The non-linear q-voter model

We introduce a non-linear variant of the voter model, the q-voter model, in which q neighbors (with possible repetition) are consulted for a voter to change opinion. If the q neighbors agree, the voter takes their opinion; if they do not have an unanimous opinion, still a voter can flip its state with probability $\epsilon$. We solve the model on a fully connected network (i.e. in mean-field) and compute the exit probability as well as the average time to reach consensus. We analyze the results in the perspective of a recently proposed Langevin equation aimed at describing generic phase transitions in systems with two ($Z_2$ symmetric) absorbing states. We find that in mean-field the q-voter model exhibits a disordered phase for high $\epsilon$ and an ordered one for low $\epsilon$ with three possible ways to go from one to the other: (i) a unique (generalized voter-like) transition, (ii) a series of two consecutive Ising-like and directed percolation transition, and (iii) a series of two transitions, including an intermediate regime in which the final state depends on initial conditions. This third (so far unexplored) scenario, in which a new type of ordering dynamics emerges, is rationalized and found to be specific of mean-field, i.e. fluctuations are explicitly shown to wash it out in spatially extended systems.Comment: 9 pages, 7 figure

Velocity fluctuations and hydrodynamic diffusion in sedimentation

We study non-equilibrium velocity fluctuations in a model for the sedimentation of non-Brownian particles experiencing long-range hydrodynamic interactions. The complex behavior of these fluctuations, the outcome of the collective dynamics of the particles, exhibits many of the features observed in sedimentation experiments. In addition, our model predicts a final relaxation to an anisotropic (hydrodynamic) diffusive state that could be observed in experiments performed over longer time ranges.Comment: 7 pages, 5 EPS figures, EPL styl

Steady-State Dynamics of the Forest Fire Model on Complex Networks

Many sociological networks, as well as biological and technological ones, can be represented in terms of complex networks with a heterogeneous connectivity pattern. Dynamical processes taking place on top of them can be very much influenced by this topological fact. In this paper we consider a paradigmatic model of non-equilibrium dynamics, namely the forest fire model, whose relevance lies in its capacity to represent several epidemic processes in a general parametrization. We study the behavior of this model in complex networks by developing the corresponding heterogeneous mean-field theory and solving it in its steady state. We provide exact and approximate expressions for homogeneous networks and several instances of heterogeneous networks. A comparison of our analytical results with extensive numerical simulations allows to draw the region of the parameter space in which heterogeneous mean-field theory provides an accurate description of the dynamics, and enlights the limits of validity of the mean-field theory in situations where dynamical correlations become important.Comment: 13 pages, 9 figure

Percolation in Hierarchical Scale-Free Nets

We study the percolation phase transition in hierarchical scale-free nets. Depending on the method of construction, the nets can be fractal or small-world (the diameter grows either algebraically or logarithmically with the net size), assortative or disassortative (a measure of the tendency of like-degree nodes to be connected to one another), or possess various degrees of clustering. The percolation phase transition can be analyzed exactly in all these cases, due to the self-similar structure of the hierarchical nets. We find different types of criticality, illustrating the crucial effect of other structural properties besides the scale-free degree distribution of the nets.Comment: 9 Pages, 11 figures. References added and minor corrections to manuscript. In pres

Halting viruses in scale-free networks

The vanishing epidemic threshold for viruses spreading on scale-free networks indicate that traditional methods, aiming to decrease a virus' spreading rate cannot succeed in eradicating an epidemic. We demonstrate that policies that discriminate between the nodes, curing mostly the highly connected nodes, can restore a finite epidemic threshold and potentially eradicate a virus. We find that the more biased a policy is towards the hubs, the more chance it has to bring the epidemic threshold above the virus' spreading rate. Furthermore, such biased policies are more cost effective, requiring less cures to eradicate the virus

Critical load and congestion instabilities in scale-free networks

We study the tolerance to congestion failures in communication networks with scale-free topology. The traffic load carried by each damaged element in the network must be partly or totally redistributed among the remaining elements. Overloaded elements might fail on their turn, triggering the occurrence of failure cascades able to isolate large parts of the network. We find a critical traffic load above which the probability of massive traffic congestions destroying the network communication capabilities is finite.Comment: 4 pages, 3 figure

Understanding the internet topology evolution dynamics

The internet structure is extremely complex. The Positive-Feedback Preference (PFP) model is a recently introduced internet topology generator. The model uses two generic algorithms to replicate the evolution dynamics observed on the internet historic data. The phenomenological model was originally designed to match only two topology properties of the internet, i.e. the rich-club connectivity and the exact form of degree distribution. Whereas numerical evaluation has shown that the PFP model accurately reproduces a large set of other nontrivial characteristics as well. This paper aims to investigate why and how this generative model captures so many diverse properties of the internet. Based on comprehensive simulation results, the paper presents a detailed analysis on the exact origin of each of the topology properties produced by the model. This work reveals how network evolution mechanisms control the obtained topology properties and it also provides insights on correlations between various structural characteristics of complex networks.Comment: 15 figure

A dissemination strategy for immunizing scale-free networks

We consider the problem of distributing a vaccine for immunizing a scale-free network against a given virus or worm. We introduce a new method, based on vaccine dissemination, that seems to reflect more accurately what is expected to occur in real-world networks. Also, since the dissemination is performed using only local information, the method can be easily employed in practice. Using a random-graph framework, we analyze our method both mathematically and by means of simulations. We demonstrate its efficacy regarding the trade-off between the expected number of nodes that receive the vaccine and the network's resulting vulnerability to develop an epidemic as the virus or worm attempts to infect one of its nodes. For some scenarios, the new method is seen to render the network practically invulnerable to attacks while requiring only a small fraction of the nodes to receive the vaccine

Heterogeneous pair approximation for voter models on networks

For models whose evolution takes place on a network it is often necessary to augment the mean-field approach by considering explicitly the degree dependence of average quantities (heterogeneous mean-field). Here we introduce the degree dependence in the pair approximation (heterogeneous pair approximation) for analyzing voter models on uncorrelated networks. This approach gives an essentially exact description of the dynamics, correcting some inaccurate results of previous approaches. The heterogeneous pair approximation introduced here can be applied in full generality to many other processes on complex networks.Comment: 6 pages, 6 figures, published versio