Tuning degree distributions: Departing from scale-free networks

Scale-free networks are characterized by a degree distribution with power-law behavior. Although scale-free networks have been shown to arise in many areas, ranging from the World Wide Web to transportation or social networks, degree distributions of other observed networks often differ from the power-law type. Data based investigations require modifications of the typical scale-free network.We present an algorithm that generates networks in which the shape of the degree distribution is tunable by modifying the preferential attachment step of the Barabási-Albert construction algorithm. The shape of the distribution is represented by dispersion measures such as the variance and the skewness, both of which are highly correlated with the maximal degree of the network and, therefore, adequately represents the influence of superspreaders or hubs. By combining our algorithm with work of Holme and Kim, we show how to generate networks with a variety of degree distributions and clustering coefficients.     

供应链型网络中双幂律分布模型

考察了供应链网络的基本特征，提出了节点到达过程是更新过程、新增入边和出边数是具有Bernoulli分布随机变量的供应链型有向网络.研究了这类网络节点的瞬态度分布和稳态平均度分布.利用更新过程理论对这类网络进行了分析，获得了网络节点瞬态度分布和网络稳态平均度分布的解析表达式.分析表明, 虽然这类网络节点的稳态度分布不存在，但是网络的稳态平均度分布具有双向幂律性. 关键词： 复杂网络 入度 出度 度分布     

Complex stock trading network among investors

We provide an empirical investigation aimed at uncovering the statistical properties of intricate stock trading networks based on the order flow data of a highly liquid stock (Shenzhen Development Bank) listed on Shenzhen Stock Exchange during the whole year of 2003. By reconstructing the limit order book, we can extract detailed information of each executed order for each trading day and demonstrate that the trade size distributions for different trading days exhibit power-law tails and that most of the estimated power-law exponents are well within the Lévy stable regime. Based on the records of order matching among investors, we can construct a stock trading network for each trading day, in which the investors are mapped into nodes and each transaction is translated as a direct edge from the seller to the buyer with the trade size as its weight. We find that all the trading networks comprise a giant component and have power-law degree distributions and disassortative architectures. In particular, the degrees are correlated with order sizes by a power-law function. By regarding the size of executed order as its fitness, the fitness model can reproduce the empirical power-law degree distribution.     

Degree and wealth distribution in a network induced by wealth

A network induced by wealth is a social network model in which wealth induces individuals to participate as nodes, and every node in the network produces and accumulates wealth utilizing its links. More specifically, at every time step a new node is added to the network, and a link is created between one of the existing nodes and the new node. Innate wealth-producing ability is randomly assigned to every new node, and the node to be connected to the new node is chosen randomly, with odds proportional to the accumulated wealth of each existing node. Analyzing this network using the mean value and continuous flow approaches, we derive a relation between the conditional expectations of the degree and the accumulated wealth of each node. From this relation, we show that the degree distribution of the network induced by wealth is scale-free. We also show that the wealth distribution has a power-law tail and satisfies the 80/20 rule. We also show that, over the whole range, the cumulative wealth distribution exhibits the same topological characteristics as the wealth distributions of several networks based on the Bouchaud-Mèzard model, even though the mechanism for producing wealth is quite different in our model. Further, we show that the cumulative wealth distribution for the poor and middle class seems likely to follow by a log-normal distribution, while for the richest, the cumulative wealth distribution has a power-law behavior.     

手机通话网络度特征分析

手机通信数据详细记录了人们的通信行为, 成为研究人们社会关系、行为模式的重要资源. 通话号码个数、通话次数和时长是手机通信网络的基本属性. 本文在复杂网络理论基础上, 应用统计的方法研究了中国西部某城市三百余万手机用户不同节假日和工作日的 四天通话数据在不同尺度下的号码度、通话度、时长度的分布以及平均号码度、 平均通话度、平均时长度的特征.研究表明, 所有尺度下, 号码度、通话度、时长度均为幂律分布, 幂指数随尺度、日期和指标的不同而不同, 在[1.3, 4] 范围内波动.总体上, 号码度幂指数大于通话度和时长度幂指数, 入度幂指数大于出度幂指数;节假日幂指数大于相应指标的工作日幂指数, 休息时段幂指数大于工作时段幂指数;与工作日相比, 节假日的平均号码度和平均通话度较小, 平均时长度较大.揭示了绝大多数用户每日只接打1个号码的电话, 节假日期间接打电话的用户数、次数、时长减少, 但平均通话时长增大的特征. 关键词： 手机通话网络 复杂网络 度分布 通话模式     

Recursive weighted treelike networks

We propose a geometric growth model for weighted scale-free networks, which is controlled by two tunable parameters. We derive exactly the main characteristics of the networks, which are partially determined by the parameters. Analytical results indicate that the resulting networks have power-law distributions of degree, strength, weight and betweenness, a scale-free behavior for degree correlations, logarithmic small average path length and diameter with network size. The obtained properties are in agreement with empirical data observed in many real-life networks, which shows that the presented model may provide valuable insight into the real systems.     

A general evolving model for growing bipartite networks

In this Letter, we propose and study an inner evolving bipartite network model. Significantly, we prove that the degree distribution of two different kinds of nodes both obey power-law form with adjustable exponents. Furthermore, the joint degree distribution of any two nodes for bipartite networks model is calculated analytically by the mean-field method. The result displays that such bipartite networks are nearly uncorrelated networks, which is different from one-mode networks. Numerical simulations and empirical results are given to verify the theoretical results.     

An evolving model of online bipartite networks

Understanding the structure and evolution of online bipartite networks is a significant task since they play a crucial role in various e-commerce services nowadays. Recently, various attempts have been tried to propose different models, resulting in either power-law or exponential degree distributions. However, many empirical results show that the user degree distribution actually follows a shifted power-law distribution, the so-called Mandelbrot’s law, which cannot be fully described by previous models. In this paper, we propose an evolving model, considering two different user behaviors: random and preferential attachment. Extensive empirical results on two real bipartite networks, Delicious and CiteULike  , show that the theoretical model can well characterize the structure of real networks for both user and object degree distributions. In addition, we introduce a structural parameter p$p$, to demonstrate that the hybrid user behavior leads to the shifted power-law degree distribution, and the region of power-law tail will increase with the increment of p$p$. The proposed model might shed some lights in understanding the underlying laws governing the structure of real online bipartite networks.     

Convergence speed of consensus problems over undirected scale-free networks

Scale-free networks and consensus behaviour among multiple agents have both attracted much attention.To investigate the consensus speed over scale-free networks is the major topic of the present work.A novel method is developed to construct scale-free networks due to their remarkable power-law degree distributions,while preserving the diversity of network topologies.The time cost or iterations for networks to reach a certain level of consensus is discussed,considering the influence from power-law parameters.They are both demonstrated to be reversed power-law functions of the algebraic connectivity,which is viewed as a measurement on convergence speed of the consensus behaviour.The attempts of tuning power-law parameters may speed up the consensus procedure,but it could also make the network less robust over time delay at the same time.Large scale of simulations are supportive to the conclusions.     

Relationship between degree-rank distributions and degree distributions of complex networks

Both the degree distribution and the degree-rank distribution, which is a relationship function between the degree and the rank of a vertex in the degree sequence obtained from sorting all vertices in decreasing order of degree, are important statistical properties to characterize complex networks. We derive an exact mathematical relationship between degree-rank distributions and degree distributions of complex networks. That is, for arbitrary complex networks, the degree-rank distribution can be derived from the degree distribution, and the reverse is true. Using the mathematical relationship, we study the degree-rank distributions of scale-free networks and exponential networks. We demonstrate that the degree-rank distributions of scale-free networks follow a power law only if scaling exponent λ>2. We also demonstrate that the degree-rank distributions of exponential networks follow a logarithmic law. The simulation results in the BA model and the exponential BA model verify our results.     

Local Events and Dynamics on Weighted Complex Networks

We examine the weighted networks grown and evolved by local events, such as the addition of new vertices and links and we show that depending on frequency of the events, a generalized power-law distribution of strength can emerge. Continuum theory is used to predict the scaling function as well as the exponents, which is in good agreement with the numerical simulation results. Depending on event frequency, power-law distributions of degree and weight can also be expected. Probability saturation phenomena for small strength and degree in many real world networks can be reproduced. Particularly, the non-trivial clustering coefficient, assortativity coefficient and degree-strength correlation in our model are all consistent with empirical evidences.     

S-curve networks and an approximate method for estimating degree distributions of complex networks

In the study of complex networks almost all theoretical models have the property of infinite growth,but the size of actual networks is finite.According to statistics from the China Internet IPv4(Internet Protocol version 4) addresses,this paper proposes a forecasting model by using S curve(logistic curve).The growing trend of IPv4 addresses in China is forecasted.There are some reference values for optimizing the distribution of IPv4 address resource and the development of IPv6.Based on the laws of IPv4 growth,that is,the bulk growth and the finitely growing limit,it proposes a finite network model with a bulk growth.The model is said to be an S-curve network.Analysis demonstrates that the analytic method based on uniform distributions(i.e.,Barab’asi-Albert method) is not suitable for the network.It develops an approximate method to predict the growth dynamics of the individual nodes,and uses this to calculate analytically the degree distribution and the scaling exponents.The analytical result agrees with the simulation well,obeying an approximately power-law form.This method can overcome a shortcoming of Baraba’si-Albert method commonly used in current network research.     

Degree distribution in discrete case

Vertex degree of many network models and real-life networks is limited to non-negative integer. By means of measure and integral, the relation of the degree distribution and the cumulative degree distribution in discrete case is analyzed. The degree distribution, obtained by the differential of its cumulative, is only suitable for continuous case or discrete case with constant degree change. When degree change is not a constant but proportional to degree itself, power-law degree distribution and its cumulative have the same exponent and the mean value is finite for power-law exponent greater than 1.     

用户需求行为对互联网动力学整体特性的影响

由Internet构成的复杂网络的动力学特性主要受到用户需求行为的影响,具备时域的统计规律性. 通过对区域群体用户需求行为的时域实验统计分析,发现用户对Web网站的访问频度及其生成的二分网络的入度分布也呈现幂律分布和集聚现象,其幂指数介于1.7到1.8之间. 建立了虚拟资源网络VRN和物理拓扑网络PTN双层模型,分析了双层模型映射机理,并对网络用户需求行为进行建模. 虚拟资源网络VRN对物理拓扑网络PTN映射过程的不同机理,模拟了Internet资源网络到物理网络的不同影响模式. 幂律分布的用户需求特性会 关键词： 复杂网络 无标度拓扑 用户需求 相变     

Avalanche dynamics of idealized neuron function in the brain on an uncorrelated random scale-free network

We study a simple model for a neuron function in a collective brain system. The neural network is composed of an uncorrelated configuration model (UCM) for eliminating the degree correlation of dynamical processes. The interaction of neurons is assumed to be isotropic and idealized. These neuron dynamics are similar to biological evolution in extremal dynamics with locally isotropic interaction but has a different time scale. The functioning of neurons takes place as punctuated patterns based on avalanche dynamics. In our model, the avalanche dynamics of neurons exhibit self-organized criticality which shows power-law behavior of the avalanche sizes. For a given network, the avalanche dynamic behavior is not changed with different degree exponents of networks, γ≥2.4 and various refractory periods referred to the memory effect, T_r. Furthermore, the avalanche size distributions exhibit power-law behavior in a single scaling region in contrast to other networks. However, return time distributions displaying spatiotemporal complexity have three characteristic time scaling regimes Thus, we find that UCM may be inefficient for holding a memory.     

基于Web 2.0的边与节点同时增长网络模型

模拟了Web2.0网络的发展过程并研究其拓扑结构,分析某门户网站实际博客数据的度分布、节点度时间变化,发现与先前的无标度网络模型有所差别.根据真实网络的生长特点,提出了边与节点同时增长的网络模型,包括随机连接及近邻互联的网络构造规则.仿真研究表明,模拟的网络更接近实际,在没有优先连接过程时,模型能得到幂率的度分布;并且网络有更大的聚类系数以及正的度相关性。     

Cumulative Laws, Team Assembling Mechanisms Determining Network Structure

A number of researching works have shed light on the field of complex networks recently. We investigate a wide range of real-world networks and find several interesting phenomena. Firstly, almost all of these networks evolve by overlapping new small graphs on former networks. Secondly, not only the degree sequence of the mature network follows a power-law distribution, but also the distribution of the cumulative occurrence times during the growing process are revealed to have a heavy tail. Existing network evolving models do not provide interpretation to these phenomena. We suggest a model based on the team assembling mechanism, which is extracted from the growing processes of real-world networks and requires simple parameters, and produces networks exhibiting these properties observed in the present study and in previous works.     

幂律指数在1与3之间的一类无标度网络

借助排队系统中顾客批量到达的概念，提出节点批量到达的Poisson网络模型.节点按照到达率为λ的Poisson过程批量到达系统.模型1，批量按照到达批次的幂律非线性增长，其幂律指数为θ(0≤θ<+∞).BA模型是在θ＝0时的特例.利用Poisson过程理论和连续化方法进行分析，发现这个网络稳态平均度分布是幂律分布，而且幂律指数在1和3之间.模型2，批量按照节点到达批次的对数非线性增长，得出当批量增长较缓慢时，稳态度分布幂律指数为3.因此，节点批量到达的Poisson网络模型不仅是BA模型的推广，也为许多幂律指数在1和2之间的现实网络提供了理论依据.     

Empirical investigation of topological and weighted properties of a bus transport network from China

Many bus transport networks(BTNs) have evolved into directed networks. A new representation model for BTNs is proposed, called directed-space P. The bus transport network of Harbin(BTN-H) is described as a directed and weighted complex network by the proposed representation model and by giving each node weights. The topological and weighted properties are revealed in detail. In-degree and out-degree distributions, in-weight and out-weight distributions are presented as an exponential law, respectively. There is a strong relation between in-weight and in-degree(also between out-weight and out-degree), which can be fitted by a power function. Degree–degree and weight–weight correlations are investigated to reveal that BTN-H has a disassortative behavior as the nodes have relatively high degree(or weight). The disparity distributions of out-degree and in-degree follow an approximate power-law. Besides, the node degree shows a near linear increase with the number of routes that connect to the corresponding station. These properties revealed in this paper can help public transport planners to analyze the status quo of the BTN in nature.     

Effects of Aging and Self-organized Criticality in a Pulse-Coupled Integrate-and-Fire Neuron Model Based on Small World Networks

Effects of aging and self-organized criticality in a pulse-coupled integrate-and-fire neuron model based on small world networks have been studied. We give the degree distribution of aging network, average shortest path length, the diameter of our network, and the clustering coefficient, and find that our neuron model displays the power-law behavior, and with the number of added links increasing, the effects of aging become smaller and smaller. This shows that if the brain works at the self-organized criticality state, it can relieve some effects caused by aging.