We study how urban quality evolves as a result of carbon dioxide emissions as urban agglomerations grow. We employ a bottom-up approach combining two unprecedented microscopic data on population and carbon dioxide emissions in the continental US. We first aggregate settlements that are close to each other into cities using the City Clustering Algorithm (CCA) defining cities beyond the administrative boundaries. Then, we use data on CO2 emissions at a fine geographic scale to determine the total emissions of each city. We find a superlinear scaling behavior, expressed by a power-law, between CO2 emissions and city population with average allometric exponent β = 1.46 across all cities in the US. This result suggests that the high productivity of large cities is done at the expense of a proportionally larger amount of emissions compared to small cities. Furthermore, our results are substantially different from those obtained by the standard administrative definition of cities, i.e. Metropolitan Statistical Area (MSA). Specifically, MSAs display isometric scaling emissions and we argue that this discrepancy is due to the overestimation of MSA areas. The results suggest that allometric studies based on administrative boundaries to define cities may suffer from endogeneity bias.

We investigate at the subscale of the neighborhoods of a highly populated city the incidence of property crimes in terms of both the resident and the floating population. Our results show that a relevant allometric relation could only be observed between property crimes and floating population. More precisely, the evidence of a superlinear behavior indicates that a disproportional number of property crimes occurs in regions where an increased flow of people takes place in the city. For comparison, we also found that the number of crimes of peace disturbance only correlates well, and in a superlinear fashion too, with the resident population. Our study raises the interesting possibility that the superlinearity observed in previous studies [Bettencourt et al., Proc. Natl. Acad. Sci. USA 104, 7301 (2007) and Melo et al., Sci. Rep. 4, 6239 (2014)] for homicides versus population at the city scale could have its origin in the fact that the floating population, and not the resident one, should be taken as the relevant variable determining the intrinsic microdynamical behavior of the system.

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 introduce a cluster growth process that provides a clear connection between equilibrium statistical mechanics and an explosive percolation model similar to the one recently proposed by Achlioptas et al. [Science 323, 1453[Science 323, (2009. We show that the following two ingredients are essential for obtaining an abrupt (first-order) transition in the fraction of the system occupied by the largest cluster: (i) the size of all growing clusters should be kept approximately the same, and (ii) the inclusion of merging bonds (i.e., bonds connecting vertices in different clusters) should dominate with respect to the redundant bonds (i.e., bonds connecting vertices in the same cluster). Moreover, in the extreme limit where only merging bonds are present, a complete enumeration scheme based on tree-like graphs can be used to obtain an exact solution of our model that displays a firstorder transition. Finally, the proposed mechanism can be viewed as a generalization of standard percolation that discloses an entirely new family of models with potential application in growth and fragmentation processes of real network systems. PACS numbers:The second-order critical point of percolation [1, 2] has been successfully used to describe a large variety of phenomena in Nature, including the sol-gel transition [3], or incipient flow through porous media [4], as well as epidemic spreading [5] and network failure [6,7,8,9]. A long standing question of practical interest has been since, how the transition could be made more abrupt and in the limit become even of first-order. In other words, what ingredient must be tuned in the basic model of random percolation to change the order of the transition?Recently Achlioptas et al.[10] proposed a new mechanism on random graphs which they termed "explosive percolation" that exhibits first-order phase transition. Their process takes place in successive steps, with bonds being added to the system in accordance to a selection rule. At each step, a set of two unoccupied bonds are chosen randomly. From these two, only the one with minimum weight becomes occupied. In Ref.[10], the weight is defined as the product of the sizes of the clusters connected by this bond (this is called "product rule"). Importantly, if the bond connects two sites that already belong to the same cluster, the weight is proportional to the square of the cluster size. Since unoccupied bonds connecting vertices in the largest cluster have the largest possible weight, these bonds will become occupied only if two of them are randomly chosen. Thus, this selection rule hinders the inclusion of bonds connecting vertices that already belong to the largest cluster. As a consequence, bonds merging two smaller clusters will be selected more frequently, resulting in the fast growth observed. Their model was then implemented on a fully connected graph, however, it was shown that the same effect takes place on 2D square lattices [11] as well as scale-free networks [12,13].In this letter we investigate what are the basic principles t...