The effect of network structure on phase transitions in queuing networks

TL;DR

This paper investigates how the structure of transportation networks influences phase transitions from uncongested to congested states, revealing that network density affects the relationship between graph structure and critical traffic load.

Contribution

It establishes a connection between graph structure and critical traffic load, showing the limitations of previous degree-based predictions for sparse networks using spectral graph theory.

Findings

Critical load depends on degree sequence in dense graphs

Higher order corrections are needed for sparse graphs

Local network structure influences phase transition thresholds

Abstract

Recently, De Martino et al have presented a general framework for the study of transportation phenomena on complex networks. One of their most significant achievements was a deeper understanding of the phase transition from the uncongested to the congested phase at a critical traffic load. In this paper, we also study phase transition in transportation networks using a discrete time random walk model. Our aim is to establish a direct connection between the structure of the graph and the value of the critical traffic load. Applying spectral graph theory, we show that the original results of De Martino et al showing that the critical loading depends only on the degree sequence of the graph -- suggesting that different graphs with the same degree sequence have the same critical loading if all other circumstances are fixed -- is valid only if the graph is dense enough. For sparse graphs, higher order corrections, related to the local structure of the network, appear.