Random walks in modular scale-free networks with multiple traps

TL;DR

This paper analytically investigates how the structure of modular scale-free networks with multiple traps influences the mean first-passage time of random walks, revealing the impact of trap placement and network topology.

Contribution

It derives an exact, general formula for MFPT on modular scale-free networks with multiple traps, independent of trap number and position, and analyzes how topology affects diffusion dynamics.

Findings

MFPT grows as a power-law with network size

Trap location significantly influences MFPT

Network topology explains differences in diffusion behavior

Abstract

Extensive empirical investigation has shown that a plethora of real networks synchronously exhibit scale-free and modular structure, and it is thus of great importance to uncover the effects of these two striking properties on various dynamical processes occurring on such networks. In this paper, we examine two cases of random walks performed on a class of modular scale-free networks with multiple traps located at several given nodes. We first derive a formula of the mean first-passage time (MFPT) for a general network, which is the mean of the expected time to absorption originating from a specific node, averaged over all non-trap starting nodes. Although the computation is complex, the expression of the formula is exact; moreover, the computational approach and procedure are independent of the number and position of the traps. We then determine analytically the MFPT for the two random walks being considered. The obtained analytical results are in complete agreement with the numerical ones. Our results show that the number and location of traps play an important role in the behavior of the MFPT, since for both cases the MFPT grows as a power-law function of the number of nodes, but their exponents are quite different. We demonstrate that the root of the difference in the behavior of MFPT is attributed to the modular and scale-free topologies of the networks. This work can deepen the understanding of diffusion on networks with modular and scale-free architecture and motivate relevant studies for random walks running on complex random networks with multiple traps.