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.     

An empirical study of Chinese language networks

Chinese is spoken by the largest number of people in the world, and it is regarded as one of the most important languages. In this paper, we explore the statistical properties of Chinese language networks (CLNs) within the framework of complex network theory. Based on one of the largest Chinese corpora, i.e. People’s Daily Corpus, we construct two networks (CLN1 and CLN2) from two different respects, with Chinese words as nodes. In CLN1, a link between two nodes exists if they appear next to each other in at least one sentence; in CLN2, a link represents that two nodes appear simultaneously in a sentence. We show that both networks exhibit small-world effect, scale-free structure, hierarchical organization and disassortative mixing. These results indicate that in many topological aspects Chinese language shapes complex networks with organizing principles similar to other previously studied language systems, which shows that different languages may have some common characteristics in their evolution processes. We believe that our research may shed some new light into the Chinese language and find some potentially significant implications.     

Cascading Failures of Complex Networks Based on Two-Step Degree

We propose a new concept, two-step degree. Defining it as the capacity of a node of complex networks, we establish a novel capacity-load model of cascading failures of complex networks where the capacity of nodes decreases during the process of cascading failures. For scale-free networks, we find that the average two-step degree increases with the increase of the heterogeneity of the degree distribution, showing that the average two- step degree can be used for measuring the heterogeneity of the degree distribution of complex networks. In addition, under the condition that the average degree of a node is given, we can design a scale-free network with the optimal robustness to random failures by maximizing the average two-step degree.     

Complex network structure of musical compositions: Algorithmic generation of appealing music

In this paper we construct networks for music and attempt to compose music artificially. Networks are constructed with nodes and edges corresponding to musical notes and their co-occurring connections. We analyze classical music from Bach, Mozart, Chopin, as well as other types of music such as Chinese pop music. We observe remarkably similar properties in all networks constructed from the selected compositions. We conjecture that preserving the universal network properties is a necessary step in artificial composition of music. Power-law exponents of node degree, node strength and/or edge weight distributions, mean degrees, clustering coefficients, mean geodesic distances, etc. are reported. With the network constructed, music can be composed artificially using a controlled random walk algorithm, which begins with a randomly chosen note and selects the subsequent notes according to a simple set of rules that compares the weights of the edges, weights of the nodes, and/or the degrees of nodes. By generating a large number of compositions, we find that this algorithm generates music which has the necessary qualities to be subjectively judged as appealing.     

Network Entropy Based on Topology Configuration and Its Computation to Random Networks

A definition of network entropy is presented, and as an example, the relationship between the value of network entropy of ER network model and the connect probability p as well as the total nodes N is discussed. The theoretical result and the simulation result based on the network entropy of the ER network are in agreement well with each other. The result indicated that different from the other network entropy reported before, the network entropy defined here has an obvious difference from different type of random networks or networks having different total nodes. Thus, this network entropy may portray the characters of complex networks better. It is also pointed out that, with the aid of network entropy defined, the concept of equilibrium networks and the concept of non-equilibrium networks may be introduced, and a quantitative measurement to describe the deviation to equilibrium state of a complex network is carried out.     

Exact solutions for network rewiring models

Evolving networks with a constant number of edges may be modelled using a rewiring process. These models are used to describe many real-world processes including the evolution of cultural artifacts such as family names, the evolution of gene variations, and the popularity of strategies in simple econophysics models such as the minority game. The model is closely related to Urn models used for glasses, quantum gravity and wealth distributions. The full mean field equation for the degree distribution is found and its exact solution and generating solution are given.     

Local affinity in heterogeneous growing networks

In this paper we present a study of the influence of local affinity in heterogeneous preferential attachment (PA) networks. Heterogeneous PA models are a generalization of the Barabási-Albert model to heterogeneous networks, where the affinity between nodes biases the attachment probability of links. Threshold models are a class of heterogeneous PA models where the affinity between nodes is inversely related to the distance between their states. We propose a generalization of threshold models where network nodes have individual affinity functions, which are then combined to yield the affinity of each potential interaction. We analyze the influence of the affinity functions in the topological properties averaged over a network ensemble. The network topology is evaluated through the distributions of connectivity degrees, clustering coefficients and geodesic distances. We show that the relaxation of the criterion of a single global affinity still leads to a reasonable power-law scaling in the connectivity and clustering distributions under a wide spectrum of assumptions. We also show that the richer behavior of the model often exhibits a better agreement with the empirical observations on real networks.     

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.     

Influence of different initial distributions on robust cooperation in scale-free networks: A comparative study

We study the evolutionary Prisoner's dilemma game on scale-free networks, focusing on the influence of different initial distributions for cooperators and defectors on the evolution of cooperation. To address this issue, we consider three types of initial distributions for defectors: uniform distribution at random, occupying the most connected nodes, and occupying the lowest-degree nodes, respectively. It is shown that initial configurations for defectors can crucially influence the cooperation level and the evolution speed of cooperation. Interestingly, the situation where defectors initially occupy the lowest-degree vertices can exhibit the most robust cooperation, compared with two other distributions. That is, the cooperation level is least affected by the initial percentage of defectors. Moreover, in this situation, the whole system evolves fastest to the prevalent cooperation. Besides, we obtain the critical values of initial frequency of defectors above which the extinction of cooperators occurs for the respective initial distributions. Our results might be helpful in explaining the maintenance of high cooperation in scale-free networks.     

Scale-free networks by super-linear preferential attachment rule

A network growth model with geographic limitation of accessible information about the status of existing nodes is investigated. In this model, the probability Π(k) of an existing node of degree k is found to be super-linear with Π(k)∼k^α and α>1 when there are links from new nodes. The numerical results show that the constructed networks have typical power-law degree distributions P(k)∼k^−γ and the exponent γ depends on the constraint level. An analysis of local structural features shows the robust emergence of scale-free network structure in spite of the super-linear preferential attachment rule. This local structural feature is directly associated with the geographical connection constraints which are widely observed in many real networks.     

Sandpile Dynamics Driven by Degree on Scale-Free Networks

We introduce a sandpile model driven by degree on scale-free networks, where the perturbation is triggered at nodes with the same degree. We numerically investigate the avalanche behaviour of sandpile driven by different degrees on scale-free networks. It is observed that the avalanche area has the same behaviour with avalanche size. When the sandpile is driven at nodes with the minimal degree, the avalanches of our model behave similarly to those of the original Bak-Tang-Wiesenfeld （BTW） model on scale-free networks. As the degree of driven nodes increases from the minimal value to the maximal value, the avalanche distribution gradually changes from a clean power law, then a mixture of Poissonian and power laws, finally to a Poisson-like distribution. The average avalanche area is found to increase with the degree of driven nodes so that perturbation triggered on higher-degree nodes will result in broader spreading of avalanche propagation.     

Emergence of scale-free behavior in networks from limited-horizon linking and cost trade-offs

We study network growth from a fixed set of initially isolated nodes placed at random on the surface of a sphere. The growth mechanism we use adds edges to the network depending on strictly local gain and cost criteria. Only nodes that are not too far apart on the sphere may be considered for being joined by an edge. Given two such nodes, the joining occurs only if the gain of doing it surpasses the cost. Our model is based on a multiplicative parameter λ that regulates, in a function of node degrees, the maximum geodesic distance that is allowed between nodes for them to be considered for joining. For n nodes distributed uniformly on the sphere, and for within limits that depend on cost-related parameters, we have found that our growth mechanism gives rise to power-law distributions of node degree that are invariant for constant . We also study connectivity- and distance-related properties of the networks.     

Prisoner's Dilemma Game on Clustered Scale-Free Networks under Different Initial Distributions

The evolutionary prisoner＇s dilemma game is investigated under different initial distributions for cooperators and defectors on scale-free networks with a tunable clustering coefficient. It is found that, on the one hand, cooperation can be enhanced with the increasing clustering coefficient when only the most connected nodes are occupied by cooperators initially. On the other hand, if cooperators just occupy the lowest-degree nodes at the beginning, then the higher the value of the clustering coefficient, the more unfavorable the environment for cooperators to survive for the increment of temptation to defect. Thereafter, we analytically argue these nontrivial phenomena by calculating the cooperation probability of the nodes with different degrees in the steady state, and obtain the critical values of initial frequency of cooperators below which cooperators would vanish finally for the two initial distributions.     

The complexity of Chinese syntactic dependency networks

This paper proposes how to build a syntactic network based on syntactic theory and presents some statistical properties of Chinese syntactic dependency networks based on two Chinese treebanks with different genres. The results show that the two syntactic networks are small-world networks, and their degree distributions obey a power law. The finding, that the two syntactic networks have the same diameter and different average degrees, path lengths, clustering coefficients and power exponents, can be seen as an indicator that complexity theory can work as a means of stylistic study. The paper links the degree of a vertex with a valency of a word, the small world with the minimized average distance of a language, that reinforces the explanations of the findings from linguistics.     

Extended Petersen networks

A new type of network growth rule is studied, which comprises nodes adding, plus a rewiring procedure, and replacing of nodes with the complete bipartite graph K2,3 acting on Petersen graph. The 3-regular extended Petersen networks have 0 clustering coefficient, short diameter, high connectivity and the Hamilton path. All these properties play an important role in the information transfer of the network, as they cannot only increase the security but also save time and system resources.     

Randomness Effect on Cooperation in Memory-Based Snowdrift Game

A memory-based snowdrift game (MBSG) on spatial small-world networks is investigated. It is found that cooperation rate versus temptation shows some step structures on small-world networks, similar to the case on regular lattices. With the increment of rewiring probability based on four-neighbourregular lattices, more steps are observable. Interestingly, it is observed that cooperation rate peaks at a specific value of temptation, which indicates that properly encouraging selfish actions may lead to better cooperative behaviours in the MBSG on small-world networks. Memory effects are also discussed for different rewiring probabilities. Furthermore, optimal regions arefound in the parameter planes. The strategy-related average degrees of individuals are helpful to understand the obtained 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.     

Robustness of heterogeneous complex networks

In this paper we study the robustness of heterogeneous preferential attachment networks. The robustness of a network measures its structural tolerance to the random removal of nodes and links. We numerically analyze the influence of the affinity parameters on a set of ensemble-averaged robustness metrics. We show that the presence of heterogeneity does not fundamentally alter the smooth nature of the fragmentation process of the models. We also show that a moderate level of locality translates into slight improvements in the robustness metrics, which prompts us to conjecture an evolutionary argument for the existence of real networks with power-law scaling in their connectivity and clustering distributions.     

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.     

Link prediction in complex networks via modularity-based belief propagation

Link prediction aims at detecting missing, spurious or evolving links in a network, based on the topological information and/or nodes' attributes of the network. Under the assumption that the likelihood of the existence of a link between two nodes can be captured by nodes' similarity, several methods have been proposed to compute similarity directly or indirectly, with information on node degree. However, correctly predicting links is also crucial in revealing the link formation mechanisms and thus in providing more accurate modeling for networks. We here propose a novel method to predict links by incorporating stochastic-block-model link generating mechanisms with node degree. The proposed method first recovers the underlying block structure of a network by modularity-based belief propagation, and based on the recovered block structural information it models the link likelihood between two nodes to match the degree sequence of the network. Experiments on a set of real-world networks and synthetic networks generated by stochastic block model show that our proposed method is effective in detecting missing, spurious or evolving links of networks that can be well modeled by a stochastic block model. This approach efficiently complements the toolbox for complex network analysis, offering a novel tool to model links in stochastic block model networks that are fundamental in the modeling of real world complex networks.