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Mean field limits of co-evolutionary signed heterogeneous networks

Marios Antonios Gkogkas, Christian Kuehn, Chuang Xu

TL;DR

This work establishes a rigorous mean field limit for co-evolutionary signed heterogeneous networks built from Kuramoto-type oscillators with time-evolving edge weights. By decoupling the network dynamics from the oscillator dynamics and encoding the graph evolution as signed digraph measures, the authors reduce the problem to a one-dimensional integral equation on the circle and formulate a generalized Vlasov equation (VE) to describe the MFL. They prove well-posedness of the characteristic flow and VE, and show that empirical measures of finite networks converge to the VE solution; they also provide a discretization scheme (lattice model) that approximates the VE, with convergence guarantees. An illustration on a binary-tree network demonstrates the applicability to sparse, structured graphs. The results advance the understanding of adaptive networks with both positive and negative interactions and lay groundwork for further analysis of absolute continuity and nonlocal adaptations in co-evolutionary systems.

Abstract

Many science phenomena are modelled as interacting particle systems (IPS) coupled on static networks. In reality, network connections are far more dynamic. Connections among individuals receive feedback from nearby individuals and make changes to better adapt to the world. Hence, it is reasonable to model myriad real-world phenomena as co-evolutionary (or adaptive) networks. These networks are used in different areas including telecommunication, neuroscience, computer science, biochemistry, social science, as well as physics, where Kuramoto-type networks have been widely used to model interaction among a set of oscillators. In this paper, we propose a rigorous formulation for limits of a sequence of co-evolutionary Kuramoto oscillators coupled on heterogeneous co-evolutionary networks, which receive both positive and negative feedback from the dynamics of the oscillators on the networks. We show under mild conditions, the mean field limit (MFL) of the co-evolutionary network exists and the sequence of co-evolutionary Kuramoto networks converges to this MFL. Such MFL is described by solutions of a generalized Vlasov equation. We treat the graph limits as signed graph measures, motivated by the recent work in [Kuehn, Xu. Vlasov equations on digraph measures, JDE, 339 (2022), 261--349]. In comparison to the recently emerging works on MFLs of IPS coupled on non-co-evolutionary networks (i.e., static networks or time-dependent networks independent of the dynamics of the IPS), our work seems the first to rigorously address the MFL of a co-evolutionary network model.

Mean field limits of co-evolutionary signed heterogeneous networks

TL;DR

This work establishes a rigorous mean field limit for co-evolutionary signed heterogeneous networks built from Kuramoto-type oscillators with time-evolving edge weights. By decoupling the network dynamics from the oscillator dynamics and encoding the graph evolution as signed digraph measures, the authors reduce the problem to a one-dimensional integral equation on the circle and formulate a generalized Vlasov equation (VE) to describe the MFL. They prove well-posedness of the characteristic flow and VE, and show that empirical measures of finite networks converge to the VE solution; they also provide a discretization scheme (lattice model) that approximates the VE, with convergence guarantees. An illustration on a binary-tree network demonstrates the applicability to sparse, structured graphs. The results advance the understanding of adaptive networks with both positive and negative interactions and lay groundwork for further analysis of absolute continuity and nonlocal adaptations in co-evolutionary systems.

Abstract

Many science phenomena are modelled as interacting particle systems (IPS) coupled on static networks. In reality, network connections are far more dynamic. Connections among individuals receive feedback from nearby individuals and make changes to better adapt to the world. Hence, it is reasonable to model myriad real-world phenomena as co-evolutionary (or adaptive) networks. These networks are used in different areas including telecommunication, neuroscience, computer science, biochemistry, social science, as well as physics, where Kuramoto-type networks have been widely used to model interaction among a set of oscillators. In this paper, we propose a rigorous formulation for limits of a sequence of co-evolutionary Kuramoto oscillators coupled on heterogeneous co-evolutionary networks, which receive both positive and negative feedback from the dynamics of the oscillators on the networks. We show under mild conditions, the mean field limit (MFL) of the co-evolutionary network exists and the sequence of co-evolutionary Kuramoto networks converges to this MFL. Such MFL is described by solutions of a generalized Vlasov equation. We treat the graph limits as signed graph measures, motivated by the recent work in [Kuehn, Xu. Vlasov equations on digraph measures, JDE, 339 (2022), 261--349]. In comparison to the recently emerging works on MFLs of IPS coupled on non-co-evolutionary networks (i.e., static networks or time-dependent networks independent of the dynamics of the IPS), our work seems the first to rigorously address the MFL of a co-evolutionary network model.
Paper Structure (27 sections, 20 theorems, 120 equations, 2 figures)

This paper contains 27 sections, 20 theorems, 120 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Macroscopic limit of interacting particle systems
  3. Highlights of this paper
  4. Main results and sketch of the proof
  5. Organization of this paper
  6. Preliminaries
  7. Generalized co-evolutionary Kuramoto network
  8. Generalized Vlasov Equation
  9. Approximation of the mean field limit
  10. An example---A model on binary tree networks
  11. Proofs of main results
  12. Proof of Theorem \ref{['thm-characteristic-1']}
  13. Proof of Theorem \ref{['thm-existence-VE']}
  14. Proof of Theorem \ref{['th-approx']}
  15. Discussions and outlook
  16. ...and 12 more sections

Key Result

Theorem A

Under certain conditions, there exists a unique mean field limit of the Kuramoto-type model coevolution-a-coevolution-b, provided the signed graph sequence $\{W_{i,j}(0)\}_{N\in\mathbb{N}}$ as well as the sequence of initial empirical measures $\{\frac{1}{N}\sum_{i=1}^N\delta_{\phi_i(0)}\}_{N\in\mat

Figures (2)

  • Figure 1: Schematic diagram of the approach of deriving MFL.
  • Figure 2: Oscillators coupled on binary trees.

Theorems & Definitions (50)

  • Theorem A
  • Definition A
  • Remark B
  • Proposition C
  • Definition D
  • Definition E
  • Definition F
  • Definition G
  • Definition H
  • Definition I
  • ...and 40 more