Small World Effects in a Harmonious Unifying Hybrid Preferential Model Networks

Small worm effects in the harmonious unifying hybrid preferential model （HUHPM） networks are studied both numerically and analytically. The idea and method of the HUHPM is applied to three typical examples of unweighted BA model, weighted BBV model, and the TDE rnodel, so-called HUHPM-BA, HUHPM-BBV and HUHPM- TDE networks. Comparing the HUHPM with current typical models above, it is found that the HUHPM networks has the smallest average path length and the biggest average clustering coefficient. The results demonstrate that the HUHPM is more suitable not only for the un-iveighted models but also for the weighted models.     

A harmonious unifying hybrid preferential model and its universal properties for complex dynamical networks

To describe the real world which is a harmonious unification world with both de- terminism and randomness, we propose a harmonious unifying hybrid preferential model (HUHPM) of a certain class of complex dynamical networks. HUHPM is gov- erned only by the total hybrid ratio d/r according to the practical need. As some typical examples, the concepts and methods of the HUHPM are applied to the un-weighted BA model proposed by Barabási et al., the weighted BBV model pro- posed by Barat et al. and the weighted TDE model proposed by Wang et al. to get the so-called HUHPM-BA network, HUHPM-BBV network and HUHPM-TDE network. These HUHPM networks are investigated both analytically and numerically. It is found that the HUHPM reveals several universal properties, which more approach to the real-world networks for both un-weighted and weighted networks and have potential for applications.     

A Weighted Evolving Network with Community Size Preferential Attachment

Community structure is an important characteristic in real complex network. It is a network consists of groups of nodes within which links are dense but among which links are sparse. In this paper, the evolving network include node, link and community growth and we apply the community size preferential attachment and strength preferential attachment to a growing weighted network model and utilize weight assigning mechanism from BBV model. The resulting network reflects the intrinsic community structure with generalized power-law distributions of nodes＇ degrees and strengths.     

Weighted Evolving Networks with Self-organized Communities

In order to describe the self-organization of communities in the evolution of weighted networks, we propose a new evolving model for weighted community-structured networks with the preferential mechanisms functioned in different levels according to community sizes and node strengths, respectively. Theoretical analyses and numerical simulations show that our model captures power-law distributions of community sizes, node strengths, and link weights, with tunable exponents of v ≥ 1, γ 〉 2, and α 〉 2, respectively, sharing large clustering coefficients and scaling clustering spectra, and covering the range from disassortative networks to assortative networks. Finally, we apply our new model to the scientific co-authorship networks with both their weighted and unweighted datasets to verify its effectiveness.     

Topological Properties and Transition Features Generated by a New Hybrid Preferential Model

A new hybrid preferential model （HPM） is proposed for generating both scale-free and small world properties. The topological transition features in the HPM from random preferential attachment to deterministic preferential attachment are investigated. It is found that the exponents γ of the power law are very sensitive to the hybrid ratio （d/r） of determination to random attachment, and γ increases as the ratio d/r increases. It is also found that there exists a threshold at d/r = 1/1, beyond which γ increases rapidly and can tend to infinity if there is no random preferential attachment （r= 0）, which implies that the power law scaling disappears completely.Moreover, it is also found that when the ratio d/r increases, the average path length L is decreased, while the average clustering coefficient C is increased. Compared to the BA model and random graph, the new HPM has both the smallest L and the biggest C, which is consistent with most real-world growing networks.     

Effect Attack on Scale-Free Networks due to Cascading Failures

Adopting the initial load of a node j to be Lj = [kj ∑mεГkm）]^α with kj and Fj being the degree of the node j and the set of its neighbouring nodes respectively, we propose a cascading model based on a local preferential redistribution rule of the load after removing a node. Assuming that a failed node leads only to a redistribution of the load passing through it to its neighbouring nodes, we explore the response of scale-free networks subject to two different attack strategies on nodes and find some interesting and counterintuitive results in our cascading model. On the one hand, unexpectedly, tile attack on the nodes with the lowest degree is more harmful than the attack on the highest degree nodes when α〈1/2. On the other hand, when α = 1/2, the effects of two attacks for the robustness against cascading failures are almost identical. In addition, the numerical simulations are also verified by the theoretical analysis. These results may be very helpful for real-life networks to protect the key nodes selected effectively and to avoid cascading-failure-induced disasters.     

Influence of medium effects on the rotating hybrid stars

A hybrid star with a pure quark core,a hadron crust and a mixed phase between the two is considered.The relativistic mean field model for hadron matter and the effective mass bag model for quark matter are used to construct the equation of state for hybrid stars.The influences of medium effects that are parameterized by the strong coupling constant have been discussed on the configuration of rotating stars.The strong coupling constant is a prominent factor that influences the properties of rotating hybrid stars.     

Co-evolution of the brand effect and competitiveness in evolving networks

The principle that ‘the brand effect is attractive’ underlies the preferential attachment. Here we show that the brand effect is just one dimension of attractiveness. Another dimension is competitiveness. We firstly introduce a general framework that allows us to investigate the competitive aspect of real networks, instead of simply preferring popular nodes. Our model accurately describes the evolution of social and technological networks. The phenomenon that more competitive nodes become richer can help us to understand the evolution of many competitive systems in nature and society. In general,the paper provides an explicit analytical expression of degree distributions of the network. In particular, the model yields a nontrivial time evolution of nodes’ properties and the scale-free behavior with exponents depending on the microscopic parameters characterizing the competition rules. Secondly, through theoretical analyses and numerical simulations, we reveal that our model has not only the universality for the homogeneous weighted network, but also the character for the heterogeneous weighted network. Thirdly, we also develop a model based on the profit-driven mechanism. It can better describe the observed phenomenon in enterprise cooperation networks. We show that the standard preferential attachment,the growing random graph, the initial attractiveness model, the fitness model, and weighted networks can all be seen as degenerate cases of our model.     

Walks on Weighted Networks

We investigate the dynamics of random walks on weighted networks. Assuming that the edge weight and the node strength are used as local information by a random walker. Two kinds of walks, weight-dependent walk and strength-dependent walk, are studied. Exact expressions for stationary distribution and average return time are derived and confirmed by computer simulations. The distribution of average return time and the mean-square displacement are calculated for two walks on the Barrat-Barthelemy-Vespignani （BBV） networks. It is found that a weight-dependent walker can arrive at a new territory more easily than a strength-dependent one.     

Small World Effects in a Harmonious Unifying Hybrid Preferential Model Networks

Small world effects in the harmonious unifying hybrid preferential model (HUHPM) networks are studied both numerically and analytically. The idea and method of the HUHPM is applied to three typical examples of unweighted BA model, weighted BBV model, and the TDE model, so-called HUHPM-BA, HUHPM-BBV and HUHPM-TDE networks. Comparing the HUHPM with current typical models above, it is found that the HUHPM networks has the smallest average path length and the biggest average clustering coefficient. The results demonstrate that the HUHPM is more suitable not only for the un-weighted models but also for the weighted models.     

Entropy and HUHPM approach for complex networks

The general formulas of the network for the power exponents of the degree distribution and the entropy are presented based on an open Liouville equation for the driven network. The proposed harmonious unifying hybrid preferential models (HUHPM) have been studied using the obtained formalism, in which the simulating results are quite consistent with the theoretical results.     

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.     

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.     

A Robustness Model of Complex Networks with Tunable Attack Information Parameter

We introduce a novel model for robustness of complex with a tunable attack information parameter. The random failure and intentional attack known are the two extreme cases of our model. Based on the model, we study the robustness of complex networks under random information and preferential information, respectively. Using the generating function method, we derive the exact value of the critical removal fraction of nodes for the disintegration of networks and the size of the giant component. We show that hiding just a small fraction of nodes randomly can prevent a scale-free network from collapsing and detecting just a small fraction of nodes preferentially can destroy a scale-free network.     

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.