Synchronization of coupled oscillators on small-world networks

We investigate the synchronous dynamics of Kuramoto oscillators and van der Pol oscillators on Watts-Strogatz type small-world networks. The order parameters to characterize macroscopic synchronization are calculated by numerical integration. We focus on the difference between frequency synchronization and phase synchronization. In both oscillator systems, the critical coupling strength of the phase order is larger than that of the frequency order for the small-world networks. The critical coupling strength for the phase and frequency synchronization diverges as the network structure approaches the regular one. For the Kuramoto oscillators, the behavior can be described by a power-law function and the exponents are obtained for the two synchronizations. The separation of the critical point between the phase and frequency synchronizations is found only for small-world networks in the theoretical models studied.     

Evolving pseudofractal networks

We present a family of scale-free network model consisting of cliques, which is established by a simple recursive algorithm. We investigate the networks both analytically and numerically. The obtained analytical solutions show that the networks follow a power-law degree distribution, with degree exponent continuously tuned between 2 and 3. The exact expression of clustering coefficient is also provided for the networks. Furthermore, the investigation of the average path length reveals that the networks possess small-world feature. Interestingly, we find that a special case of our model can be mapped into the Yule process.     

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.     

Transcriptional regulatory network topology from statistics of DNA binding sites

We show that the out-degree distribution of the gene regulation network of the budding yeast, Saccharomyces cerevisiae, can be reproduced to high accuracy from the statistics of TF binding sequences. Our observation suggests a particular microscopic mechanism for the observed universal global topology in these networks. The numerical data and analytical solution of our model disagree with a simple power-law for the experimentally obtained degree distribution in the case of yeast.     

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.     

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.     

Linked by loops: Network structure and switch integration in complex dynamical systems

Simple nonlinear dynamical systems with multiple stable stationary states are often taken as models for switchlike biological systems. This paper considers the interaction of multiple such simple multistable systems when they are embedded together into a larger dynamical “supersystem.” Attention is focused on the network structure of the resulting set of coupled differential equations, and the consequences of this structure on the propensity of the embedded switches to act independently versus cooperatively. Specifically, it is argued that both larger average and larger variance of the node degree distribution lead to increased switch independence. Given the frequency of empirical observations of high variance degree distributions (e.g., power-law) in biological networks, it is suggested that the results presented here may aid in identifying switch-integrating subnetworks as comparatively homogenous, low-degree, substructures. Potential applications to ecological problems such as the relationship of stability and complexity are also briefly discussed.     

Betweenness centrality in finite components of complex networks

We use generating function formalism to obtain an exact formula of the betweenness centrality in finite components of random networks with arbitrary degree distributions. The formula is obtained as a function of the degree and the component size, and is confirmed by simulations for Poisson, exponential, and power-law degree distributions. We find that the betweenness centralities for the three distributions are asymptotically power laws with an exponent 1.5 and are invariant to the particular distribution parameters.     

Multi-directed Eulerian growing networks

We introduce and analyze a model of a multi-directed Eulerian network, that is a directed and weighted network where a path exists that passes through all the edges of the network once and only once. Networks of this type can be used to describe information networks such as human language or DNA chains. We are able to calculate the strength and degree distribution in this network and find that they both exhibit a power law with an exponent between 2 and 3. We then analyze the behavior of the accelerated version of the model and find that the strength distribution has a double slope power-law behavior. Finally we introduce a non-Eulerian version of the model and find that the statistical topological properties remain unchanged. Our analytical results are compared with numerical simulations.     

A unified model for Sierpinski networks with scale-free scaling and small-world effect

In this paper, we propose an evolving Sierpinski gasket, based on which we establish a model of evolutionary Sierpinski networks (ESNs) that unifies deterministic Sierpinski network [Z.Z. Zhang, S.G. Zhou, T. Zou, L.C. Chen, J.H. Guan, Eur. Phys. J. B 60 (2007) 259] and random Sierpinski network [Z.Z. Zhang, S.G. Zhou, Z. Su, T. Zou, J.H. Guan, Eur. Phys. J. B 65 (2008) 141] to the same framework. We suggest an iterative algorithm generating the ESNs. On the basis of the algorithm, some relevant properties of presented networks are calculated or predicted analytically. Analytical solution shows that the networks under consideration follow a power-law degree distribution, with the distribution exponent continuously tuned in a wide range. The obtained accurate expression of clustering coefficient, together with the prediction of average path length reveals that the ESNs possess small-world effect. All our theoretical results are successfully contrasted by numerical simulations. Moreover, the evolutionary prisoner’s dilemma game is also studied on some limitations of the ESNs, i.e., deterministic Sierpinski network and random Sierpinski network.     

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.     

Scale-free networks without growth

In this paper, we proposed an ungrowing scale-free network model, indicating the growth may not be a necessary condition of the self-organization of a network in a scale-free structure. The analysis shows that the degree distributions of the present model can varying from the Poisson form to the power-law form with the decrease of a free parameter α. This model provides a possible mechanism for the evolution of some scale-free networks with fixed size, such as the friendship networks of school children and the functional networks of the human brain.     

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.     

Interpersonal interactions and human dynamics in a large social network

We study a large social network consisting of over 10⁶ individuals, who form an Internet community and organize themselves in groups of different sizes. On the basis of the users’ list of friends and other data registered in the database we investigate the structure and time development of the network. The structure of this friendship network is very similar to the structure of different social networks. However, here a degree distribution exhibiting two scaling regimes, power-law for low connectivity and exponential for large connectivity, was found. The groups size distribution and distribution of number of groups of an individual have power-law form. We found very interesting scaling laws concerning human dynamics. Our research has shown how long people are interested in a single task.     

Detecting community structure from coherent oscillation of excitable systems

Many networks are proved to have community structures. On the basis of the fact that the dynamics on networks are intensively affected by the related topology, in this paper the dynamics of excitable systems on networks and a corresponding approach for detecting communities are discussed. Dynamical networks are formed by interacting neurons; each neuron is described using the FHN model. For noisy disturbance and appropriate coupling strength, neurons may oscillate coherently and their behavior is tightly related to the community structure. Synchronization between nodes is measured in terms of a correlation coefficient based on long time series. The correlation coefficient matrix can be used to project network topology onto a vector space. Then by the K-means cluster method, the communities can be detected. Experiments demonstrate that our algorithm is effective at discovering community structure in artificial networks and real networks, especially for directed networks. The results also provide us with a deep understanding of the relationship of function and structure for dynamical networks.     

Statistical properties of weighted complex networks characterized by metaweights

We propose several characterizations of weighted complex networks by incorporating the concept of metaweight into the clustering coefficient, degree correlation, and module decomposition. These incorporations make it possible to describe weighted networks depending on how strongly we emphasize weights. Using some applications to real-world weighted networks, we demonstrate that the proposed approach provides rich information that was inaccessible by previous analyses such as the degree correlation for a specific magnitude of weights or the community structure under controlling the importance of roles of the topology and weights.     

Graph kernels,hierarchical clustering,and network community structure: experiments and comparative analysis

There has been a quickly growing interest in properties of complex networks, such as the small world property, power-law degree distribution, network transitivity, and community structure, which seem to be common to many real world networks. In this study, we consider the community property which is also found in many real networks. Based on the diffusion kernels of networks, a hierarchical clustering approach is proposed to uncover the community structure of different extent of complex networks. We test the method on some networks with known community structures and find that it can detect significant community structure in these networks. Comparison with related methods shows the effectiveness of the method.     

Transient dynamics of sparsely connected Hopfield neural networks with arbitrary degree distributions

Using probabilistic approach, the transient dynamics of sparsely connected Hopfield neural networks is studied for arbitrary degree distributions. A recursive scheme is developed to determine the time evolution of overlap parameters. As illustrative examples, the explicit calculations of dynamics for networks with binomial, power-law, and uniform degree distribution are performed. The results are good agreement with the extensive numerical simulations. It indicates that with the same average degree, there is a gradual improvement of network performance with increasing sharpness of its degree distribution, and the most efficient degree distribution for global storage of patterns is the delta function.     

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.     

Epidemic outbreaks in growing scale-free networks with local structure

The class of generative models has already attracted considerable interest from researchers in recent years and much expanded the original ideas described in BA model. Most of these models assume that only one node per time step joins the network. In this paper, we grow the network by adding n interconnected nodes as a local structure into the network at each time step with each new node emanating m new edges linking the node to the preexisting network by preferential attachment. This successfully generates key features observed in social networks. These include power-law degree distribution p_k∼k^−(3+μ), where μ=(n−1)/m is a tuning parameter defined as the modularity strength of the network, nontrivial clustering, assortative mixing, and modular structure. Moreover, all these features are dependent in a similar way on the parameter μ. We then study the susceptible-infected epidemics on this network with identical infectivity, and find that the initial epidemic behavior is governed by both of the infection scheme and the network structure, especially the modularity strength. The modularity of the network makes the spreading velocity much lower than that of the BA model. On the other hand, increasing the modularity strength will accelerate the propagation velocity.