The neighbourhood function N(t) of a graph G gives, for each t, the number of pairs of nodes such that y is reachable from x in less that t hops. The neighbourhood function provides a wealth of information about the graph (e.g., it easily allows one to compute its diameter), but it is very expensive to compute it exactly. Recently, the ANF algorithm (approximate neighbourhood function) has been proposed with the purpose of approximating NG(t) on large graphs. We describe a breakthrough improvement over ANF in terms of speed and scalability. Our algorithm, called HyperANF, uses the new HyperLogLog counters and combines them efficiently through broadword programming; our implementation uses overdecomposition to exploit multi-core parallelism. With HyperANF, for the first time we can compute in a few hours the neighbourhood function of graphs with billions of nodes with a small error and good confidence using a standard workstation. Then, we turn to the study of the distribution of the shortest paths between reachable nodes (that can be efficiently approximated by means of HyperANF), and discover the surprising fact that its index of dispersion provides a clear-cut characterisation of proper social networks vs. web graphs. We thus propose the spid (Shortest-Paths Index of Dispersion) of a graph as a new, informative statistics that is able to discriminate between the above two types of graphs. We believe this is the first proposal of a significant new non-local structural index for complex networks whose computation is highly scalable.
Given large, multimillion-node graphs (e.g., Facebook, Web-crawls, etc.), how do they evolve over time? How are they connected? What are the central nodes and the outliers? In this article we define the Radius plot of a graph and show how it can answer these questions. However, computing the Radius plot is prohibitively expensive for graphs reaching the planetary scale. There are two major contributions in this article: (a) We propose HADI (HAdoop DIameter and radii estimator), a carefully designed and fine-tuned algorithm to compute the radii and the diameter of massive graphs, that runs on the top of the Hadoop MapReduce system, with excellent scale-up on the number of available machines (b) We run HADI on several real world datasets including YahooWeb (6B edges, 1 8 of a Terabyte), one of the largest public graphs ever analyzed. Thanks to HADI, we report fascinating patterns on large networks, like the surprisingly small effective diameter, the multimodal bimodal shape of the Radius plot, and its palindrome motion over time.