A model of influence in a social network

In this paper, we study a model of influence in a social network. It is assumed that each player has an inclination to say YES or NO which, due to influence of other players, may be different from the decision of the player. The point of departure here is the concept of the Hoede–Bakker index—the notion which computes the overall decisional ‘power’ of a player in a social network. The main drawback of the Hoede–Bakker index is that it hides the actual role of the influence function, analyzing only the final decision in terms of success and failure. In this paper, we separate the influence part from the group decision part, and focus on the description and analysis of the influence part. We propose among other descriptive tools a definition of a (weighted) influence index of a coalition upon an individual. Moreover, we consider different influence functions representative of commonly encountered situations. Finally, we propose a suitable definition of a modified decisional power.

[1]  Marc Roubens,et al.  Theory and Applications of Relational Structures as Knowledge Instruments II, International Workshops of COST Action 274, TARSKI, 2002-2005, Selected Revised Papers , 2006, Theory and Applications of Relational Structures as Knowledge Instruments.

[2]  Dan S. Felsenthal,et al.  Annexations and alliances: When are blocs advantageous a priori? , 2001, Soc. Choice Welf..

[3]  René van den Brink,et al.  Measuring domination in directed networks , 2000, Soc. Networks.

[4]  J. Deegan,et al.  A new index of power for simplen-person games , 1978 .

[5]  Michel Grabisch,et al.  Games on lattices, multichoice games and the shapley value: a new approach , 2007, Math. Methods Oper. Res..

[6]  G. Laan,et al.  Axiomatization of a class of share functions for n-person games , 1998 .

[7]  C. Hoede,et al.  A theory of decisional power , 1982 .

[8]  Agnieszka Rusinowska,et al.  The Not-Preference-Based Hoede-Bakker Index , 2007 .

[9]  Martin Shubik,et al.  A Method for Evaluating the Distribution of Power in a Committee System , 1954, American Political Science Review.

[10]  L. Penrose.  The Elementary Statistics of Majority Voting , 1946 .

[11]  A Banzhaf share function for cooperative games in coalition structure , 2002 .

[12]  Harrie C. M. de Swart,et al.  Generalizing and Modifying the Hoede-Bakker Index , 2006, Theory and Applications of Relational Structures as Knowledge Instruments.

[13]  D. Felsenthal,et al.  The Bicameral Postulates and Indices of a Priori Voting Power , 1998 .

[14]  G. Owen.  VALUES OF GAMES WITH A PRIORI UNIONS , 1977 .

[15]  D. Felsenthal,et al.  The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes , 1998 .

[16]  D. Rae,et al.  Decision-Rules and Individual Values in Constitutional Choice , 1969, American Political Science Review.

[17]  E. Packel,et al.  Power, luck and the right index , 1983 .

[18]  Leon A. Petrosjan,et al.  Game Theory and Applications , 2003 .

[19]  L. Shapley.  A Value for n-person Games , 1988 .

[20]  Annick Laruelle,et al.  Assessing success and decisiveness in voting situations , 2005, Soc. Choice Welf..

[21]  Dominique Lepelley,et al.  On the voting power of an alliance and the subsequent power of its members , 2007, Soc. Choice Welf..

[22]  B. Barry,et al.  Is it Better to Be Powerful or Lucky?: Part I , 1980 .

[23]  Ron Johnston,et al.  On the Measurement of Power: Some Reactions to Laver , 1978 .

[24]  Peter Borm,et al.  Digraph Competitions and Cooperative Games , 2002 .

[25]  Lloyd S. Shapley,et al.  On authority distributions in organizations: equilibrium , 2003, Games Econ. Behav..

[26]  Peter Borm,et al.  Characterizations of the β- and the Degree Network Power Measure , 2008 .

[27]  Pradeep Dubey,et al.  Mathematical Properties of the Banzhaf Power Index , 1979, Math. Oper. Res..

[28]  G. Laan,et al.  Axiomatizations of the normalized Banzhaf value and the Shapley value , 1998 .

[29]  J. Coleman.  Individual interests and collective action : selected essays , 1988 .

[30]  G. Owen.  Modification of the Banzhaf-Coleman Index for Games with a Priori Unions , 1981 .