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.

[1]  Massimo Franceschet,et al.  PageRank , 2010, Commun. ACM.

[2]  Boris T. Polyak,et al.  Regularization-based solution of the PageRank problem for large matrices , 2012 .

[3]  Francesco Bullo,et al.  Opinion Dynamics and the Evolution of Social Power in Influence Networks , 2015, SIAM Rev..

[4]  John Scott.  What is social network analysis , 2010 .

[5]  Filip Radlinski,et al.  Query chains: learning to rank from implicit feedback , 2005, KDD '05.

[6]  John N. Tsitsiklis,et al.  On Krause's Multi-Agent Consensus Model With State-Dependent Connectivity , 2008, IEEE Transactions on Automatic Control.

[7]  Alexey S. Matveev,et al.  Opinion Dynamics in Social Networks With Hostile Camps: Consensus vs. Polarization , 2015, IEEE Transactions on Automatic Control.

[8]  Roberto Tempo,et al.  Distributed Randomized Algorithms for the PageRank Computation , 2010, IEEE Transactions on Automatic Control.

[9]  Stanley Milgram,et al.  An Experimental Study of the Small World Problem , 1969 .

[10]  Noah E. Friedkin,et al.  Two steps to obfuscation , 2014, Soc. Networks.

[11]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[12]  Noah E. Friedkin,et al.  The Problem of Social Control and Coordination of Complex Systems in Sociology: A Look at the Community Cleavage Problem , 2015, IEEE Control Systems.

[13]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[14]  M. Degroot.  Reaching a Consensus , 1974 .

[15]  Roberto Tempo,et al.  The PageRank Problem, Multiagent Consensus, and Web Aggregation: A Systems and Control Viewpoint , 2013, IEEE Control Systems.

[16]  Roberto Tempo,et al.  A new model of opinion dynamics for social actors with multiple interdependent attitudes and prejudices , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[17]  Tom A. B. Snijders,et al.  Social Network Analysis , 2011, International Encyclopedia of Statistical Science.

[18]  J. French,et al.  A formal theory of social power. , 1956, Psychological review.

[19]  Amy Nicole Langville,et al.  Google's PageRank and beyond - the science of search engine rankings , 2006 .

[20]  Chiara Ravazzi,et al.  Distributed randomized algorithms for opinion formation, centrality computation and power systems estimation: A tutorial overview , 2015, Eur. J. Control.

[21]  J. Wolfowitz.  Products of indecomposable, aperiodic, stochastic matrices , 1963 .

[22]  Randal W. Beard,et al.  Distributed Consensus in Multi-vehicle Cooperative Control - Theory and Applications , 2007, Communications and Control Engineering.

[23]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[24]  Sebastiano Vigna,et al.  PageRank: Functional dependencies , 2009, TOIS.

[25]  Asuman E. Ozdaglar,et al.  Opinion Fluctuations and Disagreement in Social Networks , 2010, Math. Oper. Res..

[26]  Sergey Brin,et al.  The Anatomy of a Large-Scale Hypertextual Web Search Engine , 1998, Comput. Networks.

[27]  Roberto Tempo,et al.  Distributed Algorithms for Computation of Centrality Measures in Complex Networks , 2015, IEEE Transactions on Automatic Control.

[28]  Norbert Wiener,et al.  The human use of human beings - cybernetics and society , 1988 .

[29]  Noah E. Friedkin,et al.  Theoretical Foundations for Centrality Measures , 1991, American Journal of Sociology.

[30]  Chiara Ravazzi,et al.  Ergodic Randomized Algorithms and Dynamics Over Networks , 2013, IEEE Transactions on Control of Network Systems.

[31]  Roberto Tempo,et al.  Novel Multidimensional Models of Opinion Dynamics in Social Networks , 2015, IEEE Transactions on Automatic Control.

[32]  Claudio Altafini,et al.  Consensus Problems on Networks With Antagonistic Interactions , 2013, IEEE Transactions on Automatic Control.

[33]  Phillip Bonacich,et al.  Eigenvector-like measures of centrality for asymmetric relations , 2001, Soc. Networks.

[34]  M. Macy,et al.  Small Worlds and Cultural Polarization , 2011 .

[35]  F. Fagnani,et al.  Scaling limits for continuous opinion dynamics systems , 2009 .