passage time (MFPT) for the trapping issue. In the limit of large network order (number of nodes), for the whole class of networks, the MFPT increases asymptotically as a power-law function of network order with the exponent obviously different for different parameter $q$, which suggests that power-law degree distribution itself is not sufficient to characterize the scaling behavior of MFPT for random walks, at least trapping problem, performed on scale-free networks.