  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.
