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Register a new userPhase transition for the Maki-Thompson rumour model on a small-world network
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We consider the Maki-Thompson model for the stochastic propagation of a rumour within a population. We extend the original hypothesis of homogenously mixed population by allowing for a small-world network embedding the model. This structure is realized starting from a $k$-regular ring and by inserting, in the average, $c$ additional links in such a way that $k$ and $c$ are tuneable parameter for the population architecture. We prove that this system exhibits a transition between regimes of localization (where the final number of stiflers is at most logarithmic in the population size) and propagation (where the final number of stiflers grows algebraically with the population size) at a finite value of the network parameter $c$. A quantitative estimate for the critical value of $c$ is obtained via extensive numerical simulations.
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In this work we propose a new extension for the Maki-Thompson rumor model which incorporates inter-group directed contacts. The model is defined on an homogeneously mixing population where the existence of two differentiated groups of individuals is assumed. While individuals of one group have an active role in the spreading process, individuals of the other group only contribute in stifling the rumor provided they would contacted. For this model we measure the impact of dissemination by studying the remaining proportion of ignorants of both groups at the end of the process. In addition we discuss some examples and possible applications.
We study the collective behavior of an Ising system on a small-world network with the interaction $J(r) propto r^{-alpha}$, where $r$ represents the Euclidean distance between two nodes. In the case of $alpha = 0$ corresponding to the uniform interaction, the system is known to possess a phase transition of the mean-field nature, while the system with the short-range interaction $(alphatoinfty)$ does not exhibit long-range order at any finite temperature. Monte Carlo simulations are performed at various values of $alpha$, and the critical value $alpha_c$ beyond which the long-range order does not emerge is estimated to be zero. Thus concluded is the absence of a phase transition in the system with the algebraically decaying interaction $r^{-alpha}$ for any nonzero positive value of $alpha$.
We study epidemic spreading according to a emph{Susceptible-Infectious-Recovered} (for short, emph{SIR}) network model known as the {em Reed-Frost} model, and we establish sharp thresholds for two generative models of {em one-dimensional small-world graphs}, in which graphs are obtained by adding random edges to a cycle. In $3$-regular graphs obtained as the union of a cycle and a random perfect matching, we show that there is a sharp threshold at $.5$ for the contagion probability along edges. In graphs obtained as the union of a cycle and of a $mathcal{G}_{n,c/n}$ ErdH{o}s-Renyi random graph with edge probability $c/n$, we show that there is a sharp threshold $p_c$ for the contagion probability: the value of $p_c$ turns out to be $sqrt 2 -1approx .41$ for the sparse case $c=1$ yielding an expected node degree similar to the random $3$-regular graphs above. In both models, below the threshold we prove that the infection only affects $mathcal{O}(log n)$ nodes, and that above the threshold it affects $Omega(n)$ nodes. These are the first fully rigorous results establishing a phase transition for SIR models (and equivalent percolation problems) in small-world graphs. Although one-dimensional small-world graphs are an idealized and unrealistic network model, a number of realistic qualitative phenomena emerge from our analysis, including the spread of the disease through a sequence of local outbreaks, the danger posed by random connections, and the effect of super-spreader events.
We study four discrete time stochastic systems on $bbN$ modeling processes of rumour spreading. The involved individuals can either have an active or a passive role, speaking up or asking for the rumour. The appetite in spreading or hearing the rumour is represented by a set of random variables whose distributions may depend on the individuals. Our goal is to understand - based on those random variables distribution - whether the probability of having an infinite set of individuals knowing the rumour is positive or not.
We present a numerical study of a reaction-diffusion model on a small-world network. We characterize the models average activity $F_T$ after $T$ time steps and the transition from a collective (global) extinct state to an active state in parameter space. We provide an explicit relation between the parameters of our model at the frontier between these states. A collective active state can be associated to a global epidemic spread, or to a persistent neuronal activity. We found that $F_T$ does not depends on disorder in the network if the transmission rate $r$ or the average coordination number $K$ are large enough. The collective extinct-active transition can be induced by changing two parameters associated to the network: $K$ and the disorder parameter $p$ (which controls the variance of $K$). We can also induce the transition by changing $r$, which controls the threshold size in the dynamics. In order to operate at the transition the parameters of the model must satisfy the relation $rK=a_p$, where $a_p$ as a function of $p/(1-p)$ is a stretched exponential function. Our results are relevant for systems that operate {it at} the transition in order to increase its dynamic range and/or to operate under optimal information-processing conditions. We discuss how glassy behaviour appears within our model.
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