摘要
自然界中的许多复杂系统都可以由相互作用的个体所组成的网络来描述,自从人类发现自然界中的多数实际网络的拓扑结构具有小世界效应和无标度特性之后,科学界便掀起了一股研究复杂网络的热潮。当前,复杂网络的研究已渗透到数理学科、工程学科、生命学科等不同的领域,涉及的网络包括交通网络、电力网络、Internet、金融网络、蛋白质作用网络等等。可以说,复杂网络理论已经成为了人们研究复杂系统的一种必不可少的工具。
对复杂网络的研究,一方面有利于人们了解真实网络的结构特点及其形成机理;另一方面也有利于人们认识复杂网络中的各种动力学过程,这对现实生活中各种真实网络结构的优化以及网络中动力学过程的控制具有重要的指导意义。
尽管在过去的几年中,来自不同领域的学者已经对复杂网络的结构及动力学过程进行了广泛的研究,然而,由于真实系统的复杂性,仍然存在许多值得进一步研究的问题,本文结合这些问题就复杂网络的演化模型、复杂网络中的病毒传播过程以及复杂网络中的级联故障等几个紧密相关的内容进行了一系列的研究。
本文的主要研究内容和创新点如下:
(1)根据真实系统的演化过程,在随机演化网络和BA无标度网络的基础上提出了一个随机-无标度混合统一演化网络模型,通过引入一个参数p将随机演化网络和BA无标度网络统一起来,当p从0调到1时,网络的结构可以从BA无标度网络连续过渡到随机演化网络。研究过程中,利用平均场理论和计算机模拟的方法计算了该网络模型的度分布,两者完全吻合,并且得到了比前人的研究更精确的结果;接着,在随机-无标度混合统一网络模型的基础上进一步提出了一个内外同时演化的混合统一网络模型,研究了该网络的度分布,得到了度分布的解析解。这些研究在一定程度上揭示了真实网络的演化机理,有助于人们理解真实网络结构的生成过程。
(2)研究了基于随机-无标度混合统一演化网络的病毒传播行为,研究发现,对于规模很大的网络来说,当参数p大于0时,网络的临界传播效率为一个大于0的正数,只有在传播效率大于该临界值的情况下病毒才能持续稳定地传播,而当参数p接近0时,网络的传播临界值也趋于0,这意味着此时即使一个很弱的传染源也能使病毒长期在网络中传播。这个结论说明网络的结构是影响病毒传播的重要因素,这对网络结构的设计具有重要的指导意义。在网络结构的设计过程中,人们可以通过调节参数p的值来改变网络的结构,使网络的传播临界值尽量增大,以阻止病毒的传播。
(3)研究了二维稀疏规则网格中考虑群体密度和个体运动的SIS及SIRS病毒传播行为。理论分析和计算机模拟发现:群体密度d、传播效率λ和个体的运动都是影响病毒传播的重要因素。根据研究结果,我们给出了预防和控制病毒传播的方案,该方案与SARS爆发时政府制定的方案一致,说明该研究对病毒的控制与防御具有重要的指导意义。
(4)提出了一个网络结构演化与病毒传播同时进行且互相作用的自适应网络模型,研究了该系统在不同条件下形成的网络结构以及病毒传播效果。研究结果表明:当重连概率等于0,而感染概率、治愈概率和恢复概率大于0时,系统保持原来的随机网络结构不变,最终染病节点的数目会演化到一个定值。当重连概率大于0,而感染概率、治愈概率和恢复概率等于0时,系统将分裂成两个独立的随机网络。当重连概率、感染概率、治愈概率和恢复概率都大于0时,系统最终会演变成一个度分布较宽的网络结构,此时最终染病节点的数目会发生振荡,并且在一定的参数范围内系统存在双稳状态(分叉现象)。研究还发现,节点的重连并不能彻底阻止病毒的传播,这说明要想阻止病毒在系统中的传播不能只靠系统中个体自发的躲避行为,更需要管理者从整体上制定其他切实可行的预防和控制措施。
(5)研究了二维规则网络和BA无标度网络中含能量耗散和扩容功能的故障传播行为。研究发现,网络中的能量耗散和容量的扩充是导致系统向自组织临界状态发展的重要因素,该结论很好地解释了电力网络故障中的自组织临界现象。此外,还进一步研究了网络级联故障的控制策略,发现只要对网络中少数关键的节点进行安全控制,就能避免网络中大规模级联故障的发生,该结论对网络级联故障的控制具有重要的应用价值。
Many complex systems in the world can be described by networks made up of interactive individuals. Since the small-world effect and scale-free character of the real network were discovered, complex network has become one of the most rapidly developing areas. At present, the research of the complex network has already permeated through different fields such as mathematics, physics, engineering and life. The networks involved includes traffic network, electric network, Internet, financial network, protein function network, etc. We can say that complex network theory has already become an essential tool for people to study complex system.
The study of complex networks, on one hand is helpful for people to find out about the structural characteristic of the real networks and their formation mechanism, on the other hand is useful for people to recognize the dynamical process in the complex networks, which is of great importance for people to optimize and control the dynamic process in the real networks.
Over the past years, scholars from different fields have carried extensive research on the structure and the dynamical process of complex networks. However, there are still a lot of problems worthing further studying for the complexity of the real system. In this thesis, several closely linked works such as the evolving models of network, the spreading of virus and the cascading errors on complex networks were carried out.
The main points and innovations of this thesis are as follows:
(1) Proposed a random-scale free unified network model based on random evolving network and scale-free network model. The random evolving network and scale-free network model are unified by a parameter p. When p is regulated from 0 to 1, the structure of network is changed from BA scale-free network to random evolving network. The mean-field method and computer simulation are separately used to study the distribution of the degree. The results are more precise than the previous results. In addition, we extended this model to a unified network, in which the evolution can take place both inside and outside of the network. This researches discovered the evolution mechanism of the real network which is helpful for people to understand the growth procedure of real network.
(2) Studied virus spreading process based on random-scale free unified network. The result shows that when the scale of the network is very large and the parameter p is positive, the threshold of the spreading efficiency will be a positive number. Only when the spreading efficiency is greater than the threshold, can viruses spread in the network continuously. When p is close to 0, the threshold will be close to 0, which means that a weak spreading source can induce the virus spread in the network continuously. This result proves that the structure of the network is an important factor influencing the virus to spread. The result is of great directive significance for people to design the network. During the process of network designing, we can change the structure of the network by regulating the parameter p, so as to prevent the spreading of virus.
(3) Proposed the SIS (susceptible-infected-susceptible) and SIRS (susceptible -infected-recovered-susceptible) virus spreading models based on two dimensional lattices, the moving of individuals and the colony density is considered during the study procedure. The theory analysis and computer simulation shows that colony density, spreading efficiency and the moving of individuals are all important factors that influent the spreading process of virus. According to the result, we provided some schemes to control the spreading of virus, and these schemes keep the same with schemes which the government makes when SARS break out.
(4) Proposed an adaptive network model, in which the evolution of the network's structure and the spreading process of the virus can go on at the same time. We investigated the network's structure and the virus' spreading effect in different conditions. The results show that when re-link probability is set to 0 and the iinfected probability, cure probability and recover probability are all above 0, the system will maintain a random network structure and the final infected rate will achieve a fixed value. When re-link probability is greater than 0 and the infected probability, cure probability and recover probability equal to 0, the system will be split into two independent random networks. When re-link probability, the infected probability, cure probability and recover probability are all above 0, the system will evolve to a broad-scale network. In this case the final infected rate will be oscillated, at the same time the bistable state is observed when the parameters are set at proper values. We also discovered that the re-link of network can not completely stop the spreading of virus, which shows that the individuals' spontaneous escaping acts can not stop virus spreading in the network.
(5) Studied errors spreading process based on two dimensional regular network and BA scale-free network. During this processes energy dissipation and capability enlarging is allowed. The result shows that energy dissipation and capability enlarging are two important factors which lead the system develop toward SOC (self-organized criticality) status. This conclusion well explained the SOC phenomenon exists in electric network. In addition, the controlling strategies for avoiding cascading errors of networks are also studied. The result shows that as far as a few key nodes are protected in the network the large cascading errors will be avoided. This conclusion is of great importance to control the cascading errors of network.
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