Epidemic spreading in a finite-precision BA model

The Barabasi-Albert (BA) model with finite-precision preferential attachment is used to build a wide range of network structures. Spreading epidemics and collective dynamics are investigated on such complex networks. Numerical simulations reveal a transition from an exponential scaling to a power-law distribution of link numbers per node along with the increase of the tuning parameter ω. The collective synchronization induced by the Susceptible-Infected-Recovered-Susceptible (SIRS) epidemiological process is shown to depend on the topological structure of the network.     

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.     

The most common friend first immunization

In this paper, a standard susceptible-infected-recovered-susceptible(SIRS) epidemic model based on the Watts–Strogatz(WS) small-world network model and the Barabsi–Albert(BA) scale-free network model is established, and a new immunization scheme — "the most common friend first immunization" is proposed, in which the most common friend's node is described as being the first immune on the second layer protection of complex networks. The propagation situations of three different immunization schemes — random immunization, high-risk immunization, and the most common friend first immunization are studied. At the same time, the dynamic behaviors are also studied on the WS small-world and the BA scale-free network. Moreover, the analytic and simulated results indicate that the immune effect of the most common friend first immunization is better than random immunization, but slightly worse than high-risk immunization. However,high-risk immunization still has some limitations. For example, it is difficult to accurately define who a direct neighbor in the life is. Compared with the traditional immunization strategies having some shortcomings, the most common friend first immunization is effective, and it is nicely consistent with the actual situation.     

Scale-free networks via attaching to random neighbors

Preferential attachment is considered one of the key factors in the formation of scale-free networks. However, complete random attachment without a preferential mechanism can also generate scale-free networks in nature, such as protein interaction networks in cells. This article presents a new scale-free network model that applies the following general mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach to random neighbors of random vertices that are already well connected. The proposed model does not require global-based preferential strategies and utilizes only the random attachment method. Theoretical analysis and numerical simulation results denote that the proposed model has steady scale-free network characteristics, and random attachment without a preferential mechanism may generate scale-free networks.     

Local preferential attachment model for hierarchical networks

In real-life networks, incomers may only connect to a few others in a local area for their limited information, and individuals in a local area are likely to have close relations. Accordingly, we propose a local preferential attachment model. Here, a local-area-network stands for a node and all its neighbors, and the new nodes perform nonlinear preferential attachment, , in local areas. The stable degree distribution and clustering-degree correlations are analytically obtained. With the increasing of α, the clustering coefficient increases, while assortativity decreases from positive to negative. In addition, by adjusting the parameter α, the model can generate different kinds of degree distribution, from exponential to power-law. The hierarchical organization, independent of α, is the most significant character of this model.     

Large Communities in a Scale-Free Network

We prove the existence of a large complete subgraph w.h.p. in a preferential attachment random graph process with an edge-step. That is, we consider a dynamic stochastic process for constructing a graph in which at each step we independently decide, with probability \(p\in (0,1)\), whether the graph receives a new vertex or a new edge between existing vertices. The connections are then made according to a preferential attachment rule. We prove that the random graph \(G_{t}\) produced by this so-called generalized linear preferential (GLP) model at time t contains a complete subgraph whose vertex set cardinality is given by \(t^\alpha \), where \(\alpha = (1-\varepsilon )\frac{1-p}{2-p}\), for any small \(\varepsilon >0\) asymptotically almost surely.     

Finite-Time Fluctuations in the Degree Statistics of Growing Networks

This paper presents a comprehensive analysis of the degree statistics in models for growing networks where new nodes enter one at a time and attach to one earlier node according to a stochastic rule. The models with uniform attachment, linear attachment (the Barabási-Albert model), and generalized preferential attachment with initial attractiveness are successively considered. The main emphasis is on finite-size (i.e., finite-time) effects, which are shown to exhibit different behaviors in three regimes of the size-degree plane: stationary, finite-size scaling, large deviations.     

On the construction of complex networks with optimal Tsallis entropy

In this work, we first formulate the Tsallis entropy in the context of complex networks. We then propose a network construction whose topology maximizes the Tsallis entropy. The growing network model has two main ingredients: copy process and random attachment mechanism (C-R model). We show that the resulting degree distribution exactly agrees with the required degree distribution that maximizes the Tsallis entropy. We also provide another example of network model using a combination of preferential and random attachment mechanisms (P-R model) and compare it with the distribution of the Tsallis entropy. In this case, we show that by adequately identifying the exponent factor q, the degree distribution can also be written in the q-exponential form. Taken together, our findings suggest that both mechanisms, copy process and preferential attachment, play a key role for the realization of networks with maximum Tsallis entropy. Finally, we discuss the interpretation of q parameter of the Tsallis entropy in the context of complex networks.     

Degree correlation in scale-free graphs

We obtain closed form expressions for the expected conditional degree distribution and the joint degree distribution of the linear preferential attachment model for network growth in the steady state. We consider the multiple-destination preferential attachment growth model, where incoming nodes at each timestep attach to β existing nodes, selected by degree-proportional probabilities. By the conditional degree distribution p(?|k), we mean the degree distribution of nodes that are connected to a node of degree k. By the joint degree distribution p(k,?), we mean the proportion of links that connect nodes of degrees k and ?. In addition to this growth model, we consider the shifted-linear preferential growth model and solve for the same quantities, as well as a closed form expression for its steady-state degree distribution.     

The origin of asymmetric behavior of money flow in the business firm network

In the business firm network, the number of in-degrees and out-degrees show the same scale-free property, however, the distribution of authorities and hubs show asymmetric behavior. Here we show the result of an analysis of the two-link structure of the network to find the origin of this asymmetric behavior. We find the tendency for big construction firms intermediating small subcontracting firms to have higher hub degrees. By measuring the strength of preferential attachment rate of new companies, we also find a abnormally strong preferential attachment for which the exponent is 1.4 with respect to out-degree when a new company forms a business partnership with a construction company. We propose a new model that reproduces the asymmetric behavior of the degrees of authorities and hubs by changing the preferential attachment rate between the in-degree and the out-degree in the business firm network.     

The emergence of scale-free networks with a seceding mechanism

In order to explore further the underlying mechanism of scale-free networks, we study stochastic secession as a mechanism for the creation of complex networks. In this evolution the network growth incorporates the addition of new nodes, the addition of new links between existing nodes, the deleting and rewiring of some existing links, and the stochastic secession of nodes. To random growing networks with preferential attachment, the model yields scale-free behavior for the degree distribution. Furthermore, we obtain an analytical expression of the power-law degree distribution with scaling exponent γ ranging from 1.1 to 9. The analytical expressions are in good agreement with the numerical simulation results.     

A model for social networks

Social networks are organized into communities with dense internal connections, giving rise to high values of the clustering coefficient. In addition, these networks have been observed to be assortative, i.e., highly connected vertices tend to connect to other highly connected vertices, and have broad degree distributions. We present a model for an undirected growing network which reproduces these characteristics, with the aim of producing efficiently very large networks to be used as platforms for studying sociodynamic phenomena. The communities arise from a mixture of random attachment and implicit preferential attachment. The structural properties of the model are studied analytically and numerically, using the k-clique method for quantifying the communities.     

Effect of the age of papers on the preferential attachment in citation networks

In this paper, we investigated the influences of the age of papers on the preferential attachment on the basis of three actual citation networks. We found that the time dependence of the attachment rate follows a uniform exponentially decreasing function, T(t)∼exp(−λt), in different citation networks. Younger papers are more likely to be cited by new ones than older papers. On the basis of the aging influences, we modified the expression for the preferential attachment, to . Our results show that the modified preferential attachment works well for citation networks.     

Nonlinear growth in weighted networks with neighborhood preferential attachment

We propose a nonlinear growing model for weighted networks with two significant characteristics: (i) the new weights triggered by new edges at each time step grow nonlinearly with time; and (ii) a neighborhood local-world exists for local preferential attachment, which is defined as one selected node and its neighbors. Global strength-driven and local weight-driven preferential attachment mechanisms are involved in our model. We study the evolution process through both mathematical analysis and numerical simulation, and find that the model exhibits a wide-range power-law distribution for node degree, strength, and weight. In particular, a nonlinear degree–strength relationship is obtained. This nonlinearity implies that accelerating growth of new weights plays a nontrivial role compared with accelerating growth of edges. Because of the specific local-world model, a small-world property emerges, and a significant hierarchical organization, independent of the parameters, is observed.     

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.     

Application of coupled map lattice with parameter q in image encryption

In this paper, a novel coupled map lattice (CML) with parameter q is applied to image encryption to get higher security. The CML with parameter q is provided with Euler method and Adams–Bashforth–Moulton predictor–corrector method. In the new CML, dynamical properties are improved because the coupled strength can decrease the periodic dynamical behaviors which are caused by finite-precision. What's more, the CML changes system parameters from one-dimensional to two-dimensional. Two-dimensional parameters and coupling strengths provide researchers a possibility to improve the performance in image encryption. Finally, from numerical simulation results, it can be found that the CML improves the effectiveness and security.     

Evolution of network from node division and generation

Aimed at lowering the effect of `rich get richer' in scale-free networks with the Barab\'{a}si and Albert model, this paper proposes a new evolving mechanism, which includes dividing and preference attachment for the growth of a network. A broad scale characteristic which is independent of the initial network topology is obtained with the proposed model. By simulating, it is found that preferential attachment causes the appearance of the scale-free characteristic, while the dividing will decrease the power-law behaviour and drive the evolution of broad scale networks.     

Error and attack tolerance of evolving networks with local preferential attachment

Networks generated by local-world evolving network model display a transition from exponential network to power-law network with respect to connectivity distribution. We investigate statistical properties of the evolving networks and the responses of these networks under random errors and intentional attacks. It has been found that local world size M has great effect on the network's heterogeneity, thus leading to transitional behaviors in network's robustness against errors and attacks. Numerical results show that networks constructed with local preferential attachment mechanism can maintain the robustness of scale-free networks under random errors and concurrently improve reliance against targeted attacks on highly connected nodes.     

Complex grid computing

This article investigates the functional properties of complex networks used as grid computing systems. Complex networks following the Erdös-Rényi model and other models with a preferential attachment rule (with and without growth) or priority to the connection of isolated nodes are studied. Regular networks are also considered for comparison. The processing load of the parallel program executed on the grid is assigned to the nodes on demand, and the efficiency of the overall computation is quantified in terms of the parallel speedup. It is found that networks with preferential attachment allow lower computing efficiency than networks with uniform link attachment. At the same time, considering only node clusters of the same size, preferential attachment networks display better efficiencies. The regular networks, on the other hand, display a poor efficiency, due to their implied larger internode distances. A correlation is observed between the topological properties of the network, specially average cluster size, and their respective computing efficiency.     

Kinetics of node splitting in evolving complex networks

We introduce a collection of complex networks generated by a combination of preferential attachment and a previously unexamined process of “splitting” nodes of degree k$k$ into k$k$ nodes of degree 1$1$. Four networks are considered, each evolves at each time step by either preferential attachment, with probability p$p$, or splitting with probability 1−p$1-p$. Two methods of attachment are considered; first, attachment of an edge between a newly created node and an existing node in the network, and secondly by attachment of an edge between two existing nodes. Splitting is also considered in two separate ways; first by selecting each node with equal probability and secondly, selecting the node with probability proportional to its degree. Exact solutions for the degree distributions are found and scale-free structure is exhibited in those networks where the candidates for splitting are chosen with uniform probability, those that are chosen preferentially are distributed with a power law with exponential cut-off.