Impact of degree heterogeneity on the behavior of trapping in Koch networks

TL;DR

This paper investigates how degree heterogeneity and correlations in Koch networks affect the mean first-passage time for trapping, revealing that in large networks, MFPT scales linearly with network size regardless of degree distribution exponent.

Contribution

The study provides an exact calculation of MFPT for trapping in correlated Koch networks, showing independence from the degree distribution exponent and contrasting with uncorrelated networks.

Findings

MFPT scales linearly with network size N

MFPT is independent of the degree distribution exponent γ

Correlations in Koch networks significantly influence trapping dynamics

Abstract

Previous work shows that the mean first-passage time (MFPT) for random walks to a given hub node (node with maximum degree) in uncorrelated random scale-free networks is closely related to the exponent $\gamma$ of power-law degree distribution $P(k)\sim k^{-\gamma}$, which describes the extent of heterogeneity of scale-free network structure. However, extensive empirical research indicates that real networked systems also display ubiquitous degree correlations. In this paper, we address the trapping issue on the Koch networks, which is a special random walk with one trap fixed at a hub node. The Koch networks are power-law with the characteristic exponent $\gamma$ in the range between 2 and 3, they are either assortative or disassortative. We calculate exactly the MFPT that is the average of first-passage time from all other nodes to the trap. The obtained explicit solution shows that in large networks the MFPT varies lineally with node number $N$, which is obviously independent of $\gamma$ and is sharp contrast to the scaling behavior of MFPT observed for uncorrelated random scale-free networks, where $\gamma$ influences qualitatively the MFPT of trapping problem.