Pagerank and opinion dynamics: missing links and extensions

The PageRank algorithm is a paradigm, originally introduced for ranking websites in the search engines results. It is based on the idea that a webpage, referred by highly ranked ("influential") webpages, should be also ranked high. Applications of the PageRank algorithms are not limited to web search and include e.g. scientometric journal ratings. More generally, PageRank is a powerful centrality measure, allowing to rank nodes of a general directed graph. It has been recently shown 1 that the classical PageRank algorithm is in fact dual to a special class of opinion dynamics introduced in 2. In this paper we clarify this duality and use it to develop an extension of the PageRank paradigm, leading to a novel wider class of centrality measures. We show that the convergence of the extended PageRank algorithm is equivalent to the stability of the "dual" opinion dynamics model. This result may help to reduce the gaps between computer sciences (algorithms for data analysis), social sciences (opinion dynamics) and systems and control theory (stability of dynamical systems) and thus belongs to the unified science that Norbert Wiener called cybernetics.