The semilinear heat equation on sparse random graphs

Using the theory of $L^p$-graphons [C. Borgs et al., preprint, arXiv:1401.2906, 2014; C. Borgs et al., preprint, arXiv:1408.0744, 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.