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For the complex networks, including scale-free, small-world, local-world and random networks, the global quantitative evaluation of attack-induced cascade is investigated in this paper by introducing the risk assessment, which integrates the probability of occurrence with the damage size of attacks on nodes. It is discovered by simulations, among the several kinds of networks, that the small-world network has the largest risk assessment of attack-induced cascade; the risk assessment of three other networks are all very low and the most protection against attack should be given to the small-world network accordingly. Furthermore, the percentage of the most fragile nodes in the scale-free network is very low, compared with that in the small-world network.

Most of modern systems are coupled by two sub-networks and therefore should be modeled as interdependent networks. The study towards robustness of interdependent networks becomes interesting and significant. In this paper, mainly by numerical simulations, the robustness of interdependent Erdös–Rényi (ER) networks and interdependent scale-Free (SF) networks coupled by two sub-networks with different average degree are investigated. First, we study the robustness of interdependent networks under random attack. Second, we study the robustness of interdependent networks under targeted attack on high or low degree nodes, and find that interdependent networks with different average degree are significantly different from those interdependent networks with equal average degree.

In this paper, we propose a pair-dependent rejection rate of packet information between routers in the framework of the minimal traffic model applied to scale-free networks. We have shown that the behavior of the transition point from the phase where the system balances the inflow of new information packets with successful delivery of the old ones to the congested phase depends on the underlying mechanism of packet rejection. It is possible to achieve larger values for the critical load by varying the rejection of the packets issued from a given node by its neighbors. We have proposed an asymmetric protocol, where we found the existence of a whole interval where the packet rejection is strongly beneficial to the overall performance of the system. We have also shown that for the dynamic protocol, the transition point is shifted toward higher values permitting the network to handle more traffic load, despite the fact that the critical load decreases when increasing the rejection parameter.

We introduce a new opinion model to study the opinion evolving in three typical networks (small-world network, scale-free network and highly clustered scale-free network) by employing Monte Carlo numerical simulation. To consider important social influence, we propose a heat bath like optimized opinion model that includes both local and global social impacts. By employing the new opinion model into three typical networks, we have found the main determinants of the evolution of the opinion, that were the weight constant of the local and global effects, the structure of the network, the size of the network and the initial density of the average opinion. The simulations show how those factors affect the evolving time of reaching the consensus quantitatively. Our model also could explain the cases of de facto standard and lock-in effect, well-known phenomenon in economics and business management.

In this paper, we construct a class of growing networks by the encoding method of the iterated function system based on a planar self-similar fractal, and demonstrate that the networks have small-world and scale-free effects.

In order to further explore the mechanism responsible for weighted complex networks, we introduce a new model that incorporates the network topology and the weights' dynamical evolutions. Our model can capture the details of weight dynamics caused not only by the addition of a new node with new links and new links between old nodes, but also the deletion of old links. We calculate analytically the distributions of both degree and strength and found that all these distributions show scale-free behavior, as confirmed in many real networks. Thus our model characterizes the real weighted complex networks more precisely.

In this paper, we first derive the analytical expressions of the degree distributions for the network with random initializing attractiveness and preferential linking by using the approach of mean-field theory. Then we discuss the justification of the scale-free behavior and give a remark about the proposed model. Finally, a series of theoretical analysis and numerical simulations for the network model are conducted. The computer simulations and the theoretical results are consistent, and display the effectiveness of the model.

Urban public transit system is a typical mixed complex network with dynamic flow, and its evolution should be a process coupling topological structure with flow dynamics, which has received little attention. This paper presents the R-space to make a comparative empirical analysis on Beijing’s flow-weighted transit route network (TRN) and we found that both the Beijing’s TRNs in the year of 2011 and 2015 exhibit the scale-free properties. As such, we propose an evolution model driven by flow to simulate the development of TRNs with consideration of the passengers’ dynamical behaviors triggered by topological change. The model simulates that the evolution of TRN is an iterative process. At each time step, a certain number of new routes are generated driven by travel demands, which leads to dynamical evolution of new routes’ flow and triggers perturbation in nearby routes that will further impact the next round of opening new routes. We present the theoretical analysis based on the mean-field theory, as well as the numerical simulation for this model. The results obtained agree well with our empirical analysis results, which indicate that our model can simulate the TRN evolution with scale-free properties for distributions of node’s strength and degree. The purpose of this paper is to illustrate the global evolutional mechanism of transit network that will be used to exploit planning and design strategies for real TRNs.

Landslides have been widely studied by geologists. However, previous studies mainly focused on the formation of landslides and never considered the effect of landslides on the structural characteristics of land-cover. Here we define the modeling of the graph topology for the land-cover, using the satellite images of the earth’s surface before and after the earthquake. We find that the land-cover network satisfies the power-law distribution, whether the land-cover contains landslides or not. However, landslides may change some parameters or measures of the structural characteristics of land-cover. The results show that the linear coefficient, modularity and area distribution are all changed after the occurence of landslides, which means the structural characteristics of the land-cover are changed.

As we know, the scale-free property of networks implies that there are a great deal of nodes with low degree and a few with high degree in networks. In this paper, we mainly stick to the analysis of the heterogeneity of social networks. Heterogeneity measure of degree sequence based on Laplacian centrality (HLC) and degree ratio (HDR) and local neighborhood (HLN) are presented. Furthermore, heterogeneities of community based on the size, edge abundant degree and density of community are explored, respectively. The heterogeneity measures of community are empirically analyzed in the real-world network in terms of these three indices, i.e., size, edge abundant degree and density of community.

This paper is the first step to a study of supply chain distribution networks based on the scale-free theory, and we attempt to analyze the growth of supply chain distribution networks. We construct the supply chain distribution network and divide it into two groups. Through empirical theory and numerical study, it is found that supply chain networks have scale-free characteristics. Additionally, other three features of supply chain distribution network are proposed in this paper. Finally, the exponent of degree distribution is given.

The efficiency of a routing strategy on complex networks can be reflected by two measurements, i.e. the system capacity and the average data packets travel time. In this paper, we propose a new routing strategy which is only based on local information of network topology. This strategy integrated the delivering capability and packets queue length of nodes for enhancing the efficiency of traffic on scale-free networks. The probability that a given node i with delivering capability C_i receives packets from its neighbors is proportional to (N_i+1)/C_i and N_i is the packets queue length of the node i. Simulation results show that there exists an optimal value by maximizing the networks delivering capability and minimizing the packet travel time. We simulated the strategy on BA network with different m (connectivity density) values and the results show that our strategy is more efficient than other local information-based routing strategies.

We review results on the scaling of the optimal path length ℓ_opt in random networks with weighted links or nodes. We refer to such networks as "weighted" or "disordered" networks. The optimal path is the path with minimum sum of the weights. In strong disorder, where the maximal weight along the path dominates the sum, we find that ℓ_opt increases dramatically compared to the known small-world result for the minimum distance ℓ_min ~ log N, where N is the number of nodes. For Erdős–Rényi (ER) networks ℓ_opt ~ N^1/3, while for scale free (SF) networks, with degree distribution P(k) ~ k^-λ, we find that ℓ_opt scales as N^{(λ - 3)/(λ - 1)} for 3 < λ < 4 and as N^1/3 for λ ≥ 4. Thus, for these networks, the small-world nature is destroyed. For 2 < λ < 3 in contrary, our numerical results suggest that ℓ_opt scales as ln^λ-1 N, representing still a small world. We also find numerically that for weak disorder ℓ_opt ~ ln N for ER models as well as for SF networks. We also review the transition between the strong and weak disorder regimes in the scaling properties of ℓ_opt for ER and SF networks and for a general distribution of weights τ, P(τ). For a weight distribution of the form P(τ) = 1/(aτ) with (τ_min < τ < τ_max) and a = ln τ_max/τ_min, we find that there is a crossover network size N^* = N^*(a) at which the transition occurs. For N ≪ N^* the scaling behavior of ℓ_opt is in the strong disorder regime, while for N ≫ N^* the scaling behavior is in the weak disorder regime. The value of N^* can be determined from the expression ℓ_∞(N^*) = ap_c, where ℓ_∞ is the optimal path length in the limit of strong disorder, A ≡ ap_c → ∞ and p_c is the percolation threshold of the network. We suggest that for any P(τ) the distribution of optimal path lengths has a universal form which is controlled by the scaling parameter Z = ℓ_∞/A where plays the role of the disorder strength and τ_c is defined by . In case P(τ) ~ 1/(aτ), the equation for A is reduced to A = ap_c. The relation for A is derived analytically and supported by numerical simulations for Erdős–Rényi and scale-free graphs. We also determine which form of P(τ) can lead to strong disorder A → ∞. We then study the minimum spanning tree (MST), which is the subset of links of the network connecting all nodes of the network such that it minimizes the sum of their weights. We show that the minimum spanning tree (MST) in the strong disorder limit is composed of percolation clusters, which we regard as "super-nodes", interconnected by a scale-free tree. The MST is also considered to be the skeleton of the network where the main transport occurs. We furthermore show that the MST can be partitioned into two distinct components, having significantly different transport properties, characterized by centrality — number of times a node (or link) is used by transport paths. One component the superhighways, for which the nodes (or links) with high centrality dominate, corresponds to the largest cluster at the percolation threshold (incipient infinite percolation cluster) which is a subset of the MST. The other component, roads, includes the remaining nodes, low centrality nodes dominate. We find also that the distribution of the centrality for the incipient infinite percolation cluster satisfies a power law, with an exponent smaller than that for the entire MST. We demonstrate the significance identifying the superhighways by showing that one can improve significantly the global transport by improving a very small fraction of the network, the superhighways.

In this paper, we present results concerning a natural extension of the class of heterogeneous preferential attachment models, a generalization of the Barabási–Albert model to heterogeneous networks. In this extended class, the network nodes enjoy a nonzero attractiveness even when their connectivity degrees are zero. We analytically show that the degree densities of models in the extended class exhibit a richer scaling behavior than their homogeneous counterparts, and that power-law scaling in their degree distribution is robust in the presence of the offset in the attachment kernel.

Many complex systems, such as software systems, are full of complexity arising from interactions among basic units (such as classes, interfaces and struts in object-oriented software systems). One of the most successful approaches to capture the underlying structural features of large-scale software systems is the investigation of hierarchical organization. However, the hierarchy of software networks has not been thoroughly investigated. In this paper, the crucial fraction (CF) in software networks has been extracted and analyzed in a set of real-world software systems. First, the classes and the relationships between them have been extracted into software networks. Then software networks have been divided into different layers, and CF of software networks has been extracted by k-core. The empirical studies in this paper reveal that software networks represent flat hierarchical structure. Finally, CF has been measured by the relevant complex network parameters respectively, and the relations between CF and overall network have been analyzed by the case studies of software networks. The results show that CF represents characteristics of scale-free, small-world, strong connectivity, and the units in CF are frequently reused and dominate the overall system.

In this paper, we proposed an edge weight method for performing a community detection on mixed scale-free networks.We use the phrase “mixed scale-free networks” for networks where some communities have node degree that follows a power law similar to scale-free networks, while some have node degree that follows normal distribution. In this type of network, community detection algorithms that are designed for scale-free networks will have reduced accuracy because some communities do not have scale-free properties. On the other hand, algorithms that are not designed for scale-free networks will also have reduced accuracy because some communities have scale-free properties. To solve this problem, our algorithm consists of two community detection steps; one is aimed at extracting communities whose node degree follows power law distribution (scale-free), while the other one is aimed at extracting communities whose node degree follows normal distribution (non scale-free). To evaluate our method, we use NMI — Normalized Mutual Information — to measure our results on both synthetic and real-world datasets comparing with both scale-free and non scale-free community detection methods. The results show that our method outperforms all other based line methods on mixed scale-free networks.

The behavior of sandpile far from self-organized critical states (SOCS) is investigated in this paper. The results indicate that noise plays an important role in the critical phenomenon. The critical phenomena are detected in sandpile far from SOCS when external or internal noise is applied. In contrast to self-organized criticality, the exponent of avalanche distribution in sandpile far from SOCS increases with the distance of the sandpile from SOCS. At the same time, the exponent of avalanche distribution in a sandpile far from SOCS is consistent with some real systems such as earthquake.

The Sierpinski tetrahedron is used to construct evolving networks, whose vertexes are all solid regular tetrahedra in the construction of the Sierpinski tetrahedron up to the stage $t$ and any two vertexes are neighbors if and only if the corresponding tetrahedra are in contact with each other on boundary. We show that such networks have the small-world and scale-free effects, but are not fractal scaling.

In this paper, we investigate the vertical-affiliation-free (VAF) evolving networks whose node set is the basic squares in the process of generating the Sierpinski carpet and edge exists between any two nodes if and only if the corresponding basic squares intersect just on their boundary. Although the VAF networks gets rid of the hierarchial organizations produced naturally by the self-similar structures of fractals, we still prove that they are scale-free and have the small-world effect.

In this paper, we construct evolving networks based on the construction of the $n$-dimensional Sierpinski pyramid by the self-similar structure. We show that such networks have scale-free and small-world effects.