随机块模型中的渗流
Percolation in the Stochastic Block Model
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中文总结 AI 辅助
本文研究社区结构随网络规模演化的随机块模型序列中的渗流，发现临界平均度通常不等于1，并提出了社区尺度分支过程方法以准确预测相变，扩展了渗流研究至更现实场景。
中文摘要 AI 辅助
随机块模型是研究具有社区结构网络的一个典型模型。然而，在该模型中的渗流研究主要局限于固定块结构的情形，尽管在真实网络中，社区结构可能随着网络的增长而演化。本文研究了随机块模型序列中的渗流，其中社区的数量和大小，以及社区内和社区间的连接概率都可能随网络规模而变化。我们使用两种方法分析了五个这样的序列：对于局部树状模型，使用线性化自洽方程；对于具有非消失聚类的模型，使用社区尺度的分支过程。我们发现临界平均度通常不等于1，即使在局部树状序列中也是如此，因为相变取决于连接如何在演化的社区结构中分布。我们还表明，当社区间连接足够稀疏时，社区尺度的分支过程能准确预测相变，即使在存在非消失聚类的情况下也是如此，而一个几何随机块模型序列则展示了当社区间连接的相关性不可忽略时该方法的局限性。这些结果将随机块模型中的渗流研究扩展到具有演化社区结构的更现实场景，并可能为推导几何长程渗流中渗流阈值的上下界提供新方法。
英文摘要
The stochastic block model is a paradigmatic model of networks with community structure. Yet percolation in the model has been studied primarily in cases with a fixed block structure, even though in real networks, the community structure may evolve as the network grows. Here we study percolation in sequences of stochastic block models in which the numbers and sizes of communities, as well as the intra- and intercommunity connection probabilities may all change with the network size. We analyze five such sequences using two methods: linearized self-consistent equations for the locally tree-like models and a branching process at the community scale for the models with nonvanishing clustering. We find that the critical average degree is not generally equal to $1$, even in locally tree-like sequences, because the transition depends on how connections are distributed across the evolving community structure. We also show that the community-scale branching process accurately predicts the transition when intercommunity connections are sufficiently sparse, even in the presence of nonvanishing clustering, while a geometric stochastic block model sequence demonstrates the limitations of this method when correlations between intercommunity connections cannot be neglected. These results extend percolation studies in the stochastic block model to more realistic scenarios with evolving community structure, and may provide new methods to derive the upper and lower bounds for the percolation threshold in geometric long-range percolation.
发表机构
- Sharon High School（沙伦高中）
- Network Science Institute, Northeastern University（东北大学网络科学研究所）
- Complexity Science Hub, Vienna（维也纳复杂性科学中心）
- Department of Physics, Northeastern University（东北大学物理系）
- Department of Mathematics, Northeastern University（东北大学数学系）
- Department of Electrical & Computer Engineering, Northeastern University（东北大学电气与计算机工程系）
机构由 AI 辅助整理，请以论文原文为准。
