# Heterogeneously Coupled Maps: hub dynamics and emergence across connectivity layers

## TL;DR

This paper rigorously analyzes the dynamics of heterogeneously connected networks, revealing how hub nodes exhibit low-dimensional behavior and how different connectivity layers influence overall network dynamics.

## Contribution

It introduces a framework to understand high-dimensional heterogeneously coupled maps, showing hub dynamics can be approximated by low-dimensional systems and explaining layer-dependent behavior.

## Key findings

- Hub nodes' dynamics are well approximated by low-dimensional systems.
- Emergence of coherence among hubs within the same connectivity layer.
- Chaotic behavior observed in nodes with few connections.

## Abstract

The aim of this paper is to rigorously study dynamics of Heterogeneously Coupled Maps (HCM). Such systems are determined by a network with heterogeneous degrees. Some nodes, called hubs, are very well connected while most nodes interact with few others. The local dynamics on each node is chaotic, coupled with other nodes according to the network structure. Such high-dimensional systems are hard to understand in full, nevertheless we are able to describe the system over exponentially large time scales. In particular, we show that the dynamics of hub nodes can be very well approximated by a low-dimensional system. This allows us to establish the emergence of macroscopic behaviour such as coherence of dynamics among hubs of the same connectivity layer (i.e. with the same number of connections), and chaotic behaviour of the poorly connected nodes. The HCM we study provide a paradigm to explain why and how the dynamics of the network can change across layers.