Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

In this paper, we study and characterize the topology of the Internet at the Autonomous System level. First, we show that the topology can be described efficiently with power-laws. The elegance and simplicity of the powerlaws provide a novel perspective into the seemingly uncontrolled Internet structure. Second, we show that power-laws appear consistently over the last 5 years. We also observe that the power-laws hold even in the most recent and more complete topology [10] with correlation coefficient above 99% for the degree power-law. In addition, we study the evolution of the power-law exponents over the 5 year interval and observe a variation for the degree based power-law of less than 10%. Third, we provide relationships between the exponents and other topological metrics.

A number of recent papers in the networking community study the distance matrix defined by the node-to-node latencies in the Internet and, in particular, provide a number of quite successful distributed approaches that embed this distance into a low-dimensional Euclidean space. In such algorithms it is feasible to measure distances among only a linear or near-linear number of node pairs; the rest of the distances are simply not available. Moreover, for applications it is desirable to spread the load evenly among the participating nodes. Indeed, several recent studies use this ’fully distributed ’ approach and achieve, empirically, a low distortion for all but a small fraction of node pairs. This is concurrent with the large body of theoretical work on metric embeddings, but there is a fundamental distinction: in the theoretical approaches to metric embeddings, full and centralized access to the distance matrix is assumed and heavily used. In this paper we present the first fully distributed embedding algorithm with provable distortion guarantees for doubling metrics (which have been proposed as a reasonable abstraction of Internet latencies), thus providing some insight into the empirical success of the recent Vivaldi algorithm [7]. The main ingredient of our embedding algorithm is an improved fully distributed algorithm for a more basic problem of triangulation, where the triangle inequality is used to infer the distances that have not been measured; this problem received a considerable attention in the networking community, and has also been studied theoretically in [19]. We use our techniques to extend ɛ-relaxed embeddings and triangulations to infinite metrics and arbitrary measures, and to improve on the approximate distance labeling scheme of Talwar [36]. 1

Uncovering the hidden regularities and organizational principles of networks arising in physical systems ranging from the molecular level to the scale of large communication infrastructures is the key issue for the understanding of their fabric and dynamical prop-erties 1,2,3,4,5. The “rich-club ” phenomenon refers to the tendency of nodes with high cen-trality, the dominant elements of the system, to form tightly interconnected communities and it is one of the crucial properties accounting for the formation of dominant communi-ties in both computer and social sciences 4,5,6,7,8. Here we provide the analytical expression and the correct null models which allow for a quantitative discussion of the rich-club phe-nomenon. The presented analysis enables the measurement of the rich-club ordering and its relation with the function and dynamics of networks in examples drawn from the bio-logical, social and technological domains. 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 func-tion and attributes, the analysis of their fabric provided evidence of several shared regularities,

We review the behavior of epidemic spreading on complex networks in which there are explicit correlations among the degrees of connected vertices. 1

In many real growing networks the mean number of connections per vertex increases with time. The Internet, the Word Wide Web, collaborations networks, and many others display this behavior. Such a growth can be called accelerated. We show that this acceleration influences distribution of connections and may determine the structure of a network. We discuss general consequences of the acceleration and demonstrate its features applying simple illustrating examples. In particular, we show that the accelerated growth fairly well explains the structure of the Word Web (the network of interacting words of human language). Also, we use the models of the accelerated growth of networks to describe a wealth condensation transition in evolving societies.

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In physical, biological, technological and social systems, interactions between units give rise to intricate networks. These—typically non-trivial—structures, in turn, critically affect the dynamics and properties of the system. The focus of most current research on complex networks is, still, on global network properties. A caveat of this approach is that the relevance of global properties hinges on the premise that networks are homogeneous, whereas most real-world networks have a markedly modular structure. Here, we report that networks with different functions, including the Internet, metabolic, air transportation and protein interaction networks, have distinct patterns of connections among nodes with different roles, and that, as a consequence, complex networks can be classified into two distinct functional classes on the basis of their link type frequency. Importantly, we demonstrate that these structural features cannot be captured by means of often studied global properties.

Web 2.0 applications have attracted a considerable amount of attention because their open-ended nature allows users to create lightweight semantic scaffolding to organize and share content. To date, the interplay of the social and semantic components of social media has been only partially explored. Here we focus on Flickr and Last.fm, two social media systems in which we can relate the tagging activity of the users with an explicit representation of their social network. We show that a substantial level of local lexical and topical alignment is observable among users who lie close to each other in the social network. We introduce a null model that preserves user activity while removing local correlations, allowing us to disentangle the actual local alignment between users from statistical effects due to the assortative mixing of user activity and centrality in the social network. This analysis suggests that users with

Understanding complex systems often requires a bottom-up approach, breaking the system into small and elementary constituents and mapping out the interactions between these components. In many cases, the myriads of components and interactions are best characterized as networks. For example, the society is a network of people connected by various links, including friendships (Milgram, 1967), collaborationships (Kochen, 1989; Wasserman & Faust, 1994), sexual contacts (Liljeros et al., 2001) or scientific co-authorships (Redner, 1998; Newman, 2001). Electronic communication relies on two very di#erent networks: the physical network wiring the routers together (Internet) (Faloutsos, Faloutsos & Faloutsos, 1999; Vazquez, Pastor-Satorras & Vespignani, 2002) and the web of homepages linked by URLs (World Wide Web) (Albert, Jeong & Barabasi, 1999; Lawrence & Giles, 1999; Broder et al., 2000). Airline, cell-phone, power-grid or business networks represent further examples of compl