PageRank Induced Topology for Real-World Networks

The discovery that real-world large networks from many different domains (sociology, biology, computer science. . . ) share the same characteristics has raised an interest in their studying ([17, 2, 15]). The associated graphs of such networks are rather sparse (the mean degree stays roughly constant when the number of nodes increases), highly clustered, and there exists short paths that can be found [11, 3]. An hierarchical structure is also revealed by a heavy tail distribution for most parameters[15]. Referring to Milgram’s experiment[14], Watts and Strogatz proposed to call highly clustered graphs with low diameter small worlds [18]. In Section 2 we describe how, by taking into account distributions of random walks in a graph G as coordinates of its n nodes in R, one can fit G with a geometrical structure. This method allows to analyze precisely and efficiently the structure of small-world-shaped very large graphs. Section 3 outlines the limits of this approach, which requires reflexive and symmetric graphs to produce relevant results. We propose in Section 4 to fit PageRank computation techniques in order to generalize the random walk method described in Section 2 to all graphs without any specific restriction. The adaptability of this technique allows us, for instance, to consider the Web graph, which is neither reflexive nor symmetric.