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The diversity of power-law-like degree distribution in social networks has been discovered in a large number of empirical studies. Analyzing the origin of power-law-like the degree distribution diversity is greatly important for understanding the law of human social interaction. In our work, the diversity of power-law-like degree distribution is demonstrated empirically in social networks. The origin of the degree distribution diversity is analyzed from the point of the social stratification and the bidirectional preferential attachment among individuals. We proposed a model to reproduce the diversity of degree distribution in social networks, and the analytic solution of the model was derived. The simulation results indicate that the evolution time of social network and the biggest social hierarchy gap among individuals may be the origin which results in the power-law-like degree distribution diversity. Therefore, the model is helpful to comprehend the law of human social interaction.

Real complex networks usually have small-world effect, scale-free features and hierarchical modularity, a construction method for deterministic hierarchical network is proposed in this paper. The network model with growth and global preferential attachment characteristic and connected by the copy network modules to establish hierarchical network model. By theoretical calculation and numerical simulation about the deterministic hierarchical complex network model, the results illustrate that the complex network model satisfy the small-world effect, scale-free feature and hierarchical modularity, the calculation results show that the size of the hierarchical network model for exponential growth with the network size increased, and the average degree of nodes is shown as linear growth; at the same time, the model of scale-free feature and the hierarchical modularity with network parameters do not have correlation which is an inherent attribute of network model itself; the clustering-degree correlations in the network model satisfy power-law characteristics and the nodes contact closely together in the modules which are connected by the Hub nodes in the complex network model. .

The last couple of years have witnessed a huge surge in usage of the Internet due to the availability of several interactive applications of Web 2.0. Growth in both web users and varieties of online objects has been observed. In this paper, we propose a growth model for online emerging user–object bipartite networks to investigate selection behaviors of web users. According to model, both sets of users and objects grow constantly but edges arrive only from user set. The network evolves by arrival of external edges brought by new users and/or internal edges from old users. These external and internal edges are attached with a combination of preferential and random attachment mechanism. We have considered different attachment procedures for external edges of new users and internal edges of old users. To validate our model, we have taken nine different real online networks with distinct objects. Our results show good agreement between empirical data and theoretical model. We report mean absolute deviation (MAD), mean square error (MSE) and root mean square error (RMSE) for the proposed growth model. We emphasize on significant inferences about selection behaviors of web users after interpreting parameters’ values of the model. We also report detailed analysis of randomness in selection of new and old users that gives interesting inferences for each of the considered online networks.

Symmetries are ubiquitous in real networks and often characterize network features and functions. Here we present a generalization of network symmetry called \emph{latent symmetry}, which is an extension of the standard notion of symmetry. They are defined in terms of standard symmetries in a reduced version of the network. One unique aspect of latent symmetries is that each one is associated with a \emph{size}, which provides a way of discussing symmetries at multiple scales in a network. We are able to demonstrate a number of examples of networks (graphs) which contain latent symmetry, including a number of real networks. In numerical experiments, we show that latent symmetries are found more frequently in graphs built using preferential attachment, a standard model of network growth, when compared to non-network like (Erd{\H o}s-R\'enyi) graphs. Finally we prove that if vertices in a network are latently symmetric, then they must have the same eigenvector centrality, similar to vertices which are symmetric in the standard sense. This suggests that the latent symmetries present in real-networks may serve the same structural and functional purpose standard symmetries do in these networks. We conclude from these facts and observations that \emph{latent symmetries} are present in real networks and provide useful information about the network potentially beyond standard symmetries as they can appear at multiple scales., Comment: 18 pages

Considering that some research and development (R&D) networks are becoming increasingly interdependent, risk not only propagates within one R&D network but also spreads to other R&D networks. This paper proposes a model of risk cascading propagation in order to explore the vulnerability of interdependent R&D networks. Firstly, we propose the generation algorithm of individual R&D networks based on the rules of preferential attachment and geographical proximity, and then build the interdependences between two R&D networks by identifying the input interdependence and shared interdependence. Secondly, we propose the model of risk propagation within one R&D network by defining the risk load and risk capacity of each enterprise, and building the rule of conditional triggering among risks. We then propose the four-stage model for describing the process of risk cascading propagation across interdependent R&D networks. Finally, we investigate the vulnerability of interdependent R&D networks to risk cascading propagation through numerical simulation. Simulation results show that the critical tolerance threshold β ∗ can effectively indicate the vulnerability of interdependent R&D networks to risk cascading propagation. When the value of the tolerance parameter β exceeds the critical tolerance threshold β ∗ , the risk cascading propagation can be entirely prevented in interdependent R&D networks. The vulnerability of interdependent R&D networks increases with the increasing heterogeneity degree of all the enterprises’ risk capacities. The interdependences between interdependent R&D networks can increase their vulnerability to risk cascading propagation. Random attack has the least impact on the vulnerability of interdependent R&D networks, whereas high-degree attack and high-capacity attack have the same impact on the vulnerability.

The inconsistent fluctuation behavior of Internet predicted by preferential attachment(PA) and Gibrat’s law requires empirical investigations on the actual system. By using the interval-tunable Gibrat’s law statistics, we find the actual fluctuation, characterized by the conditional standard deviation of the degree growth rate, changes with the interval length and displays a crossover transition from PA type to Gibrat’s law type, which has not yet been captured by any previous models. We characterize the transition dynamics quantitatively and determine the applicative range of PA and Gibrat’s law. The correlation analysis indicates the crossover transition may be attributed to the accumulative correlation between the internal links.

We introduce a variety of network models using a Poisson-based edge localization strategy, which result in clustered scale-free topologies. We first verify the success of our localization strategy by realizing a variant of the well-known Watts–Strogatz model with an inverse approach, implying a small-world regime of rewiring from a random network through a regular one. We then apply the rewiring strategy to a pure Barabasi–Albert model and successfully achieve a small-world regime, with a limited capacity of scale-free property. To imitate the high clustering property of scale-free networks with higher accuracy, we adapted the Poisson-based wiring strategy to a growing network with the ingredients of both preferential attachment and local connectivity. To achieve the collocation of these properties, we used a routine of flattening the edges array, sorting it, and applying a mixing procedure to assemble both global connections with preferential attachment and local clusters. As a result, we achieved clustered scale-free networks with a computational fashion, diverging from the recent studies by following a simple but efficient approach.

Earthquakes as spatio-temporal complex systems have been recently studied using complex network theory. Seismic networks are dynamical networks due to addition of new seismic events over time leading to establishing new nodes and links to the network. Here we have constructed Iran and Italy seismic networks based on Hybrid Model and testified the preferential attachment hypothesis for the connection of new nodes which states that it is more probable for newly added nodes to join the highly connected nodes comparing to the less connected ones. We showed that the preferential attachment is present in the case of earthquakes network and the attachment rate has a linear relationship with node degree. We have also found the seismic passive points, the most probable points to be influenced by other seismic places, using their preferential attachment values.

People’s online rating behavior can be modeled by user–object bipartite networks directly. However, few works have been devoted to reveal the hidden relations between users, especially from the perspective of signed networks. We analyze the signed monopartite networks projected by the signed user–object bipartite networks, finding that the networks are highly clustered with obvious community structure. Interestingly, the positive clustering coefficient is remarkably higher than the negative clustering coefficient. Then, a Signed Growing Network model (SGN) based on local preferential attachment is proposed to generate a user’s signed network that has community structure and high positive clustering coefficient. Other structural properties of the modeled networks are also found to be similar to the empirical networks.

The problem of uncovering the internal operating function of network models is intriguing, demanded and attractive in researches of complex networks. Notice that, in the past two decades, a great number of artificial models are built to try to answer the above mentioned task. Based on the different growth ways, these previous models can be divided into two categories, one type, possessing the preferential attachment, follows a power-law P ( k ) ∼ k − γ , 2 γ 3 . The other has exponential-scaling feature, P ( k ) ∼ α − k . However, there are no models containing above two kinds of growth ways to be presented, even the study of interconnection between these two growth manners in the same model is lacking. Hence, in this paper, we construct a class of planar and self-similar graphs motivated from a new attachment way, vertex–edge-growth network-operation, more precisely, the couple of both them. We report that this model is sparse, small world and hierarchical. And then, not only is scale-free feature in our model, but also lies the degree parameter γ ( ≈ 3 . 242 ) out the typical range. Note that, we suggest that the coexistence of multiple vertex growth ways will have a prominent effect on the power-law parameter γ , and the preferential attachment plays a dominate role on the development of networks over time. At the end of this paper, we obtain an exact analytical expression for the total number of spanning trees of models and also capture spanning trees entropy which we have compared with those of their corresponding component elements.

This paper proposes an evolution model and an analysis of the behavior of music consumers on online music streaming services. While previous studies have observed power-law degree distributions of usage in online music streaming services, the underlying behavior of users has not been well understood. Users and songs can be described using a bipartite network where an edge exists between a user node and a song node when the user has listened that song. The growth mechanism of bipartite networks has been used to understand the evolution of online bipartite networks Zhang et al. (2013). Existing bipartite models are based on a preferential attachment mechanism Laszlo Barabasi and Albert (1999) in which the probability that a user listens to a song is proportional to its current popularity. This mechanism does not allow for two types of real world phenomena. First, a newly released song with high quality sometimes quickly gains popularity. Second, the popularity of songs normally decreases as time goes by. Therefore, this paper proposes a new model that is more suitable for online music services by adding fitness and aging functions to the song nodes of the bipartite network proposed by Zhang et al. (2013). Theoretical analyses are performed for the degree distribution of songs. Empirical data from an online streaming service, Last.fm , are used to confirm the degree distribution of the object nodes. Simulation results show improvements from a previous model. Finally, to illustrate the application of the proposed model, a simplified royalty cost model for online music services is used to demonstrate how the changes in the proposed parameters can affect the costs for online music streaming providers. Managerial implications are also discussed.

We discuss the connection between the theory of motion of substance in a channel and the theory of growth of large random networks. The channel contains nodes connected by edges. The substance is located in the nodes of the channel. It can move between the nodes of the channel through the edges which connect the nodes. We study probability distributions connected to distribution of substance for selected cases of stationary motion of substance in a channel. The corresponding model equations contain as specific cases the model equations for the degree distributions of several classes of growing random networks. We consider two scenarios for stationary motion of substance in the channel. Scenario 1 is connected to stationary motion of substance in direction to nodes labeled by larger numbers. Scenario 2 is connected to stationary motion of substance in the opposite direction. Probability distributions are obtained for different cases connected to the two scenarios. The results for the Scenario 1 are connected to the theory of degree distributions of nodes of a random network growing in presence of preferential attachment (as specific case we obtain also the degree distribution for the Barabasi–Albert case). Corresponding results for Scenario 2 can be connected to the theory of growing random networks too. The kind of attachment for these results can be classified as negative preferential attachment or preferential attachment with preference of attachment to nodes of smaller degree.

This paper presents a novel evolutionary model that captures the characteristics and the evolutionary pattern of an international logistics network in the context of the Belt and Road. In this study, the logistics network is considered as a complex self-adapting system, in which a new node is added by preferential attachment, and evolution is an iterative process of interaction among factors, such as the social economy, infrastructure construction and international relations. Grounded in the continuum theory, the model is deduced with rigorous derivation, and the results show that the proposed model can accurately reflect the degree distribution of the international logistics network, which follows a shifted power law. Simulation experiments are conducted to demonstrate the interaction process, and the simulated network presents a hierarchical pattern of “several central nodes and a large number of peripheral nodes”. China, South Korea, India, Indonesia and the Russian Federation are considered central nodes and have a significant influence on the B&R international logistics network.

In this paper, based on growth and preferential attachment mechanism, we give a network generation model aiming at generating networks with community structure. There are three characteristics for the networks generated by the generation model. The first is that the community sizes can be nonuniform. The second is that there are bridge hubs in each community. The third is that the strength of community structure is adjustable. Next, we investigate rumor propagation behavior on the generated networks by performing Monte Carlo simulations to reveal the influence of bridge hubs, nonuniformity of community sizes and the strength of community structure on the dynamic behavior of the rumor propagation. We find that bridge hubs have outstanding performance in propagation speed and propagation size, and larger modularity can reduce rumor propagation. Furthermore, when the decay rate of rumor spreading β is large, the final density of the stiflers is larger if the rumor originates in larger community. Additionally, when on networks with different strengths of community structure, rumor propagation exhibits greater difference in the density of stiflers and in the peak prevalence if the decay rate β is larger.

Novel phenomenon receives similar attention as popular one. Therefore predicting novelty is as important as popularity. Emergence is the side effect of competition and ageing in evolving systems. Recent behaviour or recent link gain in networks plays an important role in emergence. We exploited this wisdom and came up with two models considering different scenarios and systems. Where recent behaviour dominates over total behaviour (total link gain) in the first one, and recent behaviour is as important as total behaviour for future link gain in the second one. It supposes that random walker walks on a network and can jump to any node, the probability of jumping or making a connection to other node is based on which node is recently more active or receiving more links. In our assumption, the random walker can also jump to the node which is already popular but recently not popular. We are able to predict emerging nodes which are generally suppressed under preferential attachment effect. To show the performance of our model we have conducted experiments on four real data sets namely, MovieLens, Netflix, Facebook and Arxiv High Energy Physics paper citation. For testing our model we used four information retrieval indices namely Precision, Novelty, Area Under Receiving Operating Characteristic (AUC) and Kendal’s rank correlation coefficient. We have used four benchmark models for validating our proposed models. Although our model does not perform better in all the cases but, it has theoretical significance in working better for recent behaviour dominated systems.

Given the same two networks and only one-to-one interlinks are allowed, apparently interdependent networks coupled by these two networks has the optimal robustness when we connect every pair of the same nodes in these two networks. According to the structure of this interdependent network with the optimal robustness, we propose a preferential attachment strategy. And by applying this preferential attachment strategy to three existing connectivity link addition strategies RA (random addition strategy), LD (low degree addition strategy) and LIDD (low inter degree–degree difference addition strategy), we find that each improved strategy is obviously better than before in improving the robustness of interdependent networks. Our findings can provide guidance on connectivity link addition strategy to improve robustness of interdependent networks against cascading failures.

Tourism is a complex dynamic system including multiple actors which are related each other composing an evolving social network. This paper presents a growing model that explains how part of the supply components in a tourism system forms a social network. Specifically, the lodgings and services in a destination are the network nodes and a link between them appears if a representative tourist hosted in the lodging visits/consumes the service during his/her stay. The specific link between both categories are determined by a random and preferential attachment rule. The analytic results show that the long-term degree distribution of services follows a shifted power-law distribution. The numerical simulations show slight disagreements with the theoretical results in the case of the one-mode degree distribution of services, due to the low order of convergence to zero of X-motifs. The model predictions are compared with real data coming from a popular tourist destination in Gran Canaria, Spain, showing a good agreement between analytical and empirical data for the degree distribution of services. The theoretical model was validated assuming four type of perturbations in the real data.

Fractal is ubiquitous in many real-world networks. Previous researches showed that the strong disassortativity between the hub-nodes on all length scales was the key principle that gave rise to the fractal architecture of networks. Although fractal property emerged in some models, there were few researches about the fractal growth model and quantitative analyses about the strength of the disassortativity for fractal model. In this paper, we proposed a novel inverse renormalization method, named Box-based Preferential Attachment (BPA), to build the fractal growth models in which the Preferential Attachment was performed at box level. The proposed models provided a new framework that demonstrated small-world-fractal transition. Also, we firstly demonstrated the statistical characteristic of connection patterns of the hubs in fractal networks. The experimental results showed that, given proper growing scale and added edges, the proposed models could clearly show pure small-world or pure fractal or both of them. It also showed that the hub connection ratio showed normal distribution in many real-world networks. At last, the comparisons of connection pattern between the proposed models and the biological and technical networks were performed. The results gave useful reference for exploring the growth principle and for modeling the connection patterns for real-world networks.

We investigate the defending of networks against virus attack. We define the resistance of a network to be the maximum number of bits required to determine the code of the module that is accessible from random walk, from which random walk cannot escape. We show that for any network G , R ( G ) = H 1 ( G ) − H 2 ( G ) , where R ( G ) is the resistance of G , H 1 ( G ) and H 2 ( G ) are the one- and two-dimensional structural information of G , respectively, and that resistance maximization is the principle for defending networks against virus attack. By using the theory, we investigate the defending of real world networks and of the networks generated by the preferential attachment and the security models. We show that there exist networks that are defensible by a small number of controllers from cascading failure of any virus attack. Our theory demonstrates that resistance maximization is the principle for defending networks against virus attacks.

Spatial structures of urban systems are best described as multifractals. To provide insight on a hidden mechanism responsible for such scaling we investigate the multiplicative random cascade (MRC) model of urban spatial structure. MRC is a multiplicative process capable of producing multifractal 2D patterns, which is expressed in a closed analytical form. We use street intersection points pattern (SIPP) data as a proxy of urban structure. Renyi’s generalized dimensions (RGD) spectra, probability distributions of intersections densities, and urban layouts generated by the MRC model are compared to the same descriptors extracted from SIPP data for six U.S. metropolitan statistical areas (MSAs). We found that the RGD spectrum of the best-fit MRC model closely matches the observed spectrum in all cases. We also found that CDF of models’ density distributions do not match closely observed density distributions, although, in some cases, differences are not large. Observed density distributions have forms that are close to the log-normal distribution which suggests that a hidden process is multiplicative. Finally, we found that the MRC model is broadly compatible with the observed urban layouts. Given our findings, we suggest that the hidden process of urban scaling is the preferential attachment dynamics — development is attracted to areas in proportion to the degree of their existing development.

Recent empirical studies have witnessed the wide existence of double power-law distribution in complex systems. However its underlying origin remains inadequately disclosed. We show this distribution can arise from the proper control of BA model. Particularly, the control policy demands each new edge to determine the range(degree ≤ k c or > k c ) of its target node by probability p before applying the preferential attachment. We analytically derive the degree distribution and demonstrate a critical control parameter p c so that normal and reversed double power-law degree distribution emerges for p > p c and p p c respectively, and a single power law is recovered when p = p c . The control policy of our model can spontaneously result from various constraints, which provides possible explanation for some real systems and indicates the validity of our model.

Link prediction in complex networks has attracted much attention in many fields. In this paper, a common neighbors plus preferential attachment index is presented to estimate the likelihood of the existence of a link between two nodes based on local information of the nearest neighbors. Numerical experiments on six real networks demonstrated the high effectiveness and efficiency of the new index compared with five well-known and widely accepted indices: the common neighbors, resource allocation index, preferential attachment index, local path index and Katz index. The new index provides competitively accurate prediction with local path index and Katz index while has less computational complexity and is more accurate than the other two indices.

Individuals in social networks tend to exhibit homophily in their social ties, namely, they prefer bonding with others of the same social group (e.g., the same nationality, gender, socio-economic background, etc.). To formalize this phenomenon in the context of an evolving social network, we consider a mixed preferential attachment (MPA), bi-populated, network model. In this model each of the two social groups has a certain level of bias applied by its members toward other group members, ranging form homophily (affinity toward the same) to heterophily (affinity toward the different). Thus, the MPA model supports the following three simple and well-accepted observations on human behavior related to forming networks: (i) the “rich get richer” mechanism, (ii) heterogeneity and a resulting minority–majority partition, and (iii) selection bias between groups. The main contribution of this paper is to propose a model for the evolution of a social network under these three assumptions and analyze its behavior. We show that the suggested mixed preferential attachment model exhibits, for each population, a (possibly different) power law degree distribution, i.e., a distribution where the fraction of nodes with degree k is proportional to k − β (note that β may be different for each group, as a function of the bias level and the minority–majority partition). We believe that this model may contribute to a better understanding of the forces that shape the groups and communities of our society.

In this paper, we propose an evolving network model with information filtering and mixed attachment mechanisms. We theoretically analyze the in-degree distribution of networks generated by the proposed model in two special cases, and prove that the in-degree distribution is power-law if the new coming vertex does not filter information. Otherwise, the in-degree distribution becomes complex which has a transition between exponential and power-law scaling. Numerical simulations are consistent with analytical results. In addition, we analyze how model parameters influence the topology of the network by means of numerical simulations, and compare values of various measures of networks generated by the proposed model with that of networks generated by uniform attachment model, Barabasi–Albert model and copying model. It shows that our model can generate networks with more diverse topological features.

Barabasi and Albert (1999) have come up with the BA model based on growth and preferential attachment to depict the generative process of scale-free network. DNSE (Dynamic Network Structure Entropy) is the index to measure the state of network structure. Based on the mechanism of the LG (Linear Growth) and PA (Preferential Attachment), the measurement of DNSE is proposed. The variation of nodes and edges in the network has effects on the network structure. The paper chooses the nodes’ degree and the shortest path-length as the dynamic evolved indices to define the DNSE. The measurement provides the aspects to describe the state of the network structure.

Many complex natural and physical systems exhibit patterns of interconnection that conform, approximately, to a network structure referred to as scale-free. Preferential attachment is one of many algorithms that have been introduced to model the growth and structure of scale-free networks. With so many different models of scale-free networks it is unclear what properties of scale-free networks are typical, and what properties are peculiarities of a particular growth or construction process. We propose a simple maximum entropy process which provides the best representation of what are typical properties of scale-free networks, and provides a standard against which real and algorithmically generated networks can be compared. As an example we consider preferential attachment and find that this particular growth model does not yield typical realizations of scale-free networks. In particular, the widely discussed “fragility” of scale-free networks is actually found to be due to the peculiar “hub-centric” structure of preferential attachment networks. We provide a method to generate or remove this latent hub-centric bias—thereby demonstrating exactly which features of preferential attachment networks are atypical of the broader class of scale-free networks. We are also able to statistically demonstrate whether real networks are typical realizations of scale-free networks, or networks with that particular degree distribution; using a new surrogate generation method for complex networks, exactly analogous the widely used surrogate tests of nonlinear time series analysis.

We extend the Time Operator and Age to Network Evolution models. Internal Age formulas and the distribution of innovations are computed for Erdős–Renyi Random Networks, for Markov Networks and Barabasi–Albert preferential Attachment Networks. The innovation probabilities are found to be proportional to the quadratic entropy (which coincides with the Tsallis entropy for entropic index q = 2 ) in all Markov networks, as well as in the linear growth mechanism. The distribution of innovations in the Barabasi–Albert model is a new probability distribution of the logarithmic type.

We investigate the preferential attachment graphs proceeding from the following two assumptions. The first one: the probability that a new vertex connects to a vertex i is proportional to an arbitrary nonnegative function f of a vertex degree k . The second assumption: a new vertex can have a random number of edges. We derive formulas for any f to determine the vertex degree distribution { Q k } in generated graphs. The inverse problem is solved: we have obtained formulas, that allow from a given distribution { Q k } to determine f (the problem of a model calibration). The formulas allowing for any f to calculate the joint distribution of vertex degrees at the ends of randomly selected edge are also obtained. Some other results are presented in the paper.

The structure of return spillovers is examined by constructing Granger causality networks using daily closing prices of 20 developed markets from 2nd January 2006 to 31st December 2013. The data is properly aligned to take into account non-synchronous trading effects. The study of the resulting networks of over 94 sub-samples revealed three significant findings. First, after the recent financial crisis the impact of the US stock market has declined. Second, spatial probit models confirmed the role of the temporal proximity between market closing times for return spillovers, i.e. the time distance between national stock markets matters. Third, preferential attachment between stock markets exists, i.e. spillover from market j to market i is more likely if A) market j influences other markets other than i, or when B) market i is influenced by other markets other than j., Comment: This work was supported by the Slovak Research and Development Agency under the contract No. APVV-0666-11

Network motifs are local structural patterns and elementary functional units of complex networks in real world, which can have significant impacts on the global behavior of these systems. Many models are able to reproduce complex networks mimicking a series of global features of real systems, however the local features such as motifs in real networks have not been well represented. We propose a model to grow scale-free networks with tunable motif distributions through a combined operation of preferential attachment and triad motif seeding steps. Numerical experiments show that the constructed networks have adjustable distributions of the local triad motifs, meanwhile preserving the global features of power-law distributions of node degree, short average path lengths of nodes, and highly clustered structures.

A dynamic weight evolution network with preferential attachment is introduced. The network includes two significant characteristics. (i) Topological growth: triggered by newly added node with M links at each time step, each new edge carries an initial weight growing nonlinearly with time. (ii) Weight dynamics: the weight between two existing nodes experiences increasing or decreasing in a nonlinear way. By using continuum theory and mean-field method, we study the strength, the degree, the weight and their distributions. We find that the distributions exhibit a power-law feature. In particular, the relationship between the degree and the strength is nonlinear, and the power-law exponents of the three are the same. All the theoretical predictions are successfully contrasted with numerical simulations.

It seems a universal phenomenon of networks that the attacks on a small number of nodes by an adversary player Alice may generate a global cascading failure of the networks. It has been shown (Li et al., 2013) that classic scale-free networks (Barabasi and Albert, 1999, Barabasi, 2009) are insecure against attacks of as small as O ( log n ) many nodes. This poses a natural and fundamental question: Can we introduce a second player Bob to prevent Alice from global cascading failure of the networks? We proposed a game in networks. We say that a network has an equilibrium game if the second player Bob has a strategy to balance the cascading influence of attacks by the adversary player Alice. It was shown that networks of the preferential attachment model (Barabasi and Albert, 1999) fail to have equilibrium games, that random graphs of the Erdos–Renyi model (Erdos and Renyi, 1959, Erdos and Renyi, 1960) have, for which randomness is the mechanism, and that homophyly networks (Li et al., 2013) have equilibrium games, for which homophyly and preferential attachment are the underlying mechanisms. We found that some real networks have equilibrium games, but most real networks fail to have. We anticipate that our results lead to an interesting new direction of network theory, that is, equilibrium games in networks.

In this paper, motivated by previous research works on the evolution of the network topology, i.e., the combination of preferential attachment and temporal effect (aging of vertices), we propose a novel model which combines the mixed attachment mechanisms (preferential attachment and uniform attachment) and the temporal effect to investigate the topological properties of the network during the evolution. Our results show that the degree distribution of the model follows the exponential-decaying form.

Modeling social tagging plays a critical role in identifying statistical regularities and structural principles common to social tagging systems. Existing modeling approaches only consider imitations or background knowledge of users. However, common interests among users are ignored. In this paper, latent interactions are applied to present the common interests, and dynamic patterns in empirical data are investigated. Furthermore, the latent interaction driven model (LIDM) is proposed to model social tagging. Experimental results show that the tag frequency distribution generated by LIDM is consistent with that in real-world data. Moreover, the latent interaction graph generated by LIDM has a higher average clustering coefficient and lower average shortest path compared with that generated by preferential attachment methods. This demonstrates that LIDM outperforms traditional methods.

Recent research works have been pursued in connection with network synchronizability under various constraints for different topological structures and evolving mechanisms. However, the fundamental question of how the synchronizability of the networks relates to accelerated growth and ad hoc property in the evolving processes remains underexplored. Here we study the ad hoc dynamical models with accelerated growth where the total number of edges increases faster than linearly with network size. By adopting three attachment mechanisms: random, rewired, and preferential attachment, we investigate the second-largest eigenvalues and the ratios of the extremal eigenvalues of coupled matrices in the accelerated models with different evolving parameters. For the new models, we demonstrate the robustness and fragility of synchronization against random and specific attacks by numerical simulations. We find that accelerated growth represents a convenient tool for improving the synchronizability of an evolving network. Furthermore, we show that not only network synchronization but also topological periodicity has robustness and fragility against random failures and specifical removal of the nodes, respectively. In particular, when ad hoc property is suggested in the evolving networks, we find that the deletion of nodes is easier to change network synchronizability and robustness compared to the addition of nodes.

The available social network models that exist today were designed primarily on the basis of the analysis of statistical properties and structural features, as well as the physical or social distances between individuals of social systems, which sometimes is not sufficient because the structure of some social networks is closely tied to individuals’ social identities. In addition, the difference in growth speed between different social networks is also neglected in these models. We propose a synthetic model that involves social identity and adjustable growth speed factors to compensate for these limitations. The model features four types of node connection mechanisms: random attachment, transitive attachment, preferential attachment and anti-preferential attachment. Experimental results indicate that the model can not only produce rich topological structures but can also match real social networks well in both their macro properties and their micro foundations. Thus, the model is helpful in understanding both the evolution of social networks and the differences and similarities among different social networks.

Two stochastic models are proposed to generate a system composed of two interdependent scale-free (SF) or Erdős–Renyi (ER) networks where interdependent nodes are connected with an exponential or power-law relation, as well as different dependence strength, respectively. Each subnetwork grows through the addition of new nodes with constant accelerating random attachment in the first model but with preferential attachment in the second model. The two subnetworks interact with multi-support and undirectional dependence links. The effects of dependence relations and strength between subnetworks are analyzed in the percolation behavior of fully interdependent networks against random failure, both theoretically and numerically, and as a result, for both relations: interdependent SF networks show a second-order percolation phase transition and the increased dependence strength decreases the robustness of the system, whereas, interdependent ER networks show the opposite results. In addition, the power-law relation between networks yields greater robustness than the exponential one at the given dependence strength.

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.

a b s t r a c t In this paper, we investigate numerically the Susceptible–Infected–Recovered–Susceptible (SIRS) epidemic model on an exponential network generated by a preferential attachment procedure. The discrete SIRS model considers two main parameters: the duration τ0 of the complete infection–recovery cycle and the duration τI of infection. A permanent source of infection I0 has also been introduced in order to avoid the vanishing of the disease in the SIRS model. The fraction of infected agents is found to oscillate with a period T ≥ τ0. Simulations reveal that the average fraction of infected agents depends on I0 and τI /τ0. A maximum of synchronization of infected agents, i.e. a maximum amplitude of periodic spreading oscillations, is found to occur when the ratio τI /τ0 is slightly smaller than 1/2. The model is in agreement with the general observation that an outbreak corresponds to high τI /τ0 values.

Preferential attachment is considered as a fundamental mechanism that contributes to the scale-free characteristics of random networks, which include growth and non-growth networks. There exist some situations of non-growth random networks, particularly for very sparse or dense networks, where preferential attachments cannot consequentially result in true scale-free features, but only in scale-free-like appearances. This phenomenon implies that, a close relationship exists between the connection density p and the scaling. In this study, we propose a self-organized model with constant network size to study the phenomenon. We show analytically and numerically that there exists a certain critical point p c . Only when p = p c , the random network evolves into steady scale-free state. Otherwise, the network exhibits a steady scale-free-like state. The closer the p approximates p c , the closer the scale-free-like distribution approximates the true scale-free distribution. Our results show that, in random network lack of growth, a preferential scheme does not necessarily lead to a scale-free state, and a formation of scale-free is a consequence of two mechanisms: (i) a preferential scheme and (ii) appropriate connection density.

In this paper we investigate the effects of social power on the evolution of opinions in model networks as well as in a number of real social networks. A continuous opinion formation model is considered and the analysis is performed through numerical simulation. Social power is given to a proportion of agents selected either randomly or based on their degrees. As artificial network structures, we consider scale-free networks constructed through preferential attachment and Watts–Strogatz networks. Numerical simulations show that scale-free networks with degree-based social power on the hub nodes have an optimal case where the largest number of the nodes reaches a consensus. However, given power to a random selection of nodes could not improve consensus properties. Introducing social power in Watts–Strogatz networks could not significantly change the consensus profile.

Copyright @ 2012 Elsevier B.V. All rights reserved. This is a preprint version of the published article which can be accessed at the link below. 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 into k nodes of degree 1. Four networks are considered, each evolves at each time step by either preferential attachment, with probability p, or splitting with probability 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. Engineering and Physical Sciences Research Council

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.

Preferential attachment is widely recognised as the principal driving force behind the evolution of many growing networks, and measuring the extent to which it occurs during the growth of a network is important for explaining its overall structure. Conventional methods require that the timeline of a growing network is known, that is, the order in which the nodes of the network appeared in time is available. But growing network datasets are commonly accompanied by missing-timelines, in which instance the order of the nodes in time cannot be readily ascertained from the data. To address this shortcoming, we propose a Markov chain Monte Carlo algorithm for measuring preferential attachment in growing networks with missing-timelines. Key to our approach is that any growing network model gives rise to a probability distribution over the space of networks. This enables a growing network model to be fitted to a growing network dataset with missing-timeline, allowing not only for the prevalence of preferential attachment to be estimated as a model parameter, but the timeline also. Parameter estimation is achieved by implementing a novel Metropolis–Hastings sampling scheme for updating both the preferential attachment parameter and timeline. A simulation study demonstrates that our method accurately measures the occurrence of preferential attachment in networks generated according to the underlying model. What is more, our approach is illustrated on a small sub-network of the United States patent citation network. Since the timeline for this example is in fact known, we are able to validate our approach against the conventional methods, showing that they give mutually consistent estimates.

In this paper, motivated by the thoughts and methods of the mixture of preferential and uniform attachments, we extend the Barabasi–Albert (BA) model, and establish a network model with the power-law accelerating growth and the mixture of the two attachment mechanisms. In our model, the number of edges generated by each newly-introduced node is proportional to the power of θ ( 0 ≤ θ 1 ) of time t , i.e., m t θ . By virtue of the continuum approach, we have deduced the degree distribution P ( k , t ) of our model with the extended power-law form P ( k , t ) = A ( t ) [ k + B ( t ) ] − γ . When the number of edges k generated by each new node is much greater than the value of B ( t ) , the degree distribution P ( k , t ) will converge to the power-law form P ( k , t ) = A ( t ) k − γ . When k is much less than the value of B ( t ) , the degree distribution P ( k , t ) will converge to the exponential-law form P ( k , t ) = A ( t ) [ B ( t ) ] γ e − γ k / B ( t ) . By virtue of numerical simulations, we also discuss the dependence of the degree distribution P ( k , t ) on the model’s parameters (where t is considered as a constant in the simulations). Finally, we investigate the possible application of our model in the spreading and evolution of epidemics in some real-world systems.

In this paper the robustness of network synchronizability against random deletion of nodes, i.e. errors, in dynamical scale-free networks was studied. To this end, two measures of network synchronizability, namely, the eigenratio of the Laplacian and the order parameter quantifying the degree of phase synchrony were adopted, and the synchronizability robustness on preferential attachment scale-free graphs was investigated. The findings revealed that as the network size decreases, the robustness of its synchronizability against random removal of nodes declines, i.e. the more the number of randomly removed nodes from the network, the worse its synchronizability. We also showed that this dependence of the synchronizability on the network size is different with that in the growing scale-free networks. The profile of a number of network properties such as clustering coefficient, efficiency, assortativity, and eccentricity, as a function of the network size was investigated in these two cases, growing scale-free networks and those with randomly removed nodes. The results showed that these processes are also different in terms of these metrics.

In this paper, the dynamical behaviors of a class of weighted local-world evolving networks with aging nodes are investigated. As the network grows, each newly added node is connected to some existing nodes through strength–age preferential attachment. Each time step, the newly added node with randomly determined initial degree is connected to some existing nodes through a local strength–age preferential attachment. By using tools from stochastic analysis and continuous approximation, it is theoretically proved that the strength distribution of the generated network has a power-law form with exponent depending nontrivially on a decay factor α ∈ [ 0 , 1 ) . In addition, the clustering coefficient, correlation and small-world properties of this class of networks are examined via simulations. It is shown that clustering coefficient is a monotone decreasing function of the decay factor, and the assortative degree correlation and small-world properties appear simultaneously. Finally, the new model is used to characterize the Internet topology as a typical application.

We study a self-organized scale-free network model generated using the Merging-and-Creation dynamics with preferential attachment. We show analytically that the introduction of preferential attachment has minimal impact on the steady-state degree distribution. However, we find also that the preferential attachment gives rise to a hierarchical modular structure and degree disassortativity, commonly found in technological networks.

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.

Acyclic digraphs arise in many natural and artificial processes. Among the broader set, dynamic citation networks represent a substantively important form of acyclic digraphs. For example, the study of such networks includes the spread of ideas through academic citations, the spread of innovation through patent citations, and the development of precedent in common law systems. The specific dynamics that produce such acyclic digraphs not only differentiate them from other classes of graphs, but also provide guidance for the development of meaningful distance measures. In this article, we develop and apply our sink distance measure together with the single-linkage hierarchical clustering algorithm to both a two-dimensional directed preferential attachment model as well as empirical data drawn from the first quarter century of decisions of the United States Supreme Court. Despite applying the simplest combination of distance measures and clustering algorithms, analysis reveals that more accurate and more interpretable clusterings are produced by this scheme., Comment: 7 pages, 5 figures. Revision: Added application to the network of the first quarter-century of Supreme Court citations. Revision 2: Significantly expanded, includes application on random model as well