A global take on congestion in urban areas

We analyze the congestion data collected by a GPS device company (TomTom) for almost 300 urban areas in the world. Using simple scaling arguments and data fitting we show that congestion during peak hours in large cities grows essentially as the square root of the population density. This result, at odds with previous publications showing that gasoline consumption decreases with density, confirms that density is indeed an important determinant of congestion, but also that we need urgently a better theoretical understanding of this phenomena. This incomplete view at the urban level leads thus to the idea that thinking about density by itself could be very misleading in congestion studies, and that it is probably more useful to focus on the spatial redistribution of activities and residences.Comment: 3 pages, 2 figure

Crossover from Scale-Free to Spatial Networks

In many networks such as transportation or communication networks, distance is certainly a relevant parameter. In addition, real-world examples suggest that when long-range links are existing, they usually connect to hubs-the well connected nodes. We analyze a simple model which combine both these ingredients--preferential attachment and distance selection characterized by a typical finite `interaction range'. We study the crossover from the scale-free to the `spatial' network as the interaction range decreases and we propose scaling forms for different quantities describing the network. In particular, when the distance effect is important (i) the connectivity distribution has a cut-off depending on the node density, (ii) the clustering coefficient is very high, and (iii) we observe a positive maximum in the degree correlation (assortativity) which numerical value is in agreement with empirical measurements. Finally, we show that if the number of nodes is fixed, the optimal network which minimizes both the total length and the diameter lies in between the scale-free and spatial networks. This phenomenon could play an important role in the formation of networks and could be an explanation for the high clustering and the positive assortativity which are non trivial features observed in many real-world examples.Comment: 4 pages, 6 figures, final versio

Transitions in spatial networks

Networks embedded in space can display all sorts of transitions when their structure is modified. The nature of these transitions (and in some cases crossovers) can differ from the usual appearance of a giant component as observed for the Erdos-Renyi graph, and spatial networks display a large variety of behaviors. We will discuss here some (mostly recent) results about topological transitions, `localization' transitions seen in the shortest paths pattern, and also about the effect of congestion and fluctuations on the structure of optimal networks. The importance of spatial networks in real-world applications makes these transitions very relevant and this review is meant as a step towards a deeper understanding of the effect of space on network structures.Comment: Corrected version and updated list of reference

Betweenness Centrality in Large Complex Networks

We analyze the betweenness centrality (BC) of nodes in large complex networks. In general, the BC is increasing with connectivity as a power law with an exponent $\eta$. We find that for trees or networks with a small loop density $\eta=2$ while a larger density of loops leads to $\eta<2$. For scale-free networks characterized by an exponent $\gamma$ which describes the connectivity distribution decay, the BC is also distributed according to a power law with a non universal exponent $\delta$. We show that this exponent $\delta$ must satisfy the exact bound $\delta\geq (\gamma+1)/2$. If the scale free network is a tree, then we have the equality $\delta=(\gamma+1)/2$.Comment: 6 pages, 5 figures, revised versio

A Path Integral Approach to Effective Non-Linear Medium

In this article, we propose a new method to compute the effective properties of non-linear disordered media. We use the fact that the effective constants can be defined through the minimum of an energy functional. We express this minimum in terms of a path integral allowing us to use many-body techniques. We obtain the perturbation expansion of the effective constants to second order in disorder, for any kind of non-linearity. We apply our method to both cases of strong and weak non-linearities. Our results are in agreement with previous ones, and could be easily extended to other types of non-linear problems in disordered systems.Comment: 7 page

Modeling the polycentric transition of cities

Empirical evidence suggest that most urban systems experience a transition from a monocentric to a polycentric organisation as they grow and expand. We propose here a stochastic, out-of-equilibrium model of the city which explains the appearance of subcenters as an effect of traffic congestion. We show that congestion triggers the unstability of the monocentric regime, and that the number of subcenters and the total commuting distance within a city scale sublinearly with its population, predictions which are in agreement with data gathered for around 9000 US cities between 1994 and 2010.Comment: 11 pages, 12 figure

Fluctuation effects in metapopulation models: percolation and pandemic threshold

Metapopulation models provide the theoretical framework for describing disease spread between different populations connected by a network. In particular, these models are at the basis of most simulations of pandemic spread. They are usually studied at the mean-field level by neglecting fluctuations. Here we include fluctuations in the models by adopting fully stochastic descriptions of the corresponding processes. This level of description allows to address analytically, in the SIS and SIR cases, problems such as the existence and the calculation of an effective threshold for the spread of a disease at a global level. We show that the possibility of the spread at the global level is described in terms of (bond) percolation on the network. This mapping enables us to give an estimate (lower bound) for the pandemic threshold in the SIR case for all values of the model parameters and for all possible networks.Comment: 14 pages, 13 figures, final versio