空间模块网络中的两种变迁
Two transitions in spatial modular networks
摘要：
了解交通网络等基础设施的弹性对我们的日常生活具有重要意义。近年来，一种同质空间网络模型被发展用于研究具有特征链接长度的空间嵌入网络，如电网和大脑。然而，尽管许多现实世界的网络是空间嵌入式的，并且它们的链路具有长度特征，例如管道、电力线或地面运输线，但它们不是同质的，而是异质的。例如，城市内部的联系密度明显高于城市之间。在这里，我们提出并研究了一个类似的现实异构空间模块化模型，利用渗流过程，以更好地了解这种网络的异质性的影响，数值和分析。该模型假设在一个城市内部有许多连接不同位置的线，而城市之间的长线是稀疏的，通常在二维平面上只直接连接几个最近的相邻城市。我们发现，这个模型经历了两个不同的连续过渡，一个是当城市彼此断开，第二个是当每个城市分裂。虽然在二维网格中的临界渗流阈值仍然是一个悬而未决的问题，我们解析地找到了该模型中的临界渗流阈值。此外，虽然均匀模型经历了一个单一的过渡，具有独特的现象称为\textit{critical stretching}，其中从随机到空间结构的几何交叉在不同的尺度发现非线性拉伸的特征长度在临界状态。在这里，我们表明，异质模型没有经历这样的现象，表明临界拉伸强烈依赖于网络结构。
英文摘要：
Understanding the resilience of infrastructures such as transportation network has significant importance for our daily life. Recently, a homogeneous spatial network model was developed for studying spatial embedded networks with characteristic link length such as power-grids and the brain. However, although many real-world networks are spatially embedded and their links have characteristics length such as pipelines, power lines or ground transportation lines they are not homogeneous but rather heterogeneous. For example, density of links within cities are significantly higher than between cities. Here we present and study numerically and analytically a similar realistic heterogeneous spatial modular model using percolation process to better understand the effect of heterogeneity on such networks. The model assumes that inside a city there are many lines connecting different locations, while long lines between the cities are sparse and usually directly connecting only a few nearest neighbours cities in a two dimensional plane. We find that this model experiences two distinct continues transitions, one when the cities disconnect from each other and the second when each city breaks apart. Although the critical threshold for site percolation in 2D grid remains an open question we analytically find the critical threshold for site percolation in this model. In addition, while the homogeneous model experience a single transition having a unique phenomenon called \textit{critical stretching} where a geometric crossover from random to spatial structure in different scales found to stretch non-linearly with the characteristic length at criticality. Here we show that the heterogeneous model does not experience such a phenomenon indicating that critical stretching strongly depends on the network structure.
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