Anomalous behavior of trapping on a fractal scale-free network

TL;DR

This paper demonstrates that trapping efficiency on fractal scale-free networks is significantly lower than on non-fractal ones, with mean first-passage time scaling superlinearly with network size, challenging previous assumptions about scale-free networks.

Contribution

It provides an analytical expression for trapping time on fractal scale-free networks, revealing a superlinear scaling and highlighting the importance of fractality in network dynamics.

Findings

MFPT scales as V^{3/2} in fractal networks

Trapping efficiency is lower in fractal than non-fractal networks

Degree distribution alone does not determine trapping behavior

Abstract

It is known that the heterogeneity of scale-free networks helps enhancing the efficiency of trapping processes performed on them. In this paper, we show that transport efficiency is much lower in a fractal scale-free network than in non-fractal networks. To this end, we examine a simple random walk with a fixed trap at a given position on a fractal scale-free network. We calculate analytically the mean first-passage time (MFPT) as a measure of the efficiency for the trapping process, and obtain a closed-form expression for MFPT, which agrees with direct numerical calculations. We find that, in the limit of a large network order $V$, the MFPT $<T>$ behaves superlinearly as $<T > \sim V^{{3/2}}$ with an exponent 3/2 much larger than 1, which is in sharp contrast to the scaling $<T > \sim V^{\theta}$ with $\theta \leq 1$, previously obtained for non-fractal scale-free networks. Our results indicate that the degree distribution of scale-free networks is not sufficient to characterize trapping processes taking place on them. Since various real-world networks are simultaneously scale-free and fractal, our results may shed light on the understanding of trapping processes running on real-life systems.