We show that complex (scale-free) network topologies 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 remarkable efficiency sustains even in highly dynamic networks. Our findings suggest that forwarding information through complex networks, such as the Internet, is possible without the overhead of existing routing protocols, and may also find practical applications in overlay networks for tasks such as application-level routing, information sharing, and data distribution.

The understanding of the immense and intricate topological structure of the World Wide Web (WWW) is a major scientific and technological challenge. This has been recently tackled by characterizing the properties of its representative graphs, in which vertices and directed edges are identified with Web pages and hyperlinks, respectively. Data gathered in large-scale crawls have been analyzed by several groups resulting in a general picture of the WWW that encompasses many of the complex properties typical of rapidly evolving networks. In this article, we report a detailed statistical analysis of the topological properties of four different WWW graphs obtained with different crawlers. We find that, despite the very large size of the samples, the statistical measures characterizing these graphs differ quantitatively, and in some cases qualitatively, depending on the domain analyzed and the crawl used for gathering the data. This spurs the issue of the presence of sampling biases and structural differences of Web crawls that might induce properties not representative of the actual global underlying graph. In short, the stability of the widely accepted statistical description of the Web is called into question. In order to provide a more accurate characterization of the Web graph, we study statistical measures beyond the degree distribution, such as degree-degree correlation functions or the statistics of reciprocal connections. The latter appears to enclose the relevant correlations of the WWW graph and carry most of the topological information of the Web. The analysis of this quantity is also of major interest in relation to the navigability and searchability of the Web.