Statistical Physics of Complex Networks.

We live in a connected world. It is of great practical importance and intellectual appeal to understand the networks surrounding us. In this work we study ranking of the nodes in complex networks. In large networks such as World Wide Web (WWW) and citation networks of scientific literature, searching by keywords is a common practice to retrieve useful information. On the WWW, apart from the contents of webpages, the topology of the network itself can be a rich source of information about their relative importance and relevancy to the search query. It is the effective utilization of this topological information [50] which advanced the Google search engine to its present position of the most popular tool on the WWW. The World-Wide Web (WWW) is characterized by a strong community structure in which communities of webpages are densely interconnected by hyperlinks. We study how such network architecture affects the average Google ranking of individual webpages in the community. Using a mean-field approximation, we quantify how the average Google rank of community's webpages depends on the degree to which it is isolated from the rest of the world in both incoming and outgoing directions, and alpha -- the only intrinsic parameter of Google's PageRank algorithm. We proceed with numerical study of simulated networks and empirical study of several internal web-communities within two US universities. The predictions of our mean-field treatment were qualitatively verified in those real-life networks. Furthermore, the value alpha = 0.15 used by Google seems to be optimized for the degree of isolation of communities as they exist in the actual WWW. We then extend Google's PageRank algorithm to citation networks of scientific literature. Unlike hyperlinks, citations cannot be updated after the point of publication. This results in strong aging characteristics of citation networks that affect the performance of the PageRank algorithm. To rectify this we modify the PageRank algorithm to a new ranking method, CiteRank, in which the starting point of random surfers is exponentially biased towards more recent publications. The ranking results are compared for two rather different citation networks: all American Physical Society publications between 1893 and 2003 and the set of high energy physics theory (hep-th) preprints. Despite major differences between these two networks, we find that their optimal parameters of the CiteRank algorithm are remarkably similar.