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We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the… (More)

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… (More)

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

We propose a class of models of social network formation based on a mathematical abstraction of the concept of social distance. Social distance attachment is represented by the tendency of peers to establish acquaintances via a decreasing function of the relative distance in a representative social space. We derive analytical results (corroborated by… (More)

—We show that complex (scale-free) network topolo-gies naturally emerge from hyperbolic metric spaces. Hyperbolic geometry facilitates maximally efficient greedy forwarding in these networks. Greedy forwarding is topology-oblivious. Nevertheless , greedy packets find their destinations with 100% probability following almost optimal shortest paths. This… (More)

We study a class of models of correlated random networks in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices. We find analytical expressions for the main topological properties of these models as a function of the distribution of hidden variables and the probability of connecting vertices.… (More)

PageRank has become a key element in the success of search engines, allowing to rank the most important hits in the top screen of results. One key aspect that distinguishes PageR-ank from other prestige measures such as in-degree is its global nature. From the information provider perspective, this makes it difficult or impossible to predict how their pages… (More)

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… (More)

Uncorrelated random scale-free networks are useful null models to check the accuracy and the analytical solutions of dynamical processes defined on complex networks. We propose and analyze a model capable of generating random uncorrelated scale-free networks with no multiple and self-connections. The model is based on the classical configuration model, with… (More)

We demonstrate that the self-similarity of some scale-free networks with respect to a simple degree-thresholding renormalization scheme finds a natural interpretation in the assumption that network nodes exist in hidden metric spaces. Clustering, i.e., cycles of length three, plays a crucial role in this framework as a topological reflection of the triangle… (More)