Synchronization is optimal in non-diagonalizable networks

TL;DR

This paper extends the master stability formalism to non-diagonalizable networks and shows that optimal synchronizability is achieved through unidirectional, hierarchical structures unless a node connects to all others.

Contribution

It introduces a generalized formalism for analyzing synchronization in all network types and characterizes the structure of networks with maximum synchronizability.

Findings

Maximum synchronizability networks are non-diagonalizable unless a node connects to all others.

Optimal networks are formed by unidirectional links with normalized input strengths.

Hierarchical structures are naturally associated with high synchronizability.

Abstract

We consider the problem of maximizing the synchronizability of oscillator networks by assigning weights and directions to the links of a given interaction topology. We first extend the well-known master stability formalism to the case of non-diagonalizable networks. We then show that, unless some oscillator is connected to all the others, networks of maximum synchronizability are necessarily non-diagonalizable and can always be obtained by imposing unidirectional information flow with normalized input strengths. The extension makes the formalism applicable to all possible network structures, while the maximization results provide insights into hierarchical structures observed in complex networks in which synchronization plays a significant role.