Synchronization in networks with multiple interaction layers

TL;DR

This paper develops a framework to analyze synchronization stability in multi-layer networks, revealing complex phenomena such as induced stability in systems where individual layers are unstable, with applications to neural and power grid systems.

Contribution

It introduces a generalized stability assessment method for multi-layer networks, extending the Master Stability Function approach to account for multiple interaction layers.

Findings

Synchronization stability can be induced by multi-layer interactions even if individual layers are unstable.

The framework is validated on networks of R"ossler oscillators with double-layer interactions.

Rich phenomena, including stability induction, emerge from the multi-layer structure.

Abstract

The structure of many real-world systems is best captured by networks consisting of several interaction layers. Understanding how a multi-layered structure of connections affects the synchronization properties of dynamical systems evolving on top of it is a highly relevant endeavour in mathematics and physics, and has potential applications to several societally relevant topics, such as power grids engineering and neural dynamics. We propose a general framework to assess stability of the synchronized state in networks with multiple interaction layers, deriving a necessary condition that generalizes the Master Stability Function approach. We validate our method applying it to a network of R\"ossler oscillators with a double layer of interactions, and show that highly rich phenomenology emerges. This includes cases where the stability of synchronization can be induced even if both layers would have individually induced unstable synchrony, an effect genuinely due to the true multi-layer structure of the interactions amongst the units in the network.