Role of fractal dimension in random walks on scale-free networks

TL;DR

This paper investigates how the fractal dimension influences the efficiency of random walks in scale-free networks, revealing that higher fractal dimensions lead to faster trapping times, with analytical results on two network families.

Contribution

It provides the first analytical study linking fractal dimension to random walk trapping times on fractal scale-free networks, including deterministic and hybrid models.

Findings

Average trapping time decreases as fractal dimension increases.

Derived explicit formulas for trapping times on two network families.

Fractal dimension is a key factor in the dynamics of random walks on these networks.

Abstract

Fractal dimension is central to understanding dynamical processes occurring on networks; however, the relation between fractal dimension and random walks on fractal scale-free networks has been rarely addressed, despite the fact that such networks are ubiquitous in real-life world. In this paper, we study the trapping problem on two families of networks. The first is deterministic, often called $(x,y)$-flowers; the other is random, which is a combination of $(1,3)$-flower and $(2,4)$-flower and thus called hybrid networks. The two network families display rich behavior as observed in various real systems, as well as some unique topological properties not shared by other networks. We derive analytically the average trapping time for random walks on both the $(x,y)$-flowers and the hybrid networks with an immobile trap positioned at an initial node, i.e., a hub node with the highest degree in the networks. Based on these analytical formulae, we show how the average trapping time scales with the network size. Comparing the obtained results, we further uncover that fractal dimension plays a decisive role in the behavior of average trapping time on fractal scale-free networks, i.e., the average trapping time decreases with an increasing fractal dimension.