Estimating the clustering coefficient in scale-free networks on lattices with local spatial correlation structure

When we study the architecture of networks of spatially extended systems the nodes in the network are subject to local correlation structures. In this case, we show that for scale-free networks the traditional way to estimate the clustering coefficient may not be meaningful. Here we explain why and propose an approach that corrects this problem.     

The web of connections between tourism companies: Structure and dynamics

Tourism destination networks are amongst the most complex dynamical systems, involving a myriad of human-made and natural resources. In this work we report a complex network-based systematic analysis of the Elba (Italy) tourism destination network, including the characterization of its structure in terms of several traditional measurements, the investigation of its modularity, as well as its comprehensive study in terms of the recently reported superedges approach. In particular, structural (the number of paths of distinct lengths between pairs of nodes, as well as the number of reachable companies) and dynamical features (transition probabilities and the inward/outward activations and accessibilities) are measured and analyzed, leading to a series of important findings related to the interactions between tourism companies. Among the several reported results, it is shown that the type and size of the companies influence strongly their respective activations and accessibilities, while their geographical position does not seem to matter. It is also shown that the Elba tourism network is largely fragmented and heterogeneous, so that it could benefit from increased integration.     

Unified index to quantifying heterogeneity of complex networks

Although recent studies have revealed that degree heterogeneity of a complex network has significant impact on the network performance and function, a unified definition of the heterogeneity of a network with any degree distribution is absent. In this paper, we define a heterogeneity index 0≤H<1 to quantify the degree heterogeneity of any given network. We analytically show the existence of an upper bound of H=0.5 for exponential networks, thus explain why exponential networks are homogeneous. On the other hand, we also analytically show that the heterogeneity index of an infinite power law network is between 1 and 0.5 if and only if its degree exponent is between 2 and 2.5. We further show that for any power law network with a degree exponent greater than 2.5, there always exists an exponential network such that both networks have the same heterogeneity index. This may help to explain why 2.5 is a critical degree exponent for some dynamic behaviors on power law 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.     

Attack vulnerability of scale-free networks due to cascading failures

In this paper, adopting the initial load of a node i to be with k_i being the degree of the node i, we propose a cascading model based on a load local redistribution rule and examine cascading failures on the typical network, i.e., the BA network with the scale-free property. We find that the BA scale-free network reaches the strongest robustness level in the case of α=1 and the robustness of the network has a positive correlation with the average degree 〈k〉, where the robustness is quantified by a transition from normal state to collapse. In addition, we further discuss the effects of two different attacks for the robustness against cascading failures on our cascading model and find an interesting result, i.e., the effects of two different attacks, strongly depending to the value α. These results may be very helpful for real-life networks to avoid cascading-failure-induced disasters.     

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.     

A complex network model based on the Gnutella protocol

Gnutella is one of the basic protocols for P2P software. In this paper, a novel network model based on Gnutella is introduced. The mechanism of this network is based on resource occupancy and search activities of peers. As for the structure, the power-law exponent of in-degree γ_in≈4.2, the length of the average shortest path 〈l〉=57.74, and the diameter of the network is 156; these topological properties of the proposed structure differ from known results.     

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.     

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.     

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.     

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.     

A model for cascading failures in scale-free networks with a breakdown probability

Considering that not all overload nodes will be removed from networks due to some effective measures to protect them, we propose a new cascading model with a breakdown probability. Adopting the initial load of a node j to be L_j=[k_j(∑_m∈Γjk_m)]^α with k_j and Γ_j being the degree of the node j and the set of its neighboring nodes, respectively, where α is a tunable parameter, we investigate the relationship between some parameters and universal robustness characteristics against cascading failures on scale-free networks. According to a new measure originated from a phase transition from the normal state to collapse, the numerical simulations show that Barabási-Albert (BA) networks reach the strongest robustness level against cascading failures when the tunable parameter α=0.5, while not relating to the breakdown probability. We furthermore explore the effect of the average degree 〈k〉 for network robustness, thus obtaining a positive correlation between 〈k〉 and network robustness. We then analyze the effect of the breakdown probability on the network robustness and confirm by theoretical predictions this universal robustness characteristic observed in simulations. Our work may have practical implications for controlling various cascading-failure-induced disasters in the real world.     

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.     

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.     

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.     

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.     

Synchrony-optimized networks of non-identical Kuramoto oscillators

In this Letter we discuss a method for generating synchrony-optimized coupling architectures of Kuramoto oscillators with a heterogeneous distribution of native frequencies. The method allows us to relate the properties of the coupling network to its synchronizability. These relations were previously only established from a linear stability analysis of the identical oscillator case. We further demonstrate that the heterogeneity in the oscillator population produces heterogeneity in the optimal coupling network as well. Two rules for enhancing the synchronizability of a given network by a suitable placement of oscillators are given: (i) native frequencies of adjacent oscillators must be anti-correlated and (ii) frequency magnitudes should positively correlate with the degree of the node they are placed at.     

Network analysis of an online community

We empirically study various network properties of an online community. The numbers of articles written by each user to the bulletin boards of each of the others are used to construct the directed and weighted network B, and gifting behaviors among users are also kept track of, to build the network G which is again directed and weighted. Detailed analysis reveals that B and G have very different network properties. In particular, whereas B contains many more bidirectional links than directed arcs, G shows the opposite characteristic. The number of writings on bulletin boards is found to decay with the distance from the hub vertex, which reflects the structural assortativeness in B. We also observe that the activities in writings and purchases are negatively correlated with each other for highly active users in B.     

A spatial weighted network model based on optimal expected traffic

We propose a spatial weighted network model based on the optimal expected traffic. The expected traffic represents the prediction of the flow created by two vertices and is calculated by the improved gravity equation. The model maximizes the total expected traffic of the network. By changing two parameters which control the fitness and the geographical constraints, the model can vary its topology and give rise to a variety of statistical properties observed in the real-world network. Notably, our study shows that a linear and a nonlinear strength-degree correlation can emerge when considering and neglecting the “transport effect”, respectively.     

An optimal network for passenger traffic

The optimal solution of an inter-city passenger transport network has been studied using Zipf’s law for the city populations and the Gravity law describing the fluxes of inter-city passenger traffic. Assuming a fixed value for the cost of transport per person per kilometer we observe that while the total traffic cost decreases, the total wiring cost increases with the density of links. As a result the total cost to maintain the traffic distribution is optimal at a certain link density which vanishes on increasing the network size. At a finite link density the network is scale-free. Using this model the air-route network of India has been generated and an one-to-one comparison of the nodal degree values with the real network has been made.