# Multiscale Dynamics of an Adaptive Catalytic Network

## TL;DR

This paper investigates the multiscale behavior of the Jain-Krishna adaptive network model, analyzing the convergence of continuous dynamics and proposing multiscale methods for understanding its complex graph evolution.

## Contribution

It provides rigorous results on the convergence of the continuous-time dynamics and explores multiscale analysis approaches for the discrete graph updates in adaptive networks.

## Key findings

- Continuous dynamics converge to equilibrium points.
- Multiple time scales depend on system parameters.
- Singular limits facilitate analysis of the model.

## Abstract

We study the multiscale structure of the Jain-Krishna adaptive network model. This model describes the co-evolution of a set of continuous-time autocatalytic ordinary differential equations and its underlying discrete-time graph structure. The graph dynamics is governed by deletion of vertices with asymptotically weak concentrations of prevalence and then re-insertion of vertices with new random connections. In this work we prove several results about convergence of the continuous-time dynamics to equilibrium points. Furthermore, we motivate via formal asymptotic calculations several conjectures regarding the discrete-time graph updates. In summary, our results clearly show that there are several time scales in the problem depending upon system parameters, and that analysis can be carried out in certain singular limits. This shows that for the Jain-Krishna model, and potentially many other adaptive network models, a mixture of deterministic and/or stochastic multiscale methods is a good approach to work towards a rigorous mathematical analysis.