  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
d-hyperbolicity. This generalized definition can be applied to graphs, which we
explore in this report. We provide strong evidence that 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 the 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 amplified by renormalization. This allows us to detect hyperbolicity in
large networks efficiently, through much smaller renormalized versions. These
observations indicate that d-hyperbolicity is a common feature of large-scale
networks. We propose that d-hyperbolicity in conjunction with other local
characteristics of networks, such as the degree distribution and clustering
coefficients, provide a more complete unifying picture of networks, and helps
classify in a parsimonious way what is otherwise a bewildering and complex
array of features and characteristics specific to each natural and man-made
network.
