Synchronization Probability in Large Random Networks

TL;DR

This paper derives a condition linking network structure and dynamics to synchronization stability in large random networks, revealing that stability probability rapidly increases past a certain randomness threshold.

Contribution

It introduces a generalized framework for synchronization stability in multi-state networks and provides a probabilistic bound for large Erdős-Rényi networks.

Findings

Stability probability increases sharply past a specific randomness threshold.

Derived a lower bound on synchronization stability probability for large networks.

Established a relationship between network size, randomness, and stability likelihood.

Abstract

In a generalized framework, where multi-state and inter-state linkages are allowed, we derive a sufficient condition for the stability of synchronization in a network of chaotic attractors. This condition explicitly relates the network structure and the local and coupling dynamics to synchronization stability. For large Erd\"{o}s-R\'{e}nyi networks, the obtained condition is translated into a lower bound on the probability of stability of synchrony. Our results show that the probability of stability quickly increases as the randomness crosses a threshold which for large networks is inversely proportional to the network size.