nwlocal - Local Algorithms for Finding Interesting...

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Local Algorithms for Finding Interesting Individuals in Large Networks Mickey Brautbar Michael Kearns Computer and Information Science, University of Pennsylvania, Philadelphia, 19104, USA brautbar@cis.upenn.edu mkearns@cis.upenn.edu Abstract: We initiate the study of local, sublinear time algorithms for finding vertices with extreme topological properties — such as high degree or clustering coefficient — in large social or other net- works. We introduce a new model, called the Jump and Crawl model, in which algorithms are permitted only two graph operations. The Jump operation returns a randomly chosen vertex, and is meant to model the ability to discover “new” vertices via keyword search in the Web, shared hobbies or interests in so- cial networks such as Facebook, and other mechanisms that may return vertices that are distant from all those currently known. The Crawl operation permits an algorithm to explore the neighbors of any currently known vertex, and has clear analogous in many modern networks. We give both upper and lower bounds in the Jump and Crawl model for the problems of finding vertices of high degree and high clustering coefficient. We consider both arbitrary graphs, and specializations in which some common assumptions are made on the global topology (such as power law degree distri- butions or generation via preferential attachment). We also examine local algorithms for some related vertex or graph properties, and discuss areas for future investigation. Keywords: social networks; graph theory; algorithms 1 Introduction The proliferation of very large social and tech- nological networks over the last decade or so — and the attendant scientific and cultural interest they have attracted — has led to the documentation of certain local topological properties that are now believed to be quite common. Perhaps beginning with earlier sociological interest in global struc- ture such as the “six degrees” phenomenon (small diameter) and structural holes, recent research has further identified local topological properties, such as individuals with extraordinarily high degree (sometimes dubbed “connectors” or “hubs”), lo- cal neighborhoods with a high degree of clustering (fraction of edges present) compared to the over- all edge density, vertices of high “centrality” for various definitions of that term, and so on. There is now a compelling dialogue in the literature be- tween empirical works documenting and refining these various notions of extreme individuals and neighborhoods, and theoretical works attempting to explain their persistent emergence via genera- tive models for network formation [11]. Given the presence of such “interesting” indi- viduals in large networks, how would we actually find them — especially considering that for many such networks (including the Web, or for non- employee researchers of online social networks such as Facebook), there may not exist an acces- sible, centralized description of the network? This
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nwlocal - Local Algorithms for Finding Interesting...

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