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.

Abstract. -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 network's 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 system's 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.The behavior of an isolated generic dynamical system in the long-term limit could be described by stable fixed points, limit cycles or chaotic attractors [1]. We have also learned in recent years that when many of such dynamical systems are coupled together, new collective phenomena emerge. In this way, the study of regular networks of dynamical systems have shed light on a number of natural phenomena ranging from earthquakes to ecosystems and living organisms [2][3][4]. One of the most fascinating phenomena in the behavior of complex dynamical systems made up of many elements is the spontaneous emergence of order and the phenomenon of collective synchronization [5], where a large number of the system's constituents forms a common dynamical pattern, despite the intrinsic differences in their individual dynamics. Of recent interest are a plenty of biological examples that have become accessible at least numerically with the advent of modern computers [6].On the other hand, it has been recently shown that many biological [7,8], social [9], and technological [10] systems exhibit an intricate pattern of interconnections in the form of complex networks [6]. This structural complexity cannot be described by the couplings of a regular network. In order to characterize topologically these complex networks, one computes the probability, P (k), that any given element of the network has k connections to other nodes. Interestingly, many real-world networks such as the Internet, protein interaction networks and social webs [11] can be well approximated by a power-law connectivity distribution, P (k) ∼

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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