Patterns often appear in a variety of large, real-world networks, and interesting physical phenomena are often explained by network topology as in the case of the bow-tie structure of the World Wide Web, or the small world phenomenon in social networks. The discovery and modelling of such regular patterns has a wide application from disease propagation to financial markets. In this work we describe a newly discovered regularly occurring striation pattern found in the PageRank ordering of adjacency matrices that encode real-world networks. We demonstrate that these striations are the result of well-known graph generation processes resulting in regularities that are manifest in the typical neighborhood distribution. The spectral view explored in this paper encodes a tremendous amount about the explicit and implicit topology of a given network, so we also discuss the interesting network properties, outliers and anomalies that a viewer can determine from a brief look at the re-ordered matrix.
The origin of PageRank was rooted in the intent to rank web pages based on their link topology @cite_5 . Although there are alternative link topology based algorithms such as HITS @cite_41 and the SALSA @cite_15 (which combines PageRank and HITS), PageRank enjoys a brand recognition due to its early integration into and association with the Google search engine @cite_37 . Many studies have examined the type and quality of output of the PageRank algorithm. Page al advised that @math based on empirical evidence @cite_5 , and Bechetti and Castillo further showed that PageRank does not fit a power-law distribution for extreme @math values @cite_23 . Pandurangan al showed that the PageRank values of the web follow a power law @cite_6 , and Volkovich al showed the correlation between various parameters of the network (in-degree, out-degree, and percentage of dangling nodes) and the overall log-log shape of the PageRank plot @cite_40 .
@cite_37 @cite_41 @cite_6 @cite_40 @cite_23 @cite_5 @cite_15
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The concept of pagerank was first started as a way for determining the ranking of Web pages by Web search engines. Based on relations in interconnected networks, pagerank has become a major tool for addressing fundamental problems arising in general graphs, especially for large information networks with hundreds of thousands of nodes. A notable notion of pagerank, introduced by Brin and Page and denoted by PageRank, is based on random walks as a geometric sum. In this paper, we consider a notion of pagerank that is based on the (discrete) heat kernel and can be expressed as an exponential sum of random walks. The heat kernel satisfies the heat equation and can be used to analyze many useful properties of random walks in a graph. A local Cheeger inequality is established, which implies that, by focusing on cuts determined by linear orderings of vertices using the heat kernel pageranks, the resulting partition is within a quadratic factor of the optimum. This is true, even if we restrict the volume of the small part separated by the cut to be close to some specified target value. This leads to a graph partitioning algorithm for which the running time is proportional to the size of the targeted volume (instead of the size of the whole graph). We study the relation between PageRank and other parameters of information networks such as in-degree, out-degree, and the fraction of dangling nodes. We model this relation through a stochastic equation inspired by the original definition of PageRank. Further, we use the theory of regular variation to prove that PageRank and in-degree follow power laws with the same exponent. The difference between these two power laws is in a multiplicative constant, which depends mainly on the fraction of dangling nodes, average in-degree, the power law exponent, and the damping factor. The out-degree distribution has a minor effect, which we explicitly quantify. Finally, we propose a ranking scheme which does not depend on out-degrees. We show that the empirical distribution of the PageRank values in a large set of Web pages does not follow a power-law except for some particular choices of the damping factor. We argue that for a graph with an in-degree distribution following a power-law with exponent between 2.1 and 2.2, choosing a damping factor around 0.85 for PageRank yields a power-law distribution of its values. We suggest that power-law distributions of PageRank in Web graphs have been observed because the typical damping factor used in practice is between 0.85 and 0.90. Since the publication of Brin and Page's paper on PageRank, many in the Web community have depended on PageRank for the static (query-independent) ordering of Web pages. We show that we can significantly outperform PageRank using features that are independent of the link structure of the Web. We gain a further boost in accuracy by using data on the frequency at which users visit Web pages. We use RankNet, a ranking machine learning algorithm, to combine these and other static features based on anchor text and domain characteristics. The resulting model achieves a static ranking pairwise accuracy of 67.3 (vs. 56.7 for PageRank or 50 for random).