Synchronization in Weighted Uncorrelated Complex Networks in a Noisy Environment: Optimization and Connections with Transport Efficiency

TL;DR

This paper investigates how to optimize synchronization in weighted uncorrelated scale-free networks under noisy conditions by adjusting link weights, revealing an optimal weight exponent and connecting the problem to transport efficiency and resistor networks.

Contribution

It introduces a specific weight scheme for networks, derives the optimal weight exponent for synchronization, and links this to transport efficiency and resistor network models.

Findings

Optimal weight exponent is approximately -1.

Numerical results confirm mean-field predictions.

Connections established between synchronization, transport efficiency, and resistor networks.

Abstract

Motivated by synchronization problems in noisy environments, we study the Edwards-Wilkinson process on weighted uncorrelated scale-free networks. We consider a specific form of the weights, where the strength (and the associated cost) of a link is proportional to $(k_{i}k_{j})^{\beta}$ with $k_{i}$ and $k_{j}$ being the degrees of the nodes connected by the link. Subject to the constraint that the total network cost is fixed, we find that in the mean-field approximation on uncorrelated scale-free graphs, synchronization is optimal at $\beta^{*}$$=$-1. Numerical results, based on exact numerical diagonalization of the corresponding network Laplacian, confirm the mean-field results, with small corrections to the optimal value of $\beta^{*}$. Employing our recent connections between the Edwards-Wilkinson process and resistor networks, and some well-known connections between random walks and resistor networks, we also pursue a naturally related problem of optimizing performance in queue-limited communication networks utilizing local weighted routing schemes.