随机图的谱界在扩散现象和渗流中的应用
Spectral Bounds in Random Graphs Applied to Spreading Phenomena and Percolation
摘要：
在本文中，我们推导出随机图的影响，取决于一个特定的矩阵，称为风险矩阵的谱半径的非渐近理论界。我们还表明，这些结果是通用的，有效的一大类随机图显示相关性在当地的规模，称为LPC随机图。特别是，它们导致在渗流，流行病学和信息级联紧密和新颖的界限。本文的主要结果指出，在次临界状态下的LPC随机图的影响是在最多的顺序为$O(\sqrt{n})$ $n$是网络的大小，和$O(n^{2/3})$在临界状态下，其中的流行病阈值是由危害矩阵的谱半径的大小相对于1。作为一个推论，它也表明，这样的界限举行的巨大的组件的大小在非均匀渗流，SIR模型在流行病学，以及为一个节点的长期影响的独立级联模型。
英文摘要：
In this paper, we derive nonasymptotic theoretical bounds for the influence in random graphs that depend on the spectral radius of a particular matrix, called the Hazard matrix. We also show that these results are generic and valid for a large class of random graphs displaying correlation at a local scale, called the LPC random graphs. In particular, they lead to tight and novel bounds in percolation, epidemiology and information cascades. The main result of the paper states that the influence in the sub-critical regime for LPC random graphs is at most of the order of $O(\sqrt{n})$ where $n$ is the size of the network, and of $O(n^{2/3})$ in the critical regime, where the epidemic thresholds are driven by the size of the spectral radius of the Hazard matrix with respect to 1. As a corollary, it is also shown that such bounds hold for the size of the giant component in inhomogeneous percolation, the SIR model in epidemiology, as well as for the long-term influence of a node in the Independent Cascade Model.
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