局域择优的复杂网络建模

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摘要

自然界和人类社会存在各种复杂系统，复杂系统可以通过复杂网络来描述。

复杂网络在通信、社会、工程技术、管理等领域应用广泛，它极大地促进了复杂

系统的发展，成为学术研究的热点之一。信息的传播与交换、病毒式营销、动态

决策、供应链管理、赠样策略等等，复杂网络的发展也为经济管理问题的研究开

辟了新的视角。但是，复杂网络的规律尚未揭开，如何严密分析复杂网络的拓扑

结构不仅具有理论意义，更加具有应用潜力。本文主要通过建立网络模型来模拟

现实网络的拓扑结构并分析其性质，得到以下结论：

1.提出一个有先行者优势的确定性网络模型。用节点度刻画节点的强弱，即

强节点的度比较大，考虑新节点有两种不同的强弱状态，提出一个确定性网络模

型。在计算度分布时运用了古典概率法。通过解析计算，得到了网络的拓扑特征

量，发现模型可以生成一个具有分形特性的无标度层次网络，且分形维数与幂指

数相等。BA 无标度网络呈现出先行者优势现象。如果用节点度表示个体拥有的资

源、能力、社会关系等等，本模型生成的网络中，越早生成的节点拥有的度越大，

这恰好对应于占先策略，即先行者优势。

2. 探索计算增长网络度分布的方法。借鉴随机服务系统处理顾客输入过程的

思想，利用随机过程来刻画增长网络的节点到达过程。给出一般的随机增长网络

模型，并运用随机过程理论解析分析其度分布，此方法简捷且适用范围较广，可

以体现出无标度网络的富者愈富现象。

3. 探讨非平稳增长网络幂指数变化情况。模拟现实社会中的一些信息网络，

提出连边对数加速增长有向网络模型、连边指数加速增长有向网络模型和节点指

数加速增长网络模型；并解析分析了连边对数加速增长有向网络模型的度分布。

探讨了非平稳增长网络的增长速度和幂指数的对应变化关系，发现幂指数与增长

速度呈反方向变化。

4. 提出一个局域连接网络模型。在许多现实网络中，受个体本身能力或者获

取信息量的局限，新个体通常只能与局域网中的个体有联系，并且局域网中的个

体之间联系比较紧密。基于此构想，提出局域连接网络模型，局域网是由网络中

随机选择的一个节点和它的所有邻居构成的，新节点在局域网中进行非线性择优

连接。模拟计算了网络的基本统计特征量，解析分析了网络的度分布和簇度相关

性。通过调节参数，模型可以生成具有不同度分布的网络。网络表现出与非线性

择优参数无关的层次性。从复杂网络、社会网分析两个角度比较分析了局域连接

网络模型的特点。

关键词：复杂网络层次网络局域连接度分布簇度相关性

III

ABSTRACT

There are many kinds of complex systems, which can be depicted as complex

networks, in nature world and human society. Complex networks are widely used in

many fields such as communications, society, engineering technology and management.

Complex networks, one of the hot spots in academic research, promote studies of

complex systems immensely. Opening up a new prospect in the study of economics and

management, complex networks have been applied to such aspects as information

propagation and exchange, viral marketing, dynamic decision, supply chain

management, sampling strategy and so on. The rules of complex network, however,

have not yet unclosed. Analyzing the topology of complex networks mathematically has

not only academic significance but also further applied potentials. In this thesis,

topologies of real-life systems are simulated via building network models, then

analyzing their properties. Main results are as following:

1. A deterministic network with first-mover advantages is proposed. Nodes’ statuses

are depicted by node degree and strong nodes have large degree. A deterministic

network model, in which new nodes have different statuses, is presented. By classical

probability, the degree distribution is calculated. The relevant network parameters are

analyzed. The network is a scale-free hierarchical one with fractal property and the

fractal dimension is equal to the power-law exponent. BA scale-free network displays

first-mover advantages. In our network, the earlier a node was created, the larger degree

it has. Regarding node degree as capacity, ability or social relations individual possesses,

earlier nodes have large degree just corresponds with take precedence, that is

first-mover advantages.

2. To explore methods to calculate the degree distribution of growth networks.

Basing on thoughts of the customer input process in random service system, nodes

incoming is considered as a random process. A general random-growth-network model

is presented and its stationary degree distribution is analyzed by the random process

theory. The method is a short-cut one, with the boarder scope of application, and can be

display the “Success to the Successful” phenomenon in the scale-free network.

3. The power-law exponent of non-stationary growth networks is discussed. To

simulating some information networks, a directed network model with logarithm

accelerating growth of edges, a directed network model with exponent accelerating

growth of edges and a network model with exponent accelerating growth of nodes is

presented. Then the degree distribution of the directed network model with logarithm

accelerating growth of edges is analytically obtained. The corresponding relations

between accelerating growth of non-stationary growth networks and its degree

distribution exponents are discussed. It is founded that the power-law degree exponent

changes in a negative direction with the accelerating growth.

4. A local attachment network model is presented. In many real-life networks,

incomers may only connect to a few others in a local area because of their limited

ability or information, and individuals in a local area are likely to have close relations.

Accordingly, we propose a local-attachment-network model. Here, a local area network

stands for one constructed by a node and all its neighbors. The new nodes perform

nonlinear preferential attachment in local areas. Main quantities are calculated in

simulation and the degree distribution and clustering-degree correlations are analytically

obtained. The model can generate different kinds of degree distributions by adjusting

the parameter. And the network displays the hierarchical organization independent of

the parameter. In two aspects, the complex network and the social network analysis,

features of our network are gained comparatively.

Key Word: complex networks, hierarchical networks, local attachment,

degree distribution, clustering-degree correlations

中文摘要

ABSTRACT

第一章绪论 ............................................................................................................. 1

§ 1.1 网络发展简介及研究意义 ..............................................................................1

§ 1.2 复杂网络在经济管理中的应用研究 ..............................................................2

§ 1.3 复杂网络特征量 ..............................................................................................3

§ 1.3.1 度分布 ....................................................................................................4

§ 1.3.2 集聚系数 ................................................................................................5

§ 1.3.3 平均路径长度 ........................................................................................5

§ 1.3.4 介数 ........................................................................................................6

§ 1.3.5 社区 ........................................................................................................6

§ 1.3.6 簇度相关性 ............................................................................................7

§ 1.3.7 度度关联性 ............................................................................................7

§ 1.4 无标度网络模型研究 ......................................................................................8

§ 1.4.1 BA（Barabási-Albert）模型 ................................................................. 8

§ 1.4.2 确定性无标度网络模型 ........................................................................8

§ 1.4.3 局域世界网络模型 ................................................................................9

§ 1.5 本文主要工作 ................................................................................................10

第二章有先行者优势的确定性网络 .......................................................................13

§ 2.1 一个确定性网络模型 ....................................................................................13

§ 2.1.1 网络的度分布 ......................................................................................14

§ 2.1.2 网络的集聚系数 ..................................................................................16

§ 2.1.3 网络的直径 ..........................................................................................17

§ 2.2 网络的分形特征 ............................................................................................17

§ 2.3 先行者优势分析 ............................................................................................18

第三章增长网络度分布与非平稳增长网络幂指数 ...............................................21

§ 3.1 增长网络度分布 ............................................................................................21

§ 3.1.1 随机增长网络模型 ..............................................................................21

§ 3.1.2 解析网络的度分布 ..............................................................................22

§ 3.1.3 举例 ......................................................................................................23

§ 3.2 非平稳增长网络幂指数 ................................................................................25

§ 3.2.1 连边对数加速增长有向网络 ..............................................................26

§ 3.2.2 连边指数加速增长有向网络 ..............................................................27

§ 3.2.3 节点指数加速增长网络 ......................................................................27

§ 3.2.4 增长速度对幂指数的影响 ..................................................................28

第四章局域连接网络 ............................................................................................... 31

§ 4.1 局域连接网络模型 ........................................................................................31

§ 4.2 网络的度分布 ................................................................................................32

§ 4.2.1





时网络的度分布........................................................................33

§ 4.2.2



 

时网络的度分布 ......................................................................35

§ 4.2.3

0, 1m



 

时网络的度分布 .............................................................. 36

§ 4.3 网络的簇度相关性 ........................................................................................37

§ 4.4 网络的度度关联性 ........................................................................................40

§ 4.5 局域连接网络的特点 ....................................................................................41

§ 4.6 社会网分析比较 BA 无标度网络和局域连接无标度网络 ........................ 42

§ 4.6.1 图形密度比较 ......................................................................................43

§ 4.6.2 群体程度中心性比较 ..........................................................................43

§ 4.6.3 群体中介性比较 ..................................................................................47

第五章结束语 ........................................................................................................... 49

参考文献 .........................................................................................................................51

在读期间公开发表的论文和承担科研项目及取得成果............................................ 58

致谢 .............................................................................................................................59

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摘要：

I摘要自然界和人类社会存在各种复杂系统，复杂系统可以通过复杂网络来描述。复杂网络在通信、社会、工程技术、管理等领域应用广泛，它极大地促进了复杂系统的发展，成为学术研究的热点之一。信息的传播与交换、病毒式营销、动态决策、供应链管理、赠样策略等等，复杂网络的发展也为经济管理问题的研究开辟了新的视角。但是，复杂网络的规律尚未揭开，如何严密分析复杂网络的拓扑结构不仅具有理论意义，更加具有应用潜力。本文主要通过建立网络模型来模拟现实网络的拓扑结构并分析其性质，得到以下结论：1.提出一个有先行者优势的确定性网络模型。用节点度刻画节点的强弱，即强节点的度比较大，考虑新节点有两种不同的强弱状态，提出一个确定性...

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