网络的网络的鲁棒性及一致性研究
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摘要
随着科学技术的飞速发展,我们赖以生存的自然环境、社会环境以及各种人造环境之间的联系变得更加紧密和错综复杂,而这些联系可以用复杂网络来描述。因而,复杂网络在很多交叉学科得到了广泛的应用,例如自然界中的生物网,社会中的各类基础设施网,以及生活中的互联网等。复杂网络又是一把双刃剑,现代社会中的很多灾难,从飓风到大规模停电,到恐怖分子的袭击,乃至世界范围的金融危机,都表明了潜在的危险和脆弱性无时无刻不存在于现代基础设施网络及这些网络的相互依赖中；多个体系统的一致性行为,不仅可以让我们更好的理解生物复杂性和群集智能的产生过程,而且可以使我们借鉴生物的智慧来设计控制,让系统呈现出所期望的涌现行为。在过去的十多年里,几乎所有关于复杂网络鲁棒性和一致性研究都是针对单个网络展开的。然而现实中的网络都是相互耦合在一起的,例如相互依赖的基础设施网以及相互作用的生物种群网等,从而形成了一个由复杂网络组成的网络(NON)。因此,NON的鲁棒性和一致性研究已经成为关系国计民生、社会安全稳定、乃至整个生存环境的重要科学技术难题。
采用渗流理论研究相互依赖网络的鲁棒性
(1)一般性框架：与基于单个孤立网络的研究不同,本文考虑现实社会中网络是相互依赖和相互作用的,得到了一个分析任意网络的网络(NON)鲁棒性的理论框架,该框架具有结构的任意性和功能的普适性。(ⅰ)结构的任意性：该系统是由任意n个网络组成,每个网络可以是任意结构,如随机网络、随机规则网络、无标度网络、格子网络,甚至小世界网络等,把每个网络视作一个节点,这些节点可以组成一个网络,该网络的拓扑结构依然可以是任意结构。(ⅱ)功能的普适性：从该一般性框架可以很容易得出之前经典的结论,单个网络或两个相互依赖网络的渗流理论只是NON结果中的特例。并且基于该框架理论分析了很多不同结构及不同攻击条件下的NON的鲁棒性,都发现了突然崩溃的一级不连续相变。这与单个网络表现出连续的二级相变显著不同。对于连续相变,一个小比例的节点失效,只能对系统造成较小的损害；而对于不连续相变,一个小比例的节点失效,有可能会造成整个NON发生完全的崩溃。由于网络之间的相互依赖关系,尤其是当网络之间的连接存在环时,NON变得异常脆弱。
(2)恶意攻击：考虑到现实生活中不仅仅存在随机失效,更存在恶意攻击,即有针对性地攻击某些节点,例如度大的节点。本文将恶意攻击的思想引入相互依赖网络的级联失效动态模型及其鲁棒性分析的研究框架,采用将恶意攻击映射到随机攻击的策略,利用生成函数和平均场的方法,研究了两个网络之间在完全相互依赖和部分相互依赖两种条件下的鲁棒性。研究结果表明：保护度大的节点在单个网络中很有效,而在相互依赖网络中却不能显著提高系统的鲁棒性。
(3)完全相互依赖NON的鲁棒性：理论分析了n个完全相互依赖网络构成树状NON鲁棒性研究的一般框架,其中形成NON的每一个网络的结构是随机(ER)网络、随机规则(RR)网络和无标度(SF)网络。研究结果表明：(ⅰ)在相同的网络个数及平均度下,RR网络组成的NON的鲁棒性最强,而SF组成的NON的鲁棒性最弱,这点和单个网络正好相反；(ⅱ)对于完全相互依赖的树状结构的NON,其相变临界值和最大聚类中有效的节点只与网络个数n有关,而与这些网络之间的依赖结构无关,并且NON的鲁棒性随着网络个数的增加而递减；(ⅲ)通过比较由ER网络和RR网络组成的NON的相变临界值,理论证明了脆弱性的根源为孤立节点和度为1的节点,该条件导致了当网络的个数足够多,在初始时刻一个节点的失效也会导致整个NON完全崩溃。
(4)部分相互依赖NON的鲁棒性：理论研究了n个部分相互依赖NON鲁棒性分析的一般框架,其中,NON的拓扑结构可以是星状结构、链状结构、环状结构和随机规则网络结构。主要以n个部分相互依赖的随机(ER)网络为例,研究了不同拓扑结构NON的渗流理论,理论结果有：(ⅰ)获得了介于一级相变与二级相变分界点的临界耦合强度,该临界值将为控制整个NON是否存在突然崩溃的一级相变,也就是整个NON的脆弱性提供理论依据；(ⅱ)对于由随机网络组成的随机规则网络,其渗流理论的结果与树状结构的NON的结果完全不同。令人惊奇的是,NON的相变临界值和最后剩余的最大聚类与网络的个数无关,这是因为网络之间的连接形成环形的缘故,NON的相变临界值和最后剩余的最大聚类与每个网络的平均度以及每个网络连接一起网络的个数相关。众所周知,单个网络中的很多结果的正确性是建立在网络节点个数无穷大的基础上,然而这里,对于NON来说,网络的个数可以是无限的、有限的、甚至是1；(ⅲ)为了进一步将研究成果转化为应用,利用国际航空网络、航海网络以及公司网络为例,研究了三个部分相互依赖网络的鲁棒性,验证了理论结果的正确性和一般性框架的普适性。
优化相互作用动态网络的一致性及其收敛速度
(1)加权模型：考虑到动态网络结构的不均匀性,也就是有一些个体节点的度比较大,有些比较小,而度大的个体的方向能够影响到更多的个体,本文利用这一动态网络的拓扑结果提出了以度为权的一致性模型。加权模型大大提高了系统的收敛速度及其一致程度。
(2)角度限定模型：实际中的个体往往不能在瞬间转动太大的角度,本文构建了角度限定的一致性模型。角度限定模型不仅更加接近现实中的生活或者物体的运动规律,更能大大提高系统的一致程度。并且在含噪音的情况下,对于给定的噪音,存在一个最优的角度限定值。因此,通过控制系统的角度限定值,可以优化系统并使系统达到最优的一致性。
(3)相互作用模型：加权模型和角度限定模型都可以用来优化单个动态网络或系统的一致性。在现实社会里网络之间相互耦合和相互作用的背景下,作者进而提出了相互作用的动态网络模型和衡量网络的网络一致性的序参量。研究结果发现不同的相互作用关系(共生、捕食和竞争)对整个网络的网络或单个网络的一致性有着不同的影响,例如共生关系下,存在两个动态网络之间的最优耦合强度,使得整个系统和单个网络同时达到最优一致性。相互作用模型结果可以直接被推广到多个网络的一致性问题中去。
总之,本文的主要贡献是：
(a)提出了研究任意NON的一般性框架。
(b)NON表现出,与单个网络连续的二级相变相比,更加脆弱的一级相变。
(c)对于完全相互依赖的树状结构的NON,其渗流临界值只与网络个数有关,而与NON的拓扑结构无关。
(d)NON脆弱性的根源是孤立节点和度为1的节点。
(e)对于环状结构的NON和随机网络组成的随机规则网络,其渗流临界值与网络个数无关。
(f)保护度大的节点在单个网络中很有效,而在相互依赖网络中却不能显著提高系统的鲁棒性。
(g)加权模型和角度限制的模型可以提高单个动态网络系统的一致性。
(h)在共生关系的动态网络中,存在最优的耦合强度,可以控制该参数使得每个网络或系统整体达到最优的一致性。
With the rapid development of science and technology, the natural environment,social systems and infrastructures, which we rely on, are closely interdependent, andbecome very complex. Network science provides useful tools to understand the cou-pled relationship and improve their efficiency and reliability. So we witness the widespreadscientific interest in interdisciplinary field of networks in many disciplines, such as bi-ological networks in the natural world, various infrastructures networks (includingpower grid systems and transportation systems etc.), and in networks serving societylike internet etc. Complex networks can be considered as being a double-edged swordin our modern society. Many disasters ranging from hurricanes to large-scale poweroutage to terrorist attacks, and even the world-wide financial crisis, show the poten-tial risk and vulnerability that exist in modern infrastructure networks and are hiddenunder the interdependence between the networks; Meanwhile, the synchronization ofmulti-agent systems, not only allows us to better understand the biological complexityand swarm intelligence generation process, but also enables us to utilize the wisdomof biologic to design and control the system to produce the desired macro behavior. Inthe past ten years, complex networks research attracted much attention of the scientistsfrom many disciplines. Almost all network research has been focused on the proper-ties of a single isolated network that does not interact with other networks. However,in the real world, the networks are coupled and interdependent. Example include var-ious infrastructures and the interaction of the biological networks. Thereby they forma complex network composed of various interdependent networks (NON). Therefore,research on robustness and synchronization of NON has become an important sci-entific and technical problem and a major challenge which is related to the people’slivelihood, security, and the whole living system. The thesis is composed on two main parts.
Using percolation theory to study the robustness of interdependent networksThe main work of this part include:
(1) A general framework: Unlike previous studies, which were based on a singeisolated network, here we consider real networks that are interdependent andinteract with each other, and develop a general theoretical framework to studythe robustness of any network of networks. This general framework featuresthe arbitrary of structure and the general percolation theory.(i) The arbitrary ofstructure. The system is composed of arbitrary n networks, each network can beof any structure, such as Erdo s-Re′nyi (ER) networks, random regular (RR) net-work, scale free (SF) networks, lattices, or even small world networks etc. Eachnetwork can be regarded as a node, these nodes form a network, the network ofnetworks (NON) have an arbitrary topology.(ii) The general percolation the-ory. From the general framework, several previous classical conclusions drawnfrom percolation theory of a single or two interdependent networks, which arejust special cases of our NON results, can be easily obtained. The theoreti-cal analysis of robustness of NON under different structures and various failureconditions were studied. One important result is that we found a discontinuoustransition that leads to abrupt collapse in interdependent networks. This is ofsignificant different from the results in a single network which shows continu-ous transition. The system becomes more vulnerable because of the dependencylinks between networks, and even more vulnerable because of the loops appear-ing in the networks. Why is the discontinuous transition more vulnerable thanthe continuous transition? When the transition is continuous, the removal of asmall fraction of nodes can cause only a small damage to the system and there isno risk of complete failure as long as the largest connected cluster in the networkhas many nodes. On the other hand, when the transition is discontinuous, the re-moval of even a single node near the critical point can cause a complete collapsein the network of networks, even when the largest connected cluster still con-sists of many nodes. Thus it becomes important to find the critical threshold forthe first order case. In this thesis, we show a theoretical solution of the critical threshold for the case of a regular (including random) network of Erdo s-Re′nyinetworks.
(2) Targeted attack. Considering the fact that in many cases attack can not randombut targeting specific high degree nodes, we introduce a targeted attack approachinto the cascading failure model and the robustness analysis framework of twointerdependent networks. The idea is to map the targeted attack to a randomfailure, and adopt the generating function and mean-field methodology. Usingthis approach we study the robustness under targeted attack of two fully inter-dependent networks and two partially interdependent networks. We find thatwhen the highly connected nodes are protected and have lower probability tofail, coupled SF networks are significantly more vulnerable compared to singlenetworks. The result implies that interdependent networks are difficult to de-fend by strategies such as protecting the high degree nodes that have been founduseful to significantly improve robustness of single networks.
(3) We first theoretically analyze a tree-like NON formed by n fully interdependentER networks, RR networks and SF networks. The results show that (i) for thesame number of networks and the same average degree, the robustness of NONcomposite of RR networks is the highest, while that of NON composite of SFnetworks is the lowest, which is the opposite of the results obtained for a singlenetwork;(ii) Independent of the topology (as long as it is a tree) but dependentof number of networks: For the tree like fully interdependent network of net-works, the percolation threshold and largest cluster depend only on the numberof networks, but not on the topology of the NON structure. And the robustnessdecreases with the number of networks.(iii) The origin of vulnerability: Ac-cording to the comparison of the critical threshold of ER NON and RR NON,we proved that the origin of vulnerability is the singly connected nodes and theisolated nodes. And this condition cause a whole collapse of a NON when thenumber of networks is large enough.
(4) Next, we analyze a star-like NON, a loop-like NON and random regular NONcomposed of n partially interdependent networks. We systemically study the robustness of several different topology of the NON and of each individual net-work. The results show that (i) Critical fraction of dependency links: Since thevulnerability of a network of networks can be measured according to type oftransition, we must find an analytical way to control the discontinuous transi-tion and determine the condition under which the discontinuous transition willcease. We examine a regular network of Erdo s-Re′nyi networks and find thatwhen there are fewer dependency links than the critical fraction of dependencylinks, the system is in continuous transition, but when there are more depen-dency links than the critical fraction of dependency links, the system in is dis-continuous transition. We present an analytical solution determining the con-dition necessary for the existence of a discontinuous transition, a solution thatwill be very helpful when designing robust interdependent infrastructures.(ii)Independent of the number of networks: The novel results presented in thismanuscript differ significantly from previous results in the field of interdepen-dent networks. Surprisingly, in contrast to the treelike network of networks inwhich the largest connected cluster depends on the number of networks n, whenthere are loops in the network of networks the largest connected cluster is un-affected by n. We demonstrate that the largest connected cluster of a regularnetwork of Erdo s-Re′nyi networks depends only on the number of networks onwhich each network depends. Although it is widely known that most results fora single network are obtained by assuming that the number of nodes goes to in-finity, our general results are correct for any number of networks, which can beinfinite, finite, or even1.(iii) In order to further translate the theoretical resultsinto application, we utilize real data corresponding to the international networkof airport, the network of seaport and the network of firm to study the robustnessof NON composite of three partially interdependent SF networks. The numeri-cal results further verify the theoretical results and the universality of the generalframework.
Optimize the synchronization and convergence speed of interacting dynamicnetworks
(1) A weighted model. In the dynamic network model (Vicsek model), although in- fluencing radius is the same, the neighbor number of each agent is different. Theagent with more neighbors might have larger influences on its neighbors, whichshould play a very important role in the dynamic process. Thus, we introducea weight related to the degree of each agent into Vicsek model when the direc-tions are updated. Moreover, in order to diversify the difference between thelarge degree agents and the small degree agents, a general exponential weightmodel is proposed. The simulation results show that this approach can accelerateconsensus process and improve convergence efficiency. Furthermore, when theexponent is increasing, the contribution of the weight is larger, the self-propelledagent system is much easier to obtain consensus even for the noise disturbance.
(2) A restricted angle model. Because the directions and positions of all the objectsare initially randomly distributed, most of the objects make sharp changes indirection that bear little similarity to behavior found in nature, and are thus im-practical when developing applications in engineering. From the point of viewof an engineer, any robot or vehicle powered by an engine can not make anacute-angle turn in a very short time period. In order to more closely resemblebehavior found in nature and to be useful in developing real-world applications,we introduce a restricted angle model. The results show that the restricted an-gle model significantly (i) improves the synchronization of Self propelled ob-ject(SPO) systems when the restricted angle decreases because reducing anglerestriction is effectively like reducing the internal noise which leads to improv-ing synchronization,(ii) demonstrates the existence of a critical restricted angleabove which the synchronization order parameter changes sharply form a largevalue to a small value, and (iii) reveals that for each noise amplitude the syn-chronization shows a peak as a function of angle restriction, so there exists anoptimal angle restriction for which one will obtain the best synchronization.
(3) The weighted model and the restricted angle model can be used to optimizethe dynamic single network or system synchronization. In the biological world,many different dynamic networks are coupled together. Under this context, herewe propose a model to describe the synchronization of interaction network com-posite of dynamic networks, and introduce a parameter to measure the synchro- nization of network of networks. We study the synchronization of two inter-acting networks under three different types of interactions including symbiosis,predation and competition, which can be explored to n dynamic networks. Theresults show that different interaction, like symbiosis, predation and competi-tion, among networks has a different impact on the synchronization of the entiresystem or each single network. For example, as for symbiotic interaction re-lationship, there exists the optimal coupling strength between the two dynamicnetworks, making the whole system and a single network achieve best synchro-nization at the same time.
In conclusion, the main contributions of this thesis include:
(a) Developed a general theoretical framework to study the robustness of any net-work formed from any interdependent networks.
(b) Compared with a single network, NON exhibits first order phase transition,which is more vulnerable.
(c) For fully interdependent treelike NON, the percolation threshold and giant com-ponent depend only on the number of networks, but not on the topology of theNON structure
(d) The origin of vulnerability is the singly connected nodes and the isolated nodes,which may cause a complete collapse of a NON when the number of networksis large enough.
(e) For the looplike NON and the random regular network of random networks, thepercolation threshold is independent with the number of networks.
(f) It is very effective to protect nodes with high degree in a single network, but itcan not significantly improve the robustness of interdependent networks.
(g) The weighted model and the restricted angle model can be used to optimize thesynchronization of a single dynamic network.
(h) Optimal coupling strength exists in the symbiotic relationship interacting dy-namic networks, which indicates that by controlling this parameter we are ableto achieve the optimal synchronization for the whole system and for each net-work.
采用渗流理论研究相互依赖网络的鲁棒性
(1)一般性框架：与基于单个孤立网络的研究不同,本文考虑现实社会中网络是相互依赖和相互作用的,得到了一个分析任意网络的网络(NON)鲁棒性的理论框架,该框架具有结构的任意性和功能的普适性。(ⅰ)结构的任意性：该系统是由任意n个网络组成,每个网络可以是任意结构,如随机网络、随机规则网络、无标度网络、格子网络,甚至小世界网络等,把每个网络视作一个节点,这些节点可以组成一个网络,该网络的拓扑结构依然可以是任意结构。(ⅱ)功能的普适性：从该一般性框架可以很容易得出之前经典的结论,单个网络或两个相互依赖网络的渗流理论只是NON结果中的特例。并且基于该框架理论分析了很多不同结构及不同攻击条件下的NON的鲁棒性,都发现了突然崩溃的一级不连续相变。这与单个网络表现出连续的二级相变显著不同。对于连续相变,一个小比例的节点失效,只能对系统造成较小的损害；而对于不连续相变,一个小比例的节点失效,有可能会造成整个NON发生完全的崩溃。由于网络之间的相互依赖关系,尤其是当网络之间的连接存在环时,NON变得异常脆弱。
(2)恶意攻击：考虑到现实生活中不仅仅存在随机失效,更存在恶意攻击,即有针对性地攻击某些节点,例如度大的节点。本文将恶意攻击的思想引入相互依赖网络的级联失效动态模型及其鲁棒性分析的研究框架,采用将恶意攻击映射到随机攻击的策略,利用生成函数和平均场的方法,研究了两个网络之间在完全相互依赖和部分相互依赖两种条件下的鲁棒性。研究结果表明：保护度大的节点在单个网络中很有效,而在相互依赖网络中却不能显著提高系统的鲁棒性。
(3)完全相互依赖NON的鲁棒性：理论分析了n个完全相互依赖网络构成树状NON鲁棒性研究的一般框架,其中形成NON的每一个网络的结构是随机(ER)网络、随机规则(RR)网络和无标度(SF)网络。研究结果表明：(ⅰ)在相同的网络个数及平均度下,RR网络组成的NON的鲁棒性最强,而SF组成的NON的鲁棒性最弱,这点和单个网络正好相反；(ⅱ)对于完全相互依赖的树状结构的NON,其相变临界值和最大聚类中有效的节点只与网络个数n有关,而与这些网络之间的依赖结构无关,并且NON的鲁棒性随着网络个数的增加而递减；(ⅲ)通过比较由ER网络和RR网络组成的NON的相变临界值,理论证明了脆弱性的根源为孤立节点和度为1的节点,该条件导致了当网络的个数足够多,在初始时刻一个节点的失效也会导致整个NON完全崩溃。
(4)部分相互依赖NON的鲁棒性：理论研究了n个部分相互依赖NON鲁棒性分析的一般框架,其中,NON的拓扑结构可以是星状结构、链状结构、环状结构和随机规则网络结构。主要以n个部分相互依赖的随机(ER)网络为例,研究了不同拓扑结构NON的渗流理论,理论结果有：(ⅰ)获得了介于一级相变与二级相变分界点的临界耦合强度,该临界值将为控制整个NON是否存在突然崩溃的一级相变,也就是整个NON的脆弱性提供理论依据；(ⅱ)对于由随机网络组成的随机规则网络,其渗流理论的结果与树状结构的NON的结果完全不同。令人惊奇的是,NON的相变临界值和最后剩余的最大聚类与网络的个数无关,这是因为网络之间的连接形成环形的缘故,NON的相变临界值和最后剩余的最大聚类与每个网络的平均度以及每个网络连接一起网络的个数相关。众所周知,单个网络中的很多结果的正确性是建立在网络节点个数无穷大的基础上,然而这里,对于NON来说,网络的个数可以是无限的、有限的、甚至是1；(ⅲ)为了进一步将研究成果转化为应用,利用国际航空网络、航海网络以及公司网络为例,研究了三个部分相互依赖网络的鲁棒性,验证了理论结果的正确性和一般性框架的普适性。
优化相互作用动态网络的一致性及其收敛速度
(1)加权模型：考虑到动态网络结构的不均匀性,也就是有一些个体节点的度比较大,有些比较小,而度大的个体的方向能够影响到更多的个体,本文利用这一动态网络的拓扑结果提出了以度为权的一致性模型。加权模型大大提高了系统的收敛速度及其一致程度。
(2)角度限定模型：实际中的个体往往不能在瞬间转动太大的角度,本文构建了角度限定的一致性模型。角度限定模型不仅更加接近现实中的生活或者物体的运动规律,更能大大提高系统的一致程度。并且在含噪音的情况下,对于给定的噪音,存在一个最优的角度限定值。因此,通过控制系统的角度限定值,可以优化系统并使系统达到最优的一致性。
(3)相互作用模型：加权模型和角度限定模型都可以用来优化单个动态网络或系统的一致性。在现实社会里网络之间相互耦合和相互作用的背景下,作者进而提出了相互作用的动态网络模型和衡量网络的网络一致性的序参量。研究结果发现不同的相互作用关系(共生、捕食和竞争)对整个网络的网络或单个网络的一致性有着不同的影响,例如共生关系下,存在两个动态网络之间的最优耦合强度,使得整个系统和单个网络同时达到最优一致性。相互作用模型结果可以直接被推广到多个网络的一致性问题中去。
总之,本文的主要贡献是：
(a)提出了研究任意NON的一般性框架。
(b)NON表现出,与单个网络连续的二级相变相比,更加脆弱的一级相变。
(c)对于完全相互依赖的树状结构的NON,其渗流临界值只与网络个数有关,而与NON的拓扑结构无关。
(d)NON脆弱性的根源是孤立节点和度为1的节点。
(e)对于环状结构的NON和随机网络组成的随机规则网络,其渗流临界值与网络个数无关。
(f)保护度大的节点在单个网络中很有效,而在相互依赖网络中却不能显著提高系统的鲁棒性。
(g)加权模型和角度限制的模型可以提高单个动态网络系统的一致性。
(h)在共生关系的动态网络中,存在最优的耦合强度,可以控制该参数使得每个网络或系统整体达到最优的一致性。
With the rapid development of science and technology, the natural environment,social systems and infrastructures, which we rely on, are closely interdependent, andbecome very complex. Network science provides useful tools to understand the cou-pled relationship and improve their efficiency and reliability. So we witness the widespreadscientific interest in interdisciplinary field of networks in many disciplines, such as bi-ological networks in the natural world, various infrastructures networks (includingpower grid systems and transportation systems etc.), and in networks serving societylike internet etc. Complex networks can be considered as being a double-edged swordin our modern society. Many disasters ranging from hurricanes to large-scale poweroutage to terrorist attacks, and even the world-wide financial crisis, show the poten-tial risk and vulnerability that exist in modern infrastructure networks and are hiddenunder the interdependence between the networks; Meanwhile, the synchronization ofmulti-agent systems, not only allows us to better understand the biological complexityand swarm intelligence generation process, but also enables us to utilize the wisdomof biologic to design and control the system to produce the desired macro behavior. Inthe past ten years, complex networks research attracted much attention of the scientistsfrom many disciplines. Almost all network research has been focused on the proper-ties of a single isolated network that does not interact with other networks. However,in the real world, the networks are coupled and interdependent. Example include var-ious infrastructures and the interaction of the biological networks. Thereby they forma complex network composed of various interdependent networks (NON). Therefore,research on robustness and synchronization of NON has become an important sci-entific and technical problem and a major challenge which is related to the people’slivelihood, security, and the whole living system. The thesis is composed on two main parts.
Using percolation theory to study the robustness of interdependent networksThe main work of this part include:
(1) A general framework: Unlike previous studies, which were based on a singeisolated network, here we consider real networks that are interdependent andinteract with each other, and develop a general theoretical framework to studythe robustness of any network of networks. This general framework featuresthe arbitrary of structure and the general percolation theory.(i) The arbitrary ofstructure. The system is composed of arbitrary n networks, each network can beof any structure, such as Erdo s-Re′nyi (ER) networks, random regular (RR) net-work, scale free (SF) networks, lattices, or even small world networks etc. Eachnetwork can be regarded as a node, these nodes form a network, the network ofnetworks (NON) have an arbitrary topology.(ii) The general percolation the-ory. From the general framework, several previous classical conclusions drawnfrom percolation theory of a single or two interdependent networks, which arejust special cases of our NON results, can be easily obtained. The theoreti-cal analysis of robustness of NON under different structures and various failureconditions were studied. One important result is that we found a discontinuoustransition that leads to abrupt collapse in interdependent networks. This is ofsignificant different from the results in a single network which shows continu-ous transition. The system becomes more vulnerable because of the dependencylinks between networks, and even more vulnerable because of the loops appear-ing in the networks. Why is the discontinuous transition more vulnerable thanthe continuous transition? When the transition is continuous, the removal of asmall fraction of nodes can cause only a small damage to the system and there isno risk of complete failure as long as the largest connected cluster in the networkhas many nodes. On the other hand, when the transition is discontinuous, the re-moval of even a single node near the critical point can cause a complete collapsein the network of networks, even when the largest connected cluster still con-sists of many nodes. Thus it becomes important to find the critical threshold forthe first order case. In this thesis, we show a theoretical solution of the critical threshold for the case of a regular (including random) network of Erdo s-Re′nyinetworks.
(2) Targeted attack. Considering the fact that in many cases attack can not randombut targeting specific high degree nodes, we introduce a targeted attack approachinto the cascading failure model and the robustness analysis framework of twointerdependent networks. The idea is to map the targeted attack to a randomfailure, and adopt the generating function and mean-field methodology. Usingthis approach we study the robustness under targeted attack of two fully inter-dependent networks and two partially interdependent networks. We find thatwhen the highly connected nodes are protected and have lower probability tofail, coupled SF networks are significantly more vulnerable compared to singlenetworks. The result implies that interdependent networks are difficult to de-fend by strategies such as protecting the high degree nodes that have been founduseful to significantly improve robustness of single networks.
(3) We first theoretically analyze a tree-like NON formed by n fully interdependentER networks, RR networks and SF networks. The results show that (i) for thesame number of networks and the same average degree, the robustness of NONcomposite of RR networks is the highest, while that of NON composite of SFnetworks is the lowest, which is the opposite of the results obtained for a singlenetwork;(ii) Independent of the topology (as long as it is a tree) but dependentof number of networks: For the tree like fully interdependent network of net-works, the percolation threshold and largest cluster depend only on the numberof networks, but not on the topology of the NON structure. And the robustnessdecreases with the number of networks.(iii) The origin of vulnerability: Ac-cording to the comparison of the critical threshold of ER NON and RR NON,we proved that the origin of vulnerability is the singly connected nodes and theisolated nodes. And this condition cause a whole collapse of a NON when thenumber of networks is large enough.
(4) Next, we analyze a star-like NON, a loop-like NON and random regular NONcomposed of n partially interdependent networks. We systemically study the robustness of several different topology of the NON and of each individual net-work. The results show that (i) Critical fraction of dependency links: Since thevulnerability of a network of networks can be measured according to type oftransition, we must find an analytical way to control the discontinuous transi-tion and determine the condition under which the discontinuous transition willcease. We examine a regular network of Erdo s-Re′nyi networks and find thatwhen there are fewer dependency links than the critical fraction of dependencylinks, the system is in continuous transition, but when there are more depen-dency links than the critical fraction of dependency links, the system in is dis-continuous transition. We present an analytical solution determining the con-dition necessary for the existence of a discontinuous transition, a solution thatwill be very helpful when designing robust interdependent infrastructures.(ii)Independent of the number of networks: The novel results presented in thismanuscript differ significantly from previous results in the field of interdepen-dent networks. Surprisingly, in contrast to the treelike network of networks inwhich the largest connected cluster depends on the number of networks n, whenthere are loops in the network of networks the largest connected cluster is un-affected by n. We demonstrate that the largest connected cluster of a regularnetwork of Erdo s-Re′nyi networks depends only on the number of networks onwhich each network depends. Although it is widely known that most results fora single network are obtained by assuming that the number of nodes goes to in-finity, our general results are correct for any number of networks, which can beinfinite, finite, or even1.(iii) In order to further translate the theoretical resultsinto application, we utilize real data corresponding to the international networkof airport, the network of seaport and the network of firm to study the robustnessof NON composite of three partially interdependent SF networks. The numeri-cal results further verify the theoretical results and the universality of the generalframework.
Optimize the synchronization and convergence speed of interacting dynamicnetworks
(1) A weighted model. In the dynamic network model (Vicsek model), although in- fluencing radius is the same, the neighbor number of each agent is different. Theagent with more neighbors might have larger influences on its neighbors, whichshould play a very important role in the dynamic process. Thus, we introducea weight related to the degree of each agent into Vicsek model when the direc-tions are updated. Moreover, in order to diversify the difference between thelarge degree agents and the small degree agents, a general exponential weightmodel is proposed. The simulation results show that this approach can accelerateconsensus process and improve convergence efficiency. Furthermore, when theexponent is increasing, the contribution of the weight is larger, the self-propelledagent system is much easier to obtain consensus even for the noise disturbance.
(2) A restricted angle model. Because the directions and positions of all the objectsare initially randomly distributed, most of the objects make sharp changes indirection that bear little similarity to behavior found in nature, and are thus im-practical when developing applications in engineering. From the point of viewof an engineer, any robot or vehicle powered by an engine can not make anacute-angle turn in a very short time period. In order to more closely resemblebehavior found in nature and to be useful in developing real-world applications,we introduce a restricted angle model. The results show that the restricted an-gle model significantly (i) improves the synchronization of Self propelled ob-ject(SPO) systems when the restricted angle decreases because reducing anglerestriction is effectively like reducing the internal noise which leads to improv-ing synchronization,(ii) demonstrates the existence of a critical restricted angleabove which the synchronization order parameter changes sharply form a largevalue to a small value, and (iii) reveals that for each noise amplitude the syn-chronization shows a peak as a function of angle restriction, so there exists anoptimal angle restriction for which one will obtain the best synchronization.
(3) The weighted model and the restricted angle model can be used to optimizethe dynamic single network or system synchronization. In the biological world,many different dynamic networks are coupled together. Under this context, herewe propose a model to describe the synchronization of interaction network com-posite of dynamic networks, and introduce a parameter to measure the synchro- nization of network of networks. We study the synchronization of two inter-acting networks under three different types of interactions including symbiosis,predation and competition, which can be explored to n dynamic networks. Theresults show that different interaction, like symbiosis, predation and competi-tion, among networks has a different impact on the synchronization of the entiresystem or each single network. For example, as for symbiotic interaction re-lationship, there exists the optimal coupling strength between the two dynamicnetworks, making the whole system and a single network achieve best synchro-nization at the same time.
In conclusion, the main contributions of this thesis include:
(a) Developed a general theoretical framework to study the robustness of any net-work formed from any interdependent networks.
(b) Compared with a single network, NON exhibits first order phase transition,which is more vulnerable.
(c) For fully interdependent treelike NON, the percolation threshold and giant com-ponent depend only on the number of networks, but not on the topology of theNON structure
(d) The origin of vulnerability is the singly connected nodes and the isolated nodes,which may cause a complete collapse of a NON when the number of networksis large enough.
(e) For the looplike NON and the random regular network of random networks, thepercolation threshold is independent with the number of networks.
(f) It is very effective to protect nodes with high degree in a single network, but itcan not significantly improve the robustness of interdependent networks.
(g) The weighted model and the restricted angle model can be used to optimize thesynchronization of a single dynamic network.
(h) Optimal coupling strength exists in the symbiotic relationship interacting dy-namic networks, which indicates that by controlling this parameter we are ableto achieve the optimal synchronization for the whole system and for each net-work.
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