Maximum Performance at Minimum Cost in Network Synchronization

TL;DR

This paper extends the master stability framework to non-diagonalizable Laplacian matrices, characterizes optimal oscillator networks for synchronization, and provides methods to construct such networks with specific topological features.

Contribution

It introduces an extension of the master stability framework for non-diagonalizable Laplacians and characterizes the solution set of optimization problems for network synchronization.

Findings

Optimal networks are often directed and non-diagonalizable.

Hierarchical networks without feedback loops are among the optimal solutions.

Explicit construction methods for optimal networks using oriented spanning trees.

Abstract

We consider two optimization problems on synchronization of oscillator networks: maximization of synchronizability and minimization of synchronization cost. We first develop an extension of the well-known master stability framework to the case of non-diagonalizable Laplacian matrices. We then show that the solution sets of the two optimization problems coincide and are simultaneously characterized by a simple condition on the Laplacian eigenvalues. Among the optimal networks, we identify a subclass of hierarchical networks, characterized by the absence of feedback loops and the normalization of inputs. We show that most optimal networks are directed and non-diagonalizable, necessitating the extension of the framework. We also show how oriented spanning trees can be used to explicitly and systematically construct optimal networks under network topological constraints. Our results may provide insights into the evolutionary origin of structures in complex networks for which synchronization plays a significant role.