遭受攻击的耦合相依网络的鲁棒性研究
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摘要
作为复杂性理论的重要分支,复杂网络理论受到来自不同研究领域学者们的广泛关注。随着信息技术的飞速发展,实证数据越来越丰富,基于大数据的定量化分析,通过理论研究、算法设计、应用实施、平台架构等多种研究方法,使得复杂网络的研究蓬勃发展。以前的研究兴趣主要集中在单个网络理论和方法的创新,而现实世界里存在的大多是耦合相依的网络。因此,对于耦合相依网络的研究具有重要的理论和实际意义。根据国内外研究进展,本文把耦合相依网络主要分为四类：I)网络之间只存在依赖边；II)网络之间存在支持-依赖的有向连边；III)网络之间只存在连接边；IV)网络之间存在连接边和依赖边。本论文分别对上述四类耦合相依网络遭受随机或蓄意攻击时,分析网络内部及网络之间的失效机制,从理论和数值模拟两方面对网络的鲁棒性进行研究。
1、随机攻击下的耦合相依网络
(1)对于具有多重支持-依赖关系的两个聚集网络在随机攻击下的鲁棒性问题,研究发现：强耦合强度下,聚集系数对网络的鲁棒性具有显著影响,弱耦合强度下,特别是在同时攻击两个网络时,聚集系数对网络的鲁棒性影响不大。对于具有多重支持-依赖关系的多个耦合网络,当遭受随机攻击时,通过定义网络中的有效节点,搭建失效机制下的模型框架,对网络的鲁棒性展开研究,研究发现：通过增加各网络中自治节点的比例,增加网络之间的供应强度,可以有效地提高网络的抗毁性,增强网络的鲁棒性。
(2)对于网络之间具有两种连边方式的两个耦合相依网络,其中相依边为反馈边,在遭受随机攻击时,通过运用渗流理论,对网络的鲁棒性进行分析。研究发现：随着耦合强度的增加,系统由二阶相变经过混合相变转变为一阶相变；随着网络之间平均度的增加,一阶相变区域逐渐变大,混合相变区域逐渐变小最终消失；随着网络内部平均度的增加,一阶相变区域变小,二阶相变区域变大,混合相变区域不变。此外,我们发展了两类数学框架来研究网络之间存在两种连边方式的多个耦合相依网络的鲁棒性问题,根据相依边为反馈边和无反馈边。以多个Erdos-Renyi (ER)网络组成的星状结构网络为例,研究发现：对于同一个耦合强度,中心网络的鲁棒性随着网络个数的增加而减弱,随着网络内部平均度或网络之间连接边平均度的增加而增强。特别地,当耦合强度取1时,得到了一阶相变点的解析表达式。对比于无反馈边,在参数相同的情况时当相依边为反馈边时,网络变得极其脆弱。我们的理论不仅适用于随机网络,而且适用于其它任何拓扑结构的网络系统。
2、蓄意攻击下的耦合相依网络
(1)对于两个部分相依网络的蓄意攻击问题,本文分别从数值模拟和理论两方面对网络的鲁棒性进行了研究。引入了含有参数α的新的攻击概率函数,当α=0,1时,得到了系统由一阶相变向二阶相变过渡的临界值qc和pc的数值解。对于多个部分相依网络的蓄意攻击问题的研究发现：当度值高的节点有较高概率失效时,网络变得更脆弱。对于由ER网络构成的完全相依树状结构网络,推演出了最大连通集团、临界值pc的数值解；对于由ER网络构成的部分相依星状结构网络,在相同的耦合强度下,随着网络个数的增加,中心网络变得更为脆弱。
(2)对于网络之间具有两种连边方式的两个耦合相依网络,其中相依边为反馈边和无反馈边,在遭受蓄意攻击时,对网络的鲁棒性进行了研究。通过数值模拟研究发现：当网络中的高度值节点具有较高概率失效时,网络的鲁棒性变弱；当网络中的低度值节点具有较高概率失效时,网络的鲁棒性增强。此外,还研究了蓄意攻击下具有多重支持-依赖关系的多个耦合相依网络的鲁棒性问题,发现：除了增加各网络中自治性节点比例、增加网络之间的供应强度之外,还可以对网络中高度值节点进行重点保护,来有效地提高网络的抗毁性,增强网络的鲁棒性。
As an important branch of complexity theory, complex networks theory attracts more and more scholars from various research. With rapid growth of information technologies, empirical data are getting increasingly rich. Based on the quantitative analysis of big data, theoretical research, algorithm design, practical application, and platform framework etc., the study of complex networks has flourished over the past few years. Previous studies mainly concentrated on the innovation of theory and method of isolated network. However, most real networks are coupled as interdepen-dent networks. Therefore, the study has some theoretical and practical significance. In this thesis, according to the coupling characteristics in real networks, we classify the coupling dependency networks into four categories:I) there only exist interdependendy links, II) there exist support-dependence links between networks, III) there exist only interconnectivity links, IV) there exist interdependent and interconnect links. When the system are under random attack or targeted attack, we study the robustness of networks with theoretical and simulating analysis, by considering the nodes failure mechanisms of internal network and between two networks.
1Random attack on coupled interdependent networks
(1) For the robustness of partially coupled clustered networks with multiple support-dependence relations under random attack, we find that, for strong coupling strength, clustering coefficient has a significant impact on the robustness of the system, but for weak coupling strength, it has little effect especially when attacking the both two networks. By defining the functional nodes in the network of networks (NON) and developing the analytical framework, we study the robustness of NON with multiple support-dependence relations under random attack. The system will be more robust by increasing the fraction of autonomous nodes, and the density of support links.
(2) We study the robustness of two interdependent and interconnected networks with feedback condition under random attack based on the percolation theory. We find that increasing the coupling strength leads to a change from a second to first through hybrid order percolation transition. As the average interconnectivity links degree increases, the first order transition region grows large, the hybrid order transition region becomes small and ultimately disappears. And as the average intralinks degree increases, the first order transition region becomes small, the second order transition region becomes large, and the hybrid order transition region remains unchanged. Additionally, we study the robustness of interdpendent and interconnected network of networks (NON) with feedback condition or no feedback condition under random attack. For a starlike network of Erdos-Renyi (ER) networks, we find that the central network becomes more vulnerable as the number of networks increase, and becomes more robust as the average inter-or intra-connectivity links degree increases. Especially when the coupling strength is equal to one, we get the analytical expression of the first order critical point. Comparing to no feedback case,the feedback condition makes the system more vulnerable. Our theory is not only applied to random network, but also applied to any network systems topology.
2Targeted attack on coupled interdependent networks
(1) We study the robustness of two partially interdependent networks under targeted attack analytically and numerically. A new targeted probability function with parameter a is introduced. When a=0,1, we obtain the analytical solutions of critical points from first to second order phase transition. The robustness of partially interdependent network of networks (NON) is also studied. When highly connected nodes have higher probability to fail, the system becomes more vulnerable. For a fully interdependent treelike network of ER networks, we get the analytical solutions of the giant component and the critical threshold pc. For a part-ially interdependent starlike network of ER networks, as the number of networks increases, the central network becomes more vulnerable.
(2) We study the robustness of two interdependent and inter conne-cted networks with feedback condition or no feedback condition under targeted attack. We find from numerical simulation that, when highly con-nected nodes have high probability to fail,the system becomes more vulnerable. But when low connected nodes have high probability to fail, the system becomes more robust. Furthermore, we study the robustness of network of networks (NON) with multiple support-dependence relations under targeted attack. Besides increasing the fraction of autonomous nodes, and the density of support links, we can still improve the robustness of networks by protecting the nodes with higher degree.
1、随机攻击下的耦合相依网络
(1)对于具有多重支持-依赖关系的两个聚集网络在随机攻击下的鲁棒性问题,研究发现：强耦合强度下,聚集系数对网络的鲁棒性具有显著影响,弱耦合强度下,特别是在同时攻击两个网络时,聚集系数对网络的鲁棒性影响不大。对于具有多重支持-依赖关系的多个耦合网络,当遭受随机攻击时,通过定义网络中的有效节点,搭建失效机制下的模型框架,对网络的鲁棒性展开研究,研究发现：通过增加各网络中自治节点的比例,增加网络之间的供应强度,可以有效地提高网络的抗毁性,增强网络的鲁棒性。
(2)对于网络之间具有两种连边方式的两个耦合相依网络,其中相依边为反馈边,在遭受随机攻击时,通过运用渗流理论,对网络的鲁棒性进行分析。研究发现：随着耦合强度的增加,系统由二阶相变经过混合相变转变为一阶相变；随着网络之间平均度的增加,一阶相变区域逐渐变大,混合相变区域逐渐变小最终消失；随着网络内部平均度的增加,一阶相变区域变小,二阶相变区域变大,混合相变区域不变。此外,我们发展了两类数学框架来研究网络之间存在两种连边方式的多个耦合相依网络的鲁棒性问题,根据相依边为反馈边和无反馈边。以多个Erdos-Renyi (ER)网络组成的星状结构网络为例,研究发现：对于同一个耦合强度,中心网络的鲁棒性随着网络个数的增加而减弱,随着网络内部平均度或网络之间连接边平均度的增加而增强。特别地,当耦合强度取1时,得到了一阶相变点的解析表达式。对比于无反馈边,在参数相同的情况时当相依边为反馈边时,网络变得极其脆弱。我们的理论不仅适用于随机网络,而且适用于其它任何拓扑结构的网络系统。
2、蓄意攻击下的耦合相依网络
(1)对于两个部分相依网络的蓄意攻击问题,本文分别从数值模拟和理论两方面对网络的鲁棒性进行了研究。引入了含有参数α的新的攻击概率函数,当α=0,1时,得到了系统由一阶相变向二阶相变过渡的临界值qc和pc的数值解。对于多个部分相依网络的蓄意攻击问题的研究发现：当度值高的节点有较高概率失效时,网络变得更脆弱。对于由ER网络构成的完全相依树状结构网络,推演出了最大连通集团、临界值pc的数值解；对于由ER网络构成的部分相依星状结构网络,在相同的耦合强度下,随着网络个数的增加,中心网络变得更为脆弱。
(2)对于网络之间具有两种连边方式的两个耦合相依网络,其中相依边为反馈边和无反馈边,在遭受蓄意攻击时,对网络的鲁棒性进行了研究。通过数值模拟研究发现：当网络中的高度值节点具有较高概率失效时,网络的鲁棒性变弱；当网络中的低度值节点具有较高概率失效时,网络的鲁棒性增强。此外,还研究了蓄意攻击下具有多重支持-依赖关系的多个耦合相依网络的鲁棒性问题,发现：除了增加各网络中自治性节点比例、增加网络之间的供应强度之外,还可以对网络中高度值节点进行重点保护,来有效地提高网络的抗毁性,增强网络的鲁棒性。
As an important branch of complexity theory, complex networks theory attracts more and more scholars from various research. With rapid growth of information technologies, empirical data are getting increasingly rich. Based on the quantitative analysis of big data, theoretical research, algorithm design, practical application, and platform framework etc., the study of complex networks has flourished over the past few years. Previous studies mainly concentrated on the innovation of theory and method of isolated network. However, most real networks are coupled as interdepen-dent networks. Therefore, the study has some theoretical and practical significance. In this thesis, according to the coupling characteristics in real networks, we classify the coupling dependency networks into four categories:I) there only exist interdependendy links, II) there exist support-dependence links between networks, III) there exist only interconnectivity links, IV) there exist interdependent and interconnect links. When the system are under random attack or targeted attack, we study the robustness of networks with theoretical and simulating analysis, by considering the nodes failure mechanisms of internal network and between two networks.
1Random attack on coupled interdependent networks
(1) For the robustness of partially coupled clustered networks with multiple support-dependence relations under random attack, we find that, for strong coupling strength, clustering coefficient has a significant impact on the robustness of the system, but for weak coupling strength, it has little effect especially when attacking the both two networks. By defining the functional nodes in the network of networks (NON) and developing the analytical framework, we study the robustness of NON with multiple support-dependence relations under random attack. The system will be more robust by increasing the fraction of autonomous nodes, and the density of support links.
(2) We study the robustness of two interdependent and interconnected networks with feedback condition under random attack based on the percolation theory. We find that increasing the coupling strength leads to a change from a second to first through hybrid order percolation transition. As the average interconnectivity links degree increases, the first order transition region grows large, the hybrid order transition region becomes small and ultimately disappears. And as the average intralinks degree increases, the first order transition region becomes small, the second order transition region becomes large, and the hybrid order transition region remains unchanged. Additionally, we study the robustness of interdpendent and interconnected network of networks (NON) with feedback condition or no feedback condition under random attack. For a starlike network of Erdos-Renyi (ER) networks, we find that the central network becomes more vulnerable as the number of networks increase, and becomes more robust as the average inter-or intra-connectivity links degree increases. Especially when the coupling strength is equal to one, we get the analytical expression of the first order critical point. Comparing to no feedback case,the feedback condition makes the system more vulnerable. Our theory is not only applied to random network, but also applied to any network systems topology.
2Targeted attack on coupled interdependent networks
(1) We study the robustness of two partially interdependent networks under targeted attack analytically and numerically. A new targeted probability function with parameter a is introduced. When a=0,1, we obtain the analytical solutions of critical points from first to second order phase transition. The robustness of partially interdependent network of networks (NON) is also studied. When highly connected nodes have higher probability to fail, the system becomes more vulnerable. For a fully interdependent treelike network of ER networks, we get the analytical solutions of the giant component and the critical threshold pc. For a part-ially interdependent starlike network of ER networks, as the number of networks increases, the central network becomes more vulnerable.
(2) We study the robustness of two interdependent and inter conne-cted networks with feedback condition or no feedback condition under targeted attack. We find from numerical simulation that, when highly con-nected nodes have high probability to fail,the system becomes more vulnerable. But when low connected nodes have high probability to fail, the system becomes more robust. Furthermore, we study the robustness of network of networks (NON) with multiple support-dependence relations under targeted attack. Besides increasing the fraction of autonomous nodes, and the density of support links, we can still improve the robustness of networks by protecting the nodes with higher degree.
引文
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