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CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 1
Accepted for publication in American Psychologist
Special Issue: Psychological Perspectives on Culture Change
Article Type: Computational Modeling
Cultural Drift, Indirect Minority Influence, Network Structure and Their
Impacts on Cultural Change and Diversity
Jiin Jung
Department of Psychology
University of Kansas
Aaron Bramson
Lab of Symbolic Cognitive Development
RIKEN Center for Biosystems Dynamics Research
William D. Crano
Department of Psychology
Claremont Graduate University
Scott E. Page
Center for the Study of Complex Systems
University of Michigan
John H. Miller
Social and Decision Sciences
Carnegie Mellon University
Author Note
Word count: 5466 (including main text, notes) February 24, 2021
Author Note
Correspondence concerning this article should be addressed to Jiin Jung, Department of
Psychology, University of Kansas, 1415 Jayhawk Blvd. Lawrence, KS 66045. Email:
jiin.jung@ku.edu.
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 2
Abstract
The present research investigates how psychological mechanisms and social network
structures generate patterns of cultural change and diversity. The two psychological
mechanisms studied here are cultural drift and indirect minority influence; the former is
parameterized by an error rate (ε) and the latter by a leniency threshold (λ). The patterns
of cultural change are examined in terms of magnitude (small vs large), speed (gradual vs
rapid), and frequency (frequent vs rare). Diversity and polarization in a society are
examined in terms of global cultural variation (inverse Simpson index) and local
neighborhood difference (Hamming distance). Key findings are that in networks with high
connectivity or local community structures (complete, scale-free, random, and modular
networks) cultural drift can produce a rapid, large, and rare pattern of cultural change
(punctuated equilibrium), whereas in lattice or small world networks, it produces a more
gradual change pattern. Indirect minority influence robustly produces a gradual, small,
and frequent pattern of cultural change (gradualism) across various network structures.
When cultural change occurs in social networks that have a modular community structure,
indirect minority influence generates a regime of cultural diversity whereas cultural drift
generates a polarized regime. Finally, cultural drift and indirect minority influence generate
distinct tipping points for social change in different network structures, but prediction of
whether and when cultural change emerges is difficult at tipping points in both cases.
Public Significance Statement
This study uses computational modeling and simulations to suggest that different patterns
of cultural change and diversity can be attributed to different psychological mechanisms.
Cultural drift generates rapid social change and polarization whereas indirect minority
influence generates gradual social change and diversity in a society. While cultural drift
proves sensitive to network modularity and connectivity, the effects of indirect minority
influence on social change are robust to variations in network structure.
Keywords: indirect minority influence; cultural drift; gradual social change;
punctuated equilibria; diversity
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 3
Cultural Drift, Indirect Minority Influence, Network Structure and Their
Impacts on Cultural Change and Diversity
Introduction
Cultures – shared coherent patterns of behaviors, norms, beliefs, and
representations – emerge within communities of individuals who interact to achieve
individual and collective goals (Swidler, 1986). These patterns arise because people’s
behavior follows established psychological principles. Shared behaviors arise owing to the
benefits from coordination and social learning. As people interact, they influence and are
influenced by others in their local communities. Coherence arises from a desire to reduce
dissonance and increase predictability.
Cultures do not, however, remain fixed. They change and adapt in response to
adaptations by their members to the environment (Bednar & Page, 2018; Greif & Laitin,
2004; Varnum & Grossmann, 2017). That change can occur gradually or exhibit abrupt
realignments (de la Sablonnière, 2017), and patterns of cultural traits can become
homogenized or diversified (Jackson et al., 2019; Serrà et al., 2012). Recent empirical work
has documented changes in socio-ecological factors related to cultural change such as
increases in socioeconomic development and increases in individualism (Grossmann &
Varnum, 2015; Santos et al., 2017), decreases in infectious diseases and decreases in gender
inequality (Varnum & Grossmann, 2016), and weakening of social norms (increasing
tolerance toward minority dissent) and diversity in children names (Jackson et al., 2019;
Twenge et al., 2010). The computational model presented here seeks to contribute to
understanding of the mechanisms by which such socio-ecological factors translate into
cultural changes.
In this paper, we construct a computational model to explore how psychological,
ecological, and structural factors may interact to produce patterns of cultural change
covering the spectrum from gradual to revolutionary. Our model, which relies on
behavioral assumptions grounded in psychological research, reveals cultural change to be a
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 4
complex process in which the speed of adjustment, be it gradual or abrupt, depends on
interactions between two psychological mechanisms: error-induced cultural drift (mistakes
in copying a cultural attribute) and leniency-based indirect minority influence (changes to
related cultural attributes to match a minority trait); with network structures (connection
patterns among people). We also examine the extent to which the level of heterogeneity or
polarization in a culture depends on these interactions.
In our model, the psychological mechanism of indirect minority influence is
controlled by a leniency threshold parameter, which captures the extent to which societal
members listen to minority dissent with an open mind. Various socio-ecological factors
such as external threat, pathogen prevalence, resource scarcity, socioeconomic development,
etc. can affect leniency levels. These simulations are designed to generate insights into
whether and how much the level of leniency toward minority positions affects the likelihood
and pattern of cultural change and diversity via interactions among numerous individuals
over a long period time (Table 1). Results from simulation experiments should be
considered as testable hypotheses (Table 2), and we hope future empirical tests revise and
modify the model specification.
We begin by reviewing key theoretical frameworks and computational models of
cultural change in the literature, highlighting how network structure may facilitate or
inhibit the psychological mechanisms of cultural change. We then present a computational
model that includes key psychological mechanisms.
Our model expands on previous models that examine psychological mechanisms and
cultural emergence. Those models show that desire for coordination, for example, can
result in polarization (Axelrod, 1997) while a desire for coordination and internal
consistency can spread small errors and increase diversity in a society (Bednar et al., 2010).
While our model includes both conformity and internal consistency effects – both
well-established psychological mechanisms of cultural emergence – we focus on the relative
impact of errors and indirect minority influence on cultural change.
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 5
Table 1
Definitions of Key Concepts and Terms
Term Definition
Culture A set of ideas, beliefs, attitudes, and behaviors that are common
among a group of people.
Culture change A cultural group adopts a new idea or attitude that eventually be-
comes accepted as normative.
Punctuated equilibrium A specific pattern of cultural change characterized by long periods
of cultural stability, which is punctuated by rare instances of sudden
rapid change.
Gradualism A specific pattern of cultural change characterized by small increments
gradually and continuously over time.
Cultural drift A process by which cultural variants disappear and rare variants be-
come more frequent through small errors in the learning and memory
processes.
Indirect minority influence A minority’s influence on majority attitude change on indirect or re-
lated issues, even though the indirect issue was never discussed in the
direct (i.e., focal) minority communication.
Leniency contract An implicit agreement that stipulates that the majority attend to a
dissenting in-group minority opinion, thereby maintaining the viability
and cohesion of the group as a whole; In turn, the minority implicitly
accepts that a change in the focal belief is unlikely.
Internal consistency A cognitive process that maintains a set of attitude, beliefs, and be-
haviors consistent with one another in the cognitive system, and rebal-
ances conflicting beliefs.
Previous models explore different network structures as well. For example, models
find that cultural drift induced by transmission processes can generate rapid social change
in both complete and random networks (De et al., 2018; Muthukrishna & Schaller, 2020)
and more gradual change in a lattice network (Nowak & Lewenstein, 1996). Indirect
minority influence can generate gradual social change and diversity in both lattice and
complete networks (Jung, Bramson, & Crano, 2018; Jung, Page, Miller, et al., 2018).
Although numerous computational models reveal causal relationships between
psychological mechanisms and cultural change, no systematic analysis has been undertaken
to demonstrate whether a particular cultural pattern or rate of change can be attributed
solely to a specific psychological mechanism or some combination of them, or if these
patterns and rates of change are primarily an effect of network characteristics. Previous
models either implemented a limited set of psychological mechanisms or considered a single
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 6
network structure. And they did not apply common measures of cultural patterns and
change, making comparisons difficult.
Thus, in this paper we undertake a more systematic approach that explores the
interplay between cultural drift, indirect minority influence, and social network structures.
Our model design allows us to compare cultural drift and indirect minority influence in
terms of their effects on cultural change and diversity in different social network structures.
We measure patterns of cultural change in terms of speed (gradual vs. rapid), magnitude
(small vs. large), and frequency (rare vs. frequent) and we measure patterns of emerging
cultures using the effective number of types1(inverse Simpson index) as well as the
Hamming distance to differentiate homogenization, polarization, and diversity regimes.
Cultural Drift, Errors, and Noise-induced Transition
Cultural drift has been considered one of the primary processes of cultural change
and variation (Bentley et al., 2004; Koerper & Stickel, 1980). Analogous to genetic drift,
cultural traits can disappear, and rare variants can become more frequent through small
errors in the learning and memory processes such as forgetting or mistakes in
communication (Campbell, 1960; Singer et al., 2019; Tavris & Aronson, 2007; Vitevitch,
2008). Numerous models have discovered psychological mechanisms that spread random
errors that can shift a culture including Sherif’s latitude of acceptance (Axelrod, 1997;
Deffuant et al., 2000; Flache & Macy, 2007; Sherif, 1963) and latitude of rejection in a
coevolving network (Centola et al., 2007).
Internal consistency is a well-established psychological mechanism for spreading
small errors throughout a society, especially within groups. Bednar et al. (2010) point out
that the Axelrod (1997) culture model failed to generate intragroup heterogeneity although
it explains intergroup differences. Drawing on Festinger (1957) cognitive dissonance theory,
1A population equally distributed across Ntypes has an effective number of types equal to N(Laakso &
Taagepera, 1979). But, for example, a population in which 70% of the population is one type and the other
30% are evenly spread across three types, the effective number of types is 1.92.
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 7
Bednar and her colleagues added an internal consistency rule to Axelrod’s model. When
random errors were introduced, conformity and internal consistency mechanisms together
spread small random errors to a system and increased diversity within cultural groups.
Both stubbornness and conformity have been identified as critical factors that can
spread small errors and rare variants to a society. In the dynamical social impact model
(Nowak & Lewenstein, 1996), agents vary individually in their level of stubbornness. When
small errors are introduced to increase the stubbornness of minority agents, minority
opinions are more likely to spread through an entire system and become majority opinions.
Recently, De et al. (2018) extended Nowak and Lewenstein’s model (1996) and found
highly conformist culture produces rapid social change. Muthukrishna and Schaller (2020)
implemented stubbornness and conformity as both individual and cross-cultural differences
and found rapid social change where a minority belief quickly spreads through a society.2
Leniency Contract and Indirect Minority Influence
A rich literature has shown that majority and minority influence processes follow
different cognitive and psychological mechanisms (Baker & Petty, 1994; Chaiken, 1980;
Crano & Alvaro, 1998; Levine & Tindale, 2015; Mackie, 1987; Martin & Hewstone, 2001;
Moscovici et al., 1980; Mugny et al., 1995; Nemeth, 1986). However, previous
computational models implemented single processes (e.g., De et al., 2018; Muthukrishna &
Schaller, 2020; Nowak & Lewenstein, 1996).
A first attempt to model the dual process perspective of minority and majority
influence focused on indirect minority influence (Jung & Bramson, 2014, 2016; Jung,
Bramson, & Crano, 2018). First described by Moscovici et al. (1969), indirect minority
influence takes place when the majority listens to a minority’s opinion about some focal
issue (e.g., the color of an image, messages about pollution, attitudes toward foreigners),
2This occurs in about 50% of their simulations when the minority belief is assigned to a maximally
stubborn agent and the rest of the agents are highly susceptible to social influence. If the minority agents
get just slightly less stubborn or the other agents are less susceptible, the likelihood of social change drops
significantly.
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 8
and although the majority doesn’t immediately change their opinion on the focal issues,
they do change their opinion in unmentioned, related issues (Moscovici et al., 1980;
Moscovici et al., 1969). For this to happen, a minority message should contain fair,
reconciliatory, and consistent points of view rather than rigid, dogmatic, and extreme
points of view (Moscovici et al., 1980). For example after listening to a minority message
advocating a policy to force automobile producers to equip their vehicles with antipollution
filters, individuals did not become more likely to support that policy, but did become more
likely to support other ways to reduce air pollution. Similar indirect minority influence was
observed in the context of attitudes toward foreigners (Moscovici et al., 1980).
Ample empirical studies have confirmed the validity of the indirect minority
influence mechanism (Brandstätter et al., 1991; De Dreu & De Vries, 1993; De Vries et al.,
1996; Pérez & Mugny, 1996), and meta-analysis indicates that minority influence evokes
greater indirect change, whereas majority influence is greater on direct/focal issues (Wood
et al., 1994).
The psychological mechanisms of indirect minority influence was explicated in
Crano’s (2012) leniency contract theory. When minority belief holders are ingroup
members and their messages are not threatening to the ingroup’s identity (Abrams &
Hogg, 1990; Chenoweth & Stephan, 2011), they are influential due to a leniency contract:
an implicit agreement stipulating that the majority will attend to a dissenting ingroup
minority opinion, thereby maintaining the viability and cohesion of the group as a whole.
In turn, the minority implicitly accepts that a change in the focal belief is unlikely.
Note that even though direct focal attitude change is unlikely, the majority’s
open-minded tolerance of the minority position creates cognitive pressure that can result in
a change of related attitudes within the same belief system, in the direction of the thrust of
the minority’s argument. As indirect changes accumulate, delayed focal change can occur
due to a tendency to maintain internal consistency within the belief system (Dalege et al.,
2018; Festinger, 1957).
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 9
The leniency contract theory has been empirically validated (Alvaro & Crano, 1996,
1997; Crano, 2010; Crano & Alvaro, 1998; Crano & Chen, 1998). For example, individuals
after listening to a minority message about euthanasia (focal issue) were more likely to
change their attitude toward abortion (unmentioned in the minority message but related to
euthanasia according to multi-dimensional analysis) to the direction of the minority
position (Alvaro & Crano, 1996, 1997), and one week after the messages were delivered, the
focal attitude also changed to the direction of the minority message (Crano & Chen, 1998).
The ability of indirect minority influence and internal consistency to create gradual
and persistent social change was validated using a computational model (Jung & Bramson,
2014, 2016; Jung, Bramson, & Crano, 2018). These cultural patterns appear without
introducing errors. A recent refinement of the model parameterizes a leniency threshold
(Table S1) in small complete networks and finds that leniency that is just slightly above a
tipping point (>0.50) is enough to ensure a maximal benefit of minority influence. When
the leniency threshold reaches the tipping point, an initial minority opinion can gradually
spread to achieve social change and then maintain approximately equal-sized factions that
switch their majority and minority status causing small yet frequent changes (Jung et al.,
2017; Jung, Page, Miller, et al., 2018).
Network Structure
The structure of a network has strong effects on information diffusion (Centola,
2010; Dodds & Watts, 2004; Moreno et al., 2004; Watts, 2002). Despite that, agent-based
models of culture change generally rely on one of two social network structures.
Computational models that address the dynamics of large societies generally rely on a
square lattice with the top and bottom connected and the sides connected so as to form a
torus (Axelrod, 1997; Bednar et al., 2010; Jung, Bramson, & Crano, 2018; Nowak &
Lewenstein, 1996; Pulick et al., 2016) while computational models of small group dynamics
rely on complete networks where every agent interacts with every other agent (Jung, Page,
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 10
Miller, et al., 2018; Singer et al., 2019). Mathematical models of large societies can also
assume complete networks for reasons of analytical tractability (De et al., 2018).
Some models do analyze more complex networks and others include network
generation mechanisms. Centola et al. (2007) included a network coevolution mechanism
starting with a square lattice, but evolving through a random network and a modular
network to a network with multiple disconnected communities. Muthukrishna and Schaller
(2020) proposed a novel approach to cultural networks: their model includes a separate
phase that generates cultural networks based on different distributions of extraversion,
which results in a family of random networks. Luhmann and Rajaram (2015) tested
information transmission mechanisms on three distinct networks: a small world network
(Watts & Strogatz, 1998), Zachary’s karate club network that has two communities, each
with a leader (hub node) and followers, and a coauthorship network with multiple
communities that vary in their internal network structure and the degree of connection
with other communities (Newman, 2006).
Each of these studies relies on idiosyncratic networks, thus it is difficult to attribute
the degree of diffusion to any particular network characteristic because these networks
either co-evolve or vary in multiple dimensions (size, density, topology, community
structure, etc.).
To address this limitation, we model a range of multi-agent network systems
(Figure 1). Different network structures can facilitate or inhibit the spread of minority
positions or rare cultural variants. In complete networks in which everyone is connected,
minority positions can spread rapidly because all agents can be exposed to minority
positions. Minority influence also can be blocked through majority influence because high
connectivity inhibits minority agents from forming cliques. Scale-free networks contain a
few hubs (highly popular and influential nodes with high degree connections) and many
low degree nodes. If rare cultural variants are adopted by hubs, they can quickly spread to
other nodes connected to them. However, because innovation depends on a small number
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 11
of agents, mainly hubs, changes may not be able to occur frequently.
(a) Complete Network (b) Scale-free Network (c) Ring Lattice Network
(d) Square Lattice Network (e) Small World Network (f) Modular Network
Figure 1
Examples of multi-agent network systems with varying structures.
In lattice networks, all agents have the same node degree. The diffusion of
information depends more on structural characteristics than node characteristics. A square
lattice has local structures that can preserve minority opinions in local cliques. A ring
lattice has a high clustering coefficient that preserves minority opinions because ideas that
are globally in the minority can be locally in the majority within clusters, but its long
average path length inhibits diffusion.
In small world networks, information can spread more effectively than in ring
lattices because they have the same high clustering coefficient as ring lattices but have
short average path lengths via long ties that can facilitate the spread of information across
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 12
the network (Granovetter, 1983; Watts & Strogatz, 1998). A small world network begins as
a ring lattice and then a proportion of its connections are rewired at random. Small world
characteristics emerge with the rewiring probability between 0.001 and 0.1 (e.g., the
rewiring probability of the small world network in Fig. 1e is 0.005). As the rewiring
probability gets larger than 0.1, the network loses small world characteristics and comes to
resemble a random network where high connectivity inhibits minority agents from forming
local clusters.
Finally, in modular networks, minority opinions can form local majority
communities and be isolated within dense communities (Weng et al., 2013); however,
sparse connections between communities may inhibit the spread and speed of information
transmission (Galstyan & Cohen, 2007; Girvan & Newman, 2002; Gleeson, 2008).
Additional network structures and properties exist, too many to canvas here, but the ones
above capture the bulk of those used in agent-based models of information transmission.
The Model
We now describe an agent-based model which was designed to permit analysis of the
interactions between psychological mechanisms and network structure on the rate of
cultural change and diversity.
Initial Setup: Agent-properties and Networks
Each simulation begins with a population of agents connected to one another to
form a social network (recall Fig. 1). Complete networks have 20 agents as nodes, and all
other networks have 200 agents as nodes. Complete networks have n2−nedges and each
node has degree n−1. For the other networks, each node has mlinks; therefore, these
networks have n×nlinks and an average node degree of 2m. However, individual nodes’
degrees vary depending on the network generation algorithm (see Table S1 for details of
model parameters).
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 13
Each agent jhas attitudes on ten issues, and each attitude itakes binary values;
thus, there are 210 = 1024 possible attitude profiles. Adjacent attitudes (i−1,i+ 1) are
considered issues related to a focal issue i(e.g., euthanasia and pro-choice; ban on
homosexual soldiers and gun control, Alvaro & Crano, 1997). The first and last attitudes
are considered as related attitudes to resolve a potential edge effect. We regard attitudes as
internally consistent or cognitively balanced if the adjacent attitudes take the same value
as the focal one (000 or 111).
The initial distribution of attitudes is the same across network types: 80%of agents
have all ten attitudes with the value of 1 (initial majority), and 20%of agents have all ten
attitudes with the value of 0 (initial minority). We use the color of the agents to represent
the average of their ten attitudes as shown in Fig. 2. The colors range from dark blue to
deep yellow with increasing numbers of 1s in their attitude list. When an agent is a centrist
with an equal number of 0s and 1s, it is colored white.
Figure 2
The visualization of agents’ mean attitude values from dark blue representing zero 1s and
moving toward yellow with increasing numbers of 1s.
Simulations and Interaction Rules
Once the initial setup configuration is completed, at each iteration agents follow an
attitude updating algorithm shown diagrammatically in Fig. 3. Cultural drift is
parameterized by an error rate (ε) and indirect minority influence with a leniency
threshold (λ), which vary from 0 to 1 in increments of 0.05 (see Table S1 in Supplementary
Materials for the model parameters). We run 100 simulation experiments for each
combination of parameters, which sum up to total of 73,800 simulation experiments.
The model progresses in discrete time steps; in each step each of the populated
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 14
Figure 3
Attitude updating algorithm
agents was selected in a random order without replacement until all agents were selected.
When selected, an agent may potentially change an attitude. To do so, it first randomly
chooses what we call a focal attitude i. Whether that attitude will change depends on the
error rate, leniency threshold, and the agent’s neighbors. The active agent’s network
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 15
neighbors are considered its ingroup, and majority and minority positions are defined in
terms of that ingroup for each agent.
Errors occur with a probability ε, and when an error occurs it simply flips the value
of the focal attitude. These errors capture spontaneous changes that can happen during
interaction, communications, and influence.
If no error occurs and there exists a minority agent in the ingroup that is consistent
in its focal and related attitudes (i−1 = i=i+ 1), then with a probability equal to the
leniency threshold λ, the active agent changes one of its related issues (i−1or i+ 1) to the
minority position. Otherwise, the active agent conforms to the majority position on the
focal issue i.
Lastly, all agents randomly pick an attitude kin their attitude set and make it and
its adjacent attitudes consistent. For example, if the three adjacent attitudes (k−1,k,
k+ 1) are {0 1 1}, {1 0 1}, or {1 1 0}, they become {1 1 1}; and if they are {0 0 1}, {0 1
0}, or {1 0 0}, they become {0 0 0}.
Patterns of Social Change
The patterns of cultural change are examined in terms of magnitude (small vs.
large), speed (gradual vs. rapid), and frequency (rare vs. frequent). Diversity and
polarization in a society are examined in terms of global cultural variation (inverse
Simpson index) and local neighborhood difference (Hamming distance).
Magnitude, Speed, and Frequency of Social Change. Simulation runs are
visualized via plots of stacked time series of the percentages of blue agents, centrist agents,
and yellow agents. Patterns of social change are examined in terms of three dimensions:
magnitude (small vs. large), speed (gradual vs. rapid), and frequency (rare vs. frequent).
These simulations display two distinct patterns of cultural change: punctuated equilibrium
and gradualism.
Punctuated equilibrium is a concept developed in evolutionary biology to refer to a
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 16
specific pattern of species evolution – the population is stable, showing little evolutionary
change for long periods of time and then rapid evolution and speciation occurs (Gould &
Eldredge, 1977). “Sudden jumps” can be attributed to rare rapid geographical events,
macro mutations, rapid episodes of gradual evolution, or isolation effects (Mayr, 1954).
Cultural punctuated changes have been understood as results of rapid environmental shifts
(Kolodny et al., 2015) or intrinsic dynamics of complex systems (Epstein, 2009).
Gradualism is often observed in human society and politics (Marquis, 1947). The
middle class was gradually created over the course of the industrial revolution. Women’s
rights have changed (and continue to change) gradually through the suffrage movement
and laws giving women equal opportunity for education, jobs, and income. Gradual
changes have also occurred for gay and transgender rights. Small frequent changes are
often observed in most democratic nations where the general public prefers one political
group to another by a small margin.
Likelihood of Social Change. Simulations stop when (a) the percent of blue
agents equals or exceeds the percent of yellow agents (i.e., the initial minority becomes the
majority), (b) the system converges to all yellow, or (c) a simulation runs for 10,000 steps.
The occurrence of social change (0=no social change, 1=social change) and the number of
steps taken to achieve social change are visualized with binary logistic regression lines and
boxplots, respectively.
Spread of Minority Opinions. The prevalence of the initial minority position
(0s, blue) at the termination of a simulation run is calculated as the percent of 0s in the
total attitude pool. In addition, the distributions of majority and minority attributes on
different networks are also visualized.
Diversity Measures
We use two diversity indices to capture patterns of heterogeneity (Bramson et al.,
2016; Bramson et al., 2013; Page, 2008, 2010). The first is the effective number of types
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 17
(Inverse Simpson Index). Originally developed as a biodiversity index to measure the
number of species in an ecosystem, it was adopted by the social sciences to measure the
effective number of parties (Laakso & Taagepera, 1979). For our model, the inverse
Simpson index ranges from 1 to 2: 1 indicates that a single type exists, i.e., no diversity,
and 2 indicates 0s and 1s are equal in frequency (i.e., maximal diversity).
The effective number of types does not differentiate diversity (two attitudes are
mixed) from polarization (two attitudes are segregated) because it only takes account the
number of types and ignores how they are distributed in a network. We therefore also use
Hamming Distance (Hamming, 1950). This measures local neighborhood diversity by
counting the number of attitudes where two agents have dissimilar values. If two agents
have exactly the same attitude composition (e.g., 0110111011), the Hamming distance for
the pair is zero. If two agents have different attitude compositions, then even if they have
the same number of 1s and 0s, the distance is positive (e.g., agents 1011111101 and
1111011011 have distance 4 because the differ in 4 positions).
To create a local diversity score, the Hamming distance of all pairs of connected
agents is calculated and averaged by each agents’ total number of neighbors, and then
divided by the number of attitudes. The average standardized Hamming distance can
range from 0 to 1; 0 indicates all agents have the same attitudes and the group is
homogenized, 0.5 indicates maximal local diversity where agents are connected with similar
and different agents, 1 would be a checkerboard-like pattern of alternating maximally
different types (which is impossible in networks with clustering).
The combination of the effective number of types and Hamming distance can
identify three types of regimes. Diversity regimes have close to two types and a large
Hamming distance, indicating blue and yellow agents are intermixed. Polarization regimes
also have close to two types but a small Hamming distance, indicating similar numbers of
blue and yellow agents are segregated. Majority regimes have closer to one type and a
small Hamming distance, indicating either blue or yellow agents are dominant.
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 18
Results
Our computational model reveals rich and complex patterns of culture and culture
change. The results from our computational experiments indicate that network structures
affect the impact of the psychological mechanisms on the likelihood and patterns of
cultural change.
First, network connectivity is a critical factor for cultural drift’s ability to produce
social change. Highly connected networks have short average path lengths so that
information transmits quickly. In highly connected networks such as complete, scale-free,
and random networks, cultural drift generates a punctuated equilibrium pattern of social
change characterized by spurts of rapid change with long periods of stability (Table 2).
For example, consider the case of cultural drift on random networks (small
clustering coefficient and short path length). With errors smaller than 0.60, conformity and
internal consistency mechanisms quickly correct errors and homogenize the attitude pool of
a small group. With a large error rate (ε > 0.60), conformity and internal consistency are
no longer able to correct all the errors, and instead rapidly spread uncorrected errors to the
group (Fig. 4e), and changes occur unpredictably at tipping points (Fig. S6A.11b &
S6A.14b). The group stays in either a blue or yellow majority regime with a small number
of persistent minority dissents for a long time, but occasionally a rapid change occurs
causing a large swing from one majority regime state to the other. This pattern is
characterized by punctuated equilibrium.
The impact of indirect minority influence on the likelihood and patterns of cultural
change are robust across different network structures. Indirect minority influence generates
a gradualism pattern: a gradual spread followed by a diversity regime with small frequent
changes.
With a low leniency threshold (λ < 0.50), the indirect minority influence mechanism
cannot spread minority opinions, even in combination with the internal consistency
mechanism that increases consistent minority dissents. Instead, the conformity mechanism
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 19
(a) Error rate = 0.45
Gradual change with large swings
Polarization Regime
(b) Leniency threshold = 0.55
Gradual change with small changes
Polarization Regime
(c) Error rate = 0.45
Gradual change with large swings
Polarization regime
(d) Leniency threshold = 0.55
Gradual change with small changes
Polarization Regime
(e) Error rate = 0.60
Punctuated equilibria
Between two stable majority regimes
(f) Leniency threshold = 0.55
Gradual change with small changes
Diversity regime
Figure 4
The patterns of social change at tipping points in a ring lattice (a & b), a small world
network with a rewiring probability of 0.005 (c & d), and a de facto random network using
a rewiring probability of 0.25 (e & f) (n= 200,m= 4).
homogenizes the attitude pool. Social change does occur when the leniency threshold
reaches just slightly over 0.50, meaning that group members are lenient toward minority
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 20
dissents and listen to minority voices at least half of the time (Fig. 4f).
We see a similar gradual effect in ring lattices and small world networks, a minority
cluster emerges and gradually expands to half the population, forming a
polarization/segregation regime (Fig. 4a & 4b). In square lattices, a minority cluster is
distributed somewhat amorphously throughout the network, increasing local diversity (Fig.
S5B.4).
Tipping points for cultural drift vary depending on the network connectivity level.
For example, as a network becomes more connected with an increasing rewiring probability
in Fig. 4, it reinforces major influence and much higher error rates are required for social
change (ε= 0.60 vs. ε= 0.45). However, tipping points for indirect minority influence are
similar regardless of the network structure or connectivity (λ= 0.55).
Network modularity (how cleanly the network divides into internally dense clusters)
appears important for successful social change because the community-based network
structure allows minority opinion holders to form protected and persistent opinion clusters.
For example, when a society is not lenient but highly prejudiced toward minority
(λ= 0.05), a community structure can protect a minority group to be able to persist
within their community relatively isolated from other communities (Fig. S7.2). As a
society becomes more lenient and reaches the tipping point (λ= 0.55), indirect minority
influence can generate gradual social change with a diverse society. In contrast, cultural
drift with an error rate of 0.55 generates punctuated equilibrium pattern of social change
with a polarized society (Fig. S7.1). When modularity decreases (e.g., through increased
rewiring in small world networks, or more intergroup connections in community structure
networks) social change becomes unlikely via cultural drift but remains possible via
indirect minority influence given a sufficient leniency threshold (Fig 4 & Fig. S7.2).
Network density (parametrized by m, the number of edges per node), is also a
critical factor for the effect of psychological mechanisms on social change. For example, in
scale-free networks, when a network is sparse (m= 1), small errors (ε= 0.10) are enough
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 21
Table 2
Key Results
Network Psychological Mechanisms Pattern of Cultural Change Regime Type
complete cultural drift punctuated equilibrium alternating majority
indirect minority influence gradualism with small frequent change diversity
scale-free cultural drift punctuated equilibrium pol./alt. majority
indirect minority influence gradualism with medium change polarization/diversity
ring lattice cultural drift gradualism with large change polarization
indirect minority influence gradualism with small change polarization
square lattice cultural drift gradualism with large change diversity
indirect minority influence gradualism with small frequent change diversity
small world cultural drift gradualism with large change polarization
indirect minority influence gradualism with medium change polarization
random cultural drift punctuated equilibrium alternating majority
indirect minority influence gradualism with small frequent change diversity
modular cultural drift punctuated equilibrium polarization
indirect minority influence gradualism with small frequent change diversity
Note. Sparse scale-free networks (m= 1) generate polarization regimes regardless of psychological mech-
anisms due to their modularity. Denser scale-free networks (m > 1) generate regime types similar to
other highly connected networks (complete, random) due to short path lengths driven by hubs.
for social change to occur (Fig. S4A.1). When errors accumulate enough to change a hub’s
position to the minority one, the many nodes linked to the hub also rapidly adopt minority
positions. In this way innovation is adopted at the cluster level, and globally minority
belief holders can form persistent cliques because they are a local majority within the
clique. As a network becomes denser (m= 4), its cluster structure decreases and its
average path length becomes shorter. This reinforces conformity and much higher error
rates (ε= 0.65) are required for social change (Fig. S4A.7).
Social change via indirect minority influence, however, is less sensitive to the density
of scale-free network, as evidenced by the parameter values (λ= 0.50 −0.55) at which social
change occurs being less affected by network density (Fig. S4B.1 & S4B.7). In a sparse
scale-free network, the minority belief agents form small persistent local-majority clusters.
Because hubs influence other hubs, accumulated attitude changes in one hub can turn the
neighboring hub to the minority position, which then spreads the minority opinions to its
followers. Because hubs are early adopters of innovation and connected to other hubs, hubs
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 22
tend to have moderate centrist positions and followers tend to have more extremist
positions (S4B.1). When network density increases, diversity increases in a society (S4B.7).
We summarize the simulation results for each network across variations of both the
error rate and the leniency threshold at their tipping points (Table 2). A complete
characterization of the results can be found in the supplementary materials.
Discussion
The current model suggests that the patterns of cultural change (in terms of
magnitude, speed, and frequency), and the level of diversity, can be predicted to a degree.
The precise time required for change to occur is difficult to predict, particularly around
tipping points in the parameters. But the further the error rates and/or leniency levels
deviate from tipping points, whether and when social change occurs become more
predictable. However, the observation that parameter tipping points vary depending on
network structure implies that successful prediction depends on knowing the network
structure of a society.
The present study unravels the seemingly conflicting results of previous
computational work by synthesizing network science and psychological mechanisms of
cultural change from different theoretical perspectives. There are several ways that the
model might be extended. First, although the current model has a fairly large number of
attitudes which are related and affect one another via indirect minority influence and
internal consistency, the cognitive structure of multiple attitudes are simple in this model,
whereas they can be quite complex and vary across cultures (Brandt et al., 2019). Second,
in this work as in earlier literature errors are introduced as a random process. However,
some information can be more easily forgotten, or less accurately communicated than other
information (Henrich, 2001; Singer et al., 2019). Future work should address the impact of
various biased error processes on cultural change.
Lastly, the present model examines the impact of leniency and error rate separately.
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 23
However, the two mechanisms can operate simultaneously and might interact. Because
indirect minority influence can spread newly introduced errors to a system, smaller tipping
points may be observed for social change. Jung and colleagues (2018) explored how
cultural drift and indirect minority influence operate simultaneously in a small complete
network, observed slightly smaller tipping points, and found that the two mechanisms
generated regime shifts between diversity and homogeneous majority. Future research is
needed to better understand this interaction effect in various network structures.
Before we can hope to apply a model of this sort to improve our ability to predict
culture change, or even gain a fuller understanding of whether such prediction is possible,
we need to determine several key factors:
1. Leniency thresholds: how lenient people are in a society toward minority opinion
holders; how often people listen to dissenting voices; how cultures, organizations, and
groups vary in their leniency toward minorities (Gelfand et al., 2011); how much
various socio-ecological factors (e.g., pathogen prevalence, resource scarcity) affect
leniency levels (Varnum & Grossmann, 2017).
2. Error rates: how frequent are the kinds of errors employed here; how many errors
people make in their communications (Vitevitch, 2008); how specific/universal speech
errors are across different cultures; if there is feedback between network structure and
error rates.
3. Network characteristics: how people in various cultures are connected with their
acquaintances, colleagues, friends, and family members; how modular the network
structures of different societies are; how ecological and cultural variations (e.g.,
relational mobility) affect the evolution of network structure.
Many complex systems are highly unpredictable: epidemics, economic bubbles, riots,
revolutions, rumor propagation, ecologies, stock markets, traffic jams, urban environments,
weather, etc. (Miller & Page, 2009). Weather prediction uses real-time large-scale data
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 24
collection from numerous weather stations, then computational forecast models divide the
physical space into high resolution grids of a few kilometers, ensembles of various forecast
models are used to reduce errors, and these simulations take hours to complete. But in
spite of the advanced state of this art, weather prediction becomes dramatically less
accurate as one increases the forecast time. Compared to weather systems, the prediction
of social systems is severely limited. In the case of epidemic transmission, for example, less
data are available for modeling and predicting transmission, people change behaviors in
response to their own or others’ illness, modes of transmission vary by illness, relevant
contact networks are not well understood at the societal level, and there are multiple illness
and comorbidities at play all the time. Although epidemic transmission has been heavily
studied by an ensemble of mathematical and agent-based models (Anderson & May, 1992;
Epstein, 2009), disease spread remains difficult to predict with even moderate accuracy.
Drawing on lessons learned from forecasting in these and other complex systems, we
need to be cautiously optimistic about forecasting any social phenomenon. Prediction of
cultural change can certainly be improved by frequently obtaining attitude data from
cultural pockets throughout a society to build time series datasets of attitude changes. In
addition to attitude data, we need to collect and assemble social network characteristics
and geospatial information to understand and properly model the lines of transmission.
Throughout these exercises, it is important to keep in mind that a single model
cannot include all relevant parameters and mechanisms. Different models can explain
different dynamics, and hence a diverse ensemble of models can reduce errors and improve
prediction (Page, 2018). Our model’s demonstration of cross-level causal links from
minority influence to social change can be interpreted in the context of an ensemble of
mathematical (Chenoweth & Belgioioso, 2019) and agent-based models that address similar
questions (De et al., 2018; Jung, Bramson, & Crano, 2018; Jung, Page, Miller, et al., 2018;
Muthukrishna & Schaller, 2020; Nowak & Vallacher, 2001).
Agent-based modeling can be particularly useful for predicting complex social
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 25
dynamics and culture change in two ways. First, ABM generates whole distributions of
outcomes with the full range of parameter values, rather than making point predictions.
With rich simulation data, researchers can examine trends in patterns of change and
heterogeneity across society at various time scales and report a distribution of more and
less likely outcomes along with the paths to those outcomes. This richer data allows one to
identify key events and conditions that differentiate among those outcomes, such as tipping
points, thresholds, black swan events, and in some cases inevitable conclusions. Generative
models such as ours are well-suited to produce this kind of rich data.
Second, ABM provides informed intuition by exploring possibilities where
experience is lacking. In social systems, the future is never just like the past, so prediction
cannot be fully based on historical data nor extrapolated from simple models. The
behaviors revealed through generative models inform us of which mechanisms are, and are
not, capable of yielding particular social dynamics. They can be used to test the outcomes
of multiple policy changes before committing to any one of them. They may never reach
the accuracy of experiments in physics or chemistry, but by leveraging the insights gained
through models like this one, we can make our educated guesses a bit more educated.
CULTURAL DRIFT AND INDIRECT MINORITY INFLUENCE 26
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