Emergence of scaling and assortative mixing through altruism

Many social networks, apart from displaying scale-free characteristics observed in some instances, possess another remarkable feature that distinguishes them from those that appear in biological and technological context—assortativity. However, little or no attention has been payed to the mechanism of assortativity in modeling these networks. Inspired by individuals’ altruistic behavior in sociology, we develop a model with a different growth mechanism called “altruistic attachment”, which can reproduce not only the power law degree distribution but degree correlations. We study in detail the statistical properties of our network model, which we also demonstrate striking differences with the BA model, and can portray real social networks more precisely.     

Organization of networks with tagged nodes and biased links: A priori distinct communities: The case of intelligent design proponents and Darwinian evolution defenders

Among the topics of opinion formation it is of interest to observe the characteristics of networks with a priori distinct communities. The citation network(s) between selected members of the Neocreationist and Intelligent Design and the Darwinian Evolution communities are unfolded through the available internet citations. The resulting adjacency matrix is not symmetric. A generalization of considerations pertaining to the case of networks with tagged nodes and biased links, directed or undirected, is presented. The main characteristic coefficients describing the structure of such networks are outlined. The structural features are discussed searching for statistical aspects of the communities. The degree distributions, each network’s assortativity, specific global and local clustering coefficients and the Average Overlap Indices are especially calculated since the distribution of elements in the rectangular submatrices represent inter-community connections. The various closed and open triangles made from nodes, distinguishing the community, are listed. The z-scores of patterns are calculated. One can distinguish between opinion leaders, followers and main rivals and briefly interpret their relationships through intuitively expected behavior in defence of an opinion. Suggestions for more elaborate models describing such communities and their subsequent structures are found in conclusions.     

Local assortativity and growth of Internet

Local assortativity has been recently proposed as a measure to analyse complex networks. It has been noted that the Internet Autonomous System level networks show a markedly different local assortativity profile to most biological and social networks. In this paper we show that, even though several Internet growth models exist, none of them produce the local assortativity profile that can be observed in the real AS networks. We introduce a new generic growth model which can produce a linear local assortativity profile similar to that of the Internet. We verify that this model accurately depicts the local assortativity profile criteria of Internet, while also satisfactorily modelling other attributes of AS networks already explained by existing models.     

Coevolutionary,coexisting learning and teaching agents model for prisoner’s dilemma games enhancing cooperation with assortative heterogeneous networks

Unlike other natural network systems, assortativity can be observed in most human social networks, although it has been reported that a social dilemma situation represented by the prisoner’s dilemma favors dissortativity to enhance cooperation. We established a new coevolutionary model for both agents’ strategy and network topology, where teaching and learning agents coexist. Remarkably, this model enables agents’ enhancing cooperation more than a learners-only model on a time-frozen scale-free network and produces an underlying assortative network with a fair degree of power-law distribution. The model may imply how and why assortative networks are adaptive in human society.     

Assortativity of complementary graphs

Newman's measure for (dis)assortativity, the linear degree correlationρ _D , is widely studied although analytic insight into the assortativity of an arbitrary network remains far from well understood. In this paper, we derive the general relation (2), (3) and Theorem 1 between the assortativity ρ _D (G) of a graph G and the assortativityρ _D (G ^c) of its complement G ^c. Both ρ _D (G) and ρ _D (G ^c) are linearly related by the degree distribution in G. When the graph G(N,p) possesses a binomial degree distribution as in the Erd?s-Rényi random graphs G _p (N), its complementary graph G _p ^c (N) = G _{1- p} (N) follows a binomial degree distribution as in the Erd?s-Rényi random graphs G _{1- p} (N). We prove that the maximum and minimum assortativity of a class of graphs with a binomial distribution are asymptotically antisymmetric: ρ _max(N,p) = -ρ _min(N,p) for N → ∞. The general relation (3) nicely leads to (a) the relation (10) and (16) between the assortativity range ρ _max(G)–ρ _min(G) of a graph with a given degree distribution and the range ρ _max(G ^c)–ρ _min(G ^c) of its complementary graph and (b) new bounds (6) and (15) of the assortativity. These results together with our numerical experiments in over 30 real-world complex networks illustrate that the assortativity range ρ _max–ρ _min is generally large in sparse networks, which underlines the importance of assortativity as a network characterizer.     

确定度分布条件下可变同配系数的算法构造与影响分析

复杂网络是现实中大量节点和边的抽象拓扑, 如何揭示网络内部拓扑对网络连通性、脆弱性等特征的影响是当前研究的热点. 本文在确定度分布的条件下, 根据Newman提出的同配系数的定义分析其影响因素. 首先在可变同配系数下分别提出了基于度分布的确定算法和基于概率分布的不确定算法, 并分别在三种不同类型的网络(Erdös-Rényi网络, Barabási-Albert网络, Email真实网络)中验证. 实验结果表明: 当网络规模达到一定程度时, 确定算法优于贪婪算法. 以此为基础, 分析了同配系数改变时聚类系数的变化, 发现两者之间存在关联性, 并从网络的微观结构变化中揭示了聚类系数变化的原因.     

Network evolution by nonlinear preferential rewiring of edges

The mathematical framework for small-world networks proposed in a seminal paper by Watts and Strogatz sparked a widespread interest in modeling complex networks in the past decade. However, most of research contributing to static models is in contrast to real-world dynamic networks, such as social and biological networks, which are characterized by rearrangements of connections among agents. In this paper, we study dynamic networks evolved by nonlinear preferential rewiring of edges. The total numbers of vertices and edges of the network are conserved, but edges are continuously rewired according to the nonlinear preference. Assuming power-law kernels with exponents α and β, the network structures in stationary states display a distinct behavior, depending only on β. For β>1, the network is highly heterogeneous with the emergence of starlike structures. For β<1, the network is widely homogeneous with a typical connectivity. At β=1, the network is scale free with an exponential cutoff.     

Assortative and disassortative networks. The effect of the topology of a complex network on the properties of dynamical processes on it

The effect of the structure of a complex network on the properties of avalanche dynamical process on it has been analyzed for the first time. It has been shown that the assortativity (disassortativity) degree, which is a structure characteristic of the network and is numerically characterized by the assortativity coefficient r, is a control parameter governing the properties of the dynamical process on the network. The structure of individual avalanches on networks with various r values has been studied. It has been shown that the number of nodes involved in an avalanche is a periodic function of the time.     

A new deterministic complex network model with hierarchical structure

We introduce a new simple pseudo tree-like network model, deterministic complex network (DCN). The proposed DCN model may simulate the hierarchical structure nature of real networks appropriately and have the unique property of ‘skipping the levels’, which is ubiquitous in social networks. Our results indicate that the DCN model has a rather small average path length and large clustering coefficient, leading to the small-world effect. Strikingly, our DCN model obeys a discrete power-law degree distribution P(k)∝k^−γ, with exponent γ approaching 1.0. We also discover that the relationship between the clustering coefficient and degree follows the scaling law C(k)∼k⁻¹, which quantitatively determines the DCN's hierarchical structure.     

Hierarchy property of traffic networks

The flourishing complex network theory has aroused increasing interest in studying the properties of real-world networks. Based on the traffic network of Chang--Zhu--Tan urban agglomeration in central China, some basic network topological characteristics were computed with data collected from local traffic maps, which showed that the traffic networks were small-world networks with strong resilience against failure; more importantly, the investigations of assortativity coefficient and average nearest-neighbour degree implied the disassortativity of the traffic networks. Since traffic network hierarchy as an important basic property has been neither studied intensively nor proved quantitatively, the authors are inspired to analyse traffic network hierarchy with disassortativity and to finely characterize hierarchy in the traffic networks by using the n-degree--n-clustering coefficient relationship. Through numerical results and analyses an exciting conclusion is drawn that the traffic networks exhibit a significant hierarchy, that is, the traffic networks are proved to be hierarchically organized. The result provides important information and theoretical groundwork for optimal transport planning.     

小世界网络与无标度网络的社区结构研究

模块性(modularity)是度量网络社区结构(community structure)的主要参数.探讨了Watts和Strogatz的小世界网络(简称W-S模型)以及Barabàsi 等的B-A无标度网络(简称B-A模型)两类典型复杂网络模块性特点.结果显示，网络模块性受到网络连接稀疏的影响，W-S模型具有显著的社区结构，而B-A模型的社区结构特征不明显.因此，应用中应该分别讨论网络的小世界现象和无标度特性.社区结构不同于小世界现象和无标度特性，并可以利用模块性区别网络类型，因此网络复杂性指标应该包括 关键词： 模块性 社区结构 小世界网络 无标度网络     

From calls to communities: a model for time-varying social networks

Social interactions vary in time and appear to be driven by intrinsic mechanisms thatshape the emergent structure of social networks. Large-scale empirical observations ofsocial interaction structure have become possible only recently, and modelling theirdynamics is an actual challenge. Here we propose a temporal network model which builds onthe framework of activity-driven time-varying networks with memory. Themodel integrates key mechanisms that drive the formation of social ties – socialreinforcement, focal closure and cyclicclosure, which have been shown to give rise to community structure andsmall-world connectedness in social networks. We compare the proposed model with areal-world time-varying network of mobile phone communication, and show that they shareseveral characteristics from heterogeneous degrees and weights to rich communitystructure. Further, the strong and weak ties that emerge from the model follow similarweight-topology correlations as real-world social networks, including the role of weakties.     

Percolation thresholds of a group of anisotropic three-dimensional fracture networks

Percolation thresholds (average number of connections per object) of two models of anisotropic three-dimensional (3D) fracture networks made of mono-disperse hexagons have been calculated numerically. The first model is when the fracture networks are comprised of two groups of fractures that are distributed in an anisotropic manner about two orthogonal mean directions, i.e., Z- and X-directions. We call this model bipolar anisotropic fracture network (BFN). The second model is when three groups of fractures are distributed about three orthogonal mean directions, that is Z-, X-, and Y-directions. In this model three families of fractures about three orthogonal mean directions are oriented in 3D space. We call this model tripolar anisotropic fracture network (TFN). The finite-size scaling method is used to predict the infinite percolation thresholds. The effect of anisotropicity on percolation thresholds in X-, Y-, and Z-directions is investigated. We have revealed that as the anisotropicity of networks increases, the percolation thresholds in X-, Y-, and Z-directions span the range of 2.3 to 2.0, where 2.3 and 2.0 are extremums of percolation thresholds for isotropic and non-isotropic orthogonal fracture networks, respectively.     

Horizontal visibility graphs transformed from fractional Brownian motions: Topological properties versus the Hurst index

Nonlinear time series analysis aims at understanding the dynamics of stochastic or chaotic processes. In recent years, quite a few methods have been proposed to transform a single time series to a complex network so that the dynamics of the process can be understood by investigating the topological properties of the network. We study the topological properties of horizontal visibility graphs constructed from fractional Brownian motions with different Hurst indexes H∈(0,1). Special attention has been paid to the impact of the Hurst index on topological properties. It is found that the clustering coefficient C decreases when H increases. We also found that the mean length L of the shortest paths increases exponentially with H for fixed length N of the original time series. In addition, L increases linearly with respect to N when H is close to 1 and in a logarithmic form when H is close to 0. Although the occurrence of different motifs changes with H, the motif rank pattern remains unchanged for different H. Adopting the node-covering box-counting method, the horizontal visibility graphs are found to be fractals and the fractal dimension d_B decreases with H. Furthermore, the Pearson coefficients of the networks are positive and the degree-degree correlations increase with degree, which indicate that the horizontal visibility graphs are assortative. With the increase of H, the Pearson coefficient decreases first and then increases, in which the turning point is around H=0.6. The presence of both fractality and assortativity in the horizontal visibility graphs converted from fractional Brownian motions is different from many cases where fractal networks are usually disassortative.     

Non-consensus Opinion Models on Complex Networks

Social dynamic opinion models have been widely studied to understand how interactions among individuals cause opinions to evolve. Most opinion models that utilize spin interaction models usually produce a consensus steady state in which only one opinion exists. Because in reality different opinions usually coexist, we focus on non-consensus opinion models in which above a certain threshold two opinions coexist in a stable relationship. We revisit and extend the non-consensus opinion (NCO) model introduced by Shao et al. (Phys. Rev. Lett. 103:01870, 2009). The NCO model in random networks displays a second order phase transition that belongs to regular mean field percolation and is characterized by the appearance (above a certain threshold) of a large spanning cluster of the minority opinion. We generalize the NCO model by adding a weight factor W to each individual’s original opinion when determining their future opinion (NCOW model). We find that as W increases the minority opinion holders tend to form stable clusters with a smaller initial minority fraction than in the NCO model. We also revisit another non-consensus opinion model based on the NCO model, the inflexible contrarian opinion (ICO) model (Li et al. in Phys. Rev. E 84:066101, 2011), which introduces inflexible contrarians to model the competition between two opinions in a steady state. Inflexible contrarians are individuals that never change their original opinion but may influence the opinions of others. To place the inflexible contrarians in the ICO model we use two different strategies, random placement and one in which high-degree nodes are targeted. The inflexible contrarians effectively decrease the size of the largest rival-opinion cluster in both strategies, but the effect is more pronounced under the targeted method. All of the above models have previously been explored in terms of a single network, but human communities are usually interconnected, not isolated. Because opinions propagate not only within single networks but also between networks, and because the rules of opinion formation within a network may differ from those between networks, we study here the opinion dynamics in coupled networks. Each network represents a social group or community and the interdependent links joining individuals from different networks may be social ties that are unusually strong, e.g., married couples. We apply the non-consensus opinion (NCO) rule on each individual network and the global majority rule on interdependent pairs such that two interdependent agents with different opinions will, due to the influence of mass media, follow the majority opinion of the entire population. The opinion interactions within each network and the interdependent links across networks interlace periodically until a steady state is reached. We find that the interdependent links effectively force the system from a second order phase transition, which is characteristic of the NCO model on a single network, to a hybrid phase transition, i.e., a mix of second-order and abrupt jump-like transitions that ultimately becomes, as we increase the percentage of interdependent agents, a pure abrupt transition. We conclude that for the NCO model on coupled networks, interactions through interdependent links could push the non-consensus opinion model to a consensus opinion model, which mimics the reality that increased mass communication causes people to hold opinions that are increasingly similar. We also find that the effect of interdependent links is more pronounced in interdependent scale free networks than in interdependent Erd?s Rényi networks.     

Enhancing robustness and synchronizability of networks homogenizing their degree distribution

A new family of networks, called entangled, has recently been proposed in the literature. These networks have optimal properties in terms of synchronization, robustness against errors and attacks, and efficient communication. They are built with an algorithm which uses modified simulated annealing to enhance a well-known measure of networks’ ability to reach synchronization among nodes. In this work, we suggest that a class of networks similar to entangled networks can be produced by changing some of the connections in a given network, or by just adding a few connections. We call this class of networks weak-entangled. Although entangled networks can be considered as a subset of weak-entangled networks, we show that both classes share similar properties, especially with respect to synchronization and robustness, and that they have similar structural properties.     

Local preferential attachment model for hierarchical networks

In real-life networks, incomers may only connect to a few others in a local area for their limited information, and individuals in a local area are likely to have close relations. Accordingly, we propose a local preferential attachment model. Here, a local-area-network stands for a node and all its neighbors, and the new nodes perform nonlinear preferential attachment, , in local areas. The stable degree distribution and clustering-degree correlations are analytically obtained. With the increasing of α, the clustering coefficient increases, while assortativity decreases from positive to negative. In addition, by adjusting the parameter α, the model can generate different kinds of degree distribution, from exponential to power-law. The hierarchical organization, independent of α, is the most significant character of this model.     

Multi-component static model for social networks

The static model was introduced to generate a scale-free network. In the model, N number of vertices are present from the beginning. Each vertex has its own weight, representing how much the vertex is influential in a system. The static model, however, is not relevant, when a complex network is composed of many modules such as communities in social networks. An individual may belong to more than one community and has distinct weights for each community. Thus, we generalize the static model by assigning a q-component weight on each vertex. We first choose a component among the q components at random and a pair of vertices is linked with a color according to their weights of the component as in the static model. A (1-f) fraction of the entire edges is connected following this way. The remaining fraction f is added with (q + 1)-th color as in the static model but using the maximum weights among the q components each individual has. The social activity with such maximum weights is an essential ingredient to enhance the assortativity coefficient as large as the ones of real social networks.Received: 27 October 2003, Published online: 17 February 2004PACS: 89.65.-s Social and economic systems - 89.75.Hc Networks and genealogical trees - 89.75.Da Systems obeying scaling laws     

Ising-like agent-based technology diffusion model: Adoption patterns vs. seeding strategies

The well-known Ising model used in statistical physics was adapted to a social dynamics context to simulate the adoption of a technological innovation. The model explicitly combines (a) an individual’s perception of the advantages of an innovation and (b) social influence from members of the decision-maker’s social network. The micro-level adoption dynamics are embedded into an agent-based model that allows exploration of macro-level patterns of technology diffusion throughout systems with different configurations (number and distributions of early adopters, social network topologies). In the present work we carry out many numerical simulations. We find that when the gap between the individual’s perception of the options is high, the adoption speed increases if the dispersion of early adopters grows. Another test was based on changing the network topology by means of stochastic connections to a common opinion reference (hub), which resulted in an increment in the adoption speed. Finally, we performed a simulation of competition between options for both regular and small world networks.