We provide evidence that networks representing social interactions, as measured and archived by electronic communication systems, such as friendship, collaboration, co-authorship, web, peer-to-peer and other kindred networks, exhibit strong hyperbolicity in the sense of Gromov (4-point delta being much smaller than the graph diameter). We outline how one may exploit hyperbolicity to reduce computations on such graphs massively. As an example, we show that all-pair distances on a (mobile call) graph with E=100s of millions of edges may be approximated in O(E) time (i.e., in tens of minutes) with small average error (well below 10% additive error), far smaller than anticipated by theory. We speculate that this result is likely due to the uncharacteristically small hyperbolic cores of these networks (i.e., the set of nodes whose betweenness centrality scales like k*N^2 with k~0.5 as N = the node size of the graph goes to infinity).