We present an advanced algorithm for the determination of watershed lines on Digital Elevation Models (DEMs), which is based on the iterative application of Invasion Percolation (IIP) . The main advantage of our method over previosly proposed ones is that it has a sub-linear time-complexity. This enables us to process systems comprised of up to 10 8 sites in a few cpu seconds. Using our algorithm we are able to demonstrate, convincingly and with high accuracy, the fractal character of watershed lines.We find the fractal dimension of watersheds to be D f = 1.211 ± 0.001 for artificial landscapes, D f = 1.10 ± 0.01 for the Alpes and D f = 1.11 ± 0.01 for the Himalaya.

We examine fluctuations in particle density in the restricted-height, conserved stochastic sandpile (CSS). In this and related models, the global particle density is a temperaturelike control parameter. Thus local fluctuations in this density correspond to disorder; if this disorder is a relevant perturbation of directed percolation (DP), then the CSS should exhibit non-DP critical behavior. We analyze the scaling of the variance Vℓ of the number of particles in regions of ℓd sites in extensive simulations of the quasistationary state in one and two dimensions. Our results, combined with a Harris-like argument for the relevance of particle-density fluctuations, strongly suggest that conserved stochastic sandpiles belong to a universality class distinct from that of DP.

We study diffusion of particles in large-scale simulations of one-dimensional stochastic sandpiles, in both the restricted and unrestricted versions. The results indicate that the diffusion constant scales in the same manner as the activity density, so that it represents an alternative definition of an order parameter. The critical behavior of the unrestricted sandpile is very similar to that of its restricted counterpart, including the fact that a data collapse of the order parameter as a function of the particle density is only possible over a very narrow interval near the critical point. We also develop a series expansion, in inverse powers of the density. for the collective diffusion coefficient in a variant of the stochastic sandpile in which the toppling rate at a site with n particles is n(n − 1), and compare the theoretical prediction with simulation results. §

We introduce a model which consists in a planar network which grows by adding nodes at a distance r from the pre-existing barycenter. Each new node position is randomly located through the distribution law P (r) ∝ 1/r γ with γ > 1. The new node j is linked to only one pre-existing node according to the probability law P; k i is the number of links of the i th node, η i is its fitness (or quality factor), and r ij is the distance. We consider in the present paper two models for η i . In one of them, the single fitness model (SFM [D. J. B. Soares, C. Tsallis, A. M. Mariz and L. R. da Silva, Europhys. Lett. 70 (2005), 70.]), we consider η i = 1 ∀ i. In the other one, the uniformly distributed fitness model (UDFM), η i is chosen to be uniformly distributed within the interval (0, 1]. We have determined numerically the degree distribution P (k). This distribution appears to be well fitted with Pis the q-exponential function naturally emerging within Tsallis nonextensive statistical mechanics. We determine, for both models, the entropic index q as a function of α A . Additionally, we determine the average topological (or chemical) distance within the network, and the time evolution of the average number of links k i . We obtain that, asymptotically, k i ∝ (t/i) β , (i coincides with the input-time of the i th node) and β(α A ) to both cases.Scale-free networks are very popular nowadays 2)-5) due to their uncountable applications in different fields of knowledge. These models are typically associated with some physical quantities that are characterized by power-law asymptotic behavior, instead of the usual exponential laws. Most of these models do not take into account the node-to-node Euclidean distance, i.e., the geographical distance. One of them which does take into account this aspect of the problem has been introduced recently 1) and discussed. In this example, as well as in others, 6), 7) strong connection has been revealed with nonextensive statistical mechanics. 8), 9) In the present paper, we follow the lines of Ref. 1) which we extend in the sense that we include now a local variable denominated fitness or quality factor, we call this variable η i . The model studied by Ref. 1) is herein referred to as the Single Fitness Model (SFM), that is, η i = 1 ∀ i, and we also study the Uniformly Distributed Fitness Model (UDFM), in * )

We perform large-scale simulations of a two-dimensional restricted height conserved stochastic sandpile, focusing on particle diffusion and mobility, and spatial correlations. Quasistationary (QS) simulations yield the critical particle density to high precision [p c = 0.7112687(2)], and show that the diffusion constant scales in the same manner as the activity density, as found previously in the one-dimensional case. Short-time scaling is characterized by subdiffusive behavior (mean-square displacement ∼ t γ with γ < 1), which is easily understood as a consequence of the initial decay of activity, ρ(t) ∼ t −δ , with γ = 1 − δ. We verify that at criticality, the activity-activity correlation function/ , as expected at an absorbing-state phase transition.