Fracturing the optimal paths

Optimal paths play a fundamental role in numerous physical applications ranging from random polymers to brittle fracture, from the flow through porous media to information propagation. Here for the first time we explore the path that is activated once this optimal path fails and what happens when this new path also fails and so on, until the system is completely disconnected. In fact numerous applications can be found for this novel fracture problem. In the limit of strong disorder, our results show that all the cracks are located on a single self-similar connected line of fractal dimension $D_{b} \approx 1.22$. For weak disorder, the number of cracks spreads all over the entire network before global connectivity is lost. Strikingly, the disconnecting path (backbone) is, however, completely independent on the disorder.Comment: 4 pages,4 figure

Large cities are less green

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 $\rm{CO}_2$ 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 $\rm{CO}_2$ emissions and city population with average allometric exponent $\beta = 1.46$ 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