复杂网络中若干模型上的传播特性研究
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摘要
复杂网络研究已经渗透到数理科学、生命科学和工程科学等众多不同的领域，成为近年来的一个研究热点。在复杂网络研究中，复杂网络的网络模型和其上的传播特性是其中的两个重要研究方向。在过去的几年里，学者们在这两个方面作了大量的研究，提出了许多切合实际的网络模型和传播模型。虽然取得了丰硕的成果，但是还有很多需要研究和解决的问题。本论文将运用统计物理、运筹学以及计算机仿真等方法，对复杂网络模型和网络上的传播特性两个方面进行研究。主要内容和创新之处概述如下：
1、研究局域世界网络中谣言传播的特性
谣言是现实社会中一种非常普遍的现象，在突发事件、乃至各种危机中作用不可低估。局域世界是很多实际网络中普遍存在的结构，例如在世界贸易网中，许多国家都着重于加强与各自区域经济合作组织（包括欧洲联盟、东南亚国家联盟、北美自由贸易区等等）内部的国家之间的经济合作和贸易关系。在人类社会网络中，每个人实际上也生活在各自的局域世界中，有自己的朋友圈。利用两个衡量谣言传播能力的指标来考察谣言在局域世界网络中的传播特性。计算机仿真结果表明，在传播概率不变的情况下，局域世界规模和免疫概率都对谣言传播具有非常重要的影响。
2、提出一个新的带有偏好的信息传播模型，研究其在小世界网络和无标度网络中的传播特性
偏好是决策分析中的一个重要依据，决策通常是以个人的偏好为基础的。对于朋友提供的同一条信息，有的人感兴趣从而愿意传播，有的人不感兴趣而不愿意传播。也就是说对于同一条信息人们是否感兴趣并且传播与他们的偏好有关，而偏好程度可以用效用函数值来进行量化。文中将效用函数引入信息传播中，提出了一个新的带有偏好的信息传播模型。利用计算机仿真来研究在小世界网络和无标度网络中信息传播的特性，主要揭示了效用值与信息传播能力的关系。
3、提出三个新的传染病传播模型，研究其在小世界网络和无标度网络中的传播特性
真实流行病学中康复个体的免疫力强弱不同，有些康复个体会失去全部免疫力变成易感个体,但是也有些康复个体保留部分的免疫力（即免疫力减弱）变成低易感个体,低易感个体如果和感染个体接触不会马上被感染,只是失去这部分免疫力变成易感个体。针对这种感染机制，首先提出了一种新的符合实际的传染病传播模型——具有低易感的传染病模型（即SIRSLS模型）。运用平均场方法推导出该模型在小世界网络和无标度网络中的传播阈值。小世界网络中具有非零的传播阈值，而对于网络规模无限大的无标度网络，传播阈值为零。另一方面在传染病的传播过程中每个时刻每个节点不可能与其所有邻点接触，假设节点具有相同的传染能力，又提出两个新的传染病传播模型——传染能力相同的SIRS模型和传染能力相同的SIRSLS模型。运用平均场方法推导出这两个模型在无标度网络中的传播阈值。不同于SIRSLS模型，对于这两个模型，即使对于网络规模无限大的无标度网络,仍存在非零的传播阈值。通过对传染病传播过程的仿真研究，发现理论结果和仿真结果都吻合得很好。
4、提出一个新的按边权择优的加权局域世界网络模型，并且研究该网络中的传染病传播特性
复杂网络模型是模拟和研究现实网络的抽象对象，包括无权网络模型和加权网络模型。无权网络模型中只是说明节点之间边的存在性，而每条边的重要性都被认为是相同的，这一点与现实网络并不完全相符，例如，科学家合作网络中，即使有合作关系，不同的合作者合作的次数是不一样的；航空网中，即使有机场，不同机场间航班的载客量也是有区别的。因此，加权网络模型能更好地描述真实网络。考虑到科学家合作网络中密集的合作成果也会吸引更多的研究者加入，也就是说是边而不是节点吸引了新的连接，提出了一个新的按边权择优的加权局域世界网络模型。利用平均场方法解析推导出边权分布、点强度分布和度分布，并且计算机仿真得到的结果和解析结果吻合。更重要的是构造的新网络模型的边权可以很好地模拟线虫神经网络和在线社会网络这两种实证网络。最后分析了该网络中传染病传播的特性。
The research of complex networks has penetrated into many different fields includingMathematical and Physical Sciences, Life Sciences and Engineering Sciences and has become ahot research topic in recent years. There are two important research directions in complexnetworks----network models and spreading properties on networks. In the past few years,researchers have carried out considerable studies on the two directions and proposed manypractical network models and spreading models. Although fruitful achievements have been made,many problems are still to be studied and solved. In this paper, complex network models andspreading properties on networks are studied by using statistical physics, operational researchand computer simulation. The main contents and originalities of this paper can be summarized asfollows:
1. Studing the rumor spreading properties on the local-world network
Rumor is a very universal phenomenon in real society and its effect cannot beunderestimated in emergency and many crises. A local-world is a ubiquitous structure in manyrealistic networks. For example, in the World Trade Web, many countries emphatically acceleratetheir economy collaborations in various regional economy cooperative organizations (such as theEuropean Union, Association of Southeast Asian Nations, North America Free Trade Area, etc).And in human social, in fact everyone lives in the local-world of their own and has their owncircle of friends. The rumor spreading properties on the local-world network are studied by usingtwo indexes which measure the rumor spreading ability. The results indicate that the local-worldscale and the immune probability both play important roles in rumor propagation under the fixedtransmission probability.
2. Proposing a new information spreading model with preference and studing the spreadingproperties on small-world networks and scale-free networks
Preference is very important and decisions are usually based on the personal preference. For the same information offered by a certain friend, some people are interested in it and are willingto spread it, but some people are not interested in it and are not willing to spread it. That is to say,whether people are interested in and propagate the information depends on their preferences,while the extent can be quantified by using the utility function value. In this paper the utilityfunction is introduced to the information dissemination and a new information spreading modelwith preference is put forward. The information spreading properties on small-world networksand scale-free networks are studied by computer simulation. The relationship between the utilityvalue and the information spreading ability is revealed mainly.
3. Proposing three new epidemic spreading models and studing the spreading properties onsmall-world networks and scale-free networks
In real epidemiology, the immunity of removed individual is different. Some removedindividuals will lose all the immunity and become susceptible individuals, but some removedindividuals will retain partial immunity (i.e., weakened immunity) and become lowersusceptible individuals. A lower susceptible individual will not be infected immediately if itmakes contact with infected individuals. It just loses the partial immunity and becomessusceptible individual. First of all, based on this mechanism of infection, a new practicalepidemic spreading model—epidemic model with lower susceptibility (i.e., SIRSLS model) isproposed. The epidemic threshold of this model on small-world networks and scale-freenetworks are deduced by using the mean field method. There exists a nonzero epidemicthreshold on small-world networks and the threshold is null if the size of scale-free networks issufficiently large. For another, in the process of epidemic spreading, each individual will notcontact all its neighbors once at each time step. Thus supposed that every individual has thesame infectivity, two new epidemic spreading models—SIRS model with identical infectivityand SIRSLS model with identical infectivity are proposed. The epidemic thresholds of the twomodels on scale-free networks are deduced through the mean field method. Different formSIRSLS model, the thresholds of the two models are nonzero even if the size of scale-freenetworks is sufficiently large. Through the computer simulation research on epidemic spreadingprocess, we find that the theoretical results and the simulation results are in good agreement.
4. Proposing a new weighted local-world network evolving model based on the edgepreferential selection and studing the epidemic spreading properties on this network
Complex network model is an abstract object to simulate and study realistic networkswhich including unweighted network model and weighted network model. In the unweightednetwork model, only the existence of edges between nodes is showed, but the important of eachedge is known as identical. It is not correspond exactly to the realistic networks. For instance, inthe scientific collaboration network, even with the cooperation relations, the number ofcoauthored publications of different parters is unequal and in the airline transportation network,even with the airport, the passenger capacity on this route between different airports is alsounequal. So, it is better to describe some real networks by using the weighted networks. Theintensive collaborations have a greater chance to attract new collaborators than occasionalconnections in the scientific collaboration network. In other words, it is the edge but not nodethat attracts new links. Therefore, based on this phenomenon, a new weighted local-networkmodel based on the edge preferential selection is proposed. The weight distribution, the strengthdistribution and the degree distribution of this model are deduced by using the mean fieldmethod. All results of the computer simulation are well consistent with the analytical results.What’s more important is that the weight dynamics of the new model can simulate somerealistic networks such as Neural network of the nematode C. Elegans and Online SocialNetwork. Finally the epidemic spreading properties on this new network are analyzed.
1、研究局域世界网络中谣言传播的特性
谣言是现实社会中一种非常普遍的现象，在突发事件、乃至各种危机中作用不可低估。局域世界是很多实际网络中普遍存在的结构，例如在世界贸易网中，许多国家都着重于加强与各自区域经济合作组织（包括欧洲联盟、东南亚国家联盟、北美自由贸易区等等）内部的国家之间的经济合作和贸易关系。在人类社会网络中，每个人实际上也生活在各自的局域世界中，有自己的朋友圈。利用两个衡量谣言传播能力的指标来考察谣言在局域世界网络中的传播特性。计算机仿真结果表明，在传播概率不变的情况下，局域世界规模和免疫概率都对谣言传播具有非常重要的影响。
2、提出一个新的带有偏好的信息传播模型，研究其在小世界网络和无标度网络中的传播特性
偏好是决策分析中的一个重要依据，决策通常是以个人的偏好为基础的。对于朋友提供的同一条信息，有的人感兴趣从而愿意传播，有的人不感兴趣而不愿意传播。也就是说对于同一条信息人们是否感兴趣并且传播与他们的偏好有关，而偏好程度可以用效用函数值来进行量化。文中将效用函数引入信息传播中，提出了一个新的带有偏好的信息传播模型。利用计算机仿真来研究在小世界网络和无标度网络中信息传播的特性，主要揭示了效用值与信息传播能力的关系。
3、提出三个新的传染病传播模型，研究其在小世界网络和无标度网络中的传播特性
真实流行病学中康复个体的免疫力强弱不同，有些康复个体会失去全部免疫力变成易感个体,但是也有些康复个体保留部分的免疫力（即免疫力减弱）变成低易感个体,低易感个体如果和感染个体接触不会马上被感染,只是失去这部分免疫力变成易感个体。针对这种感染机制，首先提出了一种新的符合实际的传染病传播模型——具有低易感的传染病模型（即SIRSLS模型）。运用平均场方法推导出该模型在小世界网络和无标度网络中的传播阈值。小世界网络中具有非零的传播阈值，而对于网络规模无限大的无标度网络，传播阈值为零。另一方面在传染病的传播过程中每个时刻每个节点不可能与其所有邻点接触，假设节点具有相同的传染能力，又提出两个新的传染病传播模型——传染能力相同的SIRS模型和传染能力相同的SIRSLS模型。运用平均场方法推导出这两个模型在无标度网络中的传播阈值。不同于SIRSLS模型，对于这两个模型，即使对于网络规模无限大的无标度网络,仍存在非零的传播阈值。通过对传染病传播过程的仿真研究，发现理论结果和仿真结果都吻合得很好。
4、提出一个新的按边权择优的加权局域世界网络模型，并且研究该网络中的传染病传播特性
复杂网络模型是模拟和研究现实网络的抽象对象，包括无权网络模型和加权网络模型。无权网络模型中只是说明节点之间边的存在性，而每条边的重要性都被认为是相同的，这一点与现实网络并不完全相符，例如，科学家合作网络中，即使有合作关系，不同的合作者合作的次数是不一样的；航空网中，即使有机场，不同机场间航班的载客量也是有区别的。因此，加权网络模型能更好地描述真实网络。考虑到科学家合作网络中密集的合作成果也会吸引更多的研究者加入，也就是说是边而不是节点吸引了新的连接，提出了一个新的按边权择优的加权局域世界网络模型。利用平均场方法解析推导出边权分布、点强度分布和度分布，并且计算机仿真得到的结果和解析结果吻合。更重要的是构造的新网络模型的边权可以很好地模拟线虫神经网络和在线社会网络这两种实证网络。最后分析了该网络中传染病传播的特性。
The research of complex networks has penetrated into many different fields includingMathematical and Physical Sciences, Life Sciences and Engineering Sciences and has become ahot research topic in recent years. There are two important research directions in complexnetworks----network models and spreading properties on networks. In the past few years,researchers have carried out considerable studies on the two directions and proposed manypractical network models and spreading models. Although fruitful achievements have been made,many problems are still to be studied and solved. In this paper, complex network models andspreading properties on networks are studied by using statistical physics, operational researchand computer simulation. The main contents and originalities of this paper can be summarized asfollows:
1. Studing the rumor spreading properties on the local-world network
Rumor is a very universal phenomenon in real society and its effect cannot beunderestimated in emergency and many crises. A local-world is a ubiquitous structure in manyrealistic networks. For example, in the World Trade Web, many countries emphatically acceleratetheir economy collaborations in various regional economy cooperative organizations (such as theEuropean Union, Association of Southeast Asian Nations, North America Free Trade Area, etc).And in human social, in fact everyone lives in the local-world of their own and has their owncircle of friends. The rumor spreading properties on the local-world network are studied by usingtwo indexes which measure the rumor spreading ability. The results indicate that the local-worldscale and the immune probability both play important roles in rumor propagation under the fixedtransmission probability.
2. Proposing a new information spreading model with preference and studing the spreadingproperties on small-world networks and scale-free networks
Preference is very important and decisions are usually based on the personal preference. For the same information offered by a certain friend, some people are interested in it and are willingto spread it, but some people are not interested in it and are not willing to spread it. That is to say,whether people are interested in and propagate the information depends on their preferences,while the extent can be quantified by using the utility function value. In this paper the utilityfunction is introduced to the information dissemination and a new information spreading modelwith preference is put forward. The information spreading properties on small-world networksand scale-free networks are studied by computer simulation. The relationship between the utilityvalue and the information spreading ability is revealed mainly.
3. Proposing three new epidemic spreading models and studing the spreading properties onsmall-world networks and scale-free networks
In real epidemiology, the immunity of removed individual is different. Some removedindividuals will lose all the immunity and become susceptible individuals, but some removedindividuals will retain partial immunity (i.e., weakened immunity) and become lowersusceptible individuals. A lower susceptible individual will not be infected immediately if itmakes contact with infected individuals. It just loses the partial immunity and becomessusceptible individual. First of all, based on this mechanism of infection, a new practicalepidemic spreading model—epidemic model with lower susceptibility (i.e., SIRSLS model) isproposed. The epidemic threshold of this model on small-world networks and scale-freenetworks are deduced by using the mean field method. There exists a nonzero epidemicthreshold on small-world networks and the threshold is null if the size of scale-free networks issufficiently large. For another, in the process of epidemic spreading, each individual will notcontact all its neighbors once at each time step. Thus supposed that every individual has thesame infectivity, two new epidemic spreading models—SIRS model with identical infectivityand SIRSLS model with identical infectivity are proposed. The epidemic thresholds of the twomodels on scale-free networks are deduced through the mean field method. Different formSIRSLS model, the thresholds of the two models are nonzero even if the size of scale-freenetworks is sufficiently large. Through the computer simulation research on epidemic spreadingprocess, we find that the theoretical results and the simulation results are in good agreement.
4. Proposing a new weighted local-world network evolving model based on the edgepreferential selection and studing the epidemic spreading properties on this network
Complex network model is an abstract object to simulate and study realistic networkswhich including unweighted network model and weighted network model. In the unweightednetwork model, only the existence of edges between nodes is showed, but the important of eachedge is known as identical. It is not correspond exactly to the realistic networks. For instance, inthe scientific collaboration network, even with the cooperation relations, the number ofcoauthored publications of different parters is unequal and in the airline transportation network,even with the airport, the passenger capacity on this route between different airports is alsounequal. So, it is better to describe some real networks by using the weighted networks. Theintensive collaborations have a greater chance to attract new collaborators than occasionalconnections in the scientific collaboration network. In other words, it is the edge but not nodethat attracts new links. Therefore, based on this phenomenon, a new weighted local-networkmodel based on the edge preferential selection is proposed. The weight distribution, the strengthdistribution and the degree distribution of this model are deduced by using the mean fieldmethod. All results of the computer simulation are well consistent with the analytical results.What’s more important is that the weight dynamics of the new model can simulate somerealistic networks such as Neural network of the nematode C. Elegans and Online SocialNetwork. Finally the epidemic spreading properties on this new network are analyzed.
引文
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