The nonlinear heat equation on W-random graphs

Abstract

For systems of coupled differential equations on a sequence of W-random graphs, we derive the continuum limit in the form of an evolution integral equation. We prove that solutions of the initial value problems (IVPs) for the discrete model converge to the solution of the IVP for its continuum limit. These results combined with the analysis of nonlocally coupled deterministic networks in [9] justify the continuum (thermodynamic) limit for a large class of coupled dynamical systems on convergent families of graphs.

References in corpus (4)

• Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators
• Coexistence of Coherence and Incoherence in Nonlocally Coupled Phase Oscillators
• Chimera states in heterogeneous networks
• Small-world networks of Kuramoto oscillators

Cited by in corpus (22)

• Graphon Mean Field Games and the GMFG Equations
• Small-world networks of Kuramoto oscillators
• The semilinear heat equation on sparse random graphs
• Bifurcations in the Kuramoto model on graphs
• Stability of twisted states in the Kuramoto model on Cayley and random graphs
• Linear Quadratic Graphon Field Games
• Consensus Formation in First-Order Graphon Models with Time-Varying Topologies
• Stability of twisted states in the continuum Kuramoto model
• Continuum limits of coupled oscillator networks depending on multiple sparse graphs
• Pattern Formation in Random Networks Using Graphons
• Graphop Mean-Field Limits and Synchronization for the Stochastic Kuramoto Model
• The mean field analysis for the Kuramoto model on graphs I. The mean field equation and transition point formulas
• Consensus and Voting on Large Graphs: An Application of Graph Limit Theory
• The mean field analysis of the Kuramoto model on graphs II. Asymptotic stability of the incoherent state, center manifold reduction, and bifurcations
• LQG Graphon Mean Field Games: Analysis via Graphon Invariant Subspaces
• The Kuramoto model on power law graphs
• Mathematical Analysis of Nonlocal PDEs for Network Generation
• Limits and consistency of non-local and graph approximations to the Eikonal equation
• Dynamics and control for multi-agent networked systems: a finite difference approach
• Capturing the critical coupling of large random Kuramoto networks with graphons
• Persistence of steady-states for dynamical systems on large networks
• Pacemakers in large arrays of oscillators with nonlocal coupling