Networked structures arise in a wide array of different contexts such as technological and transportation infrastructures, social phenomena, and biological systems. These highly interconnected systems have recently been the focus of a great deal of attention that has uncovered and characterized their topological complexity. Along with a complex topological structure, real networks display a large heterogeneity in the capacity and intensity of the connections. These features, however, have mainly not been considered in past studies where links are usually represented as binary states, i.e., either present or absent. Here, we study the scientific collaboration network and the world-wide air-transportation network, which are representative examples of social and large infrastructure systems, respectively. In both cases it is possible to assign to each edge of the graph a weight proportional to the intensity or capacity of the connections among the various elements of the network. We define appropriate metrics combining weighted and topological observables that enable us to characterize the complex statistical properties and heterogeneity of the actual strength of edges and vertices. This information allows us to investigate the correlations among weighted quantities and the underlying topological structure of the network. These results provide a better description of the hierarchies and organizational principles at the basis of the architecture of weighted networks.A large number of natural and man-made systems are structured in the form of networks. Typical examples include large communication systems (the Internet, the telephone network, the World Wide Web), transportation infrastructures (railroad and airline routes), biological systems (gene and͞or protein interaction networks), and a variety of social interaction structures (1-3). The macroscopic properties of these networks have been the subject of intense scientific activity that has highlighted the emergence of a number of significant topological features. Specifically, many of these networks show the smallworld property (4), which implies that the network has an average topological distance between the various nodes increasing very slowly with the number of nodes (logarithmically or even slower), despite showing a large degree of local interconnectedness typical of more ordered lattices. Additionally, several of these networks are characterized by a statistical abundance of ''hubs'' with a very large number of connections k compared with the average degree value ͗k͘. The empirical evidence collected from real data indicates that this distinctive feature finds its statistical characterization in the presence of scale-free degree distributions P(k), i.e., showing a power-law behavior P(k) ϳ k Ϫ␥ for a significant range of values of k (5). These topological features turn out to be extremely relevant because they have a strong impact in assessing such networks' physical properties as their robustness or vulnerability (6-9).While these findings alone might prov...

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