Cellular Models for River Networks

A cellular model introduced for the evolution of the fluvial landscape is revisited using extensive numerical and scaling analyses. The basic network shapes and their recurrence especially in the aggregation structure are then addressed. The roles of boundary and initial conditions are carefully analyzed as well as the key effect of quenched disorder embedded in random pinning of the landscape surface. It is found that the above features strongly affect the scaling behavior of key morphological quantities. In particular, we conclude that randomly pinned regions (whose structural disorder bears much physical meaning mimicking uneven landscape-forming rainfall events, geological diversity or heterogeneity in surficial properties like vegetation, soil cover or type) play a key role for the robust emergence of aggregation patterns bearing much resemblance to real river networks.Comment: 7 pages, revtex style, 14 figure

Modeling urban street patterns

Urban streets patterns form planar networks whose empirical properties cannot be accounted for by simple models such as regular grids or Voronoi tesselations. Striking statistical regularities across different cities have been recently empirically found, suggesting that a general and details-independent mechanism may be in action. We propose a simple model based on a local optimization process combined with ideas previously proposed in studies of leaf pattern formation. The statistical properties of this model are in good agreement with the observed empirical patterns. Our results thus suggests that in the absence of a global design strategy, the evolution of many different transportation networks indeed follow a simple universal mechanism.Comment: 4 pages, 5 figures, final version published in PR

Local minimal energy landscapes in river networks

The existence and stability of the universality class associated to local minimal energy landscapes is investigated. Using extensive numerical simulations, we first study the dependence on a parameter $\gamma$ of a partial differential equation which was proposed to describe the evolution of a rugged landscape toward a local minimum of the dissipated energy. We then compare the results with those obtained by an evolution scheme based on a variational principle (the optimal channel networks). It is found that both models yield qualitatively similar river patterns and similar dependence on $\gamma$. The aggregation mechanism is however strongly dependent on the value of $\gamma$. A careful analysis suggests that scaling behaviors may weakly depend both on $\gamma$ and on initial condition, but in all cases it is within observational data predictions. Consequences of our resultsComment: 12 pages, 13 figures, revtex+epsfig style, to appear in Phys. Rev. E (Nov. 2000

The Fractal Properties of Internet

In this paper we show that the Internet web, from a user's perspective, manifests robust scaling properties of the type $P(n)\propto n^{-\tau}$ where n is the size of the basin connected to a given point, $P$ represents the density of probability of finding n points downhill and $\tau=1.9 \pm 0.1$ s a characteristic universal exponent. This scale-free structure is a result of the spontaneous growth of the web, but is not necessarily the optimal one for efficient transport. We introduce an appropriate figure of merit and suggest that a planning of few big links, acting as information highways, may noticeably increase the efficiency of the net without affecting its robustness.Comment: 6 pages,2 figures, epl style, to be published on Europhysics Letter

Network Structures from Selection Principles

We present an analysis of the topologies of a class of networks which are optimal in terms of the requirements of having as short a route as possible between any two nodes while yet keeping the congestion in the network as low as possible. Strikingly, we find a variety of distinct topologies and novel phase transitions between them on varying the number of links per node. Our results suggest that the emergence of the topologies observed in nature may arise both from growth mechanisms and the interplay of dynamical mechanisms with a selection process.Comment: 4 pages, 5 figure

Pricing Rainfall Based Futures Using Genetic Programming

Rainfall derivatives are in their infancy since starting trading on the Chicago Mercantile Exchange (CME) since 2011. Being a relatively new class of financial instruments there is no generally recognised pricing framework used within the literature. In this paper, we propose a novel framework for pricing contracts using Genetic Programming (GP). Our novel framework requires generating a risk-neutral density of our rainfall predictions generated by GP supported by Markov chain Monte Carlo and Esscher transform. Moreover, instead of having a single rainfall model for all contracts, we propose having a separate rainfall model for each contract. We compare our novel framework with and without our proposed contract-specific models for pricing against the pricing performance of the two most commonly used methods, namely Markov chain extended with rainfall prediction (MCRP), and burn analysis (BA) across contracts available on the CME. Our goal is twofold, (i) to show that by improving the predictive accuracy of the rainfall process, the accuracy of pricing also increases. (ii) contract-specific models can further improve the pricing accuracy. Results show that both of the above goals are met, as GP is capable of pricing rainfall futures contracts closer to the CME than MCRP and BA. This shows that our novel framework for using GP is successful, which is a significant step forward in pricing rainfall derivatives

An Analytical and Numerical Study of Optimal Channel Networks

We analyze the Optimal Channel Network model for river networks using both analytical and numerical approaches. This is a lattice model in which a functional describing the dissipated energy is introduced and minimized in order to find the optimal configurations. The fractal character of river networks is reflected in the power law behaviour of various quantities characterising the morphology of the basin. In the context of a finite size scaling Ansatz, the exponents describing the power law behaviour are calculated exactly and show mean field behaviour, except for two limiting values of a parameter characterizing the dissipated energy, for which the system belongs to different universality classes. Two modified versions of the model, incorporating quenched disorder are considered: the first simulates heterogeneities in the local properties of the soil, the second considers the effects of a non-uniform rainfall. In the region of mean field behaviour, the model is shown to be robust to both kinds of perturbations. In the two limiting cases the random rainfall is still irrelevant, whereas the heterogeneity in the soil properties leads to new universality classes. Results of a numerical analysis of the model are reported that confirm and complement the theoretical analysis of the global minimum. The statistics of the local minima are found to more strongly resemble observational data on real rivers.Comment: 27 pages, ps-file, 11 Postscript figure

Scale-free Networks from Optimal Design

A large number of complex networks, both natural and artificial, share the presence of highly heterogeneous, scale-free degree distributions. A few mechanisms for the emergence of such patterns have been suggested, optimization not being one of them. In this letter we present the first evidence for the emergence of scaling (and smallworldness) in software architecture graphs from a well-defined local optimization process. Although the rules that define the strategies involved in software engineering should lead to a tree-like structure, the final net is scale-free, perhaps reflecting the presence of conflicting constraints unavoidable in a multidimensional optimization process. The consequences for other complex networks are outlined.Comment: 6 pages, 2 figures. Submitted to Europhysics Letters. Additional material is available at http://complex.upc.es/~sergi/software.ht

Basins of attraction on random topography

We investigate the consequences of fluid flowing on a continuous surface upon the geometric and statistical distribution of the flow. We find that the ability of a surface to collect water by its mere geometrical shape is proportional to the curvature of the contour line divided by the local slope. Consequently, rivers tend to lie in locations of high curvature and flat slopes. Gaussian surfaces are introduced as a model of random topography. For Gaussian surfaces the relation between convergence and slope is obtained analytically. The convergence of flow lines correlates positively with drainage area, so that lower slopes are associated with larger basins. As a consequence, we explain the observed relation between the local slope of a landscape and the area of the drainage basin geometrically. To some extent, the slope-area relation comes about not because of fluvial erosion of the landscape, but because of the way rivers choose their path. Our results are supported by numerically generated surfaces as well as by real landscapes

Co-evolution of density and topology in a simple model of city formation

We study the influence that population density and the road network have on each others' growth and evolution. We use a simple model of formation and evolution of city roads which reproduces the most important empirical features of street networks in cities. Within this framework, we explicitely introduce the topology of the road network and analyze how it evolves and interact with the evolution of population density. We show that accessibility issues -pushing individuals to get closer to high centrality nodes- lead to high density regions and the appearance of densely populated centers. In particular, this model reproduces the empirical fact that the density profile decreases exponentially from a core district. In this simplified model, the size of the core district depends on the relative importance of transportation and rent costs.Comment: 13 pages, 13 figure