Barabasi-Albert Research Articles

Network growth with preferential attachment and without “rich get richer” mechanism

We propose a simple preferential attachment model of growing network using the complementary probability of Barabási–Albert (BA) model, i.e. [Formula: see text]. In this network, new nodes are preferentially attached to not well connected nodes. Numerical simulations, in perfect agreement with the master equation solution, give an exponential degree distribution. This suggests that the power law degree distribution is a consequence of preferential attachment probability together with “rich get richer” phenomena. We also calculate the average degree of a target node at time t[Formula: see text] and its fluctuations, to have a better view of the microscopic evolution of the network, and we also compare the results with BA model.

Evolution of egoism on semi-directed and undirected Barabási-Albert networks

Through Monte Carlo simulations, we study the evolution of the four strategies: Ethnocentric, altruistic, egoistic and cosmopolitan in one community of individuals. Interactions and reproduction among computational agents are simulated on undirected and semi-directed Barabási–Albert (BA) networks. We study the Hammond–Axelrod (HA) model on undirected and semi-directed BA networks for the asexual reproduction case. With a small modification in the traditional HA model, our simulations showed that egoism wins, differently from other results found in the literature where ethnocentric strategy is common. Here, mechanisms such as reciprocity are absent.

Time-Dependent Variation of the Centrality Measures of the Nodes during the Evolution of a Scale-Free Network

Scale-free networks are a type of complex networks in which the degree distribution of the nodes is according to the power-law. Centrality of the nodes is a quantitative measure of the importance of the nodes according to the topological structure of the network. The commonly used centrality measures are the degree-based degree centrality and eigenvector centrality and the shortest path-based closeness centrality and betweenness centrality. We use the widely studied Barabasi-Albert (BA) model to simulate the evolution of scale-free networks. The model works by adding new nodes to the network, one at a time, with the new node connected to m of the currently existing nodes. Accordingly, nodes that have been in the network for a longer time have greater chances of acquiring more links and hence a larger degree centrality. While the degree centrality of the nodes has been observed to show a concave down pattern of increase with time; but the time-dependent variation of the other centrality measures has not been analyzed until now. In this paper, we study the time-dependent variation of degree centrality, eigenvector centrality, closeness centrality and betweenness centrality of the nodes during the evolution of a scale-free network according to the BA model

Accuracy test for link prediction in terms of similarity index: The case of WS and BA models

Link prediction is a technique that uses the topological information in a given network to infer the missing links in it. Since past research on link prediction has primarily focused on enhancing performance for given empirical systems, negligible attention has been devoted to link prediction with regard to network models. In this paper, we thus apply link prediction to two network models: The Watts–Strogatz (WS) model and Barabási–Albert (BA) model. We attempt to gain a better understanding of the relation between accuracy and each network parameter (mean degree, the number of nodes and the rewiring probability in the WS model) through network models. Six similarity indices are used, with precision and area under the ROC curve (AUC) value as the accuracy metrics. We observe a positive correlation between mean degree and accuracy, and size independence of the AUC value.

New behavior of degree distribution in connected communication networks

The behavior of the degree distribution of two interdependent Barabasi–Albert (BA) sub-networks has been investigated numerically. The final complex structure obtained after connection of the two BA subnets exhibits two different kind of degree distribution law, which depends strongly on the manner in which the connection between the two subnets has been made. When connecting two existing BA subnets, the degree distribution follows a Gaussian distribution, while ensuring that the highest frequency level is still around the average degree of the final network. Whereas, when the connection is established progressively at the same time of the formation of the two BA subnets, the degree distribution follows a power-law scaling observed in real networks. It is also found that the evolution of links formed over a time for a specific node follows the same behavior, as the BA networks.

Operational risk transmission model of banks based on a scale-free network

Operational risk has been considered as the key issue of Chinese commercial banks in recent years. Transmission mechanism of operational risk has been a vital part of risk control. In order to better control and manage the operational risk, it is helpful to further understand the whole dynamic process of the risks affecting the business. This paper proposes a transmission model based on a scale-free network that simulates the transmission process of operational risk and explains the transmission mechanism of operational risk factors. Using the Monte Carlo simulation based on the Barabasi–Albert (BA) model under different conditions, factors that affect the operational risk losses are analysed. It is found that the node degree of operational risk network as well as the selection of initial node to attack plays an important role in losses, while the scale of the network plays a small role in affecting the losses. Finally, some control recommendations based on the simulation results are presented. The result...

Dynamical interplay between epidemics and cascades in complex networks

Epidemics and cascading failure are extensively investigated. Traditionally, they are independently studied, but in practice, there are many cases where these two dynamics interact with each other and neither of their effects can be ignored. For example, consider that a digital virus is spreading in a communication network, which is transferring data in the meantime. We build a model based on the epidemiological SIR model and a local load sharing cascading failure model to study the interplay between these two dynamics. In this model, when the dynamical process stops at equilibrium, the nodes both uninfected and unfailed form several clusters. We consider the relative size of the largest one, i.e. the giant component. A phenomenon is observed in both Erdős-Rényi (ER) random networks and Barabási-Albert (BA) scale-free networks that when the infection probability is over some critical value, a giant component forms only if the tolerance parameter α is within some interval . In this interval, the size of the remained giant component first increases and then decreases. After analyzing the cause of this phenomenon, we then present in ER random networks a theoretical solution of the key values of and , which are very important when we evaluate the robustness of the network. Finally, our theory is verified by numerical simulations.

Study on cascading invulnerability of multi-coupling-links coupled networks based on time-delay coupled map lattices model

The couplings among different networks facilitate their communications, while at the same time they also bring the risk of enhancing the wide spread of cascading failures to the coupled networks. Given that there is usually the time-delay during the spread of failures and more than one coupling link a node might possess, a cascading failure model for scale-free multi-coupling-link coupled networks is built in this paper, based on time-delay coupled map lattices (CML) model, which may be wider representative than previous models. Our research shows that in BA (Barabási-Albert) scale-free coupled networks, there is a threshold hT ≈ 3: when the coupling strength is bellow this threshold, the stronger coupling strength corresponds to a lower invulnerability; and vice versa, the stronger coupling strength would bring a higher invulnerability. In addition, our studies show that the presence of time-delay not only prolongs the failure spreading time during which measures can be taken to suppress cascading failures, but also has a significant influence on the eventual cascading size, for detail, if intra-layer time-delay τ1 and inter-layer time-delay τ2 can have any values, then the multiples of the two numbers will cause larger cascading size. We hope our research can provide a reference for building high-invulnerable coupled networks or the increase of the invulnerability of the coupled networks.

Robustness of Internet under targeted attack: A cascading failure perspective

In many networks, there exist some heterogeneous nodes, for example the hosts and the routers in the Internet, and the users and the supply-grid stations in the power grid. In the previous studies, however, few cascading models were constructed in the heterogeneous networks. Considering two types of nodes, we propose a new cascading edge model and investigate the cascading dynamic behaviors in the Internet. Compared with an extending Barabási–Albert (BA) network with two types of nodes, we find that the Internet displays the stronger robust level under two targeted attacks and propose some effective methods to protect these networks. The cascading model in the Internet can be extensively applied to other real-life networks, and may provide a new perspective to study the network safety.

Opinion evolution on a BA scaling network

In this paper, the dynamics of opinion formation is investigated based on a BA (Barabási–Albert) scale-free network, using a majority–minority rule governed by parameter q. As the value of q is smoothly varied, a phase transition occurs between an ordered phase and a disordered one. By performing extensive Monte Carlo simulations, we show that the phase transition is dependent on the system size, as well as on m, the number of edges added at each time step during the growth of the BA scaling network. Additionally, some theoretical analysis is given based on mean-field theory, by neglecting fluctuations and correlations. It is observed that the theoretical results coincide with results from simulations, especially for very large m.

Research on the search ability of Brownian particles on networks with an adaptive mechanism

In this paper, we focus on the search ability of Brownian particles with an adaptive mechanism. In the adaptive mechanism, nodes are allowed to be able to change their own accepting probability according to their congestion states. Two searching-traffic models, the static one in which nodes have fixed accepting probability to the incoming particles and the adaptive one in which nodes have adaptive accepting probability to the incoming particles are presented for testing the adaptive mechanism. Instead of number of hops, we use the traveling time, which includes not only the number of hops for a particle to jump from the source node to the destination but also the time that the particle stays in the queues of nodes, to evaluate the search ability of Brownian particles. We apply two models to different networks. The experiment results show that the adaptive mechanism can decrease the network congestion and the traveling time of the first arriving particle. Furthermore, we investigate the influence of network topologies on the congestion of networks by addressing several main properties: degree distribution, average path length, and clustering coefficient. We show the reason why random topologies are more able to deal with congested traffic states than others. We also propose an absorption strategy to deal with the additional Brownian particles in networks. The experiment results on Barabási–Albert (BA) scale-free networks show that the absorption strategy can increase the probability of a successful search and decrease the average per-node particles overhead for our models.

Explosive synchronization transitions in complex neural networks

It has been recently reported that explosive synchronization transitions can take place in networks of phase oscillators [Gómez-Gardeñes et al. Phys. Rev. Lett. 106, 128701 (2011)] and chaotic oscillators [Leyva et al. Phys. Rev. Lett. 108, 168702 (2012)]. Here, we investigate the effect of a microscopic correlation between the dynamics and the interacting topology of coupled FitzHugh-Nagumo oscillators on phase synchronization transition in Barabási-Albert (BA) scale-free networks and Erdös-Rényi (ER) random networks. We show that, if natural frequencies of the oscillations are positively correlated with node degrees and the width of the frequency distribution is larger than a threshold value, a strong hysteresis loop arises in the synchronization diagram of BA networks, indicating the evidence of an explosive transition towards synchronization of relaxation oscillators system. In contrast to the results in BA networks, in more homogeneous ER networks, the synchronization transition is always of continuous type regardless of the width of the frequency distribution. Moreover, we consider the effect of degree-mixing patterns on the nature of the synchronization transition, and find that the degree assortativity is unfavorable for the occurrence of such an explosive transition.

Structural features beneath neuronal avalanches

Recently, Friedman et al performed high-resolution measurements [1] that strongly supported an universal critical character (in the sense of statistical physics) of neuronal avalanches, despite the abnormally high frequency of large events (bumps in the distributions of size and lifetime of avalanches) deforming the pure power-law behavior expected from analogies to equilibrium critical phenomena that became manifest only for smaller events. Based on simulations of the Kinouchi-Copelli (KC) model, we have shown [2] that such bumps may not be experimental artifacts and really be typical at criticality when the topology of the neural network is the Barabasi-Albert (BA) model, leading to a scale-free degree distribution of exponent -3.0. On the other hand, those simulations could not reproduce the exponents of the power-law region of the avalanche distributions in [1] (namely, -1.7 for the size distribution and -1.9 for the lifetime distribution). Besides that, the KC dynamics on BA topology revealed that the information capacity (entropy of avalanche size distribution) did not exhibit critical optimization, in contrast with an earlier experiment [3]. In this study we investigate the KC dynamics on the Uncorrelated Configuration Model (UCM) [4]. The UCM is a kind of wiring procedure of a neural topology that can lead to scale-free degree distributions with tunable exponents and can be matched to BA model. However, even so their avalanches are not identical. While the UCM also allows the appearance of bumps on the avalanche distributions, it both shows that the information capacity may be critically optimal and exhibits quantitatively accurate values of the critical exponents for small avalanches, suggesting the UCM may be descriptive of some structural features in the systems claimed to exhibit critical dynamics.

Learning Algorithm and the Cooperation Behavior of Continuous Prisoner's Dilemma Game on Complex Networks

The standard evolutionary Prisoner’s Dilemma Game (PDG), which is the most widely used model for investigating the evolution of cooperation, does not allow for changeable degrees of cooperation and the players can either cooperate or defect in each time. In this paper, we study Continuous Prisoner’s Dilemma Games (CPDG) on complex networks (or Spatial Continuous Prisoner’s Dilemma Games (SCPDG)); in which each agent locates on a vertex of the network and interacts with all his neighboring agents by making an investment. Each agent’s investment will benefit all its neighbors and incur a cost to the focus agent. The investment which can be varied continuously exhibits variable degrees of cooperation. The cooperative level of the group is characterized by the average investment of the population. Unlike the imitation of the best neighbors learning rule, the probability based imitation learning rule and the Moran process based learning rule that usually used in the standard evolutionary SPDG, we propose a new learning algorithm in the continuous condition. Three parameters to describe the imitation ability of the best neighbors, the memory length of the individual and uncertainty effect of the environment are introduced. We investigate the co-effect of these parameters on the cooperation level of the population on the Barabasi-Albert (BA) scale-free networks, the nearest-neighbor coupled networks and the Newman-Watts (NW) small-world networks. Our results may provide some new conclusion on the study of evolutionary games on networks.

Scale-Free Networks with the Same Degree Distribution: Different Structural Properties

We have analysed some structural properties of scale-free networks with the same degree distribution. Departing from a degree distribution obtained from the Barabási-Albert (BA) algorithm, networks were generated using four additional different algorithms (Molloy-Reed, Kalisky, and two new models named A and B) besides the BA algorithm itself. For each network, we have calculated the following structural measures: average degree of the nearest neighbours, central point dominance, clustering coefficient, the Pearson correlation coefficient, and global efficiency. We found that different networks with the same degree distribution may have distinct structural properties. In particular, model B generates decentralized networks with a larger number of components, a smaller giant component size, and a low global efficiency when compared to the other algorithms, especially compared to the centralized BA networks that have all vertices in a single component, with a medium to high global efficiency. The other three models generate networks with intermediate characteristics between B and BA models. A consequence of this finding is that the dynamics of different phenomena on these networks may differ considerably.

An adaptive strategy based on linear prediction of queue length to minimize congestion in Barabási—Albert scale-free networks

In this paper, we propose an adaptive strategy based on the linear prediction of queue length to minimize congestion in Barabási—Albert (BA) scale-free networks. This strategy uses local knowledge of traffic conditions and allows nodes to be able to self-coordinate their accepting probability to the incoming packets. We show that the strategy can delay remarkably the onset of congestion and systems avoiding the congestion can benefit from hierarchical organization of accepting rates of nodes. Furthermore, with the increase of prediction orders, we achieve larger values for the critical load together with a smooth transition from free-flow to congestion.

A new evolution model for B2C e-commerce market

In order to explain the formation of the business to customer e-commerce market structure, we introduce two concepts--market trading volume and user penetration into the analytical framework for e-commerce market. Based on the modification of Barabasi---Albert (BA) model, a new model which is added with fitness parameters and more reasonable growth mechanism is proposed. The model reveals a "bubble-stable" evolutionary process which is correspondent with real e-commerce market from an initial network to a scale-free one. The simulation results show that the number of websites a buyer chooses could affect the evolutionary process of user penetration distribution, but almost not affect the stable trend of the market. In addition, the initial network scale almost does not affect the nature of network, but causes market fluctuation. The model also reveals that unfair competition among websites is the reason for the formation of structure. Hence, a new method which is calculating numbers of overlap users between each pair of websites is developed to measure the competitive strength. Then, three distinct components are found in the competitive network: a small nucleus, a secondary core and a huge bulk body.

A New Evolution Mechanism Model for B2B E-Commerce Network

To study the structure and evolution mechanism of B2B e-commerce network, the authors propose a network with several layers description of B2B e-commerce market. The empirical analysis from Alibaba.com shows that degree distribution of the network structure follows power-law. Based on modified BA (Barabasi-Albert) model, a new model is proposed, adding layers, fitness parameters, and more reasonable growth mechanism. The model reveals that, the structure of the network is stable, and the average path length and clustering coefficient are small. The authors also find that the impact of parameters is not decisive, and the emergence of scale-free nature depends on the mechanism itself.

IMPROVING NETWORK TRANSPORT EFFICIENCY BY EDGE REWIRING

Considering the heterogeneous structure of scale-free networks causing low traffic capacity of network, we propose to improve the network transport efficiency by rewiring a fraction of edges for the network. In this paper, six edge rewiring strategies are discussed and extensive simulations on Barabási–Albert (BA) scale-free networks confirm the effectiveness of these strategies. From another perspective, rewiring edges for scale-free networks directly reuse the removed edges under some edge-removal strategies [Z. Liu, M. B. Hu, R. Jiang, W. X. Wang and Q. S. Wu, Phys. Rev. E76 (2007) 037101; G. Q. Zhang, D. Wang and G. J. Li, Phys. Rev. E76 (2007) 017101], and can significantly enhance the traffic capacity of the network at the expense of increasing a little average path length. After the edge rewiring process, the network structure becomes significantly homogeneous. This work is helpful for network design and network performance optimization.

Growing highly synchronizable scale-free networks

In this letter, a model for growing highly synchronizable networks is introduced. The model is based on preferential attachment in which the new nodes tip to old ones in a way to maximize the synchronization properties of the network. Criteria based on the eigenvectors corresponding to the second smallest and the largest eigenvalues of the Laplacian matrix of the connection graph were used to choose the old nodes to which a new node makes connection. Numerical simulations showed that these networks have considerably lower eigenratio (the largest eigenvalue of the Laplacian divided by the second smallest one), and hence better synchronizability, as compared to standard Barabási-Albert (BA) scale-free networks. These two models were also different in their statistical properties. BA networks demonstrated larger global and local efficiency, while modularity and assortativity of highly synchronizable networks were larger. The proposed model resulted in networks with almost power-law degree distribution (except an exponential start). Furthermore, the hub nodes in these networks had much smaller degree as compared to those of BA networks.