PageRank equation and localization in the WWW

 

ALESSANDRO CHESSA

PHYSICS DEPARTMENT, UNIVERSITY OF CAGLIARI AND LIN

 

The growth and the development of the World Wide Web has created a huge ensemble of connections whose complexity can be fruitfully described and quantified by network theory. In this scenario the PageRank algorithm has been so far a fundamental tool in classifying the importance of web pages and in disentangling the web structure. In this work we show that the PageRank in a network can be represented as the solution of a differential equation discretized over a directed graph. We can formally define a wave function related to the PageRank value and a potential function connected to the topological features of the network. By exploiting a formal relationship with the time-independent Schrödinger equation it is possible to interpret hub formation and related phenomena as a wave-like localization process in the presence of disorder and trapping potentials. We show that the topological disorder given by the unbalance of outgoing and ingoing links between pages, induces wave function and potential structuring. This allows to directly localize the pages with the largest score. Through this new representation we can now compute the PageRank without iterative techniques. Our results also clarify the role of topology in the diffusion of information within complex networks. The result opens new perspectives in the physics of networks with interdisciplinary connections and opens the way to the employment of various mathematical techniques to the analysis of self-organization in structured systems. Applications are envisaged in the World-Wide Web, traffic, social and biological networks.