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Cyclization in bipartite random graphs
Physical Review E ( IF 2.4 ) Pub Date : 2020-03-16 , DOI: 10.1103/physreve.101.032306
A. A. Lushnikov

In this paper the time evolution of a finite bipartite graph initially comprising two sorts of isolated vertices is considered. The graph is assumed to evolve by adding edges, one at a time. Each new edge connects either two linked components and forms a new component of a larger order (coalescence of graphs) or increases (by one) the number of edges in a given linked component (cycling). Any state of the graph is thus characterized by the set of occupation numbers (the numbers of linked components comprising a given numbers of vertexes of the both sorts and a given number of edges. Once the rate of appearance of an extra edge in the graph being known, the master equation governing the time evolution of the probability to find the random graph in a given state is reformulated in terms of the functional generating the probability to find the evolving graph in a given state. The exact solution of the evolution equation for the generating functional applies for analyzing the average population numbers of linked components. In the limit of large order of the graph the distribution factorizes into two multipliers, one of which is just the spectrum of linked components in the infinite bipartite graph, The second multiplier includes the dependence on the total size of the graph. Both these multipliers contain information on the emergence of the giant component that forms at a critical time.

中文翻译:

二分随机图的环化

在本文中,考虑了最初包含两种孤立顶点的有限二部图的时间演化。假设该图通过一次添加一条边来演化。每个新边都连接两个链接的组件,并形成一个更大顺序的新组件(图的聚结),或者在给定的链接组件中增加(增加一个)边的数量(循环)。因此,图形的任何状态都由一组占用编号(链接的组件的数量,包括给定数量的两种顶点和一定数量的边)来确定。已知 重新构造了控制在给定状态下找到随机图的概率随时间变化的主方程,并根据函数生成了在给定状态下找到演化图的概率。用于生成函数的演化方程的精确解适用于分析链接组件的平均总体数。在图的高阶限制中,分布分解为两个乘数,其中一个只是无限二分图中的链接成分的频谱。第二个乘数包括对图总大小的依赖性。这两个乘数都包含有关在关键时刻形成的巨型成分的出现的信息。生成函数的演化方程的精确解适用于分析链接组件的平均总体数。在图的高阶限制中,分布分解为两个乘数,其中一个只是无限二分图中的链接成分的频谱。第二个乘数包括对图总大小的依赖性。这两个乘数都包含有关在关键时刻形成的巨大成分出现的信息。生成函数的演化方程的精确解适用于分析链接组件的平均总体数。在图的高阶限制中,分布分解为两个乘数,其中一个只是无限二分图中的链接成分的频谱。第二个乘数包括对图总大小的依赖性。这两个乘数都包含有关在关键时刻形成的巨大成分出现的信息。
更新日期:2020-03-19
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