Curvature of co-links uncovers hidden thematic layers in the World Wide Web

Beyond the information stored in pages of the World Wide Web, novel types of "meta-information" are created when pages connect to each other. Such meta-information is a collective effect of independent agents writing and linking pages, hidden from the casual user. Accessing it and understanding the interrelation between connectivity and content in the World Wide Web is a challenging problem [Botafogo, R. A. & Shneiderman, B. (1991) in Proceedings of Hypertext (Assoc. Comput. Mach., New York), pp. 63-77 and Albert, R. & Barabási, A.-L. (2002) Rev. Mod. Phys. 74, 47-97]. We demonstrate here how thematic relationships can be located precisely by looking only at the graph of hyperlinks, gleaning content and context from the Web without having to read what is in the pages. We begin by noting that reciprocal links (co-links) between pages signal a mutual recognition of authors and then focus on triangles containing such links, because triangles indicate a transitive relation. The importance of triangles is quantified by the clustering coefficient [Watts, D. J. & Strogatz, S. H. (1999) Nature (London) 393, 440-442], which we interpret as a curvature [Bridson, M. R. & Haefliger, A. (1999) Metric Spaces of Non-Positive Curvature (Springer, Berlin)]. This curvature defines a World Wide Web landscape whose connected regions of high curvature characterize a common topic. We show experimentally that reciprocity and curvature, when combined, accurately capture this meta-information for a wide variety of topics. As an example of future directions we analyze the neural network of Caenorhabditis elegans, using the same methods.