Dynamics of social group competition: modeling the decline of religious affiliation.

Abstract

When social groups compete for members, the resulting dynamics may be understandable with mathematical models. We demonstrate that a simple ordinary differential equation (ODE) model is a good fit for religious shift by comparing it to a new international data set tracking religious nonaffiliation. We then generalize the model to include the possibility of nontrivial social interaction networks and examine the limiting case of a continuous system. Analytical and numerical predictions of this generalized system, which is robust to polarizing perturbations, match those of the original ODE model and justify its agreement with real-world data. The resulting predictions highlight possible causes of social shift and suggest future lines of research in both physics and sociology.

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Dynamics of Social Group Competition: Modeling the Decline of Religious Affiliation
Daniel M. Abrams and Haley A. Yaple
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA
Richard J. Wiener
Research Corporation for Science Advancement, Tucson, Arizona 85712, USA
and Department of Physics, University of Arizona, Tucson, Arizona 85721, USA
(Received 5 January 2011; published 16 August 2011)
When social groups compete for members, the resulting dynamics may be understandable with
mathematical models. We demonstrate that a simple ordinary differential equation (ODE) model is a
good fit for religious shift by comparing it to a new international data set tracking religious nonaffiliation.
We then generalize the model to include the possibility of nontrivial social interaction networks and
examine the limiting case of a continuous system. Analytical and numerical predictions of this generalized
system, which is robust to polarizing perturbations, match those of the original ODE model and justify its
agreement with real-world data. The resulting predictions highlight possible causes of social shift and
suggest future lines of research in both physics and sociology.
DOI: 10.1103/PhysRevLett.107.088701PACS numbers: 89.65.Ef, 02.50.Le, 64.60.aq, 89.75.Fb
The tools of statistical mechanics and nonlinear dynam-
ics have been used successfully not just to analyze physical
systems, but also models of social phenomena ranging
from language choice [1] to political party affiliation [2]
towar[3]and peace[4].Modelsof binarychoice dynamics
have been of particular interest. In this work, we focus on
social systems composed of two mutually exclusivegroups
in competition for members [5–11]. We compile and ana-
lyze a new data set quantifying the declining rates of
religious affiliation in a variety of regions worldwide and
present a theory to explain this trend.
People claiming no religious affiliation constitute the
fastest growing religious minority in many countries
throughout the world [12]. Americans without religious
affiliation comprise the only religious group growing in
all 50 states; in 2008 those claiming no religion rose to
15% nationwide, with a maximum in Vermont at 34% [13].
In the Netherlands nearly half the population is religiously
unaffiliated. Here we use a minimal model of competition
for members between social groups to explain historical
census data on the growth of religious nonaffiliation in 85
regions around the world. According to the model, a single
parameter quantifying the perceived utility of adhering to a
religion determines whether the unaffiliated group will
grow in a society. The model predicts that for societies in
which the perceived utility of not adhering is greater than
the utility of adhering, religion will be driven toward
extinction.
Model.—We begin by idealizing a society as partitioned
into two mutually exclusive social groups, X and Y, the
unaffiliated and those who adhere to any religion. We
assume the attractiveness of a group increases with the
number of members, which is consistent with research on
social conformity [14–17]. We further assume that
attractiveness also increases with the perceived utility of
the group, a quantity independent of group size encom-
passing many factors including the social, economic, po-
litical, and security benefits derived from membership as
well as spiritual or moral consonance with a group. Then a
simple model of the dynamics of conversion is given by [1]
dx
dt
¼ yPyxðx;uxÞ ? xPxyðx;uxÞ;
(1)
where Pyxðx;uxÞ is the probability, per unit of time, that an
individual converts from Y to X, x is the fraction of the
population in group X at time t, 0 ? ux? 1 is a measure
of X’s perceived utility, and y and uyare complementary
fractions to x and ux. We require Pxyðx;uxÞ ¼ Pyxð1 ? x;
1 ? uxÞ to obtain symmetry under exchange of x and y and
Pyxðx;0Þ ¼ 0becausenoonewillswitch toagroupwithno
utility. Moreover, since the change in the dynamics of
Eq. (1) is small for small values of Pyxð0;uxÞ, and data
presented in this Letter are consistent with negligible
probability for the birth of a new social group, for
simplicity we set Pyxð0;uxÞ ¼ 0 (see Sec. S9 in the
Supplemental Material [18]). The assumptions regarding
the attractiveness of a social group also imply that Pyxis
smooth and monotonically increasing in both arguments.
Under these assumptions, for generic Pyxðx;uxÞ Eq. (1) has
at most three fixed points, with alternating stability (see
Sec. S2 in [18]).
Equation (1) provides a general theoretical framework
that can be applied to a wide variety of physical and social
systems. Appropriate choices of the function Pyxproduce
well-known physical models, e.g., the Ising model, with
Pyx/ e??Ei=kBTHð?EiÞ þ Hð??EiÞ,where?Eiisthedif-
ference in configuration energies Ei¼ ?JP
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si¼ ?1, with H the Heaviside function and Gijthe cou-
pling matrix, or the Kuramoto model with Pyx¼ ! ? kx,
where x ¼ hsinð?ð?0Þ ? ?ð?ÞÞi?0. Here, in the context of
social group competition, we choose a functional form for
the transition probabilities consistent with the minimal
assumptions of the model: Pyxðx;uxÞ ¼ cxaux, where c
and a are constants that scale time and determine the
relative importance of x and uxin attracting converts,
respectively. If a > 1 there are three fixed points, one
each at x ¼ 0 and x ¼ 1, which are stable, and one at
0 < x < 1, which is unstable. For a < 1 the stability of
these fixed points is reversed. For a ¼ 1, there are only
two fixed points, with opposite stability.
In Figs. 1(a)–1(c) we fit the model to historical census
data from regions of Switzerland, Finland, and the
Netherlands, three of 85 worldwide locations for which
we compiled and analyzed data. The initial fraction un-
affiliated x0and the perceived utility uxwere varied to
optimizethefittoeachdata set,whilecand awere takento
be global. A broad minimum in the error near a ¼ 1
indicated that as a reasonable choice (see Sec. S4 in
[18]). Figure 1(c) shows that, if the model is accurate,
nearly 70% of the Netherlands will be nonaffiliated by
midcentury. Figure 1(d) shows the totality of the data
collected and a comparison to the prediction of Eq. (1)
with a ¼ 1, demonstrating the general agreement with our
model. Time has been rescaled in each data set and the
origin shifted so that they lie on top of one another. See
Sec. S3 in [18] for more details.
The behavior of the model can be understood analyti-
cally for a ¼ 1, in which case we have dx=dt ¼ cx
ð1 ? xÞð2ux? 1Þ, logistic growth. An analysis of the fixed
points of this equation tells us that religionwill disappear if
its perceived utility is less than that of nonaffiliation,
regardless of how large a fraction initially adheres to a
religion (i.e., if ux> 0:5). However, if a is less than but
closeto1,asmallsocialgroupcanindefinitelycoexistwith
a large social group. Even if a ? 1 it is possible that
society will reach such a state if model assumptions break
down when the population is nearly all one group.
One might ask whether our model explains data better
than a simple empirical curve. Logistic growth would be a
reasonable null hypothesis for the observed data, but here
we have provided a theoretical framework for expecting a
moregeneral growth law, Eq. (1), and have shown that data
suggest logistic growth as a particular case of the general
law. Our framework includes a rational mathematical foun-
dation for the observed growth law.
Generalizations.—Thus far, we have implicitly assumed
that society is highly interconnected, because the function
Pyxdepends on x,avariable measuringglobal participation
in group X. One might imagine a more general model
where Pyxinstead depends on a measure of the local
participation in group X among an individual’s social peers
[6,19]. To create such a generalization, we represent a
social network by a binary adjacency matrix A (Aij¼ 1
if i and j are socially linked) and we define the local mean
religious affiliation among social peers of individual i as
xi¼PN
tracked by the binary-valued vector R, where Ri¼ 1 for
unaffiliated and Ri¼ 0 for affiliated. Equation (1) then
becomes
j¼1AijRj=PN
j¼1Aij. Each individual’s affiliation is
dhRii
dt
¼ ð1 ? hRiiÞPyxðxi;uxÞ ? hRiiPyxð1 ? xi;1 ? uxÞ:
(2)
We have used angled brackets to indicate that this
equation holds only in the sense of ensemble average
over many realizations, since this is a stochastic rather
than deterministic system. In the all-to-all coupling limit,
A ¼ 1, xi¼ ? x, and Eq. (2) reduces to Eq. (1).
We also consider a further generalization to a system
with real-valued rather than binary-valued group affiliation
(so individual religiosity lies in a continuum between fully
unaffiliated and fully affiliated); such a model can be
constructed with the introduction of a spatial dimension.
The spatial coordinate?will be allowedtovary from?1 to
1 with a normalized coupling kernel Gð?;?0Þ determining
the strength of social connection between spatial coordi-
nates ? and ?0. The binary religious affiliation vector R
from the previous network model is now reinterpreted as
19001950 2000
0
0.03
0.06
194519752005
0
0.08
0.16
185019502050
0
0.4
0.8
01
0
0.6
(b)
(d)
(c)
(a)
FIG. 1 (color online).
affiliated versus time for (a) Schwyz Canton in Switzerland,
(b) the autonomous Aland islands region of Finland, (c) the
Netherlands, and (d) all 85 data sets. Dots indicate data points
from census surveys; light gray (red) dots correspond to regions
within countries and dark gray (blue) dots to entire countries.
Black lines indicate model fits. For (a)–(c), relative utilities for
the religiously unaffiliated populations as determined by model
fits were ux¼ 0:70;0:63;0:56. In (c) we extend the model’s
projection to show the expected change in concavity around
2025. For (d) time has been rescaled so data sets lie on top of one
another and the solution curve with ux¼ 0:65. Representative
data sets were chosen to show varied current rates of nonaffilia-
tion (low, medium, and high).
Fraction of population religiously un-
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a continuous real-valued function 0 ? Rð?;tÞ ? 1 that
varies spatially and temporally. Then the dynamics of R
satisfy
@R
@t
¼ ð1 ? RÞPyxðx;uxÞ ? RPyxð1 ? x;1 ? uxÞ
(3)
in analogy with the discrete system. Here x again repre-
sents the local mean religious affiliation, xð?;tÞ ¼
R1
pling kernel rather than a sum over an adjacency matrix).
We may still recover the original model Eq. (1) by
considering the special case of all-to-all coupling
Gð?;?0Þ ¼ 1=2 and spatially uniform Rð?;tÞ ¼ R0ðtÞ;
then xð?;tÞ ¼1
comes
?1Gð?;?0ÞRð?0;tÞd?0(this time an integral over a cou-
2
R1
?1Rð?0;tÞd?0¼ R0ðtÞ, and Eq. (3) be-
@R0
@t¼ð1?R0ÞPyxðR0;uxÞ?R0Pyxð1?R0;1?uxÞ;
(4)
which follows dynamics identical to Eq. (1).
Note that Eq. (2) represents a stochastic system with
binary-valued vector R, while Eq. (3) represents a deter-
ministic system for real Rð?;tÞ 2 ½0;1?, but both limit to
the same dynamics for large N if the adjacency matrix A
and coupling kernel G are chosen analogously.
We can impose perturbations to both the coupling kernel
(i.e., the social network structure) and the spatial distribu-
tion of Rð?;tÞ to examine the stability of this system and
therobustnessofour resultsfortheall-to-allcase.Onevery
destabilizing example consists of perturbing the system
towards two separate clusters. These clusters might repre-
sent a polarized society that consists of two social cliques
in which members of each are more strongly connected
to others in their clique than to members of the other
clique. Mathematically, this can be written as Gð?;?0Þ ¼
1
2þ1
that determines the amplitude of the perturbation. This
kernel implies that individuals with the same sign of ?
are more strongly coupled to one another than they are to
individuals with opposite-signed ?.
The above perturbation alone is not sufficient to change
the dynamics of the system—a uniform state Rð?;t0Þ ¼ R0
will still evolve according to the dynamics of the original
system Eq. (1).
We add a further perturbation to the spatial distribution
of religious affiliation by imposing Rð?;t0Þ ¼ R0þ
?sgn?, where ? is a small parameter. This should conspire
with the perturbed coupling kernel to maximally destabi-
lize the uniform state.
Surprisingly, an analysis of the resulting dynamics
reveals that this perturbed system must ultimately tend
to the same steady state as the unperturbed system with
? ¼ ? ¼ 0 [which follows the same dynamics as Eq. (1)].
Furthermore, the spatial perturbation must eventually de-
cay exponentially, although an initial growth is possible
(see Sec. S5 in the Supplemental Material [18]).
2?sgn?sgn?0, where ? is a small parameter (? ? 1)
The implication of this analysis is that systems that are
nearly all-to-all should behave very similarly to an all-to-
all system. In the next section we describe a numerical
experiment that tests this prediction.
Numerical experiment.—We design our experiment with
the goal of controlling the perturbation from an all-to-all
network through a single parameter. We construct a social
network consisting of two all-to-all clusters initially dis-
connected from one another, and then add links between
any two nodes in opposite clusters with probability p. Thus
p ¼ 1 corresponds to an all-to-all network that should
simulate Eq. (1), while p ¼ 0 leaves the network with
two disconnected components. Small perturbations from
all-to-all correspond to p near 1, and p can be related to the
coupling kernel perturbation parameter ? described above
as p ¼ ð1 ? ?Þ=ð1 þ ?Þ (assuming all links in the network
have equal weight). The size of each cluster is determined
by the initial condition x0as NX¼ x0N, NY¼ ð1 ? x0ÞN,
where all members of cluster X initially have R ¼ 1 and all
members of cluster Y initially have R ¼ 0.
Figure 2 compares the results of simulation of system
Eq. (2) with varying perturbations off of all-to-all. The
theoretical (all-to-all) separatrix between basins of attrac-
tion is a vertical line at ux¼ 1=2. Even when p ¼ 0:01,
when in-group connections are 100 times more numerous
than out-group connections, the steady states of the system
and basins of attraction remain essentially unchanged.
In the case of the continuous deterministic system
Eq. (3), the equivalent to Fig. 2 is extremely boring:
numerically, the steady states of the perturbed system are
indistinguishable from those of the unperturbed all-to-all
system, regardless of the value of p (see Sec. S6 and
Fig. S5 in [18]).
The only notable difference betweenthe dynamics of the
continuous networked system and the dynamics of the
original all-to-all system Eq. (1) is a time delay d apparent
before the onset of significant shift between groups (see
Fig. 3). Wewere able to find an approximate expression for
that time delay as d / ?lnp=ð2ux? 1Þ (see Sec. S7,
Figs. S6 and S7 [18]).
What we have shown by the generalization of the
model to include network structure is surprising: even if
FIG. 2.
Eq. (2) on a network with two initial clusters weakly coupled to
one another. The ratio p of out-group coupling strength to in-
group coupling strength is (a) p ¼ 0:01, (b) p ¼ 0:40,
(c) p ¼ 0:80 (N ¼ 500). Steady states are nearly identical to
the predictions of the all-to-all model Eq. (1).
Results of simulation of the discrete stochastic model
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conformity to a local majority influences group member-
ship, the existence of some out-group connections is
enough to drive one group to dominance and the other to
extinction. In the language of Refs. [6,8,10], the population
will reach the same consensus, despite the existence of
individual cliques, as it would without cliques, with only
the addition of a time delay.
In a modern secular society there are many opportunities
for out-group connections to form due to the prevalence of
socially integrated institutions—schools, workplaces, rec-
reational clubs, etc. Our analysis shows that just a few out-
group connections are sufficient to explain the good fit of
Eq. (1) to data, even though Eq. (1) implicitly assumes
all-to-all coupling.
Conclusions.—We have developed a general framework
for modeling competitive systems. When applied to physi-
cal systems, appropriate choices of the function Pyxcan
produce a variety of well-known physical models, but we
have focused on an application to competition between
social groupsandanalyzedthe behaviorofthe modelunder
modest relevant assumptions. We found that a particular
case of the solution fits census data on competition be-
tween religious and irreligious segments of modern secular
societies in 85 regions around the world. The model in-
dicates that in these societies the perceived utility of reli-
gious nonaffiliation is greater than that of adhering to a
religion, and therefore predicts continued growth of non-
affiliation, tending toward the disappearance of religion.
According to our calculations, the steady-state predictions
should remainvalid under small perturbations to the all-to-
all network structure that the model assumes, and, in fact,
the all-to-all analysis remains applicable to networks very
different fromall-to-all. Evenan idealized highlypolarized
society with a two-clique network structure follows the
dynamics of our all-to-all model closely, albeit with the
introduction of a time delay. This perturbation analysis
suggests why the simple all-to-all model fits data from
societies that undoubtedly have more complex network
structures.
The models we have presented, although greatly ideal-
ized, are significant in that they provide a new framework
for the understanding of human behavior in competitive
majority or minority social systems. We have shown good
agreement with historical data, with the surprising result
that the perceived utility of nonaffiliation is higher than the
utility of religious affiliation in all the societies we exam-
ined. We recognize that the simplifications in our models
may limit their applicability (see Sec. S8 [18]); nonethe-
less, our work suggests a line of research for social scien-
tists: perhaps standard sociological methodology can be
used to compare perceived utilities of affiliation and non-
affiliation in societies where nonaffiliation is growing.
This work was funded by Northwestern University and
The James S. McDonnell Foundation. The authors thank
P. Zuckerman for useful correspondence.
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athttp://link.aps.org/
0100
0
1
FIG. 3 (color online).
increasing perturbation off of all-to-all (N ¼ 500, x0¼ 0:1,
u ¼ 0:6). Equivalent values of the perturbation parameter ? in
order of decreasing p are ? ¼ 0, ? ¼ 0:11, ? ¼ 0:43, and
? ¼ 0:98.
Variation in the behavior of Eq. (3) with
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