Amalio F. Pacheco

Dynamics of rumor spreading in complex networks

We derive the mean-field equations characterizing the dynamics of a rumor process that takes place on top of complex heterogeneous networks. These equations are solved numerically by means of a stochastic approach. First, we present analytical and Monte Carlo calculations for homogeneous networks and compare the results with those obtained by the numerical method. Then, we study the spreading process in detail for random scale-free networks. The time profiles for several quantities are numerically computed, which allows us to distinguish among different variants of rumor spreading algorithms. Our conclusions are directed to possible applications in replicated database maintenance, peer-to-peer communication networks, and social spreading phenomena.

Synchronization of Kuramoto oscillators in scale-free networks

In this work, we study the synchronization of coupled phase oscillators on the underlying topology of scale-free networks. In particular, we assume that each networks component is an oscillator and that each interacts with the others following the Kuramoto model. We then study the onset of global phase synchronization and fully characterize the systems dynamics. We also found that the resynchronization time of a perturbed node decays as a power law of its connectivity, providing a simple analytical explanation to this interesting behavior.

Instability of scale-free networks under node-breaking avalanches

The instability introduced in a large scale-free network by the triggering of node-breaking avalanches is analyzed using the fiber-bundle model as conceptual framework. We found, by measuring the size of the giant component, the avalanche size distribution and other quantities, the existence of an abrupt transition. This test of strength for complex networks like Internet is more stringent than others recently considered like the random removal of nodes, analyzed within the framework of percolation theory. Finally, we discuss the possible implications of our results and their relevance in forecasting cascading failures in scale-free networks.

Fracture and second-order phase transitions.

Using the global fiber bundle model as a tractable scheme of progressive fracture in heterogeneous materials, we define the branching ratio in avalanches as a suitable order parameter to clarify the order of the phase transition occurring at the collapse of the system. The model is analyzed using a probabilistic approach suited to smooth fluctuations. The branching ratio shows a behavior analogous to the magnetization in known magnetic systems with second-order phase transitions. We obtain a universal critical exponent beta approximately = 0.5 independent of the probability distribution used to assign the strengths of individual fibers.

Fitness for synchronization of network motifs

We study the synchronization of Kuramotos oscillators in small parts of networks known as motifs. We first report on the system dynamics for the case of a scale-free network and show the existence of a non-trivial critical point. We compute the probability that network motifs synchronize, and find that the fitness for synchronization correlates well with motifs interconnectedness and structural complexity. Possible implications for present debates about network evolution in biological and other systems are discussed.

Turbulencelike Behavior of Seismic Time Series

We report on a stochastic analysis of Earths vertical velocity time series by using methods originally developed for complex hierarchical systems and, in particular, for turbulent flows. Analysis of the fluctuations of the detrended increments of the series reveals a pronounced transition in their probability density function from Gaussian to non-Gaussian. The transition occurs 5-10 hours prior to a moderate or large earthquake, hence representing a new and reliable precursor for detecting such earthquakes.

A model for complex aftershock sequences

The decay rate of aftershocks is commonly very well described by the modified Omori law, n(t) ∝ t−p, where n(t) is the number of aftershocks per unit time, t is the time after the main shock, and p is a constant in the range 0.9 < p < 1.5 and usually close to 1. However, there are also more complex aftershock sequences for which the Omori law can be considered only as a first approximation. One of these complex aftershock sequences took place in the eastern Pyrenees on February 18, 1996, and was described in detail by Correig et al. [1997]. In this paper, we propose a new model inspired by dynamic fiber bundle models to interpret this type of complex aftershock sequences with sudden increases in the rate of aftershock production not directly related to the magnitude of the aftershocks (as in the epidemic-type aftershock sequences). The model is a simple, discrete, stochastic fracture model where the elements (asperities or barriers) break because of static fatigue, transfer stress according to a local load-sharing rule and then are regenerated. We find a very good agreement between the model and the Eastern Pyrenees aftershock sequence, and we propose that the key mechanism for explaining aftershocks, apart from a time-dependent rock strength, is the presence of dynamic stress fluctuations which constantly reset the initial conditions for the next aftershock in the sequence.

Self-Organized Criticality in a Fibre-Bundle type model

The dynamics of a bre-bundle-type model with equal load sharing rule is numerically studied. The system, formed by N elements, is driven by a slow increase of the load upon it which is removed in a novel way through internal transfers to the elements broken during avalanches. When an avalanche ends, failed elements are regenerated with strengths taken from a probability distribution. For a large N and certain restrictions on the distribution of individual strengths, the system reaches a self-organized critical state where the spectrum of avalanche sizes is a power law with an exponent ’1:5. c 1999 Elsevier Science B.V. All rights reserved.

A minimalist model of characteristic earthquakes

In a spirit akin to the sandpile model of self- organized criticality, we present a simple statistical model of the cellular-automaton type which simulates the role of an asperity in the dynamics of a one-dimensional fault. This model produces an earthquake spectrum similar to the characteristic-earthquake behaviour of some seismic faults. This model, that has no parameter, is amenable to an alge- braic description as a Markov Chain. This possibility illumi- nates some important results, obtained by Monte Carlo sim- ulations, such as the earthquake size-frequency relation and the recurrence time of the characteristic earthquake.

Probabilistic approach to time-dependent load-transfer models of fracture

A probabilistic method for solving time-dependent load-transfer models of fracture is developed. It is applicable to any rule of load redistribution, i.e, local, hierarchical, etc. In the new method, the fluctuations are generated during the breaking process (annealed randomness) while in the usual method, the random lifetimes are fixed at the beginning (quenched disorder). Both approaches are equivalent.