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In this paper we study the model of opinion dynamics with individuals on nodes of a scale-free network, introducing an opinion evolution mechanism and according to the nodes' degrees defining broad-sense-concept leaders on the network. We compare the strength of opinion influence between leaders and followers. With computer simulations of opinion dynamics, we found that the more complex is the scale-free network, the easier for leaders to make their opinion accepted by the masses.

In this paper, we introduce the concept of intrinsic link, which is used to describe the intrinsic interactions between the individuals in complex systems. Furthermore, we present a model for the evolution of complex networks, in which the system dynamics motivated by four mechanisms: the addition of new nodes with preferential attachment, the addition of new nodes with intrinsic attachment, the addition of new links with preferential attachment and the addition of new intrinsic links. The model yields scale-free behavior for the degree distributions as confirmed in many real networks. With continumm theory, we get the analytical expressions of the degree distribution and the scaling exponent γ. The analytical expressions are in good agreement with the numerical simulation results.

To efficiently enhance the synchronizability of a scale-free network by adding some edges, we numerically study the effect of edge-adding on the spectrum of the network Laplacian matrix. Based on the relation between the largest eigenvalue of the Laplacian matrix and the largest degree of the scale-free network, we show that adding a new edge to the node of largest degree will generally weaken the synchronizability of a scale-free network. We consequently propose a method to effectively enhance the network synchronizability by attaching the new edge to a node whose nearest-neighbors have small degrees. The effect of the new method is analyzed and demonstrated with comparisons.

Network navigation is one of the main problems in large communication networks. We propose a new routing strategy in which some smart nodes in networks deliver messages to next hops on the paths towards destinations according to Yan's algorithm while the other nodes just deliver messages randomly. We test our routing strategy in a large scale-free network. Simulations show that the average delivery time decreases with increase of number of smart nodes, while the maximal network capacity increases with number of smart nodes in the network. Moreover our strategy is much more efficient when employed with target selection than with random selection of the smart nodes.

Traffic capacity is critical for various networks and strongly depends on the distribution of link's bandwidth resources. In this paper, we propose a betweenness-based bandwidth allocation strategy in which the bandwidth of each link l_ij is allocated proportionally to the product (1 + B_i)^α(1 + B_j)^α, where α is a tunable parameter, and B_i and B_j are the betweenness of node i and node j, respectively. The optimal value of α is achieved by extensive simulations and slightly increases with the network size. Our new bandwidth allocation strategy achieves the highest traffic capacity when compared with the average bandwidth allocation strategy and the previously proposed degree-based bandwidth allocation strategy. Our work will be beneficial for network service providers to improve the traffic capacity by efficiently allocating or reallocating the overall finite link's bandwidth resources of networks such as the Internet, urban transport networks and airway networks.

According to the dynamical characteristics of the local redistribution of the load on a removal node, by the reconnection of the neighboring edge of the most vulnerable node, we propose an effective method to improve the network robustness against cascading failures. Under two constraints, i.e. keeping the degree of each node unchanged and fixing the total protective cost of a network, we investigate the efficiency of the swap method on scale-free networks and analyze the correlation between the optimized network and the Pearson correlation coefficient. We numerically show that effective swapping of the small part of connections can dramatically improve the network robust level against cascading failures and find that the optimized networks obtained by the swap method exhibit an extremely disassortative degree–degree correlation, that is, the disassortativity decreases the robustness of the optimized network against cascading failures. While the extent of the disassortative mixing is decided by the parameters in the cascading model. In addition, we also compare the average path length and the diameter of the optimized and the original networks.

As two-layer or multi-layer network model can more accurately reveal many real structures of complex systems such as peer-to-peer (P2P) networks on IP networks, to better understand the traffic dynamics and improve the network traffic capacity, we propose to efficiently construct the structure of upper logical layer network which can be possibly implemented. From the beginning, we assume that the logical layer network has the same structure as the lower physical layer network, and then we use link-removal strategy in which a fraction of links with maximal product (k_i* k_j) are removed from the logical layer, where k_i and k_j are the degrees of node i and node j, respectively. Traffic load is strongly redistributed from center nodes to noncenter nodes. The traffic capacity of whole complex system is enhanced several times at the expense of a little average path lengthening. In two-layer network model, the physical layer network structure is unchanged and the shortest path routing strategy is used. The structure of upper layer network can been constructed freely under our own methods. This mechanism can be employed in many real complex systems to improve the network traffic capacity.

The link congestion based traffic model can more accurately reveal the traffic dynamics of many real complex networks such as the Internet, and heuristically optimizing each link's weight for the shortest path routing strategy can strongly improve the traffic capacity of network. In this work, we propose an optimal routing strategy in which the weight of each link is regulated incrementally to enhance the network traffic capacity by minimizing the maximum link betweenness of any link in the network. We also estimate more suitable value of the tunable parameter β for the efficient routing strategy under the link congestion based traffic model. The traffic load of network can be significantly balanced at the expense of increasing a bit average path length or average traffic load.

Global static routing is one kind of important routing algorithms for complex networks, especially in large communication networks. In this paper, we propose a heuristic global static routing algorithm to mitigate traffic congestion on two-layer complex networks. The proposed routing algorithm extends the relevant static weighted routing algorithm in the literature [Y. Zhou, Y. F. Peng, X. L. Yang and K. P. Long, Phys. Sci.84, 055802 (2011)]. Our routing path is constructed from a proper assignment of edge weights by considering the static information of both layers and an adjustable parameter α. When this routing algorithm is adopted on BA–BA two-layer networks with an appropriate parameter α, it can achieve the maximum network traffic capacity compared with the shortest path (SP) routing algorithm and the static weighted routing algorithm.

Many real-world networks are essentially heterogeneous, where the nodes have different abilities to gain connections. Such networks are difficult to be embedded into low-dimensional Euclidean space if we ignore the heterogeneity and treat all the nodes equally. In this paper, based on a newly defined heterogeneous distance and a generalized network distance under the constraints of network and triangle inequalities, respectively, we propose a new heterogeneous multidimensional scaling method (HMDS) to embed different networks into proper Euclidean spaces. We find that HMDS behaves much better than the traditional multidimensional scaling method (MDS) in embedding different artificial and real-world networks into Euclidean spaces. Besides, we also propose a method to estimate the appropriate dimensions of Euclidean spaces for different networks, and find that the estimated dimensions are quite close to the real dimensions for those geometrical networks under study. These methods thus can help to better understand the evolution of real-world networks, and have practical importance in network visualization, community detection, link prediction and localization of wireless sensors.

In this paper, an empirical analysis is done on the information flux network (IFN) statistical properties of genetic algorithms (GA) and the results suggest that the node degree distribution of IFN is scale-free when there is at least some selection pressure, and it has two branches as node degree is small. Increasing crossover, decreasing the mutation rate or decreasing the selective pressure will increase the average node degree, thus leading to the decrease of scaling exponent. These studies will be helpful in understanding the combination and distribution of excellent gene segments of the population in GA evolving, and will be useful in devising an efficient GA.

Scale-free networks in which the degree displays a power-law distribution can be classified into assortative, disassortative, and neutral networks according to their degree–degree correlation. The second-order centrality proposed in a distributed computation manner is quick-calculated and accurate to identify critical nodes. We explore the second-order centrality correlation (SOC) for each type of the scale-free networks. The SOC–SOC correlation in assortative network and neutral network behaves similarly to the degree–degree correlation, while it behaves an apparent difference in disassortative networks. Experiments show that the invulnerability of most of scale-free networks behaves similarly under the node removal ordering by SOC centrality and degree centrality, respectively. The netscience network and the Yeast network behave a little differently because they are native disconnecting networks.

We suggest an optimal form of traffic awareness already introduced as a routing protocol which combines structural and local dynamic properties of the network to determine the followed path between source and destination of the packet. Instead of using the shortest path, we incorporate the "efficient path" in the protocol and we propose a new parameter α that controls the contribution of the queue in the routing process. Compared to the original model, the capacity of the network can be improved more than twice when using the optimal conditions of our model. Moreover, the adjustment of the proposed parameter allows the minimization of the travel time.

Cascading failure is ubiquitous in many networked infrastructure systems, such as power grids, Internet and air transportation systems. In this paper, we extend the cascading failure model to a scale-free network with tunable clustering and focus on the effect of clustering coefficient on system robustness. It is found that the network robustness undergoes a nonmonotonic transition with the increment of clustering coefficient: both highly and lowly clustered networks are fragile under the intentional attack, and the network with moderate clustering coefficient can better resist the spread of cascading. We then provide an extensive explanation for this constructive phenomenon via the microscopic point of view and quantitative analysis. Our work can be useful to the design and optimization of infrastructure systems.

Routing strategy is essential for high transport efficiency on realistic networked complex systems. Beginning from the consideration of finite and diversiform node delivery capacity distributions, a general node capacity allocation mechanism with a tunable parameter $\alpha$ is presented. A node capacity, based routing strategy is proposed to improve the network traffic capacity. Compared with the traditional shortest path routing (SPR) and the efficient routing (ER) methods, it suggests that routing strategy should be chosen heuristically according to the limited capacity resource distribution, instead of using one certain method for all cases. With proper range of parameter $\alpha$, the new routing strategy achieves the highest traffic capacity and other preferable measure metrics including network diameter, average path length, efficient betweenness, average packet travel time and average traffic load. The theoretical analysis for traffic capacity has a good correspondence to the simulation results. This work studies routing mechanisms from a very practical perspective, and helps network researchers to understand the traffic dynamics on complex networks comprehensively.

Existing routing strategies such as the global dynamic routing [X. Ling, M. B. Hu, R. Jiang and Q. S. Wu, Phys. Rev. E81, 016113 (2010)] can achieve very high traffic capacity at the cost of extremely long packet traveling delay. In many real complex networks, especially for real-time applications such as the instant communication software, extremely long packet traveling time is unacceptable. In this work, we propose to assign a finite Time-to-Live (TTL) parameter for each packet. To guarantee every packet to arrive at its destination within its TTL, we assume that a packet is retransmitted by its source once its TTL expires. We employ source routing mechanisms in the traffic model to avoid the routing-flaps induced by the global dynamic routing. We compose extensive simulations to verify our proposed mechanisms. With small TTL, the effects of packet retransmission on network traffic capacity are obvious, and the phase transition from flow free state to congested state occurs. For the purpose of reducing the computation frequency of the routing table, we employ a computing cycle $T_c$ within which the routing table is recomputed once. The simulation results show that the traffic capacity decreases with increasing $T_c$. Our work provides a good insight into the understanding of effects of packet retransmission with finite packet lifetime on traffic capacity in scale-free networks.

In traffic networks, it is quite important to assign proper packet delivering capacities to the routers with minimum cost. In this respect, many allocation models based on static and dynamic properties have been proposed. In this paper, we are interested in the impact of limiting the packet delivering capacities already allocated to the routers; each node is assigned a packet delivering capacity limited by the maximal capacity $C_{max}$ of the routers. To study the limitation effect, we use two basic delivering capacity allocation models; static delivering capacity allocation (SDCA) and dynamic delivering capacity allocation (DDCA). In the SDCA, the capacity allocated is proportional to the node degree, and for DDCA, it is proportional to its queue length. We have studied and compared the limitation of both allocation models under the shortest path (SP) routing strategy as well as the efficient path (EP) routing protocol. In the SP case, we noted a similarity in the results; the network capacity increases with increasing $C_{max}$. For the EP scheme, the network capacity stops increasing for relatively small packet delivering capability limit $C_{max}$ for both allocation strategies. However, it reaches high values under the limited DDCA before the saturation. We also find that in the DDCA case, the network capacity remains constant when the traffic information available to each router was updated after long period times $\tau$.

In human societies, personal heterogeneity may affect the strategy adoption capability of the individuals. In this paper, we study the effects of heterogeneous learning ability on the evolution of cooperation by introducing heterogeneous imitation capability of players. We design a pre-factor ${\omega }_x$ to represent the heterogeneous learning ability of players, which is related to the degree of players. And a parameter $\alpha$ is used to tune the learning levels. If $\alpha >0$, the learning ability of players decreases and the low-degree player has the higher reduction level, but if $\alpha <0$, the learning ability of low-degree players enhances to a higher level. By carrying out extensive simulations, it reveals that the evolution of cooperation is influenced significantly by introducing player’s heterogeneous learning ability and can be promoted under the right circumstances. This finding sheds some light on the important effect of individual heterogeneity on the evolutionary game.

We explore the impact of positive news on rumor spreading in this paper. It is a fact that most of the rumors related to hot events or emergencies can be propagated rapidly on the hotbed of online social networks. In Chinese words, it is better to divert rather than block. Therefore, we propose the spreading model $IS{S}_{P}R$ in which positive news is a good factor to guide rumor spreading. Based on transition probability method, we have got the spreading parameters of the $IS{S}_{P}R$ model by running the rumor spreading process in online social networks with scale-free characteristics. The results give a good proof that improving the activity of the positive news spreader $S_P$ derived from the $IS{S}_{P}R$ model can guide and restrain the spreading speed of rumor smoothly.

We study the effects of the reciprocal links on the dynamics of direct Boolean networks with scale-free topology (SFRBNs). By means of the method of the Derrida Plot, we have investigated the SFRBNs characterized by different values of average degree and different values of reciprocity in order to test the behavioral regimes of the system. The following step was to perform numerical simulation with the quenched Kauffman model to study the dynamical properties of critical SFRBNs with ${〈k〉}_c=2$. The distribution of the number of different attractors, the period of the cyclic attractors, the transient duration and the fraction of the frozen nodes, have been studied as a function of the reciprocity and network size. The results presented reveal that reciprocity seems to have no direct effect on the changing of the behavioral regime of SFRBNs with given value of $〈k〉$. On the contrary, we observed that reciprocal links have a profound effect on the dynamic of critical SFRBNs.