# The nonlinear heat equation on dense graphs and graph limits

arXiv: https://arxiv.org/abs/1302.5804 · HTML: https://mlsanity.com/paper/1302.5804/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Feb 2013.

## Built on by (most cited first, 9 of 9)

- [DIFFormer: Scalable (Graph) Transformers Induced by Energy Constrained Diffusion](https://mlsanity.com/paper/2301.09474/) (2023)
- [Dissecting the Diffusion Process in Linear Graph Convolutional Networks](https://mlsanity.com/paper/2102.10739/) (2021)
- [Geometric Knowledge Distillation: Topology Compression for Graph Neural Networks](https://mlsanity.com/paper/2210.13014/) (2022)
- [Supercharging Graph Transformers with Advective Diffusion](https://mlsanity.com/paper/2310.06417/) (2023)
- [Learning Divergence Fields for Shift-Robust Graph Representations](https://mlsanity.com/paper/2406.04963/) (2024)
- [Transformers from Diffusion: A Unified Framework for Neural Message Passing](https://mlsanity.com/paper/2409.09111/) (2024)
- [On the Convergence and Size Transferability of Continuous-depth Graph Neural Networks](https://mlsanity.com/paper/2510.03923/) (2025)
- [DFGET: Displacement-Field Assisted Graph Energy Transmitter for Gland Instance Segmentation](https://mlsanity.com/paper/2312.07584/) (2023)
- [Graphon Particle Systems, Part II: Dynamics of Distributed Stochastic Continuum Optimization](https://mlsanity.com/paper/2407.02765/) (2024)

## Abstract (arXiv)

We use the combination of ideas and results from the theory of graph limits and nonlinear evolution equations to provide a rigorous mathematical justification for taking continuum limit for certain nonlocally coupled networks and to extend this method to cover many complex networks, for which it has not been applied before. Specifically, for dynamical networks on convergent sequences of simple and weighted graphs, we prove convergence of solutions of the initial-value problems for discrete models to those of the limiting continuous equations. In addition, for sequences of simple graphs converging to {0, 1}-valued graphons, it is shown that the convergence rate depends on the fractal dimension of the boundary of the support of the graph limit. These results are then used to study the regions of continuity of chimera states and the attractors of the nonlocal Kuramoto equation on certain multipartite graphs. Furthermore, the analytical tools developed in this work are used in the rigorous justification of the continuum limit for networks on random graphs that we undertake in a companion paper (Medvedev, 2013). As a by-product of the analysis of the continuum limit on deterministic and random graphs, we identify the link between this problem and the convergence analysis of several classical numerical schemes: the collocation, Galerkin, and Monte-Carlo methods. Therefore, our results can be used to characterize convergence of these approximate methods of solving initial-value problems for nonlinear evolution equations with nonlocal interactions.