Growth, collapse, and self-organized criticality in complex networks

TL;DR

This paper investigates how complex networks of nonlinear oscillators can maintain synchronization as they grow, revealing a tendency toward self-organized criticality characterized by power-law distributed collapse events.

Contribution

It demonstrates that growing complex networks naturally evolve into a critical state where small changes can cause large-scale synchronization collapses, highlighting a universal phenomenon across models.

Findings

Synchronization collapse follows a power-law distribution.

Networks tend to reach a critical state during growth.

Eigenvector analysis uncovers the mechanism behind collapse.

Abstract

To understand how certain dynamical behaviors can or cannot persist as the underlying network grows is a problem of increasing importance in complex dynamical systems as well as sustainability science and engineering. We address the question of whether a complex network of nonlinear oscillators can maintain its synchronization stability as it expands or grows. A network in the real world can never be completely synchronized due to noise and/or external disturbances. This is especially the case when, mathematically, the transient synchronous state during the growth process becomes marginally stable, as a local perturbation can trigger a rapid deviation of the system from the vicinity of the synchronous state. In terms of the nodal dynamics, a large scale avalanche over the entire network can be triggered in the sense that the individual nodal dynamics diverge from the synchronous state in a cascading manner within a short time period. Because of the high dimensionality of the networked system, the transient process for the system to recover to the synchronous state can be extremely long. Introducing a tolerance threshold to identify the desynchronized nodes, we find that, after an initial stage of linear growth, the network typically evolves into a critical state where the addition of a single new node can cause a group of nodes to lose synchronization, leading to synchronization collapse for the entire network. A statistical analysis indicates that, the distribution of the size of the collapse is approximately algebraic (power law), regardless of the fluctuations in the system parameters. This is indication of the emergence of self-organized criticality. We demonstrate the generality of the phenomenon of synchronization collapse using a variety of complex network models, and uncover the underlying dynamical mechanism through an eigenvector analysis.