We show results for the contact process on Barabasi networks. The contact process is a model for an epidemic spreading without permanent immunity that has an absorbing state. For finite lattices, the absorbing state is the true stationary state, which leads to the need for simulation of quasi-stationary states, which we did in two ways: reactivation by inserting spontaneous infected individuals, or by the quasi-stationary method, where we store a list of active states to continue the simulation when the system visits the absorbing state. The system presents an absorbing phase transition where the critical behavior obeys the Mean Field exponents $β=1$, $γ'=0$, and $ν=2$. However, the different quasi-stationary states present distinct finite-size logarithmic corrections. We also report the critical thresholds of the model as a linear function of the network connectivity inverse $1/z$, and the extrapolation of the critical threshold function for $z \to \infty$ yields the basic reproduction number $R_0=1$ of the complete graph, as expected. Decreasing the network connectivity leads to the increase of the critical basic reproduction number $R_0$ for this model.
The ability to sample random networks that can accurately represent real social contact networks is essential to the study of viral epidemics. The Barabási-Albert model and its extensions attempt to capture reality by generating networks with power-law degree distributions, but properties of the resulting distributions (e.g. minimum, average, and maximum degree) are often unrealistic of the social contacts the models attempt to capture. I propose a novel extension of the Barabási-Albert model, which I call the "dual-Barabási-Albert" (DBA) model, that attempts to better capture these properties…
Barabási-Albert model describes many different natural networks, often yielding sensible explanations to the subjacent dynamics. However, finite size effects may prevent from discerning among different underlying physical mechanisms and from determining whether a particular finite system is driven by Barabási-Albert dynamics. Here we propose master equations for the evolution of the degrees, links and triangles distributions, solve them both analytically and by numerical iteration, and compare with numerical simulations. The analytic solutions for all these distributions predict the network ev…
We consider two consensus formation models coupled to Barabasi-Albert networks, namely the Majority Vote model and Biswas-Chatterjee-Sen model. Recent works point to a non-universal behavior of the Majority Vote model, where the critical exponents have a dependence on the connectivity while the effective dimension $D_\mathrm{eff} = 2β/ν+ γ/ν$ of the lattice is unity. We considered a generalization of the scaling relations in order to include logarithmic corrections. We obtained the leading critical exponent ratios $1/ν$, $β/ν$, and $γ/ν$ by finite size scaling data collapses, as well as the lo…
On directed Barabasi-Albert networks with two and seven neighbours selected by each added site, the Ising model was seen not to show a spontaneous magnetisation. Instead, the decay time for flipping of the magnetisation followed an Arrhenius law for Metropolis and Glauber algorithms, but for Wolff cluster flipping the magnetisation decayed exponentially with time. On these networks the Majority-vote model with noise is now studied through Monte Carlo simulations. However, in this model, the order-disorder phase transition of the order parameter is well defined in this system. We calculate the …
Using Monte Carlo simulations, we study the evolution of contigent cooperation and ethnocentrism in the one-move game. Interactions and reproduction among computational agents are simulated on {\it undirected} and {\it directed} Barabási-Albert (BA) networks. We first replicate the Hammond-Axelrod model of in-group favoritism on a square lattice and then generalize this model on {\it undirected} and {\it directed} BA networks for both asexual and sexual reproduction cases. Our simulations demonstrate that irrespective of the mode of reproduction, ethnocentric strategy becomes common even thoug…
In usual scale-free networks of Barabasi-Albert type, a newly added node selects randomly m neighbors from the already existing network nodes, proportionally to the number of links these had before. Then the number N(k) of nodes with k links each decays as 1/k^gamma where gamma=3 is universal, i.e. independent of m. Now we use a limited directedness in the construction of the network, as a result of which the exponent gamma decreases from 3 to 2 for increasing m.
In this article, we show that if each node of the Barabási-Albert (BA) network is characterized by the generalized degree $q$, i.e. the product of their degree $k$ and the square root of their respective birth time, then the distribution function $F(q,t)$ exhibits dynamic scaling $F(q,t\rightarrow \infty)\sim t^{-1/2}ϕ(q/t^{1/2})$ where $ϕ(x)$ is the scaling function. We verified it by showing that a series of distinct $F(q,t)$ vs $q$ curves for different network sizes $N$ collapse onto a single universal curve if we plot $t^{1/2}F(q,t)$ vs $q/t^{1/2}$ instead. Finally, we show that the BA net…
Using a mean-field network formulation of the Bass innovation diffusion model and exact results by Fotouhi and Rabbat on the degree correlations of Barabasi-Albert networks, we compute the times of the diffusion peak and compare them with those on scale-free networks which have the same scale-free exponent but different assortativity properties. We compare our results with those obtained by Caldarelli et al. for the SIS epidemic model with the spectral method applied to adjacency matrices. It turns out that diffusion times on finite Barabasi-Albert networks are at a minimum. This may be due to…
With the evolution of social networks, the network structure shows dynamic nature in which nodes and edges appear as well as disappear for various reasons. The role of a node in the network is presented as the number of interactions it has with the other nodes. For this purpose a network is modeled as a graph where nodes represent network members and edges represent a relationship among them. Several models for evolution of social networks has been proposed till date, most widely accepted being the Barabási-Albert \cite{Network science} model that is based on \emph{preferential attachment} of …
We present a modified diffusive epidemic process that has a finite threshold on scale-free graphs. The diffusive epidemic process describes the epidemic spreading in a non-sedentary population, and it is a reaction-diffusion process. In the diffusion stage, the individuals can jump between connected nodes, according to different diffusive rates for the infected and susceptible individuals. In the reaction stage, the contagion can happen if there is an infected individual sharing the same node, and infected individuals can spontaneously recover. Our main modification is to turn the number of in…
We develop and test a rewiring method (originally proposed by Newman) which allows to build random networks having pre-assigned degree distribution and two-point correlations. For the case of scale-free degree distributions, we discretize the tail of the distribution according to the general prescription by Dorogovtsev and Mendes. The application of this method to Barabasi-Albert (BA) networks is possible thanks to recent analytical results on their correlations, and allows to compare the ensemble of random networks generated in the configuration model with that of "real" networks obtained fro…
Recently, Yin et al. [Eur. Phys. J. B 49, 205 (2006)] introduced an efficient small-world network traffic model using preferential next-nearest neighbor routing strategy with the so-called path iteration avoidance (PIA) rule to study the jamming transition of internet. Here we study their model without PIA rule by a mean-field analysis which carefully divides the message packets into two types. Then, we argue that our mean-field analysis is also applicable in the presence of PIA rule in the limit of a large number of nodes in the network. Our analysis gives an explicit expression of the critic…
The sampling of scale-free networks in Molecular Biology is usually achieved by growing networks from a seed using recursive algorithms with elementary moves which include the addition and deletion of nodes and bonds. These algorithms include the Barabasi-Albert algorithm. Later algorithms, such as the Duplication-Divergence algorithm, the Solé algorithm and the iSite algorithm, were inspired by biological processes underlying the evolution of protein networks, and the networks they produce differ essentially from networks grown by the Barabasi-Albert algorithm. In this paper the mean field an…
The explosive growth of Web 2.0, which was characterized by the creation of online social networks, has reignited the study of factors that could help us understand the growth and dynamism of these networks. Various generative network models have been proposed, including the Barabasi-Albert and Watts-Strogatz models. In this study, we revisit the problem from a perspective that seeks to compare results obtained from these generative models with those from real networks. To this end, we consider the dating network Skout Inc. An analysis is performed on the topological characteristics of the net…
The efficiency of a large hierarchical organisation is simulated on Barabasi-Albert networks, when each needed link leads to a loss of information. The optimum is found at a finite network size, corresponding to about five hierarchical layers, provided a cost for building the network is included in our optimization.
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.
We consider nonequilibrium phase transitions in weighted scale-free networks, in which highly connected nodes, which are created earlier in time are partially immunized. For epidemic spreading we solve the dynamical mean-field equations and discuss finite-size scaling theory. The theoretical predictions are confronted with the results of large scale Monte Carlo simulations on the weighted Barabási-Albert network. Local scaling exponents are found different at a typical site and at a node with very large connectivity.
In the study of complex networks almost all theoretical models have the property of infinite growth, but the size of actual networks is finite. According to statistics from the China Internet IPv4 (Internet Protocol version 4) addresses, this paper proposes a forecasting model by using S curve (Logistic curve). The growing trend of IPv4 addresses in China is forecasted. There are some reference value for optimizing the distribution of IPv4 address resource and the development of IPv6. Based on the laws of IPv4 growth, that is, the bulk growth and the finitely growing limit, it proposes a finit…
We propose a method to derive the stationary size distributions of a system, and the degree distributions of networks, using maximisation of the Gibbs-Shannon entropy. We apply this to a preferential attachment-type algorithm for systems of constant size, which contains exit of balls and urns (or nodes and edges for the network case). Knowing mean size (degree) and turnover rate, the power law exponent and exponential cutoff can be derived. Our results are confirmed by simulations and by computation of exact probabilities. We also apply this entropy method to reproduce existing results like th…
Inspired by dengue and yellow fever epidemics, we investigated the contact process (CP) in a multiscale network constituted by one-dimensional chains connected through a Barabási-Albert scale-free network. In addition to the CP dynamics inside the chains, the exchange of individuals between connected chains (travels) occurs at a constant rate. A finite epidemic threshold and an epidemic mean lifetime diverging exponentially in the subcritical phase, concomitantly with a power law divergence of the outbreak's duration, were found. A generalized scaling function involving both regular and SF com…
We introduce a simple model of a growing system with $m$ competing communities. The model corresponds to the phenomenon of defeats suffered by social groups living in isolation. A nonequilibrium phase transition is observed when at critical time $t_c$ the first isolated cluster occurs. In the one-dimensional system the volume of the new phase, i.e. the number of the isolated individuals, increases with time as $Z \sim t^3$. For a large number of possible communities the critical density of filled space equals to $ρ_c = (m/N)^{1/3}$ where $N$ is the system size. A similar transition is observed…
