论文标题
具有标记的稀疏随机图的大偏差，并应用于相互作用扩散
Large deviations for marked sparse random graphs with applications to interacting diffusions
论文作者
论文摘要
我们考虑了明显的稀疏Erdös-rényi随机图的经验邻域分布，该图通过装饰带有i.i.d. \ i.i.d. \随机元素的稀疏Erdös-rényi随机图获得的边缘和顶点获得。我们证明，该模型的经验邻域分布在局部弱收敛的框架中满足了较大的偏差原理。我们依靠Delgosha和Anantharam〜（2019）引入的BC-凝集的概念，该概念是对Bordenave和Caputo〜（2015）先前作品的启发。我们的主要技术贡献是一个近似结果，它允许一个人从具有离散空间的标记的图形传递到一般波兰空间中的标记。作为此处开发的结果的应用，我们证明了由梯度演化驱动并在稀疏Erdös-rényi随机图顶部定义的相互作用的大偏差原理。特别是，我们的结果适用于随机库拉莫托模型。我们获得具有给定数量边缘数量的稀疏均匀随机图的类似结果。
We consider the empirical neighborhood distribution of marked sparse Erdös-Rényi random graphs, obtained by decorating edges and vertices of a sparse Erdös-Rényi random graph with i.i.d.\ random elements taking values on Polish spaces. We prove that the empirical neighborhood distribution of this model satisfies a large deviation principle in the framework of local weak convergence. We rely on the concept of BC-entropy introduced by Delgosha and Anantharam~(2019) which is inspired on the previous work by Bordenave and Caputo~(2015). Our main technical contribution is an approximation result that allows one to pass from graph with marks in discrete spaces to marks in general Polish spaces. As an application of the results developed here, we prove a large deviation principle for interacting diffusions driven by gradient evolution and defined on top of sparse Erdös-Rényi random graphs. In particular, our results apply for the stochastic Kuramoto model. We obtain analogous results for the sparse uniform random graph with given number of edges.
