The exponential degree distribution in complex networks: Non-equilibrium network theory, numerical simulation and empirical data

The exponential degree distribution has been found in many real world complex networks, based on which, the random growing process has been introduced to analyze the formation principle of such kinds of networks. Inspired from the non-equilibrium network theory, we construct the network according to two mechanisms: growing and adjacent random attachment. By using the Kolmogorov-Smirnov Test (KST), for the same number of nodes and edges, we find the simulation results are remarkably consistent with the predictions of the non-equilibrium network theory, and also surprisingly match the empirical databases, such as the Worldwide Marine Transportation Network (WMTN), the Email Network of University at Rovira i Virgili (ENURV) in Spain and the North American Power Grid Network (NAPGN). Our work may shed light on interpreting the exponential degree distribution and the evolution mechanism of the complex networks.     

一个描述合作网络顶点度分布的模型

讨论一类社会合作网络以及一些与其拓扑结构相似的技术网络的度分布.建议一个最简化模型，通过解析的方法说明这些网络演化的共同动力学机理，而且说明顶点的度分布和项目度分布之间具有密切的一致关系，而项目所含的顶点数分布对度分布的影响较小；对模型的更一般情况进行数值模拟，说明上述结论具有一定的普遍性.这个模型显示这类广义的合作网络一般具有处于幂函数和指数函数这两种极端情况之间的度分布.简要介绍对一些实际合作网络做统计研究的结果，说明本模型的合理性. 关键词： 合作网络 度分布 项目度分布 项目含顶点数     

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.     

Modeling complex networks with accelerating growth and aging effect

Numerous empirical studies have revealed that a large number of real networks exhibit the property of accelerating growth, i.e. network size (nodes) increases superlinearly with time. Examples include the size of social networks, the output of scientists, the population of cities, and so on. In the literature, these real systems are widely represented by complex networks for analysis, and many network models have been proposed to explain the observed properties in these systems such as power-law degree distribution. However, most of these models (e.g. the well-known BA model) are based on linear growth of these systems. In this paper, we propose a network model with accelerating growth and aging effect, resulting in an emergence of super hubs which is consistent with the empirical observation in citation networks.     

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.     

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.     

Impact of degree heterogeneity on the behavior of trapping in Koch networks

Previous work shows that the mean first-passage time (MFPT) for random walks to a given hub node (node with maximum degree) in uncorrelated random scale-free networks is closely related to the exponent γ of power-law degree distribution P(k) ～ k(-γ), which describes the extent of heterogeneity of scale-free network structure. However, extensive empirical research indicates that real networked systems also display ubiquitous degree correlations. In this paper, we address the trapping issue on the Koch networks, which is a special random walk with one trap fixed at a hub node. The Koch networks are power-law with the characteristic exponent γ in the range between 2 and 3, they are either assortative or disassortative. We calculate exactly the MFPT that is the average of first-passage time from all other nodes to the trap. The obtained explicit solution shows that in large networks the MFPT varies lineally with node number N, which is obviously independent of γ and is sharp contrast to the scaling behavior of MFPT observed for uncorrelated random scale-free networks, where γ influences qualitatively the MFPT of trapping problem.     

A Collaboration Network Model with Multiple Evolving Factors

To describe the empirical data of collaboration networks,several evolving mechanisms have been proposed,which usually introduce different dynamics factors controlling the network growth.These models can reasonably reproduce the empirical degree distributions for a number of well-studied real-world collaboration networks.On the basis of the previous studies,in this work we propose a collaboration network model in which the network growth is simultaneously controlled by three factors,including partial preferential attachment,partial random attachment and network growth speed.By using a rate equation method,we obtain an analytical formula for the act degree distribution.We discuss the dependence of the act degree distribution on these different dynamics factors.By fitting to the empirical data of two typical collaboration networks,we can extract the respective contributions of these dynamics factors to the evolution of each networks.     

基于社团结构的城市地铁网络模型研究

本文运用复杂网络理论, 对我国北京、上海、广州和深圳等城市的地铁网络进行了实证研究. 分别研究了地铁网络的度分布、聚类系数和平均路径长度. 研究表明, 该网络具有高的聚类系数和短的平均路径长度, 显示小世界网络的特征, 其度分布并不严格服从幂律分布或指数分布, 而是呈多段的分布, 显示层次网络的特征. 此外, 它还具有重叠的社团结构特征. 基于实证研究的结果, 提出一种基于社团结构的交通网络模型, 并对该模型进行了模拟分析, 模拟结果表明, 该模型的模拟结果与实证研究结果相符. 此外, 该模型还能解释其他类型的复杂网络(如城市公共汽车交通网络)的网络特性. 关键词： 复杂网络 地铁网络 小世界 社团     

The eight-fold way for optical quasicrystals

Network theory is increasingly employed to study the structure and behaviour of social, physical and technological systems — including civil infrastructure. Many of these systems are interconnected and the interdependencies between them allow disruptive events to propagate across networks, enabling damage to spread far beyond the immediate footprint of disturbance. In this research we experiment with a model to characterise the configuration of interdependencies in terms of direction, redundancy, and extent, and we analyse the performance of interdependent systems with a wide range of possible coupling modes. We demonstrate that networks with directed dependencies are less robust than those with undirected dependencies, and that the degree of redundancy in inter-network dependencies can have a differential effect on robustness depending on the directionality of the dependencies. As interdependencies between many real-world systems exhibit these characteristics, it is likely that many such systems operate near their critical thresholds. The vulnerability of an interdependent network is shown to be reducible in a cost effective way, either by optimising inter-network connections, or by hardening high degree nodes. The results improve understanding of the influence of interdependencies on system performance and provide insight into how to mitigate associated risks.     

有倾向性重连产生的反匹配网络

在具有网络结构的系统中度关联属性对于动力学行为具有重要的影响, 所以产生适当度关联网络的方法对于大量网络系统的研究具有重要的作用. 尽管产生正匹配网络的方法已经得到很好的验证, 但是产生反匹配网络的方法还没有被系统的讨论过. 重新连接网络中的边是产生度关联网络的一个常用方法. 这里我们研究使用重连方法产生反匹配无标度网络的有效性. 我们的研究表明, 有倾向的重连可以增强网络的反匹配属性. 但是有倾向重连不能使皮尔森度相关系数下降到-1, 而是存在一个依赖于网络参数的最小值. 我们研究了网络的主要参数对于网络度相关系数的影响, 包括网络尺寸, 网络的连接密度和网络节点的度差异程度. 研究表明在网络尺寸大的情况下和节点度差异性强的情况下, 重连的效果较差. 我们研究了真实Internet网络, 发现模型产生的网络经过重连不能达到真实网络的度关联系数.     

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.     

Two-population dynamics in a growing network model

We introduce a growing network evolution model with nodal attributes. The model describes the interactions between potentially violent V and non-violent N agents who have different affinities in establishing connections within their own population versus between the populations. The model is able to generate all stable triads observed in real social systems. In the framework of rate equations theory, we employ the mean-field approximation to derive analytical expressions of the degree distribution and the local clustering coefficient for each type of nodes. Analytical derivations agree well with numerical simulation results. The assortativity of the potentially violent network qualitatively resembles the connectivity pattern in terrorist networks that was recently reported. The assortativity of the network driven by aggression shows clearly different behavior than the assortativity of the networks with connections of non-aggressive nature in agreement with recent empirical results of an online social system.     

Scale-free properties of information flux networks in genetic algorithms

In this study, we present empirical analysis of statistical properties of mating networks in genetic algorithms (GAs). Under the framework of GAs, we study a class of interaction network model—information flux network (IFN), which describes the information flow among generations during evolution process. The IFNs are found to be scale-free when the selection operator uses a preferential strategy rather than a random. The topology structure of IFN is remarkably affected by operations used in genetic algorithms. The experimental results suggest that the scaling exponent of the power-law degree distribution is shown to decrease when crossover rate increases, but increase when mutation rate increases, and the reason may be that high crossover rate leads to more edges that are shared between nodes and high mutation rate leads to many individuals in a generation possessing low fitness. The magnitude of the out-degree exponent is always more than the in-degree exponent for the systems tested. These results may provide a new viewpoint with which to view GAs and guide the dissemination process of genetic information throughout a population.     

Optimization in gradient networks

Gradient networks can be used to model the dominant structure of complex networks. Previous work has focused on random gradient networks. Here we study gradient networks that minimize jamming on substrate networks with scale-free and Erdos-Renyi structure. We introduce structural correlations and strongly reduce congestion occurring on the network by using a Monte Carlo optimization scheme. This optimization alters the degree distribution and other structural properties of the resulting gradient networks. These results are expected to be relevant for transport and other dynamical processes in real network systems.     

Sample-to-sample fluctuations in real-network ensembles

Network modeling based on ensemble averages tacitly assumes that the networks meant to be modeled are typical in the ensemble. Previous research on network eigenvalues, which govern a range of dynamical phenomena, has shown that this is indeed the case for uncorrelated networks with minimum degree ≥ 3. Here, we focus on real networks, which generally have both structural correlations and low-degree nodes. We show that: (i) the ensemble distribution of the dynamically most important eigenvalues can be not only broad and far apart from the real eigenvalue but also highly structured, often with a multimodal rather than a bell-shaped form; (ii) these interesting properties are found to be due to low-degree nodes, mainly those with degree ≤ 3, and network communities, which is a common form of structural correlation found in real networks. In addition to having implications for ensemble-based approaches, this shows that low-degree nodes may have a stronger influence on collective dynamics than previously anticipated from the study of computer-generated networks.     

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.     

Effect of the social influence on topological properties of user-object bipartite networks

Social influence plays an important role in analyzing online users’ collective behaviors[Salganik et al., Science 311, 854 (2006)]. However, the effect of the socialinfluence from the viewpoint of theoretical model is missing. In this paper, by takinginto account the social influence and users’ preferences, we develop a theoretical modelto analyze the topological properties of user-object bipartite networks, including thedegree distribution, average nearest neighbor degree and the bipartite clusteringcoefficient, as well as topological properties of the original user-object networks andtheir unipartite projections. According to the users’ preferences and the global rankingeffect, we analyze the theoretical results for two benchmark data sets, Amazon andBookcrossing, which are approximately consistent with the empirical results. This worksuggests that this model is feasible to analyze topological properties of bipartitenetworks in terms of the social influence and the users’ preferences.     

On the dimensioning of cellular OFDMA networks

In this paper, we address the issue of cellular OFDMA network dimensioning. Network design consists of evaluating cell coverage and capacity and may involve many parameters related to environment, system configuration, and quality of service (QoS) requirements. In order to quickly study the impact of each of these parameters, analytical formulas are needed. The key function for network dimensioning is the Signal to Interference Ratio (SIR) distribution. We thus analyze in an original way the traditional issue of deriving outage probabilities in OFDMA cellular networks. Our study takes into account the joint effect of path-loss, shadowing, and fast fading effects. Starting from the Mean Instantaneous Capacity (MIC), we derive the effective SIR distribution as a function of the number of sub-carriers per sub-channel. Our formula, based on a fluid model approach, is easily computable and can be obtained for a mobile station (MS) located at any distance from its serving base station (BS). We validate our approach by comparing all results to Monte Carlo simulations performed in a hexagonal network, and we show how our analytical study can be used to analyze outage capacity, coverage holes, and network densification. The proposed framework is a powerful tool to study performances of cellular OFDMA networks (e.g. WiMAX, LTE).     

Fault-tolerant topology in the wireless sensor networks for energy depletion and random failure

Nodes in the wireless sensor networks(WSNs) are prone to failure due to energy depletion and poor environment,which could have a negative impact on the normal operation of the network. In order to solve this problem, in this paper, we build a fault-tolerant topology which can effectively tolerate energy depletion and random failure. Firstly, a comprehensive failure model about energy depletion and random failure is established. Then an improved evolution model is presented to generate a fault-tolerant topology, and the degree distribution of the topology can be adjusted. Finally, the relation between the degree distribution and the topological fault tolerance is analyzed, and the optimal value of evolution model parameter is obtained. Then the target fault-tolerant topology which can effectively tolerate energy depletion and random failure is obtained. The performances of the new fault tolerant topology are verified by simulation experiments. The results show that the new fault tolerant topology effectively prolongs the network lifetime and has strong fault tolerance.