Random walks on bifractal networks

Abstract

It has recently been shown that networks possessing scale-free and fractal properties may exhibit a bifractal nature, in which local structures are described by two different fractal dimensions. In this study, we investigate random walks on such fractal scale-free networks (FSFNs) by examining the walk dimension $d_{\text{w}}$ and the spectral dimension $d_{\text{s}}$, to understand how the bifractality affects their dynamical properties. The walk dimension is found to be unaffected by the difference in local fractality of an FSFN and remains constant regardless of the starting node of a random walk, whereas the spectral dimension takes two values, $d_{\text{s}}^{\text{min}}$ and $d_{\text{s}}^{\text{max}}(>d_{\text{s}}^{\text{min}})$, depending on the starting node. The dimension $d_{\text{s}}^{\text{min}}$ characterizes the return probability of a random walker starting from an infinite-degree hub node in the thermodynamic limit, while $d_{\text{s}}^{\text{max}}$ describes that of a random walker starting from a finite-degree non-hub node infinitely distant from hub nodes and is equal to the global spectral dimension $D_{\text{s}}$. The existence of two local spectral dimensions is a direct consequence of the bifractality of the FSFN. Furthermore, analytical expressions of $d_{\text{w}}$, $d_{\text{s}}^{\text{min}}$, and $d_{\text{s}}^{\text{max}}$ are presented for FSFNs formed by the generator model and the giant components of critical scale-free random graphs, and are numerically confirmed.

11 pages, 3 figures

References in corpus (27)

• Self-similarity of complex networks
• Random walks and diffusion on networks
• Origins of fractality in the growth of complex networks
• Multifractal analysis of financial markets
• Percolation Critical Exponents in Scale-Free Networks
• Scaling theory of transport in complex networks
• Fractal and Transfractal Recursive Scale-Free Nets
• Random walks on graphs: ideas, techniques and results
• Multifractal methodology
• Statistical ensemble of scale-free random graphs
• Ordinary Percolation with Discontinuous Transitions
• Random walks on complex trees
• Determination of multifractal dimensions of complex networks by means of the sandbox algorithm
• Multifractality of complex networks
• Fractal and multifractal properties of a family of fractal networks
• Multifractal analysis of complex networks
• Correlations in connected random graphs
• Monte-Carlo simulation study of the two-stage percolation transition in enhanced binary trees
• Fractal and multifractal analysis of complex networks: Estonian network of payments
• Generating-function approach for bond percolations in hierarchical networks
• Origin of the hub spectral dimension in scale-free networks
• Renormalization group for link percolation on planar hyperbolic manifolds
• A general model of hierarchical fractal scale-free networks
• Renormalization group theory of percolation on pseudo-fractal simplicial and cell complexes
• A Fixed-Mass multifractal approach for unweighted complex networks
• A computationally-efficient sandbox algorithm for multifractal analysis of large-scale complex networks with tens of millions of nodes
• Bifractality of fractal scale-free networks