Analytical approach to directed sandpile models on the Apollonian network

We investigate a set of directed sandpile models on the Apollonian network, which are inspired on the work by Dhar and Ramaswamy (PRL \textbf{63}, 1659 (1989)) for Euclidian lattices. They are characterized by a single parameter $q$, that restricts the number of neighbors receiving grains from a toppling node. Due to the geometry of the network, two and three point correlation functions are amenable to exact treatment, leading to analytical results for the avalanche distributions in the limit of an infinite system, for $q=1,2$. The exact recurrence expressions for the correlation functions are numerically iterated to obtain results for finite size systems, when larger values of $q$ are considered. Finally, a detailed description of the local flux properties is provided by a multifractal scaling analysis.Comment: 7 pages in two-column format, 10 illustrations, 5 figure

Targeted Recovery as an Effective Strategy against Epidemic Spreading

We propose a targeted intervention protocol where recovery is restricted to individuals that have the least number of infected neighbours. Our recovery strategy is highly efficient on any kind of network, since epidemic outbreaks are minimal when compared to the baseline scenario of spontaneous recovery. In the case of spatially embedded networks, we find that an epidemic stays strongly spatially confined with a characteristic length scale undergoing a random walk. We demonstrate numerically and analytically that this dynamics leads to an epidemic spot with a flat surface structure and a radius that grows linearly with the spreading rate.Comment: 6 pages, 5 figure

A worldwide model for boundaries of urban settlements

The shape of urban settlements plays a fundamental role in their sustainable planning. Properly defining the boundaries of cities is challenging and remains an open problem in the Science of Cities. Here, we propose a worldwide model to define urban settlements beyond their administrative boundaries through a bottom-up approach that takes into account geographical biases intrinsically associated with most societies around the world, and reflected in their different regional growing dynamics. The generality of the model allows to study the scaling laws of cities at all geographical levels: countries, continents, and the entire world. Our definition of cities is robust and holds to one of the most famous results in Social Sciences: Zipf's law. According to our results, the largest cities in the world are not in line with what was recently reported by the United Nations. For example, we find that the largest city in the world is an agglomeration of several small settlements close to each other, connecting three large settlements: Alexandria, Cairo, and Luxor. Our definition of cities opens the doors to the study of the economy of cities in a systematic way independently of arbitrary definitions that employ administrative boundaries

Dynamics in the Fitness-Income plane: Brazilian states vs World countries

In this paper we introduce a novel algorithm, called Exogenous Fitness, to calculate the Fitness of subnational entities and we apply it to the states of Brazil. In the last decade, several indices were introduced to measure the competitiveness of countries by looking at the complexity of their export basket. Tacchella et al (2012) developed a non-monetary metric called Fitness. In this paper, after an overview about Brazil as a whole and the comparison with the other BRIC countries, we introduce a new methodology based on the Fitness algorithm, called Exogenous Fitness. Combining the results with the Gross Domestic Product per capita (GDPp), we look at the dynamics of the Brazilian states in the Fitness-Income plane. Two regimes are distinguishable: one with high predictability and the other with low predictability, showing a deep analogy with the heterogeneous dynamics of the World countries. Furthermore, we compare the ranking of the Brazilian states according to the Exogenous Fitness with the ranking obtained through two other techniques, namely Endogenous Fitness and Economic Complexity Index

Non-Local Product Rules for Percolation

Despite original claims of a first-order transition in the product rule model proposed by Achlioptas et al. [Science 323, 1453 (2009)], recent studies indicate that this percolation model, in fact, displays a continuous transition. The distinctive scaling properties of the model at criticality, however, strongly suggest that it should belong to a different universality class than ordinary percolation. Here we introduce a generalization of the product rule that reveals the effect of non-locality on the critical behavior of the percolation process. Precisely, pairs of unoccupied bonds are chosen according to a probability that decays as a power-law of their Manhattan distance, and only that bond connecting clusters whose product of their sizes is the smallest, becomes occupied. Interestingly, our results for two-dimensional lattices at criticality shows that the power-law exponent of the product rule has a significant influence on the finite-size scaling exponents for the spanning cluster, the conducting backbone, and the cutting bonds of the system. In all three cases, we observe a continuous variation from ordinary to (non-local) explosive percolation exponents.Comment: 5 pages, 4 figure

How dense can one pack spheres of arbitrary size distribution?

We present the first systematic algorithm to estimate the maximum packing density of spheres when the grain sizes are drawn from an arbitrary size distribution. With an Apollonian filling rule, we implement our technique for disks in 2d and spheres in 3d. As expected, the densest packing is achieved with power-law size distributions. We also test the method on homogeneous and on empirical real distributions, and we propose a scheme to obtain experimentally accessible distributions of grain sizes with low porosity. Our method should be helpful in the development of ultra-strong ceramics and high performance concrete.Comment: 5 pages, 5 figure

IMDB network revisited: unveiling fractal and modular properties from a typical small-world network

We study a subset of the movie collaboration network, imdb.com, where only adult movies are included. We show that there are many benefits in using such a network, which can serve as a prototype for studying social interactions. We find that the strength of links, i.e., how many times two actors have collaborated with each other, is an important factor that can significantly influence the network topology. We see that when we link all actors in the same movie with each other, the network becomes small-world, lacking a proper modular structure. On the other hand, by imposing a threshold on the minimum number of links two actors should have to be in our studied subset, the network topology becomes naturally fractal. This occurs due to a large number of meaningless links, namely, links connecting actors that did not actually interact. We focus our analysis on the fractal and modular properties of this resulting network, and show that the renormalization group analysis can characterize the self-similar structure of these networks.Comment: 12 pages, 9 figures, accepted for publication in PLOS ON

Transverse instability of dunes

The simplest type of dune is the transverse one, which propagates with invariant profile orthogonally to a fixed wind direction. Here we show numerically and with a linear stability analysis that transverse dunes are unstable with respect to along-axis perturbations in their profile and decay on the bedrock into barchan dunes. Any forcing modulation amplifies exponentially with growth rate determined by the dune turnover time. We estimate the distance covered by a transverse dune before fully decaying into barchans and identify the patterns produced by different types of perturbation.Comment: 4 pages, 3 figures; To appear in Physical Review Letter