Research On Robustness And Synchronization Of Network Of Networks

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.