双层网、群落、级联事件等复杂网络研究---邹盛荣-博士毕业论文.doc

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1、 摘 要 III摘 要以还原论方法论为基础的近代物理学取得了极大的成功，并且导致了一系列现代科学技术的产生与发展。然而，典型的复杂系统，例如生命系统和社会系统，由于存在自组织和自适应的特点，全体不等于部分的简单组合，从根本上不适用于还原论。如何建立复杂系统的定量科学系统，是整个科学界长期以来的关注热点之一。近二十年来，复杂系统研究受到了分外的重视。上世纪末作为复杂系统描述工具出现的复杂网络的研究热潮对此也起了推动作用。我们课题组也从事了复杂系统和复杂网络的研究。本论文的前两部分报告我们对复杂网络中的两个热点问题，即关联双层网和群落划分的研究；后两部分分别报告我们对一类复杂系统中的重要现象雪崩，

2、以及一个重要复杂系统科研论文发表系统的研究。典型的复杂系统可以被划分为许多层次和子系统，每个层次和子系统中存在许多种互相作用的基本单元，这些层次和子系统之间又存在复杂的关联。这使得对系统整体的研究具有很大困难。人们自然地首先研究一种基本单元及其相互作用，暂时抛开它们与其它基本单元的相互作用。这就是为什么到目前为止大多数研究的复杂网络都仅包括一种节点及其相互作用。这种网络可以称为“层状网”。显然，一个典型的复杂系统必须用许多互相复杂关联的层状网来描述。这样的更复杂网络被称为“超网络”、“网络的网络”或者“互关联网络”。它们已经成为一个公认的复杂网络研究重要方向。例如，2003年9月在罗马召开了主

3、题为“Growing Networks and Graphs in Statistical Physics, Finance, Biology and Social Systems”的国际会议,提出了十大急待研究的复杂网络问题，网络的网络位列其中之一。然而，直接研究网络的网络还存在很大困难。作为起步，许多人研究少数几层互相关联的层状网。本论文第一部分报道我们对一类关联双层网的研究。以往的研究中，一个城市公交或铁路系统常被映射为两个空间：P空间和L空间。在这两个空间中，每个节点都代表公交或铁路站点，P空间的连边表示这两个站点之间至少有一路汽车或火车提供了直达服务，在L空间中，两个节点间的连边表示

4、它们至少是一条线路上相邻的站点。在我们所知的情况下，尽管关于两个单独空间的网络拓扑性质研究已很多, 但是到目前为止还没有关于P空间和L空间拓扑关联关系的研究。和Kurant相关文章中的观念相似，我们认为P空间描述了交通流，而L空间近似地描述了承载这些交通流的道路。研究两个空间的拓扑性质关联对理解交通基础和交通流的关系肯定大有益处。借助于一些合理的简化假设，我们解析得到了同一交通工具的L空间和P空间构成的双层网中两种重要拓扑性质（度和集群系数）的关联关系，所得结论与中国铁路系统及三个中国城市的公交系统实证研究结果很好地相符。这些结果表明：在统计平均的意义上，L空间中的重要枢纽点在P空间中也同样重

5、要，次要的节点在P空间中也是次要的，这个结果符合我们的直觉。类似地，L空间的稠密部分在P空间中也同样稠密。这种双层网的描述方法应该对其它许多实际系统也有效，因此上述对两层网之间拓扑性质关系的理解对设计、控制许多系统将会非常有益。这部分研究已经在Phys. Lett. A上发表，本人（本论文作者、学位申请者）为第一作者。大量实证研究已经表明，大多数复杂网络具有所谓的“群落”结构，即网络节点由于种种原因结合成集团，集团内部连接紧密，而集团之间连接稀疏。因此在复杂网络研究中，群落结构划分或识别是最重要的问题之一。在现有的群落划分算法中，群落几乎都是根据网络的拓扑结构来划分的。依据节点功能或其他因素的

6、划分识别受到注意还是不久以前的事。也就是说，有时群落的识别不光是依赖于拓扑结构，还依赖于其他信息。例如，一个研究组的成员往往在他们的工作场所以及其他地方经常讨论，组成一个思想交流的紧密圈子，但他们并不总是在一起发表文章（大多数是和导师一起发表论文）。当我们建立一个科研合作网络时，通常的定义是把作者作为节点，两个作者之间的连边表示一篇文章的共同作者关系。那么，通过这种拓扑结构得到的群落划分很可能与实际的科研合作组很不同。这是因为一些科学研究组成员可能和组外的其他人发表文章，如果这些组外文章包含很多作者从而产生很多合作边的话，他们很可能就会隶属于另外一个稠密的群落。问题是：假定我们不知道作者的隶属

7、课题组，而又认为课题组是合理的群落，那么在课题组结构影响网络拓扑结构的前提下，仅仅通过拓扑结构信息是否能合理地划分群落？就我们所知，目前还只有极少的文章涉及到此类问题。本论文的第三章用一个例子说明：对于认为课题组是合理群落的情况，在四种有特点的算法中，加权的Newman算法是最好的选择，使用它可以仅根据拓扑信息得到最接近于课题组的群落划分。这些思想对其他实际的复杂网络的群落划分也有很好的借鉴作用。这部分研究已经在Chin. Phys. B上发表，本人（本论文作者、学位申请者）为第一作者。一个得到很多领域科学家的长时间关注的现象是复杂系统中的“雪崩”或“级联故障”。“级联故障”这个名称比“雪崩”

8、包含了更多的信息。称之为级联是因为事件的传播过程象一个链式的反应，即一代事件引起下一代事件的发生。称之为故障是因为这样级联的事件经常产生大大小小的灾难。真实的山上雪崩或泥石流、电网的大停电事故是一些常见的产生灾难的级联故障例子。因为有些级联故障并不会产生灾难，我们在对级联故障作一般性的讨论时，可称这种现象为“级联事件”。为什么级联事件经常会产生大大小小的灾难？我们认为一般有两种因素，第一种是大多数的级联事件会增长得越来越多，一个事件产生了更多的下一代事件。即使个别事件并不有害，但无节制的事件增长也常常会引起灾难。例如兔子的无限制繁殖也引起灾难。第二个原因是级联事件发生的频率变得越来越高，即两代

9、事件之间的间隔变得越来越短，比如在真实雪崩、泥石流和电网的大停电事故中，灾难是恐怖的正是因为事件的发生越来越快以至于完全没时间和能力做出反应。依照这些思想，我们可能将级联事件分为两种类型：一种是具有“内禀加速机制”的级联事件，它们往往导致灾难，另一种是不具有“内禀加速机制”的级联事件，它通常并不引起灾难。我们建立了一个简化的沙堆模型和一个简化的逻辑斯谛模型来展示两种类型的级联事件的特征规律，并且提供了分别与这些不同特征规律符合的实证观测。这些新思想可能对长期以来的雪崩研究以全新的启示。这部分研究已经在Physica A上发表，本人（本论文作者、学位申请者）为第一作者。近十年来，越来越多的物理学

10、家认识到复杂系统的重要性并投入到这项研究中，社会系统是复杂系统之一，成为吸引科学家的一项重要研究。人类的科学活动，包括科学研究的文章发表，也是非常重要的社会系统，同样得到了重视和研究。在社会系统研究中，往往会有这样的现象，即社会学家有好的思想，但不能定量地描述其中的规律。数理科学家可能得到这些思想的定量表述，并且导致新的普遍规律的发现。第五章报道了我们从全新角度对科研文章发表这个复杂系统作的调研。直观地来看，一个高水平的杂志接受高水平作者的文章，并可能邀请更高水平的作者做期刊的编委。这可以看成期刊、作者和编委会成员之间的一种类型的合作和竞争。我们给出了一个关于期刊、作者和编委会成员的交互作用模

11、型。在模型中，期刊接受作者的发文和邀请作者做编委的概率都是正态分布，然而第二个正态分布的最可几值有一定漂移，使得期刊有更高水平的作者成为它的编委。实证中我们调研了15个不同影响因子的期刊，结果与模型的解析结果相符。这个关于期刊、作者和编委会成员函数关系的定量描述对科学家来说可能是重要的，对科研管理部门可能也有参考作用。这部分研究已经在Chin. Phys. Lett.上发表，本人（本论文作者、学位申请者）为第一作者。关键词：复杂系统，复杂网络；双层网；群落划分；级联事件；科研模型 Abstract IXAbstractBased on Reductionism, modern physics

12、has obtained great success, which is contributing to the production and development of modern scientific technology. However, typical complex systems, like biological system and social system, being a whole not equal to the simple combination of the sub parts due to their features of self-organizati

13、on and self-adoption, are essentially not in accordance with Reductionism. Therefore, how to set up quantitative scientific system in complex system has for long been a hot concern in scientific community. The study of complex system, been particularly stressed in recent twenty years, has been great

14、ly promoted at the end of the last century by the study tide of complex network, a description tool of complex system. Our project group also devote ourselves to the study of complex system and complex network. The first two parts of the thesis address two hot issues of complex network: associated b

15、i-layer network and community division. The next two parts concentrate specifically on avalanche, one major phenomenon in a complex system, and scientific research article publication system, an important complex system. One typical system can be divided into many intricately related layers and subs

16、ystems, in which lots of interacting basic units exist. Thus, it is a hard and strenuous work to have a thorough study about the system as a whole. People always naturally focus on one type of basic unit at first, without taking into any account of their relationship with other basic units. And this

17、 is the main reason why most studies about complex network up to now cover only one kind of node and their interactions. The network is named layered network. Apparently, a typical complex system must be described from the perspective of many complicatedly interrelated layered networks. This more so

18、phisticated network is called super network, or network of network, or interrelated network. They have been universally known as the most important complex network research direction. For example, in September 2003, an International conference, held in Rome, concentrating on the topic “Growing Netwo

19、rks and Graphs in Statistical Physics, Finance, Biology and Social Systems”, posed the top ten most urgent research issues, one of which is the network of network. But there are still some difficulties lying in direct study of network of network. Taking for the starting stage, many researchers broug

20、ht about study on a small number of interrelated layered networks. The first part of the thesis covers our study on bi-layer network.In previous researches, the bus-transport system or the railway system of a city can be mapped into two spaces: Space P and Space L. In the two spaces, each node repre

21、sents either a bus-stop or a railway station. In space P one edge indicates that at least one bus or train route provides direct service, while in space L, the line between two nodes is a sign, to say the least, that they are the adjoining stops of one route. As we know, in spite of the fact that th

22、ere are many researches about topological property of two independent spaces, till now, there is still a blank in consideration about relationship between topological property of space P and topological property of space L. Similar to concepts in Kurants relevant articles, we hold the idea that spac

23、e P describes traffic flow and space L approximately represents the road to embody the traffic flow, in other words, they are the infrastructure of traffic flow. Undoubtedly, study about the relationship between topological properties of the two spaces is conducive to have a good knowledge of relati

24、onship between transport infrastructure and traffic flow. With the aid of some rational assumptions, we come to the comprehension of correlation of two types of topological property: degree and clustering coefficient, of the same transport vehicle in double layer network composite of space L and spa

25、ce P. The conclusion is corresponding to the result of empirical study about Railway System in China and public traffic system of three Chinese cities. The result, which is in keeping with our tuition, demonstrates that, on statistical average, weighty hubs in space L has the same vital role in spac

26、e P, and secondary nodes are also subordinate in space P. Similarly, space L and space P share the same dense part. This kind of description method of bi-layer network is also effective in other practical systems. Therefore, the above understanding about relationship between topological properties o

27、f two networks is useful for design and control of some systems. Study about this part has been published in Phys. Lett. A, with myself as the first author.A large amount of empirical study have proved that most of complex networks have the so-called “community” structure, which is composed of net n

28、odes for some reasons, compact internally and sparse externally. Consequently, “community” structure division or identification is one of the most important issues in the study of complex network. In present “community” division algorithms, they are almost divided or identified according to topologi

29、cal structure. However, in some cases, “community” identification does not only rely on structure, but also on some other information. For example, fellows in a research group usually discuss very often in their common working place and many other places, forming a dense idea communication community

30、; however, often they do not publish papers together (e.g., each of them only publishes papers with their common supervisor). So, if we set up a co-authorship bipartite network in ordinary way by defining the authors as actors and an edge between two authors as the co-authorship in a paper, the comm

31、unities identified only by topological information are often very different from the practical research cooperation groups. The main reason for this is that some members of the researching group may publish papers with some scientists outside the group. Then they may belong to another dense part of

32、the network if their out-group papers are jointly done by a great many authors, inducing more edges. The question is, having assumption that we have no idea to which project group one author belongs and project group is a rational community, then, is it possible to divide the community reasonably on

33、ly by topological structure information, which is under influence of the division of project groups. According to the study we have done, there are few articles touching upon this issue at present. The third chapter of the thesis adopts one example to illustrate, based on the assumption that project

34、 group is a rational community, that weighted Newman algorithm is the best choice, for it can obtain the most close community division of the project group on the basis of topological information. These ideas have reference value to community division of other practical complex networks. Study about

35、 this part has been published in Chin. Phys. B, with myself as the first author.A widely existing phenomenon, “avalanche” or “cascade failure”, has attracted attentions for a long time. The name of “cascade failure” bears more information than “avalanche” does. It is called “cascade” because the tra

36、nsmission process of an event undergoes the chain reactions, i.e., one generation events induce the next. It is called “failure” due to the disaster or catastrophe is has brought about. Snow avalanche or landslide avalanche, and blackout induced by cascade failures in power grids can serve as exampl

37、es. Since some cascades do not cause disasters, we address this kind of cascading phenomenon as “cascade events” (CE). Why CE often cause disaster or catastrophe? We find that there are two important factors. The first one is that in most cases one event produces more than one next generation events

38、. An uncontrolled increase of CE may bring about disaster even if the single event is harmless. The second important factor is that the frequency of CE becomes higher and higher, namely, time duration between two neighboring generations decreases so rapidly that a lash-up becomes impossible, such as

39、 in the courses of snow avalanche, landslide avalanche, and the power blackout. Disaster and catastrophe are frightening because they happened in such a short time that people completely have no time or ability to react to them. Accordingly, we first classify the cascading phenomena into two categor

40、ies: the cascading disasters that result in large-scale functional failures and the cascading events that do not lead to disasters. We present a simplified sandpile model and a heuristic logistic map to show the characteristics of two kinds of CE. Our present findings contribute to the understanding

41、 of the transition between different cascading events, providing a basis for further understanding of the transitions among more general critical events. Study about this part has been published in Physica A , with myself as the first author.In recent decade, more and more physicists have realized t

42、he importance of complex system study and threw themselves into the research. Social system, as one of the most complicated complex systems, has become a promising research object, attracting interest of many physics. Human beings scientific activities, including scientific paper publication, being

43、a very important social system, has been attached great importance and studied broadly. In social system studies, it often happens that a sociologist has a brilliant qualitative idea, but cannot present a quantitative description. Mathematical experts may have a quantitative description to the thoug

44、hts, which would contribute to uncovering new general laws. The fifth chapter covers an investigation, made from a totally different angle, into the complex system of the scientific research article publication. It is intuitionist that a higher quality scientific journal accepts articles from higher

45、 academic level authors and invites even higher level authors being its editorial board members. This can be viewed as a kind of cooperation and competition between the journals, authors and editorial board members. Recently, quantitative study on complex systems becomes an attracting research area

46、for physicists. Human scientific activity, including scientific paper publication, is one of the important complex systems therefore deserves an investigation. In this thesis we present a very simple model describing interaction between journals, authors and editorial board members. In the model bot

47、h the probabilities, with which a journal accepts papers from an author or invites him being an editorial board member, obey normal distributions. However, the most probable value of the later distribution shifts so that the journal can have higher level authors being its editorial board members. Th

48、e model analytic conclusion is in rather good agreement with the empirical observation in 15 real world journals with different influence factors. We successfully obtained a quantitative expression of an intuitional idea by a simple and appropriate model. This quantitative expression describes the r

49、elation functions between journals, authors and editorial board members which is important for scientists and, may be, also for the scientific administrations. Study about this part has been published in Chin. Phys. Lett. , with myself as the first author.Keywords: Complex System；Complex Network；Bilayer Network；Community division；Cascade Event；Science Research model