The Semilinear Heat Equation on Sparse Random Graphs
The Semilinear Heat Equation on Sparse Random Graphs
复制标题
稀疏随机图上的半线性热方程
DOI：
10.1137/16m1075831
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
G. Medvedev
中科院分区：
文献类型：
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作者：
Dmitry S. Kaliuzhnyi;G. Medvedev
利用L^p-图子理论[C. Borgs等人，预印本，arXiv：1401.2906，2014; C. Borgs等人，预印本，arXiv：1408.0744，2014]，我们推导并严格证明了稀疏随机图上微分方程系统的连续极限。具体来说，我们表明，离散模型的初值问题的解决方案可以近似的适当的非局部扩散方程。我们的研究结果适用于一系列不同的物理，生物，社会和经济网络的空间扩展的动力学模型。重要的是，我们的假设涵盖了许多重要的现实网络中的网络拓扑结构。特别地，我们推导出了幂律图上耦合动力系统的连续极限。后者是这项工作的主要动机。
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.