Origin of the hub spectral dimension in scale-free networks

TL;DR

This paper investigates the origin of the hub spectral dimension in scale-free networks by applying a renormalization group approach to understand how network heterogeneity influences random walk dynamics.

Contribution

It provides a theoretical explanation for the emergence of the hub spectral dimension using RG analysis, linking it to the conservation of degree distribution during network renormalization.

Findings

The hub spectral dimension arises from the scale-free degree distribution.

RG transformation preserves the power-law degree distribution.

Anomalous random walk behavior is explained by degree conservation.

Abstract

The return-to-origin probability and the first passage time distribution are essential quantities for understanding transport phenomena in diverse systems. The behaviors of these quantities typically depend on the spectral dimension $d_s$. However, it was recently revealed that in scale-free networks these quantities show a crossover between two power-law regimes characterized by $ d_s $ and the so-called hub spectral dimension $d_s^{\textrm{(hub)}}$ due to the heterogeneity of connectivities of each node. To understand the origin of $d_s^{\textrm{(hub)}}$ from a theoretical perspective, we study a random walk problem on hierarchical scale-free networks by using the renormalization group (RG) approach. Under the RG transformation, not only the system size but also the degree of each node changes due to the scale-free nature of the degree distribution. We show that the anomalous behavior of random walks involving the hub spectral dimension $d_s^{\textrm{(hub)}}$ is induced by the conservation of the power-law degree distribution under the RG transformation.