We introduce a numerical method to solve epidemic models on the underlying topology of complex networks. The approach exploits the mean-field-like rate equations describing the system and allows us to work with very large system sizes, where Monte Carlo simulations are useless due to memory needs. We then study the susceptible-infected-removed epidemiological model on assortative networks, providing numerical evidence of the absence of epidemic thresholds. Besides, the time profiles of the populations are analyzed. Finally, we stress that the present method would allow us to solve arbitrary epidemiclike models provided that they can be described by mean-field rate equations.

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.PACS number(s): 89.75.Fb,05.70.Jk Typeset using REVT E X

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.

Abstract. In a spirit akin to the sandpile model of selforganized 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 algebraic description as a Markov Chain. This possibility illuminates some important results, obtained by Monte Carlo simulations, such as the earthquake size-frequency relation and the recurrence time of the characteristic earthquake.

Abstract. The decay rate of aftershocks is commonly very well described by the modified Omori law, n(t) cr 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 inspiLed 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.