Diffusion on Vicsek fractals: Efficiency and the Wiener index.

Abstract

Understanding how hierarchical topology governs diffusion is essential for explaining transport efficiency in biological networks and designing engineered materials. While prior studies have characterized anomalous diffusion and derived analytical expressions for distance-based indices on fractals, the quantitative relationship between graph-theoretic structure and transport dynamics remains unclear, particularly whether fractal organization offers unique properties beyond those predictable from network size or connectivity. We address this gap by deriving a closed form expression for the Wiener index of the Vicsek fractal and performing systematic diffusion simulations across generations 0 to 4 (up to 3125 nodes). Comparative analysis with 2D lattices, random regular graphs, and the T fractal isolates fractal-specific effects from generic network properties, while spectral gap analysis provides mechanistic insight into mixing dynamics. We identify three phenomena that appear characteristic of self-similar architecture: (1) progressive transport efficiency improvement, where hierarchical redundancy increasingly compensates for path elongation; (2) structured heterogeneity, exhibiting near-maximal entropy alongside persistent spatial gradients; and (3) sustained flux growth, reflecting concentration gradients within recursive substructures. Both the Vicsek and T fractal demonstrate these behaviors, while lattices and random graphs do not, suggesting they may be generic features of deterministic self-similar networks. Although our analysis is limited to five generations and the results are suggestive rather than definitive, the combination of analytical derivations, numerical simulations, and multitopology comparison provides evidence that hierarchical self-similarity enables scalable transport not through distance minimization but through redundancy that mitigates bottleneck-induced delays. These findings offer insight into biological transport systems and suggest design principles for artificial hierarchical materials.