Error and attack tolerance of evolving networks with local preferential attachment

Networks generated by local-world evolving network model display a transition from exponential network to power-law network with respect to connectivity distribution. We investigate statistical properties of the evolving networks and the responses of these networks under random errors and intentional attacks. It has been found that local world size M has great effect on the network's heterogeneity, thus leading to transitional behaviors in network's robustness against errors and attacks. Numerical results show that networks constructed with local preferential attachment mechanism can maintain the robustness of scale-free networks under random errors and concurrently improve reliance against targeted attacks on highly connected nodes.     

Transport between multiple users in complex networks

We study the transport properties of model networks such as scale-free and Erd?s-Rényi networks as well as a real network. We consider few possibilities for the trnasport problem. We start by studying the conductance G between two arbitrarily chosen nodes where each link has the same unit resistance. Our theoretical analysis for scale-free networks predicts a broad range of values of G, with a power-law tail distribution $\Phi_{\rm SF}(G)\sim G^{-g_G}$ , where g_G=2λ-1, and λ is the decay exponent for the scale-free network degree distribution. The power-law tail in Φ_SF(G) leads to large values of G, thereby significantly improving the transport in scale-free networks, compared to Erd?s-Rényi networks where the tail of the conductivity distribution decays exponentially. We develop a simple physical picture of the transport to account for the results. The other model for transport is the max-flow model, where conductance is defined as the number of link-independent paths between the two nodes, and find that a similar picture holds. The effects of distance on the value of conductance are considered for both models, and some differences emerge. We then extend our study to the case of multiple sources ans sinks, where the transport is defined between two groups of nodes. We find a fundamental difference between the two forms of flow when considering the quality of the transport with respect to the number of sources, and find an optimal number of sources, or users, for the max-flow case. A qualitative (and partially quantitative) explanation is also given.     

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.     

Anomalous transport in scale-free networks

To study transport properties of scale-free and Erdos-Rényi networks, we analyze the conductance G between two arbitrarily chosen nodes of random scale-free networks with degree distribution P(k)-k(-lambda) in which all links have unit resistance. We predict a broad range of values of G, with a power-law tail distribution phi(SF)(G)-G(-g(G)), where g(G)=2lambda-1, and confirm our predictions by simulations. The power-law tail in phi(SF)(G) leads to large values of G, signaling better transport in scale-free networks compared to Erdos-Rényi networks where the tail of the conductivity distribution decays exponentially. Based on a simple physical "transport backbone" picture we show that the conductances of scale-free and Erdos-Rényi networks are well approximated by ck(A)k(B)/(k(A)+k(B)) for any pair of nodes A and B with degrees k(A) and k(B), where c emerges as the main parameter characterizing network transport.     

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.     

一种信息传播促进网络增长的网络演化模型

为了研究信息传播过程对复杂网络结构演化的影响,提出了一种信息传播促进网络增长的网络演化模型,模型包括信息传播促进网内增边、新节点通过局域世界建立第一条边和信息传播促进新节点连边三个阶段,通过多次自回避随机游走模拟信息传播过程,节点根据路径节点的节点度和距离与其选择性建立连接。理论分析和仿真实验表明,模型不仅具有小世界和无标度特性,而且不同参数下具有漂移幂律分布、广延指数分布等分布特性,呈现小变量饱和、指数截断等非幂律现象,同时,模型可在不改变度分布的情况下调节集聚系数,并能够产生从同配到异配具有不同匹配模式的网络.     

Growing community networks with local events

The study of community networks has attracted considerable attention recently. In this paper, we propose an evolving community network model based on local processes, the addition of new nodes intra-community and new links intra- or inter-community. Employing growth and preferential attachment mechanisms, we generate networks with a generalized power-law distribution of nodes’ degrees.     

Random pseudofractal networks with competition

In this paper, we present a simple rule which assigns fitness to each edge to generate random pseudofractal networks (RPNs). This RPN model is both scale-free and small-world. We obtain the theoretical results that the power-law exponent is γ=2+1/(1+α) for the tunable parameter α>-1, and that the degree distribution is of an exponential form for others. Analytical results also show that an RPN has a large clustering coefficient and can process hierarchical structure as C(k)∼k^-1 that is in accordance with many real networks. And we prove that the mean distance L(N) scales slower logarithmically with network size N. In particular, we explain the effect of nodes with degree 2 on the clustering coefficient. These results agree with numerical simulations very well.     

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.     

A Generalized Voter Model on Complex Networks

We study a generalization of the voter model on complex networks, focusing on the scaling of mean exit time. Previous work has defined the voter model in terms of an initially chosen node and a randomly chosen neighbor, which makes it difficult to disentangle the effects of the stochastic process itself relative to the network structure. We introduce a process with two steps, one that selects a pair of interacting nodes and one that determines the direction of interaction as a function of the degrees of the two nodes and a parameter α which sets the likelihood of the higher degree node giving its state to the other node. Traditional voter model behaviors can be recovered within the model, as well as the invasion process. We find that on a complete bipartite network, the voter model is the fastest process. On a random network with power law degree distribution, we observe two regimes. For modest values of α, exit time is dominated by diffusive drift of the system state, but as the high-degree nodes become more influential, the exit time becomes dominated by frustration effects dependent on the exact topology of the network.     

The emergence of scale-free networks with a seceding mechanism

In order to explore further the underlying mechanism of scale-free networks, we study stochastic secession as a mechanism for the creation of complex networks. In this evolution the network growth incorporates the addition of new nodes, the addition of new links between existing nodes, the deleting and rewiring of some existing links, and the stochastic secession of nodes. To random growing networks with preferential attachment, the model yields scale-free behavior for the degree distribution. Furthermore, we obtain an analytical expression of the power-law degree distribution with scaling exponent γ ranging from 1.1 to 9. The analytical expressions are in good agreement with the numerical simulation results.     

A new local-world evolving network model

In some real complex networks, only a few nodes can obtain the global information about the entire networks, but most of the nodes own only local connections therefore own only local information of the networks. A new local-world evolving network model is proposed in this paper. In the model, not all the nodes obtain local network information, which is different from the local world network model proposed by Li and Chen (LC model). In the LC model, each node has only the local connections therefore owns only local information about the entire networks. Theoretical analysis and numerical simulation show that adjusting the ratio of the number of nodes obtaining the global information of the network to the total number of nodes can effectively control the valuing range for the power-law exponent of the new network. Therefore, if the topological structure of a complex network, especially its exponent of power-law degree distribution, needs controlling, we just add or take away a few nodes which own the global information of the network.     

推广的失活网络动力学同步优化

在不改变网络度分布的条件下,研究了推广的失活网络的同步行为. 应用特征值比R来衡量网络的同步能力,发现同步能力可以通过改变结构参数——激活节点数M来进行优化.特征值比R随M的变化非常敏感,激活节点数M越大,特征值比R越小,同步能力就越强,且在一定范围内遵循R～M^-2.0的幂律关系.通过引入结构微扰,该网络的同步能力也可以得到有效优化. 关键词： 推广的失活网络 同步 特征值比 优化     

Stochastic epidemics and rumours on finite random networks

In this paper, we investigate the stochastic spread of epidemics and rumours on networks. We focus on the general stochastic (SIR) epidemic model and a recently proposed rumour model on networks in Nekovee et al. (2007) [3], and on networks with different random structures, taking into account the structure of the underlying network at the level of the degree–degree correlation function. Using embedded Markov chain techniques and ignoring density correlations between neighbouring nodes, we derive a set of equations for the final size of the epidemic/rumour on a homogeneous network that can be solved numerically, and compare the resulting distribution with the solution of the corresponding mean-field deterministic model. The final size distribution is found to switch from unimodal to bimodal form (indicating the possibility of substantial spread of the epidemic/rumour) at a threshold value that is higher than that for the deterministic model. However, the difference between the two thresholds decreases with the network size, n, following a n^−1/3 behaviour. We then compare results (obtained by Monte Carlo simulation) for the full stochastic model on a homogeneous network, including density correlations at neighbouring nodes, with those for the approximating stochastic model and show that the latter reproduces the exact simulation results with great accuracy. Finally, further Monte Carlo simulations of the full stochastic model are used to explore the effects on the final size distribution of network size and structure (using homogeneous networks, simple random graphs and the Barabasi–Albert scale-free networks).     

微博双向“关注”网络节点中心性及传播 影响力的分析

根据新浪微博的实际数据, 建立了两个基于双向“关注”的用户关系网络, 通过分析网络拓扑统计特征, 发现二者均具有小世界、无标度特征. 通过对节点度、紧密度、介数和k-core 四个网络中心性指标进行实证分析, 发现节点度服从分段幂率分布; 介数相比其他中心性指标差异性最为显著; 两个网络均具有明显的层次性, 但不是所有度值大的节点核数也大; 全局范围内各中心性指标之间存在着较强的相关性, 但在度值较大的节点群这种相关性明显减弱. 此外, 借助基于传染病动力学的SIR信息传播模型来分析四种指标在刻画节点传播能力方面的差异性, 仿真结果表明, 选择具有不同中心性指标的初始传播节点, 对信息传播速度和范围均具有不同影响; 紧密度和k-core较其他指标可以更加准确地描述节点在信息传播中所处的网络核心位置, 这有助于识别信息传播拓扑网络中的关键节点.     

加权网络权重自相似评判函数及其社团结构检测

提出了权重自相似性加权网络社团结构评判函数,并基于该函数提出一种谱分析算法检测社团结构,结果表明算法能将加权网络划分为同一社团内边权值分布均匀,而社团间边权值分布随机的社团结构.通过建立具有社团结构的加权随机网络分析了该算法的准确性,与WEO和WGN算法相比,在评判权重自相似的阈值系数取较小时,该算法具有较高的准确性.对于一个具有n个节点和c个社团的加权网络,社团结构检测的复杂度为O(cn2/2).通过设置评判权重自相似的阈值系数,可检测出能反映节点联系稳定性的层化性社团结构.这与传统意义上只将加权网络划分为社团中边权值较大而社团间边权值较小的标准不同,从另一个角度更好地提取了加权网络的结构信息.     

Resilience of the internet to random breakdowns

A common property of many large networks, including the Internet, is that the connectivity of the various nodes follows a scale-free power-law distribution, P(k) = ck(-alpha). We study the stability of such networks with respect to crashes, such as random removal of sites. Our approach, based on percolation theory, leads to a general condition for the critical fraction of nodes, p(c), that needs to be removed before the network disintegrates. We show analytically and numerically that for alpha0.99.     

Detecting community structure in complex networks based on a measure of information discrepancy

Properties of complex networks, such as small-world property, power-law degree distribution, network transitivity, and network- community structure which seem to be common to many real-world networks have attracted great interest among researchers. In this study, global information of the networks is considered by defining the profile of any node based on the shortest paths between it and all the other nodes in the network; then a useful iterative procedure for community detection based on a measure of information discrepancy and the popular modular function Q is presented. The new iterative method does not need any prior knowledge about the community structure and can detect an appropriate number of communities, which can be hub communities or non-hub communities. The computational results of the method on real networks confirm its capability.     

Modelling of weighted evolving networks with community structures

Many social and biological networks consist of communities–groups of nodes within which links are dense but among which links are sparse. It turns out that most of these networks are best described by weighted networks, whose properties and dynamics depend not only on their structures but also on the link weights among their nodes. Recently, there are considerable interests in the study of properties as well as modelling of such networks with community structures. To our knowledge, however, no study of any weighted network model with such a community structure has been presented in the literature to date. In this paper, we propose a weighted evolving network model with a community structure. The new network model is based on the inner-community and inter-community preferential attachments and preferential strengthening mechanism. Simulation results indicate that this network model indeed reflect the intrinsic community structure, with various power-law distributions of the node degrees, link weights, and node strengths.     

Surveying network community structure in the hidden metric space

Most real-world networks from various fields share a universal topological property as community structure. In this paper, we propose a node-similarity based mechanism to explore the formation of modular networks by applying the concept of hidden metric spaces of complex networks. It is demonstrated that network community structure could be formed according to node similarity in the underlying hidden metric space. To clarify this, we generate a set of observed networks using a typical kind of hidden metric space model. By detecting and analyzing corresponding communities both in the observed network and the hidden space, we show that the values of the fitness are rather close, and the assignments of nodes for these two kinds of community structures detected based on the fitness parameter are extremely matching ones. Furthermore, our research also shows that networks with strong clustering tend to display prominent community structures with large values of network modularity and fitness.