网络逾渗与级联故障
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摘要
近年来,随着科学技术的发展,复杂网络科学开始受到越来越多的关注。这个学科新兴的原因是多方面的。首先,日常生活中就有各种各样网络,例如通信网络、互联网、人际网,以及人体内的神经、代谢等网络。发展科技的目的是为了解释自然、服务生活,所以科技的车轮是不会遗忘这片未知的土地的。其次,在以物理、化学、生物为首的自然科学的迅速发展中,已经大量涉及到经典网格模型所不能描述的物质相互作用特性,所以更为复杂和实际的描述方式——复杂网络,也由此走上了科学的历史舞台。
网络这个概念很宽泛,可以从有形的交通网络到无形的人际关系网络,可以从社会学、计算机科学到生物学、物理学。一般地,多体相互作用系统均可视为一个网络。复杂网络科学的观点正是从这些形形色色的系统中抽象出一般的网络特征,进而进行系统结构与动力学性质的研究。这种研究方式可以抛开系统的具体组成(如人、细胞、计算机、网页、粒子等),研究元素间的相互作用过程(如合作、代谢、通信、链接、作用力等)。从另一个角度来看,作为一个多体相互作用系统,网络也可以被纳入统计物理的范畴。随着计算机技术的发展,海量数据的收集与处理已经成为可能,这为统计物理理论的验证、应用和发展提供了一个新舞台。在现今网络科学的研究中,传统的统计物理理论已经得到了大量的应用和发展,并取得了卓越的成绩。诸如系综理论、相变理论、平均场方法,以及Ising模型、逾渗模型等已经成为一些复杂网络研究方向的基本方法与模型。
网络科学的一个重要研究课题就是网络的结构特点及其稳定性,即一个网络的连接性质及其在外部扰动下的变化特点。这方面的研究成果可以解释很多实际问题,例如何种结构的电力网络会在一些电站发生故障时,仍能保障全局的电力供应。如果我们能对这个问题有更深层次的认识,就可以通过合理的网络规划来避免类似2003年意大利全国大停电的事件发生,从而避免大量的经济损失。在这方面的研究中,也大量地应用了统计物理中的相变理论。找到描述网络结构的序参量,通过调节控制变量,我们就可以研究网络从功能态到瘫痪态的变化。网络结构稳定性的研究,也就抽象成系统相变过程的研究了。另外,逾渗模型与沙堆模型也常被用来研究网络中故障的传播,即级联故障过程。
近期对网络结构与级联故障的研究中,节点之间的相依性以相依边的形式被引入网络逾渗模型。这种边的引入,使得网络更加脆弱,并且当相依边数量较多时,系统展现出不连续相变。我们知道经典网络逾渗的相变是连续的,所以这个不连续逾渗相变模型得到了大量的关注。我们的研究抛开了相依边与连接边的独立假设,进而考虑了两种边的重叠对网络逾渗过程的影响。研究发现,节点之间的相依并不总是使得相变不连续。在较高的重叠率下,即使网络中含有大量的相依边,逾渗过程依然呈现出连续相变。因此,当重叠率由低到高变化时,系统展现出两种相变过程,即从一阶相变转化成二阶相变。这也说明相依边数量并不是系统相变类型的唯一决定因素,相依节点之间的连接性也是一个重要因素。利用生成函数技术,我们解析得到的序参量、相变点、三相点等值都与模拟结果精确吻合。
另外,之前对含有相依边的网络逾渗的研究都假设相依边是双向的,即相依边连接的两个节点相互依赖。而实际中,依赖也常以不对称的形式出现,例如经济网络中,一个小公司依赖于一个大公司,而反过来却不一定成立。考虑到这一点,我们研究了偏向依赖对网络鲁棒性的影响。在我们的模型中,相依边连接的两个节点的依赖并不一定是相互的,而是由其度之间的关系决定。总体上说,一个节点更倾向于依赖一个度比自身大的节点,而独立于一个度比自身小的节点。研究发现网络的鲁棒性受这种不对称依赖的影响很小。如果网络中没有对称相依,即使网络中存在大量的不对称相依,网络的鲁棒性也不会明显下降。但是一旦网络中有对称相依,那么网络的鲁棒性就会迅速下降。此外,在这个模型中也发现了两种相变过程,即一阶与二阶相变。当网络中没有对称相依时,系统比较鲁棒,逾渗过程呈现出二阶相变。当网络中含有一定比例的对称相依时,系统会变得很脆弱,同时逾渗过程也转变为一阶相变。对于该模型,我们也给出了相应的理论分析,理论结果与模拟结果吻合得很好。
除以上两个工作外,本文还将对复杂网络的基本概念与参量,以及其求解方法进行了介绍与总结。并系统地回顾和总结了网络逾渗的求解方法与应用。
In recent years, much researches have been carried out to explore the structural properties and dynamics of complex networks. Why so much focus on complex net-works? Firstly, networks are ubiquitous in almost every aspect of our lives, such as com-munication networks, Internet, relationship networks, neural networks and metabolic networks. Understanding how these networks work is highly meaningful for improving the qualities of our lives, that's exactly what the scientists like to do. Secondly, in mod-ern physics and other related disciplines, networks with simple topologies can no longer meet the requirements of describing the structural relationships among the components of a system. Therefore, complex networks must be studied to give a new tool to deal with these problems.
There are many networks of interest to scientists that are composed of individual parts or components linked together in some way. For example, the Internet is a col-lection of computers linked by data connections. As also noted, human societies are collections of people linked by acquaintance or social interaction. In a concept of com-plex networks, they all can be presented as a network, the components of the system being the network nodes and the connections or interactions the links. In this way, the systems can be studied mathematically. In another perspective, these networks are all many-body systems, which can be studied using statistical mechanics. Facing the gi-gantic network information, the physics approach may be the most advantageous for the understanding of the structures and dynamics of networks. Actually, many approaches used today in complex networks are a directed generalization of the classical methods in statistical mechanics, such as ensemble theory, phase transition theory, mean filed method, Ising model and percolation model.
One of the important topics in complex networks is the robustness of networks. This study focuses on the vulnerability of a network after a fraction of nodes are re-moved. This helps us gain a deeper understanding of the vulnerability of the real net-works, such as the blackout that affected much of Italy on2003. On the other hand, this study can provide us some methods to make the networks more robust, which may avoid similar incidents happening again. Theoretically, we often use phase transition theory to explains the existence of the giant component of a network after a fraction of nodes are removed. In addition, sandpile model and percolation model are also used to study the cascading failures on networks.
In recent studies, dependence links have been proposed to the percolation model and used to study the robustness of the networks with such links, which shows that the networks are more vulnerable than the classical networks containing only connectiv-ity links. This model usually demonstrates a first order phase transition, rather than the second order phase transition found in the classical network percolation. Consid-ering the real situation that the interdependent nodes are usually connected, we study the cascading dynamics of networks when the dependence links partially overlap with the connectivity links. We find that the percolation transitions are not always sharpened by making nodes interdependent. For a high fraction of overlapping, the network is robust for random failures, and the percolation transition is second order, while for a low fraction of overlapping, the percolation process shows a first order phase transition. This work demonstrates that the crossover between two types of transitions does not only depend on the density of the dependence links but also on the overlapping of the connectivity and dependence links. Using generating functional techniques, we present exact solutions for the size of the giant component and the critical point, which are in good agreement with the simulations.
In addition, in order to properly model the dependence of nodes in real networks, we have studied the cascading failures on a network with asymmetric dependence. Instead of the unconditional dependence of two node connected by the dependence links, the de-pendence threshold is used to determine the dependence of nodes. In our model, a node tends to be dependent on a node with a larger degree. This could represent some real re-lations among interacting agents in a networked system. We find that when the networks contain only the asymmetric dependence links, the systems are robust and demonstrate a second order phase transition. Both simulation and analytical results reveal the exis-tence of the crossover between the first and second order percolation transitions in our model. We also develop an approach to study percolation on such networks, which is in agreement with the simulation results well.
Besides, we will review the basic concepts of complex networks in this article, and give a summary of the theoretical approaches to study the percolation and cascading process on networks.
网络这个概念很宽泛,可以从有形的交通网络到无形的人际关系网络,可以从社会学、计算机科学到生物学、物理学。一般地,多体相互作用系统均可视为一个网络。复杂网络科学的观点正是从这些形形色色的系统中抽象出一般的网络特征,进而进行系统结构与动力学性质的研究。这种研究方式可以抛开系统的具体组成(如人、细胞、计算机、网页、粒子等),研究元素间的相互作用过程(如合作、代谢、通信、链接、作用力等)。从另一个角度来看,作为一个多体相互作用系统,网络也可以被纳入统计物理的范畴。随着计算机技术的发展,海量数据的收集与处理已经成为可能,这为统计物理理论的验证、应用和发展提供了一个新舞台。在现今网络科学的研究中,传统的统计物理理论已经得到了大量的应用和发展,并取得了卓越的成绩。诸如系综理论、相变理论、平均场方法,以及Ising模型、逾渗模型等已经成为一些复杂网络研究方向的基本方法与模型。
网络科学的一个重要研究课题就是网络的结构特点及其稳定性,即一个网络的连接性质及其在外部扰动下的变化特点。这方面的研究成果可以解释很多实际问题,例如何种结构的电力网络会在一些电站发生故障时,仍能保障全局的电力供应。如果我们能对这个问题有更深层次的认识,就可以通过合理的网络规划来避免类似2003年意大利全国大停电的事件发生,从而避免大量的经济损失。在这方面的研究中,也大量地应用了统计物理中的相变理论。找到描述网络结构的序参量,通过调节控制变量,我们就可以研究网络从功能态到瘫痪态的变化。网络结构稳定性的研究,也就抽象成系统相变过程的研究了。另外,逾渗模型与沙堆模型也常被用来研究网络中故障的传播,即级联故障过程。
近期对网络结构与级联故障的研究中,节点之间的相依性以相依边的形式被引入网络逾渗模型。这种边的引入,使得网络更加脆弱,并且当相依边数量较多时,系统展现出不连续相变。我们知道经典网络逾渗的相变是连续的,所以这个不连续逾渗相变模型得到了大量的关注。我们的研究抛开了相依边与连接边的独立假设,进而考虑了两种边的重叠对网络逾渗过程的影响。研究发现,节点之间的相依并不总是使得相变不连续。在较高的重叠率下,即使网络中含有大量的相依边,逾渗过程依然呈现出连续相变。因此,当重叠率由低到高变化时,系统展现出两种相变过程,即从一阶相变转化成二阶相变。这也说明相依边数量并不是系统相变类型的唯一决定因素,相依节点之间的连接性也是一个重要因素。利用生成函数技术,我们解析得到的序参量、相变点、三相点等值都与模拟结果精确吻合。
另外,之前对含有相依边的网络逾渗的研究都假设相依边是双向的,即相依边连接的两个节点相互依赖。而实际中,依赖也常以不对称的形式出现,例如经济网络中,一个小公司依赖于一个大公司,而反过来却不一定成立。考虑到这一点,我们研究了偏向依赖对网络鲁棒性的影响。在我们的模型中,相依边连接的两个节点的依赖并不一定是相互的,而是由其度之间的关系决定。总体上说,一个节点更倾向于依赖一个度比自身大的节点,而独立于一个度比自身小的节点。研究发现网络的鲁棒性受这种不对称依赖的影响很小。如果网络中没有对称相依,即使网络中存在大量的不对称相依,网络的鲁棒性也不会明显下降。但是一旦网络中有对称相依,那么网络的鲁棒性就会迅速下降。此外,在这个模型中也发现了两种相变过程,即一阶与二阶相变。当网络中没有对称相依时,系统比较鲁棒,逾渗过程呈现出二阶相变。当网络中含有一定比例的对称相依时,系统会变得很脆弱,同时逾渗过程也转变为一阶相变。对于该模型,我们也给出了相应的理论分析,理论结果与模拟结果吻合得很好。
除以上两个工作外,本文还将对复杂网络的基本概念与参量,以及其求解方法进行了介绍与总结。并系统地回顾和总结了网络逾渗的求解方法与应用。
In recent years, much researches have been carried out to explore the structural properties and dynamics of complex networks. Why so much focus on complex net-works? Firstly, networks are ubiquitous in almost every aspect of our lives, such as com-munication networks, Internet, relationship networks, neural networks and metabolic networks. Understanding how these networks work is highly meaningful for improving the qualities of our lives, that's exactly what the scientists like to do. Secondly, in mod-ern physics and other related disciplines, networks with simple topologies can no longer meet the requirements of describing the structural relationships among the components of a system. Therefore, complex networks must be studied to give a new tool to deal with these problems.
There are many networks of interest to scientists that are composed of individual parts or components linked together in some way. For example, the Internet is a col-lection of computers linked by data connections. As also noted, human societies are collections of people linked by acquaintance or social interaction. In a concept of com-plex networks, they all can be presented as a network, the components of the system being the network nodes and the connections or interactions the links. In this way, the systems can be studied mathematically. In another perspective, these networks are all many-body systems, which can be studied using statistical mechanics. Facing the gi-gantic network information, the physics approach may be the most advantageous for the understanding of the structures and dynamics of networks. Actually, many approaches used today in complex networks are a directed generalization of the classical methods in statistical mechanics, such as ensemble theory, phase transition theory, mean filed method, Ising model and percolation model.
One of the important topics in complex networks is the robustness of networks. This study focuses on the vulnerability of a network after a fraction of nodes are re-moved. This helps us gain a deeper understanding of the vulnerability of the real net-works, such as the blackout that affected much of Italy on2003. On the other hand, this study can provide us some methods to make the networks more robust, which may avoid similar incidents happening again. Theoretically, we often use phase transition theory to explains the existence of the giant component of a network after a fraction of nodes are removed. In addition, sandpile model and percolation model are also used to study the cascading failures on networks.
In recent studies, dependence links have been proposed to the percolation model and used to study the robustness of the networks with such links, which shows that the networks are more vulnerable than the classical networks containing only connectiv-ity links. This model usually demonstrates a first order phase transition, rather than the second order phase transition found in the classical network percolation. Consid-ering the real situation that the interdependent nodes are usually connected, we study the cascading dynamics of networks when the dependence links partially overlap with the connectivity links. We find that the percolation transitions are not always sharpened by making nodes interdependent. For a high fraction of overlapping, the network is robust for random failures, and the percolation transition is second order, while for a low fraction of overlapping, the percolation process shows a first order phase transition. This work demonstrates that the crossover between two types of transitions does not only depend on the density of the dependence links but also on the overlapping of the connectivity and dependence links. Using generating functional techniques, we present exact solutions for the size of the giant component and the critical point, which are in good agreement with the simulations.
In addition, in order to properly model the dependence of nodes in real networks, we have studied the cascading failures on a network with asymmetric dependence. Instead of the unconditional dependence of two node connected by the dependence links, the de-pendence threshold is used to determine the dependence of nodes. In our model, a node tends to be dependent on a node with a larger degree. This could represent some real re-lations among interacting agents in a networked system. We find that when the networks contain only the asymmetric dependence links, the systems are robust and demonstrate a second order phase transition. Both simulation and analytical results reveal the exis-tence of the crossover between the first and second order percolation transitions in our model. We also develop an approach to study percolation on such networks, which is in agreement with the simulation results well.
Besides, we will review the basic concepts of complex networks in this article, and give a summary of the theoretical approaches to study the percolation and cascading process on networks.
引文
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