J. Aracena, J. Demongeot, and E. Goles, On limit cycles of monotone functions with symmetric connection graph, Theoretical Computer Science, vol.322, issue.2, pp.237-244, 2004.
DOI : 10.1016/j.tcs.2004.03.010

C. Asavathiratham, The influence model, IEEE Control Systems Magazine, vol.21, issue.6, 2000.
DOI : 10.1109/37.969135

V. Bala and S. Goyal, Conformism and diversity under social learning, Economic Theory, vol.17, issue.1, pp.101-120, 2001.
DOI : 10.1007/PL00004094

V. Bala and S. Goyal, Learning from Neighbours, Review of Economic Studies, vol.65, issue.3, pp.595-621, 1998.
DOI : 10.1111/1467-937X.00059

A. V. Banerjee, A Simple Model of Herd Behavior, The Quarterly Journal of Economics, vol.107, issue.3, pp.797-817, 1992.
DOI : 10.2307/2118364

A. V. Banerjee and D. Fudenberg, Word-of-mouth learning, Games and Economic Behavior, vol.46, issue.1, pp.1-22, 2004.
DOI : 10.1016/S0899-8256(03)00048-4

R. L. Berger, A Necessary and Sufficient Condition for Reaching a Consensus Using DeGroot's Method, Journal of the American Statistical Association, vol.30, issue.374, pp.415-419, 1981.
DOI : 10.1080/01621459.1981.10477662

S. Broadbent and J. Hammersley, Percolation processes I. Crystals and mazes, Proceedings of the Cambridge Philosophical Society, pp.629-641, 1957.

A. Calvó-armengol and M. O. Jackson, Like Father, Like Son: Social Network Externalities and Parent-Child Correlation in Behavior, American Economic Journal: Microeconomics, vol.1, issue.1, pp.124-150, 2009.
DOI : 10.1257/mic.1.1.124

B. Celen and S. Kariv, Distinguishing Informational Cascades from Herd Behavior in the Laboratory, American Economic Review, vol.94, issue.3, pp.484-497, 2004.
DOI : 10.1257/0002828041464461

M. H. Degroot, Reaching a Consensus, Journal of the American Statistical Association, vol.38, issue.345, pp.118-121, 1974.
DOI : 10.1287/mnsc.15.2.B61

P. Demarzo, D. Vayanos, and J. Zwiebel, Persuasion Bias, Social Influence, and Unidimensional Opinions, The Quarterly Journal of Economics, vol.118, issue.3, pp.909-968, 2003.
DOI : 10.1162/00335530360698469

G. Ellison, Learning, Local Interaction, and Coordination, Econometrica, vol.61, issue.5, pp.1047-1072, 1993.
DOI : 10.2307/2951493

G. Ellison and D. Fudenberg, Rules of Thumb for Social Learning, Journal of Political Economy, vol.101, issue.4, pp.612-643, 1993.
DOI : 10.1086/261890

G. Ellison and D. Fudenberg, Word-of-Mouth Communication and Social Learning, The Quarterly Journal of Economics, vol.110, issue.1, pp.93-125, 1995.
DOI : 10.2307/2118512

N. E. Friedkin and E. C. Johnsen, Social influence and opinions, The Journal of Mathematical Sociology, vol.2, issue.3-4, pp.193-206, 1990.
DOI : 10.2307/2089854

N. E. Friedkin and E. C. Johnsen, Social positions in influence networks, Social Networks, vol.19, issue.3, pp.209-222, 1997.
DOI : 10.1016/S0378-8733(96)00298-5

D. Gale and S. Kariv, Bayesian learning in social networks, Games and Economic Behavior, vol.45, issue.2, pp.329-346, 2003.
DOI : 10.1016/S0899-8256(03)00144-1

B. Golub and M. O. Jackson, Na??ve Learning in Social Networks and the Wisdom of Crowds, American Economic Journal: Microeconomics, vol.2, issue.1, pp.112-149, 2010.
DOI : 10.1257/mic.2.1.112

M. Grabisch and A. Rusinowska, A model of influence in a social network, Theory and Decision, vol.53, issue.1, pp.69-96, 2010.
DOI : 10.1007/s11238-008-9109-z
URL : https://hal.archives-ouvertes.fr/halshs-00344457

M. Grabisch and A. Rusinowska, Measuring influence in command games, Social Choice and Welfare, vol.48, issue.2, pp.177-209, 2009.
DOI : 10.1007/s00355-008-0350-8
URL : https://hal.archives-ouvertes.fr/halshs-00269084

M. Grabisch and A. Rusinowska, Influence functions, followers and command games, Games and Economic Behavior, vol.72, issue.1, pp.123-138, 2011.
DOI : 10.1016/j.geb.2010.06.003
URL : https://hal.archives-ouvertes.fr/halshs-00344823

M. Grabisch and A. Rusinowska, A model of influence with an ordered set of possible actions, Theory and Decision, vol.64, issue.4, pp.635-656, 2010.
DOI : 10.1007/s11238-009-9150-6
URL : https://hal.archives-ouvertes.fr/hal-00519413

M. Grabisch and A. Rusinowska, A model of influence with a continuum of actions, Journal of Mathematical Economics, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00666821

M. Grabisch and A. Rusinowska, Different approaches to influence based on social networks and simple games Collective Decision Making: Views from Social Choice and Game Theory, Series Theory and Decision Library C, pp.185-209, 2010.

M. Grabisch, J. Marichal, R. Mesiar, and E. Pap, Aggregation Functions. Number 127 in Encyclopedia of Mathematics and its Applications, 2009.
DOI : 10.1017/cbo9781139644150
URL : https://hal.archives-ouvertes.fr/halshs-00445120

C. Hoede and R. Bakker, A theory of decisional power, The Journal of Mathematical Sociology, vol.48, issue.2, pp.309-322, 1982.
DOI : 10.1080/0022250X.1982.9989927

X. Hu and L. S. Shapley, On authority distributions in organizations: equilibrium, Games and Economic Behavior, vol.45, issue.1, pp.132-152, 2003.
DOI : 10.1016/S0899-8256(03)00130-1

X. Hu and L. S. Shapley, On authority distributions in organizations: controls, Games and Economic Behavior, vol.45, issue.1, pp.153-170, 2003.
DOI : 10.1016/S0899-8256(03)00023-X

E. Ising, Beitrag zur Theorie des Ferromagnetismus, Zeitschrift f??r Physik, vol.6, issue.4, pp.253-258, 1925.
DOI : 10.1007/BF02980577

M. O. Jackson, The Economics of Social Networks, 2008.
DOI : 10.1017/CBO9781139052269.003

R. L. Keeney and H. Raiffa, Decision with Multiple Objectives, 1976.

H. Kesten, Percolation Theory for Mathematicians, 1982.
DOI : 10.1007/978-1-4899-2730-9

A. Kirman, Ants, Rationality, and Recruitment, The Quarterly Journal of Economics, vol.108, issue.1, pp.137-156, 1993.
DOI : 10.2307/2118498
URL : http://qje.oxfordjournals.org/cgi/content/short/108/1/137

A. Kirman, C. Oddou, and S. Weber, Stochastic Communication and Coalition Formation, Econometrica, vol.54, issue.1, pp.129-138, 1986.
DOI : 10.2307/1914161

U. Krause, A discrete nonlinear and nonautonomous model of consensus formation, Communications in Difference Equations. Amsterdam: Gordon and Breach, 2000.

W. Lenz, Beiträge zum verständnis der magnetischen eigenschaften in festen körpern, Physikalische Zeitschrift, vol.21, pp.613-615, 1920.

D. López-pintado, Diffusion in complex social networks, Games and Economic Behavior, vol.62, issue.2, pp.573-590, 2008.
DOI : 10.1016/j.geb.2007.08.001

D. López-pintado, Influence networks, Games and Economic Behavior, vol.75, issue.2, 2010.
DOI : 10.1016/j.geb.2012.01.008

D. López-pintado and D. J. Watts, Social Influence, Binary Decisions and Collective Dynamics, Rationality and Society, vol.20, issue.4, pp.399-443, 2008.
DOI : 10.1177/1043463108096787

J. Lorenz, A stabilization theorem for dynamics of continuous opinions, Physica A: Statistical Mechanics and its Applications, vol.355, issue.1, pp.217-223, 2005.
DOI : 10.1016/j.physa.2005.02.086

M. Mueller-frank, A Fixed Point Convergence Theorem in Euclidean Spaces and its Application to Non-Bayesian Learning in Social Networks, SSRN Electronic Journal, 2010.
DOI : 10.2139/ssrn.1690925

R. B. Potts, Some generalized order-disorder transformations, Mathematical Proceedings, pp.106-109, 1952.
DOI : 10.1103/PhysRev.60.252

S. *. Lemma-5 and S. *. , Suppose that C is a regular terminal class, with lower and upper bounds Then A i cannot have an influential player j ? N \ S * whenever i ? S *

?. Suppose-i, ?. N. Take, and S. ?. , This is always possible since necessarily one of the S k 's does not contain j (for if j ? S k for all k = 1, . . . , p, then j ? S * ) This implies 1 > A i (1 N \j ) ? A i, S ? ), which contradicts (*)

A. Suppose-now-that-i-?-s-*, ?. *. Take, and S. ?. , Again this is always possible since j ? S * implies j ? S k ? K k for some k. Then A i (1 S ? ) ? A i (1 j ) > 0

S. *. , S. *. , S. *. , S. ?. , ?. S. et al., one can obtain a sufficient condition (B) for forbidding any normal regular terminal class not reduced to the trivial terminal classes. Condition (B): for any, By applying the above lemma to every pair

S. *. , S. *. Such-that-i-?-s-*, and ?. , which is in fact Lemma 5: j influential in A i rules out every regular class with lower and upper bounds, Proof of Theorem 3 We consider the following rule R ??arc going into S * ) or i ? S * , j ? S * (arc outgoing from S * )

S. *. Sufficiency, S. *. , and I. E. , with 1 ? |S * | < n ? 1 and 2 ? |S * | < n. If S * has an ingoing arc, it is ruled out by R ?? . If it has no ingoing arc, then by hypothesis every node outside S * is linked to S * by a path. Since N \ S * is always nonempty, taking any node i in N \ S * , there is a path from S * to i

S. ?. Sufficiency:-take, ?. S. , ?. , ?. S. , S. et al., Then A i (1 S ) ? A i (1 N \S ) < 1, hence S is not a terminal class by Theorem 1 (ii) Now, suppose that S Then 0 < A i (1 S ? ) ? A i (1 S ), which implies that S is not a terminal class. Necessity: Suppose that S is not a terminal state. Then by Theorem 1 (ii), either there is some i ? S such that A i (1 S ) < 1 or some i ? S such that A i (1 S ) > 0. Suppose that for some

A. Also and S. ??-?-s-?, exhaustivity of A i that A i (1 S ? ) = 1, a contradiction with the definition of S ? . Therefore, A i (1 N \S ? ) > 0. Finally, we have for all S ?? ) = 1, and the claim is proved. Suppose now that for some i ? S, A i (1 S ) > 0. Take the smallest S ? ? S such that A i (1 S ? ) > 0. Then S ? is influential for i. Indeed, A i (1 S ? ) > 0, and by exhaustivity of A i we must have A i

S. ?. Sufficiency:-take, ?. S. , S. ?. , and S. ?. , Then A i (1 S ) ? A i (1 N \S ? ) < 1, hence [S, S ? K] cannot be a Boolean terminal class, for any K. Now, suppose that