AI 中文总结
研究非齐次随机图，通过基于贪婪算法的重整化变换，分析其迭代情况，表明从适当缩放的连接函数开始，迭代重整化图收敛到双参数随机图族，还探讨了轻尾和重尾情形下的不同缩放及占优情况。
AI 中文摘要
我们考虑非齐次随机图，其中顶点被赋予独立同分布的随机权重，不同顶点对以顶点权重的二元函数为概率独立地由一条边连接，单个顶点以顶点权重的一元函数为概率独立地由一个自环连接。我们应用一种重整化变换，根据贪婪算法将顶点聚合成大小相等的组。我们分析了重整化变换迭代时的情况。特别地，我们表明从适当缩放的连接函数开始，迭代重整化图收敛到一个双参数随机图族，在一个普适类中充当吸引子。我们考虑了轻尾情形，其缩放极限是齐次的厄尔多斯 - 雷尼随机图，以及重尾情形，其缩放极限是具有稳定无限均值随机权重和指数断开函数的非齐次随机图。两种情形需要不同的缩放。哪种情形占优取决于连接函数的选择和随机权重的分布律选择。
英文摘要
We consider inhomogeneous random graphs in which vertices are assigned i.i.d.\ random weights, pairs of distinct vertices are connected by an edge independently with a probability that is a bi-variate function of the weights of the vertices, and single vertices are connected to themselves by a self-loop independently with a probability that is a uni-variate function of the weight of the vertex. We apply a renormalisation transformation in which vertices are aggregated into groups of equal size according to a greedy algorithm, namely, distinct groups of aggregated vertices are connected by an aggregated edge if and only if there is at least one edge connecting two constituent vertices across the groups, while a group of aggregated vertices is connected to itself by an aggregated self-loop if and only if there is at least one self-loop at an internal vertex or one edge connecting a pair of distinct internal vertices. We analyse what happens when the renormalisation transformation is iterated. In particular, we show that, starting from appropriately scaled connection functions, the iterated renormalised graphs converge to a two-parameter family of random graphs, acting as an attractor in a universality class. We consider a light-tailed regime, for which the scaling limit is a homogeneous Erdős--Rényi random graph, and a heavy-tailed regime, for which the scaling limit is an inhomogeneous random graph with stable infinite-mean random weights and an exponential disconnection function. Different scalings are needed for the two regimes. Which of the two regimes prevails depends on the choice of the connection functions and the choice of the law of the random weights.
Comments31 pages
