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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.107.088701 PACS 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

18501950 2050

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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Pareschi, and G. Toscani (Birkha ¨user, Boston, 2010).

[12] P. Zuckerman, in Cambridge Companion to Atheism,

edited by M. Martin (Cambridge University Press,

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[13] B.A. Kosmin and A. Keysar, ‘‘American Nones: The

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[18] See Supplemental Material

supplemental/10.1103/PhysRevLett.107.088701 for more

details on methods and analysis.

[19] S. Galam, Physica (Amsterdam) 238A, 66 (1997).

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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