arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 0902.0888physics.soc-phcond-mat.dis-nn
收藏

社区之间的沟通和相互关系

Communication and correlation among communities

M. Ostilli, J. F. F. Mendes

摘要:

给定一个网络和一个划分的社区,我们考虑的问题“社区如何相互影响”和“当两个给定的社区进行通信”。具体来说,我们解决这些问题的背景下,小世界网络,其中一个任意淬火图和随机添加的远程连接。我们证明,在社区,叠加原理适用,并产生了一个自然的推广的有效场论已经提出的[物理评论E 78,031102](n=1),其中(n>1)包括在一种有效的TAP(无源,安德森和帕尔默)方程,其中每个社区发挥的作用,微观自旋。在有限温度或零温度下计算这些方程的相对渗透率,其中该方法提供了有效的渗流理论,给我们以上问题的答案。与n=1的情况不同,社区之间的不对称性可以通过方程的TAP类结构导致许多亚稳态,在社区之间的负捷径的情况下,亚稳态的数量可以随着n以指数方式快速增长。作为例子,我们考虑的n Viana-Bray社区模型和n一维小世界社区模型。尽管是最简单的,这些模型在网络理论中的相关性,例如在社交网络中,是至关重要的,并且直到现在还没有分析解决方案。本文还讨论了逾渗与网络分维之间的联系。最后,作为一个逆问题,我们展示了如何从相对的相似性,一个自然和有效的方法来检测一个通用网络的社区结构。有关主要结果的简短介绍,请参见arXiv:0812.0608。

英文摘要:

Given a network and a partition in communities, we consider the issues "how communities influence each other" and "when two given communities do communicate". Specifically, we address these questions in the context of small-world networks, where an arbitrary quenched graph is given and long range connections are randomly added. We prove that, among the communities, a superposition principle applies and gives rise to a natural generalization of the effective field theory already presented in [Phys. Rev. E 78, 031102] (n=1), which here (n>1) consists in a sort of effective TAP (Thouless, Anderson and Palmer) equations in which each community plays the role of a microscopic spin. The relative susceptibilities derived from these equations calculated at finite or zero temperature, where the method provides an effective percolation theory, give us the answers to the above issues. Unlike the case n=1, asymmetries among the communities may lead, via the TAP-like structure of the equations, to many metastable states whose number, in the case of negative short-cuts among the communities, may grow exponentially fast with n. As examples we consider the n Viana-Bray communities model and the n one-dimensional small-world communities model. Despite being the simplest ones, the relevance of these models in network theory, as e.g. in social networks, is crucial and no analytic solution were known until now. Connections between percolation and the fractal dimension of a network are also discussed. Finally, as an inverse problem, we show how, from the relative susceptibilities, a natural and efficient method to detect the community structure of a generic network arises. For a short presentation of the main result see arXiv:0812.0608.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

↑