American Institute of Mathematical Sciences

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March  2017, 12(1): 59-92. doi: 10.3934/nhm.2017003

Modelling heterogeneity and an open-mindedness social norm in opinion dynamics

Clinton Innes , Razvan C. Fetecau and  Ralf W. Wittenberg

Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC V5A 1S6, Canada

Received  December 2015 Revised  April 2016 Published  February 2017

We study heterogeneous interactions in a time-continuous bounded confidence model for opinion formation. The key new modelling aspects are to distinguish between open-minded and closed-minded behaviour and to include an open-mindedness social norm. The investigations focus on the equilibria supported by the proposed new model; particular attention is given to a novel class of equilibria consisting of multiple connected opinion clusters, which does not occur in the absence of heterogeneity. Various rigorous stability results concerning these equilibria are established. We also incorporate the effect of media in the model and study its implications for opinion formation.

Keywords: Opinion dynamics, bounded confidence, heterogeneous societies, group connectivity, opinion equilibria.
Mathematics Subject Classification: Primary: 91D30, 37N99; Secondary: 70F99.
Citation: Clinton Innes, Razvan C. Fetecau, Ralf W. Wittenberg. Modelling heterogeneity and an open-mindedness social norm in opinion dynamics. Networks and Heterogeneous Media, 2017, 12 (1) : 59-92. doi: 10.3934/nhm.2017003
References:
[1]

R. P. Abelson, Mathematical models of the distribution of attitudes under controversy, in Contributions to Mathematical Psychology (eds. N. Frederiksen and H. Gulliksen), Holt, Reinhart & Winston, New York, 1964, 142-160. doi: 10.1007/978-88-470-1766-5.

[2]

F. Abergel, A. Chakraborti, B.K. Chakrabarti and M. Mitra (eds.), Econophysics of Order-driven Markets, Springer-Verlag, Milan, 2011. doi: 10.1007/978-88-470-1766-5.

[3]

K. Arceneaux and M. Johnson, Does media fragmentation produce mass polarization? selective exposure and a new era of minimal effects, APSA Annual Meeting Paper.

[4]

R. Axelrod, The dissemination of culture: A model with local convergence and global polarization, Journal of Conflict Resolution, 41 (1997), 203-226.  doi: 10.1177/0022002797041002001.

[5]

E. Ben-Naim, Opinion dynamics: Rise and fall of political parties, Europhys. Lett., 69 (2005), 671-677.  doi: 10.1209/epl/i2004-10421-1.

[6]

E. Berscheid, Opinion change and communicator-communicatee similarity and dissimilarity, Journal of Personality and Social Psychology, 4 (1966), 670-680.  doi: 10.1037/h0021193.

[7]

L. Boudin, A. Mercier and F. Salvarani, Conciliatory and contradictory dynamics in opinion formation, Physica A, 391 (2012), 5672-5684.  doi: 10.1016/j.physa.2012.05.070.

[8]

L. Boudin, R.Monaco and F. Salvarani, Kinetic model for multidimensional opinion formation, Physical Review E, 81 (2010), 036109, 9pp. doi: 10.1103/PhysRevE.81.036109.

[9]

J.W. Brehm and D. Lipsher, Communicator-communicatee discrepancy and perceived communicator trustworthiness, Journal of Personality, 27 (1959), 352-361. 

[10]

S. Camazine, J. -L. Deneubourg, N.R. Franks, J. Sneyd, G. Theraulaz and E. Bonabeau, Self-organization in Biological Systems, Princeton Studies in Complexity, Princeton University Press, Princeton, NJ, 2003.

[11]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[12]

G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents, Advances in Complex Systems, 3 (2000), 87-98.  doi: 10.1142/S0219525900000078.

[13]

G. Deffuant, F. Amblard, G. Weisbuch and T.Faure, How can extremism prevail? A study based on the relative agreement interaction model, Journal of Artificial Societies and Social Simulation, 5.

[14]

M.H. DeGroot, Reaching a consensus, Journal of the American Statistical Association, 69 (1974), 118-121.  doi: 10.1080/01621459.1974.10480137.

[15]

W. Doise, Intergroup relations and polarization of individual and collective judgments, Journal of Personality and Social Psychology, 12 (1969), 136-143.  doi: 10.1037/h0027571.

[16]

B. Düring, P. Markowich, J.-F. Pietschmann and M.-T. Wolfram, Boltzmann and Fokker—Planck equations modelling opinion formation in the presence of strong leaders, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 465 (2009), 3687-3708.  doi: 10.1098/rspa.2009.0239.

[17]

L. Festinger, A theory of social comparison processes, Human Relations, 7 (1954), 117-140.  doi: 10.1177/001872675400700202.

[18]

L. Festinger, A Theory of Cognitive Dissonance, Stanford University Press, Stanford, USA, 1957.

[19]

N. Friedkin and E. Johnsen, Social influence and opinions, Journal of Mathematical Sociology, 15 (1990), 193-206.  doi: 10.1080/0022250X.1990.9990069.

[20]

G. Fu, W. Zhang and Z. Li, Opinion dynamics of modified Hegselmann-Krause model in a group-based population with heterogeneous bounded confidence, Physica A, 419 (2015), 558-565.  doi: 10.1016/j.physa.2014.10.045.

[21]

S. Galam and S. Moscovici, Towards a theory of collective phenomena: Consensus and attitude changes in groups, European Journal of Social Psychology, 21 (1991), 49-74.  doi: 10.1002/ejsp.2420210105.

[22]

G. H. Golub and C.F. VanLoan, Matrix Computations, 3rd edition, Johns Hopkins University Press, Baltimore, MD, USA, 1996.

[23]

P. Groeber, J. Lorenz and F. Schweitzer, Dissonance minimization as a microfoundation of social influence in models of opinion formation, Journal of Mathematical Sociology, 38 (2014), 147-174.  doi: 10.1080/0022250X.2012.724486.

[24]

J. T. Hamilton, All the News That's Fit to Sell: How the Market Transforms Information into News, Princeton University Press, Princeton, NJ, 2006.

[25]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: Models, analysis and simulation, Journal of Artificial Societies and Social Simulation, 5.

[26]

R. Hegselmann and U. Krause, Opinion dynamics under the influence of radical groups, charismatic leaders, and other constant signals: A simple unifying model, Networks and Heterogenous Media, 10 (2015), 477-509.  doi: 10.3934/nhm.2015.10.477.

[27]

M.A. Hogg, J.C. Turner and B. Davidson, Polarized norms and social frames of reference: A test of the self categorization theory of group polarization, Basic and Applied Psychology, 11 (1990), 77-100.  doi: 10.1207/s15324834basp1101_6.

[28]

C. Innes, Quantifying the Effect of Open-Mindedness on Opinion Dynamics and Advertising Optimization, Master's thesis, Simon Fraser University, 2014.

[29]

A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd edition, Cambridge Texts in Applied Mathematics, Cambridge University Press, New York, 2009.

[30]

P.-E. Jabin and S. Motsch, Clustering and asymptotic behavior in opinion formation, Journal of Differential Equations, 257 (2014), 4165-4187.  doi: 10.1016/j.jde.2014.08.005.

[31]

D. Jones, K. Ferraiolo and J. Byrne, Selective media exposure and partisan differences about {S}arah Palin's candidacy, Politics and Policy, 39 (2011), 195-221.  doi: 10.1111/j.1747-1346.2011.00288.x.

[32]

G. Kou, Y. Zhao, Y. Peng and Y. Shi, Multi-level opinion dynamics under bounded confidence PLoS ONE, 7 (2012), e43507. doi: 10.1371/journal.pone.0043507.

[33]

M.S. Levendusky, Why do partisan media polarize viewers?, American Journal of Political Science, 57 (2013), 611-623.  doi: 10.1111/ajps.12008.

[34]

H. Liang, Y. Yang and X. Wang, Opinion dynamics in networks with heterogeneous confidence and influence, Physica A, 392 (2013), 2248-2256.  doi: 10.1016/j.physa.2013.01.008.

[35]

J. Lorenz, Heterogeneous bounds of confidence: Meet, discuss and find consensus!, Complexity, 15 (2010), 43-52.  doi: 10.1002/cplx.20295.

[36]

J. Lorenz, Continuous opinion dynamics under bounded confidence: A survey, International Journal of Modern Physics C, 18 (2007), 1819-1838.  doi: 10.1142/S0129183107011789.

[37]

T. V. Martins, M. Pineda and R. Toral, Mass media and repulsive interactions in continuous-opinion dynamics, Europhysics Letters, 91 (2010), 48003. doi: 10.1209/0295-5075/91/48003.

[38]

J. -D. Mathias, S. Huet and G. Deffuant, Bounded confidence model with fixed uncertainties and extremists: The opinions can keep fluctuating indefinitely, Journal of Artificial Societies and Social Simulation, 19 2016. doi: 10.18564/jasss.2967.

[39]

A. Mirtabatabaei, P. Jia and F. Bullo, Eulerian opinion dynamics with bounded confidence and exogenous inputs, SIAM Journal on Applied Dynamical Systems, 13 (2014), 425-446.  doi: 10.1137/130934040.

[40]

R. Mitchell and S. Nicholas, Knowledge creation in groups: The value of cognitive diversity, transactive memory and open-mindedness norms, The Electronic Journal of Knowledge Management, 4 (2006), 67-74. 

[41]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, Journal of Statistical Physics, 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.

[42]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.

[43]

G. Naldi, L. Pareschi and G. Toscani, Mathematical Modeling of Collective Behavior in SocioEconomic and Life Sciences, Birkhäuser, Boston, 2010. doi: 10.1007/978-0-8176-4946-3.

[44]

M. Pineda and G.M. Buendia, Mass media and heterogeneous bounds of confidence in continuous opinion dynamics, Physica A, 420 (2015), 73-84.  doi: 10.1016/j.physa.2014.10.089.

[45]

C. R. Sunstein, Going to Extremes: How Like Minds Unite and Divide, Oxford University Press, New York, 2009.

[46]

C. Taber and M. Lodge, Motivated skepticism in the evaluation of political beliefs, American Journal of Political Science, 50 (2006), 755-769.  doi: 10.1111/j.1540-5907.2006.00214.x.

[47]

O. Taussky, A recurring theorem on determinants, American Mathematical Monthly, 56 (1949), 672-676.  doi: 10.2307/2305561.

[48]

G. R. Terranova, J. A. Revelli and G. J. Sibona, Active speed role in opinion formation of interacting moving agents, Europhysics Letters, 105 (2014), 30007. doi: 10.1209/0295-5075/105/30007.

[49]

D. Tjosvold and M. Morishima, Grievance resolution: Perceived goal interdependence and interaction patterns, Relations Industrielles, 54 (1999), 527-548.  doi: 10.7202/051253ar.

[50]

D. Tjosvold and M. Poon, Dealing with scarce resources: Open-minded interaction for resolving budget conflicts, Group & Organization Management, 23 (1998), 237-258.  doi: 10.1177/1059601198233003.

[51]

D. Tjosvold and H.F. Sun, Openness among Chinese in conflict: Effects of direct discussion and warmth on integrative decision making, Journal of Applied Social Psychology, 33 (2003), 1878-1897.  doi: 10.1111/j.1559-1816.2003.tb02085.x.

[52]

R.P. Vallone, L. Ross and M.R. Lepper, The hostile media phenomenon: biased perception and perceptions of media bias in coverage of the Beirut massacre, Journal of Personality and Social Psychology, 49 (1985), 577-585.  doi: 10.1037/0022-3514.49.3.577.

[53]

R. S. Varga, Matrix Iterative Analysis, vol. 27 of Springer Series in Computational Mathematics, expanded edition, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-642-05156-2.

show all references

References:
[1]

R. P. Abelson, Mathematical models of the distribution of attitudes under controversy, in Contributions to Mathematical Psychology (eds. N. Frederiksen and H. Gulliksen), Holt, Reinhart & Winston, New York, 1964, 142-160. doi: 10.1007/978-88-470-1766-5.

[2]

F. Abergel, A. Chakraborti, B.K. Chakrabarti and M. Mitra (eds.), Econophysics of Order-driven Markets, Springer-Verlag, Milan, 2011. doi: 10.1007/978-88-470-1766-5.

[3]

K. Arceneaux and M. Johnson, Does media fragmentation produce mass polarization? selective exposure and a new era of minimal effects, APSA Annual Meeting Paper.

[4]

R. Axelrod, The dissemination of culture: A model with local convergence and global polarization, Journal of Conflict Resolution, 41 (1997), 203-226.  doi: 10.1177/0022002797041002001.

[5]

E. Ben-Naim, Opinion dynamics: Rise and fall of political parties, Europhys. Lett., 69 (2005), 671-677.  doi: 10.1209/epl/i2004-10421-1.

[6]

E. Berscheid, Opinion change and communicator-communicatee similarity and dissimilarity, Journal of Personality and Social Psychology, 4 (1966), 670-680.  doi: 10.1037/h0021193.

[7]

L. Boudin, A. Mercier and F. Salvarani, Conciliatory and contradictory dynamics in opinion formation, Physica A, 391 (2012), 5672-5684.  doi: 10.1016/j.physa.2012.05.070.

[8]

L. Boudin, R.Monaco and F. Salvarani, Kinetic model for multidimensional opinion formation, Physical Review E, 81 (2010), 036109, 9pp. doi: 10.1103/PhysRevE.81.036109.

[9]

J.W. Brehm and D. Lipsher, Communicator-communicatee discrepancy and perceived communicator trustworthiness, Journal of Personality, 27 (1959), 352-361. 

[10]

S. Camazine, J. -L. Deneubourg, N.R. Franks, J. Sneyd, G. Theraulaz and E. Bonabeau, Self-organization in Biological Systems, Princeton Studies in Complexity, Princeton University Press, Princeton, NJ, 2003.

[11]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[12]

G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents, Advances in Complex Systems, 3 (2000), 87-98.  doi: 10.1142/S0219525900000078.

[13]

G. Deffuant, F. Amblard, G. Weisbuch and T.Faure, How can extremism prevail? A study based on the relative agreement interaction model, Journal of Artificial Societies and Social Simulation, 5.

[14]

M.H. DeGroot, Reaching a consensus, Journal of the American Statistical Association, 69 (1974), 118-121.  doi: 10.1080/01621459.1974.10480137.

[15]

W. Doise, Intergroup relations and polarization of individual and collective judgments, Journal of Personality and Social Psychology, 12 (1969), 136-143.  doi: 10.1037/h0027571.

[16]

B. Düring, P. Markowich, J.-F. Pietschmann and M.-T. Wolfram, Boltzmann and Fokker—Planck equations modelling opinion formation in the presence of strong leaders, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 465 (2009), 3687-3708.  doi: 10.1098/rspa.2009.0239.

[17]

L. Festinger, A theory of social comparison processes, Human Relations, 7 (1954), 117-140.  doi: 10.1177/001872675400700202.

[18]

L. Festinger, A Theory of Cognitive Dissonance, Stanford University Press, Stanford, USA, 1957.

[19]

N. Friedkin and E. Johnsen, Social influence and opinions, Journal of Mathematical Sociology, 15 (1990), 193-206.  doi: 10.1080/0022250X.1990.9990069.

[20]

G. Fu, W. Zhang and Z. Li, Opinion dynamics of modified Hegselmann-Krause model in a group-based population with heterogeneous bounded confidence, Physica A, 419 (2015), 558-565.  doi: 10.1016/j.physa.2014.10.045.

[21]

S. Galam and S. Moscovici, Towards a theory of collective phenomena: Consensus and attitude changes in groups, European Journal of Social Psychology, 21 (1991), 49-74.  doi: 10.1002/ejsp.2420210105.

[22]

G. H. Golub and C.F. VanLoan, Matrix Computations, 3rd edition, Johns Hopkins University Press, Baltimore, MD, USA, 1996.

[23]

P. Groeber, J. Lorenz and F. Schweitzer, Dissonance minimization as a microfoundation of social influence in models of opinion formation, Journal of Mathematical Sociology, 38 (2014), 147-174.  doi: 10.1080/0022250X.2012.724486.

[24]

J. T. Hamilton, All the News That's Fit to Sell: How the Market Transforms Information into News, Princeton University Press, Princeton, NJ, 2006.

[25]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: Models, analysis and simulation, Journal of Artificial Societies and Social Simulation, 5.

[26]

R. Hegselmann and U. Krause, Opinion dynamics under the influence of radical groups, charismatic leaders, and other constant signals: A simple unifying model, Networks and Heterogenous Media, 10 (2015), 477-509.  doi: 10.3934/nhm.2015.10.477.

[27]

M.A. Hogg, J.C. Turner and B. Davidson, Polarized norms and social frames of reference: A test of the self categorization theory of group polarization, Basic and Applied Psychology, 11 (1990), 77-100.  doi: 10.1207/s15324834basp1101_6.

[28]

C. Innes, Quantifying the Effect of Open-Mindedness on Opinion Dynamics and Advertising Optimization, Master's thesis, Simon Fraser University, 2014.

[29]

A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd edition, Cambridge Texts in Applied Mathematics, Cambridge University Press, New York, 2009.

[30]

P.-E. Jabin and S. Motsch, Clustering and asymptotic behavior in opinion formation, Journal of Differential Equations, 257 (2014), 4165-4187.  doi: 10.1016/j.jde.2014.08.005.

[31]

D. Jones, K. Ferraiolo and J. Byrne, Selective media exposure and partisan differences about {S}arah Palin's candidacy, Politics and Policy, 39 (2011), 195-221.  doi: 10.1111/j.1747-1346.2011.00288.x.

[32]

G. Kou, Y. Zhao, Y. Peng and Y. Shi, Multi-level opinion dynamics under bounded confidence PLoS ONE, 7 (2012), e43507. doi: 10.1371/journal.pone.0043507.

[33]

M.S. Levendusky, Why do partisan media polarize viewers?, American Journal of Political Science, 57 (2013), 611-623.  doi: 10.1111/ajps.12008.

[34]

H. Liang, Y. Yang and X. Wang, Opinion dynamics in networks with heterogeneous confidence and influence, Physica A, 392 (2013), 2248-2256.  doi: 10.1016/j.physa.2013.01.008.

[35]

J. Lorenz, Heterogeneous bounds of confidence: Meet, discuss and find consensus!, Complexity, 15 (2010), 43-52.  doi: 10.1002/cplx.20295.

[36]

J. Lorenz, Continuous opinion dynamics under bounded confidence: A survey, International Journal of Modern Physics C, 18 (2007), 1819-1838.  doi: 10.1142/S0129183107011789.

[37]

T. V. Martins, M. Pineda and R. Toral, Mass media and repulsive interactions in continuous-opinion dynamics, Europhysics Letters, 91 (2010), 48003. doi: 10.1209/0295-5075/91/48003.

[38]

J. -D. Mathias, S. Huet and G. Deffuant, Bounded confidence model with fixed uncertainties and extremists: The opinions can keep fluctuating indefinitely, Journal of Artificial Societies and Social Simulation, 19 2016. doi: 10.18564/jasss.2967.

[39]

A. Mirtabatabaei, P. Jia and F. Bullo, Eulerian opinion dynamics with bounded confidence and exogenous inputs, SIAM Journal on Applied Dynamical Systems, 13 (2014), 425-446.  doi: 10.1137/130934040.

[40]

R. Mitchell and S. Nicholas, Knowledge creation in groups: The value of cognitive diversity, transactive memory and open-mindedness norms, The Electronic Journal of Knowledge Management, 4 (2006), 67-74. 

[41]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, Journal of Statistical Physics, 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.

[42]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.

[43]

G. Naldi, L. Pareschi and G. Toscani, Mathematical Modeling of Collective Behavior in SocioEconomic and Life Sciences, Birkhäuser, Boston, 2010. doi: 10.1007/978-0-8176-4946-3.

[44]

M. Pineda and G.M. Buendia, Mass media and heterogeneous bounds of confidence in continuous opinion dynamics, Physica A, 420 (2015), 73-84.  doi: 10.1016/j.physa.2014.10.089.

[45]

C. R. Sunstein, Going to Extremes: How Like Minds Unite and Divide, Oxford University Press, New York, 2009.

[46]

C. Taber and M. Lodge, Motivated skepticism in the evaluation of political beliefs, American Journal of Political Science, 50 (2006), 755-769.  doi: 10.1111/j.1540-5907.2006.00214.x.

[47]

O. Taussky, A recurring theorem on determinants, American Mathematical Monthly, 56 (1949), 672-676.  doi: 10.2307/2305561.

[48]

G. R. Terranova, J. A. Revelli and G. J. Sibona, Active speed role in opinion formation of interacting moving agents, Europhysics Letters, 105 (2014), 30007. doi: 10.1209/0295-5075/105/30007.

[49]

D. Tjosvold and M. Morishima, Grievance resolution: Perceived goal interdependence and interaction patterns, Relations Industrielles, 54 (1999), 527-548.  doi: 10.7202/051253ar.

[50]

D. Tjosvold and M. Poon, Dealing with scarce resources: Open-minded interaction for resolving budget conflicts, Group & Organization Management, 23 (1998), 237-258.  doi: 10.1177/1059601198233003.

[51]

D. Tjosvold and H.F. Sun, Openness among Chinese in conflict: Effects of direct discussion and warmth on integrative decision making, Journal of Applied Social Psychology, 33 (2003), 1878-1897.  doi: 10.1111/j.1559-1816.2003.tb02085.x.

[52]

R.P. Vallone, L. Ross and M.R. Lepper, The hostile media phenomenon: biased perception and perceptions of media bias in coverage of the Beirut massacre, Journal of Personality and Social Psychology, 49 (1985), 577-585.  doi: 10.1037/0022-3514.49.3.577.

[53]

R. S. Varga, Matrix Iterative Analysis, vol. 27 of Springer Series in Computational Mathematics, expanded edition, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-642-05156-2.

Figure 1.  Schematic diagram of opinion interactions along the opinion continuum $[-1,1]$ in the presence of a social norm of open-mindedness (note that opinion space is shown on the vertical axis in all other figures). Green circles represent warm and open-minded individuals (in $O$), while diamonds correspond to individuals in $C$, those having colder and less open communication styles. (i) The interactions on the left show the influences acting on the open-minded agent at $x_1$, which experiences attractive social forces (green arrows) from both open-and closed-minded individuals with opinions within the bound of confidence $\epsilon$; the influence function $\phi(|x - x_1|) = 1_{[x_1 - \epsilon, x_1 + \epsilon]}$ is shown in green. (ii) The social forces acting on the closed-minded individual with non-moderate opinion $x_2$ (large red diamond; note $|x_2| > X_c$), in the presence of a social norm of open-mindedness, are shown on the right. Interactions with other members of $C$, both moderate (blue diamond at $x_3$) and extremist (red diamond), reinforce the opinion $x_2$ and drive it closer to the nearest extreme at 1 (blue and red arrows). However, given an open-mindedness social norm (model (6)), the interaction with an agent in $O$ within its bound of confidence (green circle near $x_2$) induces an open-minded response and attractive interaction (green arrow) according to the influence function shown in red. In the absence of a social norm of open-mindedness (model (5)), this interaction would instead also drive the individual at $x_2$ to the extreme at 1.

Further notes: The individual with opinion $x_3$ (blue diamond) is in $C$, but has a sufficiently moderate opinion ($|x_3| < X_c$) to respond in an open-minded manner. In particular, it interacts with and is attracted to the agent at $x_2$ (arrow not shown); however, due to its intrinsically colder communication style it fails to attract the closed-minded individual at $x_2$, so that the influence is not reciprocal. Lastly, the distance of the opinion of the open-minded individual at $x_4$ to each of the opinions $x_1$, $x_2$ and $x_3$ is greater than the bound of confidence $\epsilon$, so it does not interact with any of these three other individuals.

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Figure 2.  Evolution to equilibria of solutions to model (4) with (6) with $N=80$ for some choices of the parameters $m$ (number of open-minded agents), $X_c$ (critical threshold for extreme-seeking dynamics) and $\epsilon$ (bound of confidence), and with both type NS (non-symmetric) and type S (symmetric) initial conditions. The plots illustrate various qualitatively different outcomes as discussed in the text: (a) $m=80$ ($X_c$ irrelevant), $\epsilon=0.2$, type S; (b) $m=54$, $X_c=1/3$, $\epsilon=1.5010$, type NS; (c) $m=54$, $X_c=1/3$, $\epsilon=0.7909$, type NS; (d) $m=0$, $X_c=0$ $\epsilon=2$, type S; (e) $m=8$, $X_c=2/3$, $\epsilon=0.5222$, type NS; (f) $m=8$, $X_c=2/3$, $\epsilon=1.0020$, type S.
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Figure 3.  Numerical illustration of local stability: At time $t=50$ a perturbation of size $\eta $ is applied to the 5-cluster solution in Figure 2(f); the perturbed system is then evolved to $t=100$. (a) For $\eta =0.02$, the solution returns to the original 5-cluster configuration; (b) with $\eta =0.03$, the system approaches a 4-cluster state.
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Figure 8.  Bifurcation with respect to the location of the media source. The filled circles represent the locations of equilibrium clusters, while the dashed line indicates the media bias; parameter values are $X_c = 0$ and $\epsilon= 1.2$, with (a) $m=0$, (b) $m=26$, and (c) $m=54$. In the absence of open-minded individuals (a), the media has no effect, as all closed-minded individuals approach extremist views. Plots (b) and (c) correspond to non-symmetric (type NS) initial data, with inserts that show the results obtained from symmetric (type S) initializations. Asymmetry seems to enhance the effect of the media, in particular when the number of open-minded individuals is relatively low ($m=26$, plot (b)).
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Figure 4.  Dependence of the number and stability of cluster equilibria on $X_c$ (threshold for extreme-seeking dynamics) and $\epsilon$ (bound of confidence): (a) $m = 80$ (all open-minded), type S initial data; (b) $m = 0$ (none open-minded), $\epsilon = 0.2$, type S data; (c) $m = 8$, $X_c = 2/3$, type NS data (type S data in insert). Cluster locations for linearly stable equilibria are represented by blue filled circles, while neutrally stable equilibria are denoted by red stars.
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Figure 5.  Extremist consensus arising as bifurcations in parameter space. Within each plot, the same non-symmetric (type NS) initial conditions are used to generate equilibria for all parameter values. The inserts show equilibria obtained from symmetric initial data (type S), where no extremist consensus can arise. (a) Dependence on $X_c$ with $m=26$, $\epsilon=1.2$; (b) dependence on $m$ with $X_c=1/3$, $\epsilon=2$; (c) dependence on $\epsilon$ with $m=54$, $X_c=1/3$. Symbols are as in Figure 4.
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Figure 6.  Effect of a social norm of open-mindedness on opinion convergence. The initial data is of type NS (non-symmetric); parameters are set at $N = 80$, $\epsilon=2$ and $X_c=0$, and symbols are as in Figure 4. Increasing the proportion $m/N$ of open-minded individuals has a negligible effect on the opinion distribution when there is no social norm of open-mindedness (plot (a)), but increases agreement significantly when a social norm of open-mindedness is present (plot (b)).
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Figure 7.  Evolution to equilibria of the opinion model with media (19)-(20) with $(6)$ with Type NS (non-symmetric) initial data, and parameter values $m=26$ (number of open-minded agents), $X_c=0$ (threshold for extreme-seeking dynamics) and $\epsilon=1.2$ (bound of confidence). The circles indicate the media location $\mu$. (a) $\mu=1$: all individuals approach the extremist media bias. (b) $\mu=0.1$: open-minded individuals converge to the media bias, while the opinions of non-open-minded individuals form two clusters balanced between attraction to open-minded agents and to the extremes.
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