Transport Processes on Homogeneous Planar Graphs with Scale-Free Loops

TL;DR

This paper investigates how network geometry influences two diffusion processes—information packet transport with queuing and electron tunneling—on a homogeneous planar graph with scale-free loops, revealing distinct noise behaviors and the impact of topology.

Contribution

It introduces a detailed analysis of transport processes on a specific class of planar graphs with scale-free loops, highlighting differences in noise, correlations, and flow dynamics.

Findings

Long-range correlations develop in packet traffic due to local queuing interactions.

Noise fluctuations follow distinct scaling laws in two universality classes.

Betweenness inhomogeneity influences return times and flow distribution.

Abstract

We consider the role of network geometry in two types of diffusion processes: transport of constant-density information packets with queuing on nodes, and constant voltage-driven tunneling of electrons. The underlying network is a homogeneous graph with scale-free distribution of loops, which is constrained to a planar geometry and fixed node connectivity $k=3$. We determine properties of noise, flow and return-times statistics for both processes on this graph and relate the observed differences to the microscopic process details. Our main findings are: (i) Through the local interaction between packets queuing at the same node, long-range correlations build up in traffic streams, which are practically absent in the case of electron transport; (ii) Noise fluctuations in the number of packets and in the number of tunnelings recorded at each node appear to obey the scaling laws in two distinct universality classes; (iii) The topological inhomogeneity of betweenness plays the key role in the occurrence of broad distributions of return times and in the dynamic flow. The maximum-flow spanning trees are characteristic for each process type.