Encouraging Moderation: Clues from a Simple Model of Ideological Conflict

Some of the most pivotal moments in intellectual history occur when a new ideology sweeps through a society, supplanting an established system of beliefs in a rapid revolution of thought. Yet in many cases the new ideology is as extreme as the old. Why is it then that moderate positions so rarely prevail? Here, in the context of a simple model of opinion spreading, we test seven plausible strategies for deradicalizing a society and find that only one of them significantly expands the moderate subpopulation without risking its extinction in the process.

Figure 1

Model structure (see text for definitions of $A$, $B$, $AB$, and $A_c$). Labels on the arrows indicate the allowed affiliation(s) of a speaker that converts a listener from one subpopulation to another in the direction of the arrow.

Figure 2

The equilibrium values of $n_A$ (red plus signs), $n_B$ (blue dots), and $n_{AB}$ (magenta open circles) for the basic model as functions of $p$, assuming an initial population with $(n_A,n_B)=(0,1-p)$. The vertical dashed line marks the critical value $p_c=1-\sqrt{3}/2\approx 0.134$. At values of $p$ greater than $p_c$, $n_B$ and $n_{AB}$ are zero and $n_A=1-p$.

Figure 3

The equilibrium fraction of $B$ believers remaining after the basic model is run on the giant connected component of (a) the U. S. network of corporate board memberships in 1994 [24], (b) four coauthorship networks in the physics division of arXiv.org [25], and (c) the friendship networks of the location-based social networking websites Gowalla and Brightkite [26]. Note the abrupt transitions in $n_B$; compare with the corresponding curve in Fig. 2.

Figure 4

(a)–(c): Mean-field results obtained analytically for generalizations of the basic model with (a) stubborn moderates, (b) evangelical moderates, and (c) nonsocial deradicalization (see the text for details). The final equilibrium values of $n_A$ (red plus signs), $n_B$ (blue dots), and $n_{AB}$ (magenta open circles) for the initial condition $(n_A,n_B)=(0,1-p)$ are plotted as a function of the new parameter ($s$, $r$, or $u$) in the corresponding generalized model. Of the three strategies shown—and in fact for all seven considered in the Supplemental Material [23]—only nonsocial deradicalization allows for the growth of the moderate fraction up to $1-p$ without risking its extinction. (d)–(f): Representative simulation results for the discrete-time versions of the models with (d) stubborn moderates, (e) evangelical moderates, and (f) nonsocial deradicalization when these models are run on the arXiv.org coauthorship network for high energy physics theory (hep-th) [25]. The plots show the equilibrium fractions of $n_A$ (red plus signs), $n_B$ (blue dots), and $n_{AB}$ (magenta open circles) obtained. Each simulation is started from the state in which a random but highly interconnected fraction $p$ of the population is committed to a belief in $A$ and the rest believe $B$. The simulation is then run for ${10}^8$ time steps after which the values of $n_A$, $n_B$, and $n_{AB}$ are tabulated. The constant fractions of zealots in the six panels of this figure are (a) 0.1, (b) 0.1, (c) 0.05, (d) 0.035, (e) 0.02, and (f) 0.02.