BA模型的数学基础——纪念李国平院士吴新谋教授诞辰100周年

该文基于马氏链的概念和技巧, 给出了BA无标度网络模型稳态度分布存在性的严格证明, 并且从数学上重新推导了度分布的精确解析表达式. 此处所用的方法具有一定的普适性, 适用于更一般的无标度型复杂网络模型.     

随机增长网络的度分布

增长和择优机制是无标度网络中的两种重要的演化机制,在分析BA模型的基础上,提出了一种新的节点增长方式,即考虑了新增节点的连边数是随机变量的情况,从而建立了随机增长网络模型,并利用随机过程理论得到了在这种增长方式下网络的度分布,结果表明这个网络是无标度网络。     

无标度网络演化过程中的结构熵

熵是度量复杂系统无序性的重要物理量,而且现实中的大多数网络都呈现出无标度网络的特性.在网络的节点熵和结构熵概念的基础上,给出了BA模型的网络结构熵演化的解析结论和数值模拟.从解析结论和数值模拟可以得到,网络结构熵随网络大小以对数的速度增长;但在同样规模下,无标度网络的结构熵小于随机网络的结构熵.     

基于二项分布随机增长的无标度网络

提出—个具有随机增长的无标度网络模型．该模型的演化规则仍然是BA模型的增长和择优连接,但是每一时间间隔添加到网络中的边数是—个具有二项分布的随机变量．通过率方程方法,本文证明了该网络的度分布具有幂律尾部,该模型生成了—个无标度网络．     

M-G-P型复杂网络中有寿命节点的进入退出机制

现实中复杂网络结构复杂,形式多样,处在高度动态变化的过程.为了更好地理解真实网络的演化,基于复杂网络的特性进行分析,建立了Poissotn连续时间增长节点具有寿命的M-G-P型复杂网络模型,模型中包括:新节点加入、节点老化和老节点退出等,基于齐次马尔可夫链对模型的度分布进行计算,得出M-G-P型网络的度分布符合幂律分布,模型和BA模型一样能产生指数γ=3的无标度网络,验证了导致无标度网络度分布特征起关键性作用的是链接的偏好特性.     

加权无标度网络中的疾病传播

提出具有加权传播率和非线性传染能力的SIR模型和SIS模型,通过平均场方法证明了这两个模型在加权无标度网络中可以存在非零的传播阈值,从而传播率需要跨越更大的传播阈值才能流行.并且得到的结果在特殊情况下可退化为已有的一些经典结论.     

BA模型的数学基础

该文基于马氏链的概念和技巧,给出了BA无标度网络模型稳态度分布存在性的严格证明,并且从数学上重新推导了度分布的精确解析表达式.此处所用的方法具有一定的普适性,适用于更一般的无标度型复杂网络模型.     

考虑个体保护意识的关联信用风险传染机理研究

针对关联信用风险及其传染这一热点和难点问题,本文基于复杂网络异质平均场理论,运用风险传播动力学SIR经典模型,探讨风险信息促成的个体保护意识对关联信用风险传染的影响机理,并在BA无标度网络中进行数值仿真分析。研究结果表明:被感染个体数量、个体反应强度、有保护意识的易感个体比例与关联信用风险传染阈值正相关;考虑个体保护意识、增强易感个体反应强度以及提高有保护意识的易感个体比例能够有效抑制关联信用风险的传染速度和传染规模,并且能够延缓关联信用风险高峰期的到来。     

探讨几类具有完全图核心的网络模型的优美性

在复杂网络研究中,人们需要建立网络模型,无标度图就是这样的一种网络模型.我们发现具有完全图核心的网络模型可以演变成无标度图.具有完全图核心的几种网络模型的优美性得到研究.     

标度指数不大于2的无标度网络的若干性质

将标度指数不大于2的无标度网络称为亚标度网络.通过引入度秩函数研究了亚标度网络的最大度、平均度以及拓扑结构的非均匀性,通过与标度指数大于2的无标度网络对比,揭示了亚标度网络若干特殊性质.     

增长网络的形成机理和度分布计算

关于增长网络的形成机理,着重介绍由线性增长与择优连接组成的BA模型, 以及加速增长模型.此外,我们提出了一个含反择优概率删除旧连线的模型,这个模型能自组织演化成scale-free(SF)网络.关于计算SF网络的度分布,简要介绍文献上常用的基于连续性理论的动力学方法(包括平均场和率方程)和基于概率理论的主方程方法.另外,我们基于马尔可夫链理论还首次尝试了数值计算方法.这一方法避免了复杂方程的求解困难,所以较有普适性,因此可用于研究更为复杂的网络模型.我们用这种数值计算方法研究了一个具有对数增长的加速增长模型,这个模型也能自组织演化成SF网络.     

A social network model with proximity prestige property

In general, many real-world networks not only possess scale-free and high clustering coefficient properties, but also have a fast information transmission capability. However, the existing network models are unable to well present the intrinsic fast information transmission feature. The initial infected nodes and the network topology are two factors that affect the information transmission capability. By using preferential attachment to high proximity prestige nodes and triad formation, we provide a proximity prestige network model, which has scale-free property and high clustering coefficient. Simulation results further indicate that the new model also possesses tunable information transmission capability archived by adjusting its parameters. Moreover, comparing with the BA scale-free network, the proximity prestige network PPNet05 achieves a higher transmission capability when messages travel based on SIR and SIS models. Our conclusions are directed to possible applications in rumor or information spreading mechanisms.     

Vulnerability of complex networks

We consider normalized average edge betweenness of a network as a metric of network vulnerability. We suggest that normalized average edge betweenness together with is relative difference when certain number of nodes and/or edges are removed from the network is a measure of network vulnerability, called vulnerability index. Vulnerability index is calculated for four synthetic networks: Erdős–Rényi (ER) random networks, Barabási–Albert (BA) model of scale-free networks, Watts–Strogatz (WS) model of small-world networks, and geometric random networks. Real-world networks for which vulnerability index is calculated include: two human brain networks, three urban networks, one collaboration network, and two power grid networks. We find that WS model of small-world networks and biological networks (human brain networks) are the most robust networks among all networks studied in the paper.     

Naming Game with Multiple Hearers

A new model called Naming Game with Multiple Hearers (NGMH) is proposed in this paper. A naming game over a population of individuals aims to reach consensus on the name of an object through pair-wise local interactions among all the individuals. The proposed NGMH model describes the learning process of a new word, in a population with one speaker and multiple hearers, at each interaction towards convergence. The characteristics of NGMH are examined on three types of network topologies, namely ER random-graph network, WS small-world network, and BA scale-free network. Comparative analysis on the convergence time is performed, revealing that the topology with a larger average (node) degree can reach consensus faster than the others over the same population. It is found that, for a homogeneous network, the average degree is the limiting value of the number of hearers, which reduces the individual ability of learning new words, consequently decreasing the convergence time; for a scale-free network, this limiting value is the deviation of the average degree. It is also found that a network with a larger clustering coefficient takes longer time to converge; especially a small-word network with smallest rewiring possibility takes longest time to reach convergence. As more new nodes are being added to scale-free networks with different degree distributions, their convergence time appears to be robust against the network-size variation. Most new findings reported in this paper are different from that of the single-speaker/single-hearer naming games documented in the literature.     

随机BA模型的度分布

本文将马氏链首达概率方法应用于一个随机BA模型,得到这个模型度分布的精确表达式,并严格证明了度分布的存在性,同时说明择优连接对无标度特性的产生至关重要。     

The dynamics of spreading and immune strategies of sexually transmitted diseases on scale-free network

We examine epidemic threshold and dynamics for sexually transmitted diseases (STDs) spread using a multiple susceptible-infected-removed-susceptible ODE model on scale-free networks. We derive the threshold for the epidemic to be zero in infinite scale-free network. For a hard cut off scale-free network, we also prove the stability of disease-free equilibrium and the persistence of STDs infection. The effects of two immunization schemes, including proportional scheme and targeted vaccination, are studied and compared. We find that targeted strategy compare favorably to a proportional scheme in terms of effectiveness. Theory and simulations both prove that an appropriate condom using has prominent effect to control STDs spread on scale-free networks.     

Spreading of epidemics on scale-free networks with nonlinear infectivity

In this paper, we study the spreading of epidemics on scale-free networks with infectivity which is nonlinear in the connectivity of nodes. We will show that the nonlinear infectivity is more appropriate than constant or linear ones, and give the epidemic threshold of the SIS model on a scale-free network with nonlinear infectivity. In addition, we compare the effects of nonlinear infectivity on the epidemic threshold with two other cases on infinite and finite scale-free networks, and find some new results, such as: with unit recovery rate and nonlinear irrational infectivity, the epidemic threshold is always positive; and the epidemic threshold can increase with network size on finite networks, contrary to the findings in all previous work.     

The different cooperative behaviors on a kind of scale-free networks with identical degree sequence

The power-law degree distribution of scale-free networks plays an important role in the bloom of cooperation in the evolutionary games performed on them. In this paper we apply prisoner’s dilemma and public goods game on a family of scale-free networks with the same degree sequence, and show that power-law behavior alone does not determine the cooperative behavior in scale-free networks. Instead, we present that the direct connections among large-degree nodes have a crucial influence on the evolution of cooperation in the scale-free network family.