On the Hyperbolicity of Large-Scale Networks
Through detailed analysis of scores of publicly available data sets corresponding to a wide range of large-scale networks, from communication and road networks to various forms of social networks, we explore a little-studied geometric characteristic of real-life networks, namely their hyperbolicity. In smooth geometry, hyperbolicity captures the notion of negative curvature; within the more abstract context of metric spaces, it can be generalized as -hyperbolicity or negative curvature in the large. This generalized denition can be applied to graphs, which we explore in this report. We provide strong evidence that large-scale communication and social networks exhibit this fundamental property, and through extensive computations we quantify the degree of hyperbolicity of each network in comparison to its diameter. By contrast, and as evidence of the validity of the methodology, applying the same methods to graphs of road networks shows that they are not hyperbolic, which is as expected. Finally, we present practical computational means for detection of hyperbolicity and show how the test itself may be scaled to much larger graphs than those we examined via renormalization group methodology. Using well-understood mechanisms, we provide evidence through synthetically generated graphs that hyperbolicity is preserved and indeed amplied by renormalization. This allows us to detect hyperbolicity in large networks eciently, through much smaller renormalized versions. These observations indicate that -hyperbolicity is a common feature of large-scale networks, from IP-layer connectivity to citation, collaboration, co-authorship, and friendship graphs. We propose that -hyperbolicity in conjunction with other local characteristics of networks, such as the degree distribution and clustering coecients,