The Nonlinear Heat Equation on Dense Graphs and Graph Limits
The Nonlinear Heat Equation on Dense Graphs and Graph Limits
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稠密图和图极限上的非线性热方程
DOI：
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发表时间：
2013
影响因子：
2
通讯作者：
G. Medvedev
中科院分区：
文献类型：
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作者：
G. Medvedev
我们使用图极限理论和非线性演化方程的思想和结果的结合，提供了一个严格的数学理由采取连续极限的某些非局部耦合网络，并将这种方法扩展到许多复杂的网络，它还没有被应用之前。具体来说，对于简单的和加权图的收敛序列的动态网络，我们证明了离散模型的初值问题的解的收敛性的极限连续方程。此外，对于收敛到{0，1}-值图子的简单图序列，证明了收敛速度依赖于图极限支撑边界的分形维数.这些结果被用来研究嵌合态的连续区域和非局部Kuramoto方程在某些多部图上的吸引子。此外，在这项工作中开发的分析工具被用于严格证明随机图上网络的连续极限，我们在一篇配套论文中进行了证明（Medvedev，2013）。
作为确定性和随机图的连续极限分析的副产品，我们确定了这个问题和几个经典的数值方案的收敛性分析之间的联系：配置，Galerkin，和蒙特-卡罗方法。因此，我们的结果可以用来描述这些近似方法的收敛性与非局部相互作用的非线性发展方程的初值问题。
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.