Modeling worldwide highway networks

This Letter addresses the problem of modeling the highway systems of different countries by using complex networks formalism. More specifically, we compare two traditional geographical models with a modified geometrical network model where paths, rather than edges, are incorporated at each step between the origin and the destination vertices. Optimal configurations of parameters are obtained for each model and used for the comparison. The highway networks of Australia, Brazil, India, and Romania are considered and shown to be properly modeled by the modified geographical model.     

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.     

Comparison of co-occurrence networks of the Chinese and English languages

Co-occurrence networks of Chinese characters and words, and of English words, are constructed from collections of Chinese and English articles, respectively. Four types of collections are considered, namely, essays, novels, popular science articles, and news reports. Statistical parameters of the networks are studied, including diameter, average degree, degree distribution, clustering coefficient, average shortest path length, as well as the number of connected subnetworks. It is found that the character and word networks of each type of article in the Chinese language, and the word network of each type of article in the English language all exhibit scale-free and small-world features. The statistical parameters of these co-occurrence networks are compared within the same language and across the two languages. This study reveals some commonalities and differences between Chinese and English languages, and among the four types of articles in each language from a complex network perspective. In particular, it is shown that expressions in English are briefer than those in Chinese in a certain sense.     

Measuring distances between complex networks

A previously introduced concept of higher order neighborhoods in complex networks, [R.F.S. Andrade, J.G.V. Miranda, T.P. Lobão, Phys. Rev. E 73 (2006) 046101] is used to define a distance between networks with the same number of nodes. With such measure, expressed in terms of the matrix elements of the neighborhood matrices of each network, it is possible to compare, in a quantitative way, how far apart in the space of neighborhood matrices two networks are. The distance between these matrices depends on both the network topologies and the adopted node numberings. While the numbering of one network is fixed, a Monte Carlo algorithm is used to find the best numbering of the other network, in the sense that it minimizes the distance between the matrices. The minimal value found for the distance reflects differences in the neighborhood structures of the two networks that arise only from distinct topologies. This procedure ends up by providing a projection of the first network on the pattern of the second one. Examples are worked out allowing for a quantitative comparison for distances among distinct networks, as well as among distinct realizations of random networks.     

Jamming transition in air transportation networks

In this work we present a model of an air transportation traffic system from the complex network modelling viewpoint. In the network, every node corresponds to a given airport, and two nodes are connected by means of flight routes. Each node is weighted according to its load capacity, and links are weighted according to the Euclidean distance that separates each pair of nodes. Local rules describing the behaviour of individual nodes in terms of the surrounding flow have been also modelled, and a random network topology has been chosen in a baseline approach. Numerical simulations describing the diffusion of a given number of agents (aircraft) in this network show the onset of a jamming transition that distinguishes an efficient regime with null amount of airport queues and high diffusivity (free phase) and a regime where bottlenecks suddenly take place, leading to a poor aircraft diffusion (congested phase). Fluctuations are maximal around the congestion threshold, suggesting that the transition is critical. We then proceed by exploring the robustness of our results in neutral random topologies by embedding the model in heterogeneous networks. Specifically, we make use of the European air transportation network formed by 858 airports and 11 170 flight routes connecting them, which we show to be scale-free. The jamming transition is also observed in this case. These results and methodologies may introduce relevant decision-making procedures in order to optimize the air transportation traffic.     

Dynamics-driven evolution to structural heterogeneity in complex networks

The mutual influence of dynamics and structure is a central issue in complex systems. In this paper we study by simulation slow evolution of network under the feedback of a local-majority-rule opinion process. If performance-enhancing local mutations have higher chances of getting integrated into its structure, the system can evolve into a highly heterogeneous small-world with a global hub (whose connectivity is proportional to the network size), strong local connection correlations and power-law-like degree distribution. Networks with better dynamical performance are achieved if structural evolution occurs much slower than the network dynamics. Structural heterogeneity of many biological and social dynamical systems may also be driven by various dynamics-structure coupling mechanisms.     

Analysis of the airport network of India as a complex weighted network

Transportation infrastructure of a country is one of the most important indicators of its economic growth. Here we study the Airport Network of India (ANI) which represents India’s domestic civil aviation infrastructure as a complex network. We find that ANI, a network of domestic airports connected by air links, is a small-world network characterized by a truncated power-law degree distribution and has a signature of hierarchy. We investigate ANI as a weighted network to explore its various properties and compare them with their topological counterparts. The traffic in ANI, as in the World-wide Airport Network (WAN), is found to be accumulated on interconnected groups of airports and is concentrated between large airports. In contrast to WAN, ANI is found to be having disassortative mixing which is offset by the traffic dynamics. The analysis indicates possible mechanism of formation of a national transportation network, which is different from that on a global scale.     

A Law of Gravitation for Complex Networks

Inspiring Newton＇s law of universal gravitation and empirical studies, we propose a concept of virtual network mass and network gravitational force in complex networks. Then a network gravitational model for complex networks is presented. In the model, each node in the network is described with its position, edges （links） and virtual network mass. The proposed model is examined by experiments to show its potential applications.     

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.     

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.     

Synchronization in complex networks

Synchronization processes in populations of locally interacting elements are the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understanding synchronization phenomena in natural systems now take advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also take an overview of the new emergent features coming out from the interplay between the structure and the function of the underlying patterns of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.     

Adaptive synchronization of weighted complex dynamical networks with coupling time-varying delays

This Letter considers the problem of controlling a weighted complex dynamical network with coupling time-varying delay toward an assigned evolution. Adaptive controllers have been designed for nodes of the controlled network. Analytical results show that the states of the weighted dynamical network can globally asymptotically synchronize onto a desired orbit under the designed controllers. In comparison with the common linear feedback controllers, the adaptive controllers have strong robustness against asymmetric coupling matrix, time-varying weights, delays, and noise. Numerical simulations illustrated by a nearest-neighbor coupling network verify the effectiveness of the proposed controllers.     

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.     

Information traffic in scale-free networks with fluctuations in packet generation rate

We study the information traffic in scale-free networks where the information generation rate varies with time as a periodic function. We observe that when the fluctuation in packet generation rate increases, the average transit time increases and network performance degrades. In order to improve the transportation efficiency in this situation, we propose a new routing method called mixed routing. It operates in two modes: (1) when the packet generation rate is small, the shortest paths are used to deliver the packets to the destination; (2) when the packet generation rate is large, the traffic loads in central nodes are redistributed to other non-central nodes, using the so-called efficient routing method. We find that the time shifting between the two modes is very critical for the routing performance. Consequently, we provide an efficient method to determine the critical times to shift the routing modes for achieving good network performance.     

Effect of edge removal on topological and functional robustness of complex networks

We study the robustness of several network models subject to edge removal. The robustness is measured by the statistics of network breakdowns, where a breakdown is defined as the destroying of the total connectedness of a network, rather than the disappearance of the giant component. We introduce a simple traffic dynamics as the function of a network topology, and the total connectedness can be destroyed in the sense of either the topology or the function. The overall effect of the topological breakdown and the functional breakdown, as well as the relative importance of the topological robustness and the functional robustness, are studied under two edge removal strategies.     

Adaptive synchronization of weighted complex dynamical networks through pinning

This paper considers the problem of controlling weighted complex dynamical networks by applying adaptive control to a fraction of network nodes. We investigate the local and global synchronization of the controlled dynamical network through the construction of a master stability function and a Lyapunov function. Analytical results show that a certain number of nodes can be controlled by using adaptive pinning to ensure the synchronization of the entire network. We present numerical simulations to verify the effectiveness of the proposed scheme. In comparison with feedback pinning, the proposed pinning control scheme is robust when tested by noise, different weighting and coupling structures, and time delays.     

Adaptive Synchronization between Two Different Complex Networks with Time-Varying Delay Coupling

A new general network model for two complex networks with time-varying delay coupling is presented. Then we investigate its synchronization phenomena. The two complex networks of the model differ in dynamic nodes, the number of nodes and the coupling connections. By using adaptive controllers, a synchronization criterion is derived. Numerical examples are given to demonstrate the effectiveness of the obtained synchronization criterion. This study may widen the application range of synchronization, such as in chaotic secure communication.     

Cross-impact balances: Applying pair interaction systems and multi-value Kauffman nets to multidisciplinary systems analysis

Cross-impact analysis is the name for a familiar method for multidisciplinary systems analysis in social sciences and management sciences, especially in technology foresight, technology assessment and scenario planning. A recently proposed form of cross-impact analysis, CIB, may be of interest for physicists, sociophysicists and complex network researchers because the CIB concept reveals considerable relations to some concepts of these research fields.This article describes the basics of CIB analysis framework, its applications in the social sciences, and its relations to the equilibrium points of pair interaction systems, random graphs, and generalized Kauffman nets.Therefore CIB can be seen as a merger of concepts originating in utterly different scientific fields. This may prove to be fruitful for both sides: For sociophysicists as an example of the application of complex network concepts in the social sciences and for cross-impact practitioners as a source of theoretical insights in the background of their tool.     

Differential System＇s Nonlinear Behaviour of Real Nonlinear Dynamical Systems

Chaos attractor behaviour is usually preserved if the four basic arithmetic operations, i.e. addition, subtraction, multiplication, division, or their compound, are applied. First-order differential systems of one-dimensional real discrete dynamical systems and nonautonomous real continuous-time dynamical systems are also dynamical systems and their Lyapunov exponents are kept, if they are twice differentiable. These two conclusions are shown here by the definitions of dynamical system and Lyapunov exponent. Numerical simulations support our analytical results. The conclusions can apply to higher order differential systems if their corresponding order differentials exist.     

Network robustness to targeted attacks. The interplay of expansibility and degree distribution

We study the property of certain complex networks of being both sparse and highly connected, which is known as “good expansion” (GE). A network has GE properties if every subset S of nodes (up to 50% of the nodes) has a neighborhood that is larger than some “expansion factor” φ multiplied by the number of nodes in S. Using a graph spectral method we introduce here a new parameter measuring the good expansion character of a network. By means of this parameter we are able to classify 51 real-world complex networks — technological, biological, informational, biological and social — as GENs or non-GENs. Combining GE properties and node degree distribution (DD) we classify these complex networks in four different groups, which have different resilience to intentional attacks against their nodes. The simultaneous existence of GE properties and uniform degree distribution contribute significantly to the robustness in complex networks. These features appear solely in 14% of the 51 real-world networks studied here. At the other extreme we find that ∼40% of all networks are very vulnerable to targeted attacks. They lack GE properties, display skewed DD — exponential or power-law — and their topologies are changed more dramatically by targeted attacks directed at bottlenecks than by the removal of network hubs.