We propose and study a simple model for the evolution of political opinion through a population. The model includes a nonlinear term that causes individuals with more extreme views to be less receptive to external influence. Such a term was suggested in 1981 by Cobb in the context of a scalar-valued diffusion equation, and recent empirical studies support this modeling assumption. Here, we use the same philosophy in a network-based model. This allows us to incorporate the pattern of pairwise social interactions present in the population. We show that the model can admit two distinct stable steady states. This bi-stability property is seen to support polarization and can also make the long-term behavior of the system extremely sensitive to the initial conditions and to the precise connectivity structure. Computational results are given to illustrate these effects.

1.
R.
Axelrod
, “
The dissemination of culture: A model with local convergence and global polarization
,”
J. Confl. Resolut.
41
,
203
226
(
1997
).
2.
M. H.
DeGroot
, “
Reaching a consensus
,”
J. Am. Stat. Assoc.
69
,
118
121
(
1974
).
3.
N. E.
Friedkin
and
E. C.
Johnsen
, “
Social influence and opinions
,”
J. Math. Sociol.
15
,
193
206
(
1990
).
4.
R.
Hegselmann
and
U.
Krause
, “
Opinion dynamics and bounded confidence: Models, analysis, and simulation
,”
J. Artif. Soc. Soc. Simul.
5
,
2
(
2002
).
5.
G.
Albi
,
L.
Pareschi
,
G.
Toscani
, and
M.
Zanella
, “Recent advances in opinion modeling: Control and social influence,” in Active Particles (Springer, 2017), Vol. 1, pp. 49–98.
6.
B. D. O.
Anderson
and
M.
Ye
, “
Recent advances in the modelling and analysis of opinion dynamics on influence networks
,”
Int. J. Autom. Comput.
16
,
129
149
(
2019
).
7.
V. D.
Blondel
,
J. M.
Hendrickx
, and
J. N.
Tsitsiklis
, “
On Krause’s multi-agent consensus model with state-dependent connectivity
,”
IEEE Trans. Autom. Control
54
,
2586
2597
(
2009
).
8.
H.
Haibo
, “
Competing opinion diffusion on social networks
,”
R. Soc. Open Sci.
4
,
171160
(
2017
).
9.
N.
Perra
and
L. E. C.
Rocha
, “
Modelling opinion dynamics in the age of algorithmic personalisation
,”
Sci. Rep.
9
,
7261
(
2019
).
10.
L.
Cobb
, “Stochastic differential equations for the social sciences,” in Mathematical Frontiers of the Social and Policy Sciences, edited by L. Cobb and R. M. Thrall (Westview Press, 1981).
11.
M. J.
Brandt
,
A. M.
Evans
, and
J. T.
Crawford
, “
The unthinking or confident extremist? Political extremists are more likely than moderates to reject experimenter-generated anchors
,”
Psychol. Sci.
26
,
189
202
(
2015
).
12.
L.
Zmigrod
,
P. J.
Rentfrow
, and
T. W.
Robbins
, “
The partisan mind: Is extreme political partisanship related to cognitive inflexibility?
,”
J. Exp. Psychol. Gen.
149
,
407
(
2020
).
13.
In this work, we find it natural to use ϵ for the parameter that takes the form ϵ2 in Ref. 10. Also, since we will be using capitals to denote matrices, we use θ rather than G for the long-time mean.
14.
D. J.
Higham
, “
An algorithmic introduction to numerical simulation of stochastic differential equations
,”
SIAM Rev.
43
,
525
546
(
2001
).
15.
P. E.
Kloeden
and
E.
Platen
,
Numerical Solution of Stochastic Differential Equations
(
Springer-Verlag
,
Berlin
,
1992
).
16.
F. C.
Klebaner
,
Introduction to Stochastic Calculus with Applications
(
Imperial College Press
,
London
,
1998
).
17.
J. D.
Murray
, Mathematical Biology I. An Introduction, 3rd ed., Interdisciplinary Applied Mathematics Vol. 17 (Springer, New York, 2002).
18.
D. J.
Watts
and
S. H.
Strogatz
, “
Collective dynamics of ‘small-world’ networks
,”
Nature
393
,
440
442
(
1998
).
19.
A. L.
Barabási
and
R.
Albert
, “
Emergence of scaling in random networks
,”
Science
286
,
509
512
(
1999
).
20.
A.
Taylor
and
D. J.
Higham
, “
CONTEST: A controllable test matrix toolbox for MATLAB
,”
ACM Trans. Math. Softw.
35
,
26:1
26:17
(
2009
).
You do not currently have access to this content.