# The theory of Turing patterns on time varying networks

## TL;DR

This paper investigates how Turing-like pattern formation occurs on time-varying networks, revealing a novel instability mechanism influenced by network dynamics and providing analytical predictions for pattern onset.

## Contribution

It introduces a new analytical framework for understanding Turing pattern formation on dynamic networks, extending classical models to time-dependent structures.

## Key findings

- Network dynamics can induce Turing-like instabilities.
- Analytical predictions match numerical simulations.
- Pattern onset depends on network evolution frequency.

## Abstract

The process of pattern formation for a multi-species model anchored on a time varying network is studied. A non homogeneous perturbation superposed to an homogeneous stable fixed point can amplify, as follows a novel mechanism of instability, reminiscent of the Turing type, instigated by the network dynamics. By properly tuning the frequency of the imposed network evolution, one can make the examined system behave as its averaged counterpart, over a finite time window. This is the key observation to derive a closed analytical prediction for the onset of the instability in the time dependent framework. Continuously and piecewise constant periodic time varying networks will be analysed, to set the ground for the proposed approach. The extension to non periodic settings will also be discussed.