The semilinear heat equation on sparse random graphs

Abstract

Using the theory of $L^p$-graphons (Borgs et al, 2014), we derive and rigorously justify the continuum limit for systems of differential equations on sparse random graphs. Specifically, we show that the solutions of the initial value problems for the discrete models can be approximated by those of an appropriate nonlocal diffusion equation. Our results apply to a range of spatially extended dynamical models of different physical, biological, social, and economic networks. Importantly, our assumptions cover network topologies featured in many important real-world networks. In particular, we derive the continuum limit for coupled dynamical systems on power law graphs. The latter is the main motivation for this work.

References in corpus (3)

• An $L^p$ theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions
• The nonlinear heat equation on W-random graphs
• The Kuramoto model on power law graphs

Cited by in corpus (8)

• Graphon Mean Field Games and the GMFG Equations
• Bifurcations in the Kuramoto model on graphs
• Continuum limits of coupled oscillator networks depending on multiple sparse graphs
• Pattern Formation in Random Networks Using Graphons
• Uncertainty Quantification for Fisher-Kolmogorov Equation on Graphs with Application to Patient-Specific Alzheimer Disease
• Continuum limit for interacting systems on adaptive networks
• The Kuramoto model on power law graphs
• Continuum limit of $p$-Laplacian evolution problems on graphs:$L^q$ graphons and sparse graphs