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 roots of Google's PageRank can be traced back to several early, and equally remarkable, ranking techniques. The network structure of a hyperlinked environment can be a rich source of information about the content of the environment, provided we have effective means for understanding it. We develop a set of algorithmic tools for extracting information from the link structures of such environments, and report on experiments that demonstrate their effectiveness in a variety of context on the World Wide Web. The central issue we address within our framework is the distillation of broad search topics, through the discovery of “authorative” information sources on such topics. We propose and test an algorithmic formulation of the notion of authority, based on the relationship between a set of relevant authoritative pages and the set of “hub pages” that join them together in the link structure. Our formulation has connections to the eigenvectors of certain matrices associated with the link graph; these connections in turn motivate additional heuristrics for link-based analysis. Recent work on modeling the Web graph has dwelt on capturing the degree distributions observed on the Web. Pointing out that this represents a heavy reliance on "local" properties of the Web graph, we study the distribution of PageRank values (used in the Google search engine) on the Web. This distribution is of independent interest in optimizing search indices and storage. We show that PageRank values on the Web follow a power law. We then develop detailed models for the Web graph that explain this observation, and moreover remain faithful to previously studied degree distributions. We analyze these models, and compare the analyses to both snapshots from the Web and to graphs generated by simulations on the new models. To our knowledge this represents the first modeling of the Web that goes beyond fitting degree distributions on the Web. 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. The importance of a Web page is an inherently subjective matter, which depends on the readers interests, knowledge and attitudes. But there is still much that can be said objectively about the relative importance of Web pages. This paper describes PageRank, a mathod for rating Web pages objectively and mechanically, effectively measuring the human interest and attention devoted to them. We compare PageRank to an idealized random Web surfer. We show how to efficiently compute PageRank for large numbers of pages. And, we show how to apply PageRank to search and to user navigation. Abstract Today, when searching for information on the World Wide Web, one usually performs a query through a term-based search engine. These engines return, as the query's result, a list of Web sites whose contents match the query. For broad topic queries, such searches often result in a huge set of retrieved documents, many of which are irrelevant to the user. However, much information is contained in the link-structure of the World Wide Web. Information such as which pages are linked to others can be used to augment search algorithms. In this context, Jon Kleinberg introduced the notion of two distinct types of Web sites: hubs and authorities . Kleinberg argued that hubs and authorities exhibit a mutually reinforcing relationship : a good hub will point to many authorities, and a good authority will be pointed at by many hubs. In light of this, he devised an algorithm aimed at finding authoritative sites. We present SALSA, a new stochastic approach for link structure analysis, which examines random walks on graphs derived from the link structure. We show that both SALSA and Kleinberg's mutual reinforcement approach employ the same meta-algorithm. We then prove that SALSA is equivalent to a weighted in-degree analysis of the link-structure of World Wide Web subgraphs, making it computationally more efficient than the mutual reinforcement approach. We compare the results of applying SALSA to the results derived through Kleinberg's approach. These comparisons reveal a topological phenomenon called the TKC effect (Tightly Knit Community) which, in certain cases, prevents the mutual reinforcement approach from identifying meaningful authorities.