Co-evolving dynamic networks

IF 1.5 1区 数学 Q2 STATISTICS & PROBABILITY
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引用次数: 0

Abstract

Co-evolving network models, wherein dynamics such as random walks on the network influence the evolution of the network structure, which in turn influences the dynamics, are of interest in a range of domains. While much of the literature in this area is currently supported by numerics, providing evidence for fascinating conjectures and phase transitions, proving rigorous results has been quite challenging. We propose a general class of co-evolving tree network models driven by local exploration, started from a single vertex called the root. New vertices attach to the current network via randomly sampling a vertex and then exploring the graph for a random number of steps in the direction of the root, connecting to the terminal vertex. Specific choices of the exploration step distribution lead to the well-studied affine preferential attachment and uniform attachment models, as well as less well understood dynamic network models with global attachment functionals such as PageRank scores (Chebolu and Melsted, in: SODA, 2008). We obtain local weak limits for such networks and use them to derive asymptotics for the limiting empirical degree and PageRank distribution. We also quantify asymptotics for the degree and PageRank of fixed vertices, including the root, and the height of the network. Two distinct regimes are seen to emerge, based on the expected exploration distance of incoming vertices, which we call the ‘fringe’ and ‘non-fringe’ regimes. These regimes are shown to exhibit different qualitative and quantitative properties. In particular, networks in the non-fringe regime undergo ‘condensation’ where the root degree grows at the same rate as the network size. Networks in the fringe regime do not exhibit condensation. Further, non-trivial phase transition phenomena are shown to arise for: (a) height asymptotics in the non-fringe regime, driven by the subtle competition between the condensation at the root and network growth; (b) PageRank distribution in the fringe regime, connecting to the well known power-law hypothesis. In the process, we develop a general set of techniques involving local limits, infinite-dimensional urn models, related multitype branching processes and corresponding Perron–Frobenius theory, branching random walks, and in particular relating tail exponents of various functionals to the scaling exponents of quasi-stationary distributions of associated random walks. These techniques are expected to shed light on a variety of other co-evolving network models.

共同演化的动态网络
摘要 协同演化网络模型,即网络上的动态(如随机行走)影响网络结构的演化,而网络结构又反过来影响动态。虽然该领域的大部分文献目前都有数值支持,为引人入胜的猜想和相变提供了证据,但证明严格的结果却相当具有挑战性。我们提出了一类由局部探索驱动的共同演化树状网络模型,该模型从称为根的单个顶点开始。通过随机抽样一个顶点,然后沿着根的方向探索图的随机步数,连接到终端顶点,从而将新顶点连接到当前网络。探索步数分布的特定选择导致了研究较多的仿射优先附着和均匀附着模型,以及较少被人了解的具有全局附着函数(如 PageRank 分数)的动态网络模型(Chebolu 和 Melsted,载于:SODA,2008 年)。我们获得了此类网络的局部弱极限,并利用它们推导出了极限经验度和 PageRank 分布的渐近线。我们还量化了固定顶点(包括根顶点)的度和 PageRank 以及网络高度的渐近线。根据进入顶点的预期探索距离,我们发现出现了两种截然不同的状态,我们称之为 "边缘 "和 "非边缘 "状态。这两种状态表现出不同的定性和定量特性。特别是,非边缘系统中的网络会发生 "浓缩",根度的增长速度与网络规模的增长速度相同。边缘系统中的网络则不会出现凝结现象。此外,非三维相变现象还表现在以下方面(a) 非边缘系统中的高度渐近线,由根部凝聚与网络增长之间的微妙竞争驱动;(b) 边缘系统中的 PageRank 分布,与众所周知的幂律假说相关联。在此过程中,我们开发了一套通用技术,涉及局部极限、无限维瓮模型、相关的多类型分支过程和相应的佩伦-弗罗贝尼斯理论、分支随机游走,特别是将各种函数的尾部指数与相关随机游走的准静态分布的缩放指数联系起来。这些技术有望为其他各种共同演化的网络模型提供启示。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Probability Theory and Related Fields
Probability Theory and Related Fields 数学-统计学与概率论
CiteScore
3.70
自引率
5.00%
发文量
71
审稿时长
6-12 weeks
期刊介绍: Probability Theory and Related Fields publishes research papers in modern probability theory and its various fields of application. Thus, subjects of interest include: mathematical statistical physics, mathematical statistics, mathematical biology, theoretical computer science, and applications of probability theory to other areas of mathematics such as combinatorics, analysis, ergodic theory and geometry. Survey papers on emerging areas of importance may be considered for publication. The main languages of publication are English, French and German.
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