On the decentralized navigation of multiple packages on transportation networks

TL;DR

This study uses numerical simulations to analyze how long-range shortcuts in transportation networks affect package flow, revealing a critical point for congestion and identifying an optimal network configuration for resilience.

Contribution

It introduces a detailed analysis of congestion transition in spatial networks with long-range shortcuts, highlighting an optimal shortcut distribution parameter for improved resilience.

Findings

Identifies a critical package creation probability $p_c$ leading to congestion transition.

Shows $p_c$ follows a power-law decay with network size, $p_c \\sim L^{-\\gamma}$.

Finds an optimal $\\alpha$ value (~1.7) for network resilience against congestion.

Abstract

We investigate by numerical simulation and finite-size analysis the impact of long-range shortcuts on a spatially embedded transportation network. Our networks are built from two-dimensional ($d=2$) square lattices to be improved by the addition of long-range shortcuts added with probability $P(r_{ij})\sim r_{ij}^{-\alpha}$ [J. M. Kleinberg, Nature 406, 845 (2000)]. Considering those improved networks, we performed numerical simulation of multiple discrete package navigation and found a limit for the amount of packages flowing through the network. Such limit is characterized by a critical probability of creating packages $p_{c}$, where above this value a transition to a congested state occurs. Moreover, $p_{c}$ is found to follow a power-law, $p_{c}\sim L^{-\gamma}$, where $L$ is the network size. Our results indicate the presence of an optimal value of $\alpha_{\rm min}\approx1.7$, where the parameter $\gamma$ reaches its minimum value and the networks are more resilient to congestion for larger system sizes.