The Combined Effect of Connectivity and Dependency Links on Percolation of Networks
The Combined Effect of Connectivity and Dependency Links on Percolation of Networks
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DOI：
10.1007/s10955-011-0333-5
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发表时间：
2011-11-01
影响因子：
1.6
通讯作者：
Havlin, Shlomo
中科院分区：
文献类型：
--
作者：
Bashan, Amir;Havlin, Shlomo
渗流理论在统计物理和数学领域得到广泛研究，并在各个领域都有应用。然而，研究的重点是仅具有一种类型的链路（连接性链路）的系统。我们回顾了最近开发的数学框架，用于分析具有两种类型链接（连接性链接和依赖链接）的网络现实场景的渗透属性。这种形式主义被应用于研究也包括依赖链接的鄂尔多斯-仁义（ER）网络。对于由大小为 s 的依赖簇组成的平均度 (k) 的 ER 网络，属于巨型组件 P-无穷大的节点分数由下式给出： P-无穷大 = p(s-1)[1-exp(-(k) over bar )pP(无穷大)](s)，其中 1 - p 是随机删除节点的初始分数。在这里，我们将形式主义应用于随机规则（RR）网络的研究，并找到了渗流过程中巨型组件大小的公式：P-infinity=p(s-1)(1-r(k))(s)，其中r是r=p(s)(r(k-1)-1)(1-r(k))+1的解，k是节点的度。当 s = 1 时，这些一般结果分别与 ER 和 RR 网络中的已知渗流方程一致，没有依赖链接。与 s = 1 不同，渗流跃迁为二阶，而 s > 1 则为一阶。比较 ER 和 RR 网络的渗透行为，我们发现它们的弹性存在显着差异。我们通过分析和数值证明，在连接度较低或依赖集群较大的 ER 网络中，即使删除有限数量（零分数）的无限网络节点也会触发一系列故障，从而使整个网络支离破碎。具体来说，对于任何给定的 s，都存在一个临界度值 k(min)，使得 (k) 超过 bar 的 ER 网络
Percolation theory is extensively studied in statistical physics and mathematics with applications in diverse fields. However, the research is focused on systems with only one type of links, connectivity links. We review a recently developed mathematical framework for analyzing percolation properties of realistic scenarios of networks having links of two types, connectivity and dependency links. This formalism was applied to study Erdos-Renyi (ER) networks that include also dependency links. For an ER network with average degree (k) over bar that is composed of dependency clusters of size s, the fraction of nodes that belong to the giant component, P-infinity, is given by P-infinity = p(s-1)[1-exp(-(k) over bar )pP(infinity)](s) where 1 - p is the initial fraction of randomly removed nodes. Here, we apply the formalism to the study of random-regular (RR) networks and find a formula for the size of the giant component in the percolation process: P-infinity=p(s-1)(1-r(k))(s) where r is the solution of r=p(s)(r(k-1)-1)(1-r(k))+1, and k is the degree of the nodes. These general results coincide, for s = 1, with the known equations for percolation in ER and RR networks respectively without dependency links. In contrast to s = 1, where the percolation transition is second order, for s > 1 it is of first order. Comparing the percolation behavior of ER and RR networks we find a remarkable difference regarding their resilience. We show, analytically and numerically, that in ER networks with low connectivity degree or large dependency clusters, removal of even a finite number ( zero fraction) of the infinite network nodes will trigger a cascade of failures that fragments the whole network. Specifically, for any given s there exists a critical degree value, k(min), such that an ER network with (k) over bar