Math Reveals Secrets of Synchronization in Complex Systems

Summary: Researchers delved deep into the mysteries of synchronization in complex systems, uncovering how certain elements effortlessly fall into or out of sync. This dance of coordination can be observed from humans clapping in rhythm to the synchronicity of heart cells.

By studying “walks” through networks, the team discovered the role of convergent walks in diminishing the quality of synchronization. These findings could revolutionize our understanding of everything from power grid stability to brain functions and social media dynamics.

Key Facts:

  1. Synchronization is a widespread phenomenon, seen in everything from clapping crowds to heart cell activity.
  2. The structure of a network, specifically the prevalence of convergent walks, determines its synchronization quality.
  3. These insights can be applied in diverse fields, including stabilizing power grids or preventing neural synchronizations that lead to epilepsy.

Source: University of Sydney

Computer scientists and mathematicians working in complex systems at the University of Sydney and the Max Planck Institute for Mathematics in the Sciences in Germany have developed new methods to describe what many of us take for granted – how easy, or hard, it can be to fall in and out of sync.

Synchronised phenomena are all around us, whether it is human clapping and dancing, or the way fireflies flash, or how our neurons and heart cells interact. However, it is something not fully understood in engineering and science.

This shows nodes.
Walks are sequences of connected hops between individual elements or nodes in the network. Credit: Neuroscience News

Associate Professor Joseph Lizier, expert in complex systems at the University of Sydney, said: “We know the feeling of dancing in step to the ‘Nutbush’ in a crowd – or the awkward feeling when people lose time clapping to music. Similar processes occur in nature, and it is vital that we better understand how falling in and out of sync actually works.

“Being in sync in a system can be very good; you want your heart cells to all beat together rather than fibrillate. But being in sync can also be very bad; you don’t want your brain cells to all fire together in an epileptic seizure.”

Associate Professor Lizier and colleagues at the Max Planck Institute in Leipzig, Germany have published new research on synchronisation in the Proceedings of the National Academy of Sciences of the United States of America (PNAS).

The paper sets out the mathematics of how the network structure connecting a set of individual elements controls how well they can synchronise their activity.

It is a critical insight into how these systems operate, because in most real-world systems, no one individual element controls all the others. And nor can any individual directly see and react to all the others: they are only connected through a network.

Associate Professor Lizier, from the Centre of Complex Systems and the School of Computer Science in the Faculty of Engineering, said: “Our results open new opportunities for designing network structures or interventions in networks.

“This could be super useful in stabilising electricity in power grids, vital for the transition to renewables, or to avoid neural synchronisation in the brain, which can trigger epilepsy.”

To understand how these systems work, the researchers studied what are known as “walks” through a network in a complex system. Walks are sequences of connected hops between individual elements or nodes in the network.

Associate Professor Lizier said: “Our maths examines pair­ed walks: where you start at one node and set off on two walks with randomly chosen hops between nodes for a specified number of steps. Those two walks might end up at the same node (convergent walks) or at different nodes (diverg­­­­ent walks).

“Our main finding is that the more commonly paired walks on a network are convergent, the worse the quality of synchronisation on that network structure would be.”

This is good news for the brain, where synchronisation is not desirable as it can cause epilepsy . The brain’s highly modular structure means it has a high proportion of convergent walks, which naturally push it away from epilepsy.

“We can even draw an analogy to social media with the echochamber phenomenon,” said co-author Jürgen Jost, whose group also works on social network dynamics.

“Here we see sub-groups reinforcing their own messages, via convergent walks within their own group, but not necessarily synchronising to the wider population.”

The findings represent a major step forward in the theory of how the structure of complex networks affects their dynamics or how they compute, such as how brain structure underpins cognition.


Funding: The research was supported by the Australian Research Council Discovery Early Career Researcher Award (DECRA) grant DE160100630, the The University of Sydney, Sydney Research Accelerator (SOAR) award, the Alexander von Humboldt Foundation and the NSF Grant Division of Mathematical Sciences (DMS)-0804454 Differential Equations in Geometry. The research used the University of Sydney’s high-performance computing cluster Artemis in generating the paper’s results for this paper. 

The researchers declare no competing interests. 

About this computational neuroscience research news

Author: Luisa Low
Source: University of Sydney
Contact: Luisa Low – University of Sydney
Image: The image is credited to Neuroscience News

Original Research: Closed access.
Analytic relationship of relative synchronizability to network structure and motifs” by Joseph Lizier et al. PNAS


Analytic relationship of relative synchronizability to network structure and motifs

Synchronization phenomena on networks have attracted much attention in studies of neural, social, economic, and biological systems, yet we still lack a systematic understanding of how relative synchronizability relates to underlying network structure. Indeed, this question is of central importance to the key theme of how dynamics on networks relate to their structure more generally.

We present an analytic technique to directly measure the relative synchronizability of noise-driven time-series processes on networks, in terms of the directed network structure. We consider both discrete-time autoregressive processes and continuous-time Ornstein–Uhlenbeck dynamics on networks, which can represent linearizations of nonlinear systems.

Our technique builds on computation of the network covariance matrix in the space orthogonal to the synchronized state, enabling it to be more general than previous work in not requiring either symmetric (undirected) or diagonalizable connectivity matrices and allowing arbitrary self-link weights.

More importantly, our approach quantifies the relative synchronization specifically in terms of the contribution of process motif (walk) structures.

We demonstrate that in general the relative abundance of process motifs with convergent directed walks (including feedback and feedforward loops) hinders synchronizability. We also reveal subtle differences between the motifs involved for discrete or continuous-time dynamics.

Our insights analytically explain several known general results regarding synchronizability of networks, including that small-world and regular networks are less synchronizable than random networks.

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