Albert Scale-free Networks Research Articles

Evolutionary prisoner's dilemma game on Barabási–Albert scale-free networks

An evolutionary prisoner's dilemma game (PDG) with players located on Barabási–Albert scale-free networks (BASFN) with average connectivity 8 is studied in detail. The players are pure strategists and can adopt two strategies: either defect or cooperate. Several alternative update rules determining the evolution of each player's strategy are considered. Using Monte Carlo (MC) simulations we implemented both synchronous and asynchronous update dynamics to calculate the average density of cooperators ρ C as a function of the temptation-to-defect b in the equilibrium state. For the sake of comparison, evolution of cooperative behavior on random regular graphs (RRG) and regular lattices (RL) with the same total size and average connectivity as BASFN are also investigated. We find the overall result that cooperation is sometimes inhibited and sometimes enhanced on the BASFN, with respect to the cases on the RRG and RL. The differences depend on the detailed evaluation function of the players’ success (average payoffs of the players in the present case), the different update rules that determine a player's future strategy, the synchronous and asynchronous events of strategy-updating, and also on the temptation to defect.

Novel hybrid mitigation strategy for improving the resiliency of hierarchical networks subjected to attacks

This paper studies the resiliency of hierarchical networks when subjected to random errors, static attacks, and cascade attacks. The performance is compared with existing Erdös–Rényi (ER) random networks and Barabasi and Albert (BA) scale-free networks using global efficiency as the common performance metric. The results show that critical infrastructures modeled as hierarchical networks are intrinsically efficient and are resilient to random errors, however they are more vulnerable to targeted attacks than scale-free networks. Based on the response dynamics to different attack models, we propose a novel hybrid mitigation strategy that combines discrete levels of critical node reinforcement with additional edge augmentation. The proposed modified topology takes advantage of the high initial efficiency of the hierarchical network while also making it resilient to attacks. Experimental results show that when the level of damage inflicted on a critical node is low, the node reinforcement strategy is more effective, and as the level of damage increases, the additional edge augmentation is highly effective in maintaining the overall network resiliency.

Epidemic spreading in lattice-embedded scale-free networks

We study geographical effects on the spread of diseases in lattice-embedded scale-free networks. The geographical structure is represented by the connecting probability of two nodes that is related to the Euclidean distance between them in the lattice. By studying the standard susceptible-infected model, we found that the geographical structure has great influences on the temporal behavior of epidemic outbreaks and the propagation in the underlying network: the more geographically constrained the network is, the more smoothly the epidemic spreads, which is different from the clearly hierarchical dynamics that the infection pervades the networks in a progressive cascade across smaller-degree classes in Barabási–Albert scale-free networks.

Detrended Fluctuation Analysis on Correlations of Complex NetworksUnder Attack and Repair Strategy

We analyze the correlation properties of the Erdös–Rényirandom graph (RG) and the Barabási–Albert scale-free network(SF) under the attack and repair strategy with detrendedfluctuation analysis (DFA). The maximum degree kmax,representing the local property of the system, shows similarscaling behaviors for random graphs and scale-free networks. Thefluctuations are quite random at short time scales but displaystrong anticorrelation at longer time scales under the same systemsize N and different repair probability pre. The averagedegree ⟨k⟩, revealing the statistical property ofthe system, exhibits completely different scaling behaviors forrandom graphs and scale-free networks. Random graphs displaylong-range power-law correlations. Scale-free networks areuncorrelated at short time scales; while anticorrelated at longertime scales and the anticorrelation becoming stronger with theincrease of pre.

Discrete simulation of the dynamics of spread of extreme opinions in a society

We propose a discrete model for how opinions about a given “extreme” subject, about which various groups of a population have different degrees of enthusiasm for or susceptibility to, such as fanaticism, extreme social and political positions, and terrorism, may spread. The model, in a certain limit, is the discrete analogue of a deterministic continuum model suggested by others. We carry out extensive computer simulation of the model by utilizing it on lattices with infinite- or short-range interactions, and on symmetric and hierarchical (or directed) Barabási–Albert scale-free networks. Several interesting features of the model are demonstrated, and comparison is made with the deterministic continuum model.

Immunization dynamics on a two-layer network model

We introduced a two-layer network model for the study of the immunization dynamics in epidemics. Spreading of an epidemic is modeled as an excitatory process in a Watts–Strogatz small-world network (infection layer) while immunization by prevention of the disease as a dynamic process in a Barabási–Albert scale-free network (prevention layer). It is shown that prevention indeed turns periodic rages of an epidemic into small fluctuations, and in a certain situation, actually plays an adverse role and helps the disease survive. We argue that the presence of two different characteristic time scales contributes to the immunization dynamics observed.

EFFECTS OF AGING AND LINKS REMOVAL ON EPIDEMIC DYNAMICS IN SCALE-FREE NETWORKS

We study the combined effects of aging and links removal on epidemic dynamics in the Barabási–Albert scale-free networks. The epidemic is described by a susceptible-infected-refractory (SIR) model. The aging effect of a node introduced at time ti is described by an aging factor of the form (t-ti)-β in the probability of being connected to newly added nodes in a growing network under the preferential attachment scheme based on popularity of the existing nodes. SIR dynamics is studied in networks with a fraction 1-p of the links removed. Extensive numerical simulations reveal that there exists a threshold pc such that for p≥pc, epidemic breaks out in the network. For p<pc, only a local spread results. The dependence of pc on β is studied in detail. The function pc(β) separates the space formed by β and p into regions corresponding to local and global spreads, respectively.

Error and attack tolerance of complex networks

Communication/transportation systems are often subjected to failures and attacks. Here we represent such systems as networks and we study their ability to resist failures (attacks) simulated as the breakdown of a group of nodes of the network chosen at random (chosen accordingly to degree or load). We consider and compare the results for two different network topologies: the Erdös–Rényi random graph and the Barabási–Albert scale-free network. We also discuss briefly a dynamical model recently proposed to take into account the dynamical redistribution of loads after the initial damage of a single node of the network.

OPINION FORMATION ON A DETERMINISTIC PSEUDO-FRACTAL NETWORK

The Sznajd model of socio-physics, with only a group of people sharing the same opinion can convince their neighbors, is applied to a scale-free random network modeled by a deterministic graph. We also study a model for elections based on the Sznajd model and the exponent obtained for the distribution of votes during the transient agrees with those obtained for real elections in Brazil and India. Our results are compared to those obtained using a Barabási–Albert scale-free network.

AVERAGE DISTANCE IN GROWING TREES

Two kinds of evolving trees are considered here: the exponential trees, where subsequent nodes are linked to old nodes without any preference, and the Barabási–Albert scale-free networks, where the probability of linking to a node is proportional to the number of its pre-existing links. In both cases, new nodes are linked to m=1 nodes. The average node–node distance d is calculated numerically in evolving trees as dependent on the number of nodes N. The results for N not less than a thousand are averaged over a thousand of growing trees. The results on the mean node–node distance d for large N can be approximated by d=2 ln (N)+c1 for the exponential trees, and d= ln (N)+c2 for the scale-free trees, where ci are constant. We also derive iterative equations for d and its dispersion for the exponential trees. The simulation and the analytical approach give the same results.

Mean field solution of the Ising model on a Barabási–Albert network

The mean field solution of the Ising model on a Barabási–Albert scale-free network with ferromagnetic coupling between linked spins is presented. The critical temperature T c for the ferromagnetic to paramagnetic phase transition (Curie temperature) is infinite and the effective critical temperature for a finite size system increases as the logarithm of the system size in agreement with recent numerical results of Aleksiejuk, Holyst and Stauffer.

Ferromagnetic phase transition in Barabási–Albert networks

Ising spins put onto a Barabási–Albert scale-free network show an effective phase transition from ferromagnetism to paramagnetism upon heating, with an effective critical temperature increasing as the logarithm of the system size. Starting with all spins up and upon equilibration pinning the few most-connected spins down nucleates the phase with most of the spins down.