Recently, the informatics revolution has made possible the analysis of a wide range of large scale, rapidly evolving networks such as transportation, technological, social and biological networks 1,2,3,4,5 . While these networks are extremely different from each other in their function and attributes, the analysis of their fabric provided evidence of several shared regularities, suggesting general and common self-organizing principles beyond the specific details of the individual systems. In this context, the statistical physics approach has been exploited as a very 1

A large number of complex systems find a natural abstraction in the form of weighted networks whose nodes represent the elements of the system and the weighted edges identify the presence of an interaction and its relative strength. In recent years, the study of an increasing number of large-scale networks has highlighted the statistical heterogeneity of their interaction pattern, with degree and weight distributions that vary over many orders of magnitude. These features, along with the large number of elements and links, make the extraction of the truly relevant connections forming the network's backbone a very challenging problem. More specifically, coarse-graining approaches and filtering techniques come into conflict with the multiscale nature of large-scale systems. Here, we define a filtering method that offers a practical procedure to extract the relevant connection backbone in complex multiscale networks, preserving the edges that represent statistically significant deviations with respect to a null model for the local assignment of weights to edges. An important aspect of the method is that it does not belittle small-scale interactions and operates at all scales defined by the weight distribution. We apply our method to realworld network instances and compare the obtained results with alternative backbone extraction techniques. disordered systems | multiscale phenomena | filtering | visualization I n recent years, a huge amount of data on large-scale social, biological, and communication networks, meticulously collected and catalogued, has become available for scientific analysis and study. Examples can be found in all domains; from technological to social systems and transportation networks on a local and global scale, and down to the microscopic scale of biochemical networks (1-3). Common traits of these networks can be found in the statistical properties characterized by large-scale heterogeneity with statistical observables such as nodes' degree and traffic varying over a wide range of scales (4). The sheer size and multiscale nature of these networks make very difficult the extraction of the relevant information that would allow a reduced representation while preserving the key features we want to highlight. A typical example is seen in the visualization of networks. Although, in general, it is possible to create wonderful images of large-scale heterogeneous networks, the amount of valuable information gathered is in most cases very little because of the redundant intricacy generated by the overwhelming number of connections. Problems such as the extraction of the relevant backbone or the isolation of the statistically relevant structures/signal that would allow reduced but meaningful representations of the system are indeed major challenges in the analysis of large-scale networks.In complex weighted networks, the discrimination of the right trade-off between the level of network reduction and the amount of relevant information preserved in the new representation faces us with additional problems....

The principle that 'popularity is attractive' underlies preferential attachment, which is a common explanation for the emergence of scaling in growing networks. If new connections are made preferentially to more popular nodes, then the resulting distribution of the number of connections possessed by nodes follows power laws, as observed in many real networks. Preferential attachment has been directly validated for some real networks (including the Internet), and can be a consequence of different underlying processes based on node fitness, ranking, optimization, random walks or duplication. Here we show that popularity is just one dimension of attractiveness; another dimension is similarity. We develop a framework in which new connections optimize certain trade-offs between popularity and similarity, instead of simply preferring popular nodes. The framework has a geometric interpretation in which popularity preference emerges from local optimization. As opposed to preferential attachment, our optimization framework accurately describes the large-scale evolution of technological (the Internet), social (trust relationships between people) and biological (Escherichia coli metabolic) networks, predicting the probability of new links with high precision. The framework that we have developed can thus be used for predicting new links in evolving networks, and provides a different perspective on preferential attachment as an emergent phenomenon.

Economy, and consequently trade, is a fundamental part of human social organization which, until now, has not been studied within the network modeling framework. Here we present the first, to the best of our knowledge, empirical characterization of the world trade web, that is, the network built upon the trade relationships between different countries in the world. This network displays the typical properties of complex networks, namely, scale-free degree distribution, the small-world property, a high clustering coefficient, and, in addition, degree-degree correlation between different vertices. All these properties make the world trade web a complex network, which is far from being well described through a classical random network description. DOI: 10.1103/PhysRevE.68.015101 PACS number͑s͒: 89.75.Hc, 87.23.Ge, 05.70.Ln, 89.65.Gh The world is facing a challenging era. Social, political, and economic arrangements initiated after the end of the Second World War are now culminating in the recognition of globalization, a process which has been accelerated by the new technological advances. When applied to the international economic order, globalization involves control of capital flow and liberalization of trade. As a consequence, economies around the world are becoming more and more interrelated, in other words, the world is becoming a global village ͓1,2͔. In this scenario, trade plays a central role as one of the most important interaction channels between countries ͓3͔. The relevance of the international trade system goes beyond the fundamental exchange of goods and services. For instance, it can also be the channel for crises spreading ͓4͔. A good example is found in the recent Asiatic crisis, which shows how economic perturbations originating in a country can somehow propagate elsewhere in the world ͓5,6͔. Thus, it seems natural to analyze the world trade system at a global level, every country being important, regardless of its size or wealth. Despite the extremely complex nature of the problem, relevant structural information can be extracted from modeling the system as a network, where countries are represented as vertices and trade channels as links between these vertices. In this way, the global trade system can be examined under a topological point of view. This analysis will reveal complex properties that cannot be explained by the classical random graph theory, first studied by Erdös and Rényi ͓7͔.Complex networks have been the subject of an intense research activity over the last years ͓8,9͔. Examples range from metabolic networks, where cell functionality is sustained by the network structure, to technological webs, where topology determines the system's ability to transmit information ͓10-13͔. The term complex network typically refers to networks showing the following properties: ͑i͒ scale-free ͑SF͒ degree distribution, P(k)ϳk Ϫ␥ with 2Ͻ␥ р3, where the degree k is defined as the number of edges emanating from a vertex, ͑ii͒ the small-world property ͓14͔, which states that the average path length...