Simpler grammar, larger vocabulary: How population size affects language

Abstract

Languages with many speakers tend to be structurally simple while small communities sometimes develop languages with great structural complexity. Paradoxically, the opposite pattern appears to be observed for non-structural properties of language such as vocabulary size. These apparently opposite patterns pose a challenge for theories of language change and evolution. We use computational simulations to show that this inverse pattern can depend on a single factor: ease of diffusion through the population. A population of interacting agents was arranged on a network, passing linguistic conventions to one another along network links. Agents can invent new conventions, or replicate conventions that they have previously generated themselves or learned from other agents. Linguistic conventions are either Easy or Hard to diffuse, depending on how many times an agent needs to encounter a convention to learn it. In large groups, only linguistic conventions that are easy to learn, such as words, tend to proliferate, whereas small groups where everyone talks to everyone else allow for more complex conventions, like grammatical regularities, to be maintained. Our simulations thus suggest that language, and possibly other aspects of culture, may become simpler at the structural level as our world becomes increasingly interconnected.

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Research
Cite this article: Reali F, Chater N,
Christiansen MH. 2018 Simpler grammar,
larger vocabulary: How population size affects
language. Proc. R. Soc. B 285: 20172586.
http://dx.doi.org/10.1098/rspb.2017.2586
Received: 16 November 2017
Accepted: 2 January 2018
Subject Category:
Neuroscience and cognition
Subject Areas:
computational biology, evolution, cognition
Keywords:
cultural evolution, language change, social
structure, population size, language complexity
Author for correspondence:
Morten H. Christiansen
e-mail: christiansen@cornell.edu
Electronic supplementary material is available
online at https://dx.doi.org/10.6084/m9.
figshare.c.3971847.
Simpler grammar, larger vocabulary: How
population size affects language
Florencia Reali1, Nick Chater2and Morten H. Christiansen3,4
1
Department of Psychology, Universidad de los Andes, G230, Cra. 1 Nro. 18A-12, Bogota
´11001000, Colombia
2
Behavioural Science Group, Warwick Business School, University of Warwick, Coventry CV4 7AL, UK
3
Department of Psychology, Cornell University, Uris Hall, Ithaca, NY 14853, USA
4
The Interacting Minds Centre and School for Culture and Communication, Aarhus University, 8000 Aarhus,
Denmark
FR, 0000-0003-3524-3873; NC, 0000-0002-9745-0686; MHC, 0000-0002-3850-0655
Languages with many speakers tend to be structurally simple while small
communities sometimes develop languages with great structural complexity.
Paradoxically, the opposite pattern appears to be observed for non-structural
properties of language such as vocabulary size. These apparently opposite
patterns pose a challenge for theories of language change and evolution.
We use computational simulations to show that this inverse pattern can
depend on a single factor: ease of diffusion through the population. A popu-
lation of interacting agents was arranged on a network, passing linguistic
conventions to one another along network links. Agents can invent new con-
ventions, or replicate conventions that they have previously generated
themselves or learned from other agents. Linguistic conventions are either
Easy or Hard to diffuse, depending on how many times an agent needs to
encounter a convention to learn it. In large groups, only linguistic conventions
that are easy to learn, such as words, tend to proliferate, whereas small groups
where everyone talks to everyone else allow for more complex conventions,
like grammatical regularities, to be maintained. Our simulations thus suggest
that language, and possibly other aspects of culture, may become simpler at
the structural level as our world becomes increasingly interconnected.
1. Introduction
It has often been observed [1–4] that the properties of human languages appear
to be influenced by the size and degree of isolation of the linguistic community.
Small, isolated linguistic communities often develop languages with great
structural complexity, elaborate and opaque morphology, rich patterns of
agreement and many irregularities [1– 5], and it has been argued that such
‘mature’ features of languages require long interactions in small, close-knit
societies [6–8]. By contrast, languages with large communities of speakers,
such as Mandarin or English, appear to be structurally simpler. Language com-
positionality has been shown to be inversely correlated to irregularities and
nonlinear morphology [3]: regular languages are more frequent in large-sized
communities, while irregular, morphologically complex languages tend to
arise in small-sized ones. Computer simulations have shown that linguistically
‘marked’, and hence complex, patterns arise more easily in small populations
[9,10] and that compositional structures tend to emerge more extensively for
larger groups [11]. The causal role of the size of the linguistic community is,
moreover, further indicated by the historical tendency towards structural
simplification as a language gains an ever-larger community of speakers [12].
But an apparently opposite pattern appears to be observed in relation to
non-structural properties language: languages with large linguistic commu-
nities tend to have larger vocabularies of content words. For example, the
vocabulary of wide-spread languages, such as English, appears to have
&2018 The Author(s) Published by the Royal Society under the terms of the Creative Commons Attribution
License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original
author and source are credited.

grown rapidly in historical times, and is typically estimated
to have many hundreds of thousands of words, including
those with highly specialized and technical meanings [13].
Despite their frequent spectacular structural complexity,
languages spoken by small bands of hunter –gatherers are
typically assumed to have smaller vocabularies, although
reliable data for such languages are difficult to gather [14].
An analysis of Polynesian languages indicates, moreover,
that larger linguistic communities both create more new
words and lose fewer existing words over time [15]. These
contrasting patterns pose a challenge for theories based on
the cultural evolution of language. Recently, theorists have
suggested the erosion of complexity in larger language com-
munities arises from the greater proportion of second
language learners [1,16]. But why do such arguments for sim-
plification not also apply to the lexicon?
One possibility is that structural and lexical aspects of
language might diffuse through different mechanisms. For
example, adult– child interactions might be the primary vehicle
for regularizing morphology or syntax (see [17,18] for contrast-
ing perspectives) and adult–adult interactions might be the
primary vehicle for lexical innovations. Moreover, there may
be differential impactsof language contact on structural andlex-
ical aspects of language: lexical items diffuse across languages
more readily [19]. Such effects might be amplified to the
extent that structural and lexical aspects of language share a
fixed communicative burden, so that, for example, simple
morphology must be compensated by a larger vocabulary.
While such factors may play a role, we focus on a more par-
simonious alternative: that the opposite relationships between
population size and lexical versus structural complexity
depend on a single parameter: ease of diffusion. Structural
aspects of language diffuse slowly because they are difficult to
learn, are absorbed slowly and piecemeal by first language
learners, and often present persistent challenges for second
languagelearners [20]. Words, by contrast, can oftenbe acquired
from just a few exposures [21]. An account based on ease-of-
learning suggests that an increasing population of speakers
should, even within the category of words, lead to an increasing
prevalence of easy-to-learn words (e.g. concrete words) over
hard-to-learn words (e.g. abstract words). A recent corpus
analysis of two centuries of American English does, indeed,
show an increasing proportion of concrete words [22].
To illustrate this scenario, we divide properties of language,
as a first approximation, into two basic categories—Easy and
Hard—requiring, respectively, few or many exposures to be
acquired by a new speaker. Easy properties of the language
can rapidly be transmitted across the linguistic community.
As the community grows in size, so does the number of
members who can spontaneously modify or invent new
Easy properties (such as lexical items) that can diffuse across
the community. Hence, large communities will end up with
large inventories of Easy features. Conversely, in large linguis-
tic communities, speakers will have minimal interactions with
many other speakers, so that typical interactions between indi-
viduals will be too limited to transmit the Hard linguistic
property successfully.
If correct, this simple mechanism should apply to cultural
evolution more broadly. Indeed, new and structurally complex,
and difficult to acquire, cultural forms develop in small, tight-
knit communities who interact intensely, as in the birth of
bebop in 1940sNew York [23], or thelindy hop at the Savoy Ball-
room in 1930s Harlem [24]. By contrast, mass cultural forms
tend to be structurally simple and easily learned. For example,
we are now exposed to, and recall, a huge number of popular
tunes; but most are harmonically and melodically simple; and
statistical analysis suggests that modern popular music appears
to be gradually getting simpler over the decades [25].
Can these intuitions be made precise by computer simu-
lation? Building on prior preliminary work [26], we created
a novel innovate-and-propagate (IAP) process, operating
over populations of simulated agents. Agents are arranged
on a network, so that agents connected by a link on the net-
work can ‘converse’ and hence, potentially pass linguistic
‘conventions’ to one other. Each agent is not only able to
‘invent’ entirely new conventions but can also replicate con-
ventions that they have previously generated themselves or
learned from other agents (i.e. agents to which they are con-
nected by links in the network). When an agent produces a
convention (whether novel or a replication), it propagates
that convention to one of its neighbours.
Our simulations show that the size of the network can
potentially have opposite effects on the richness of different
aspects of the language. A simple quantitative change—the
ease of learning of an item—responds qualitatively in entirely
different ways to population size. Linguistic innovations that
are relatively easy to learn (such as new lexical items or modifi-
cations to existing ones) increase in number as a linguistic
community grows, because the number of potential innovators
increases and innovations can spread more rapidly. By contrast,
small linguistic communities favour linguistic innovations that
are hard to learn (such as, we suggest, structural changes in
the language), because they require multiple interactions
between individual speakers for their continued existence.
2. Simulations
(a) The model
To capture the dynamics of individuals interacting with one
another, either conversing by way of old conventions or
inventing new ones, we use a modified version of the Chinese
restaurant process [27], which we call the IAP process. The
Chinese restaurant process is a widely used probabilistic
model defining the frequency distribution over a potentially
limitless number of types (e.g. linguistic conventions,
words, categories). It embodies the assumption that the
‘rich-get-richer’—the probability of a token of an existing
type is proportional to its current frequency (i.e. the chance
of the new diner sitting down at a given table is proportional
to the number of diners already at that table), while also
allowing the creation of new types (i.e. a diner being seated
at a previously unoccupied table).
In our extension to the IAP, we view each agent as
corresponding to a ‘restaurant’ with a finite, but infinitely
extendable, number of ‘tables’, i.e. conventions. Each time
the agent generates a convention, it chooses an existing con-
vention with a probability proportional to the number of
previous tokens of that convention; this is equivalent to seating
each new customer in the restaurant at a table in proportion to
the number of customers already seated at that table. But it is
also possible that an entirely novel convention will be gener-
ated (a new table in the restaurant is created, and the new
customer becomes the first person sitting at that table). This
occurs with probability 1/(Mþ1) (where Mis the number
of current restaurant customers – stored tokens).
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2

As described thus far, each agent generates conventions
entirely independently, not sharing those conventions with
the rest of the linguistic community. IAP introduces a
simple extension of the Chinese restaurant process to deal
with this. At each iteration, every agent ‘utters’ a convention
and passes it to a randomly chosen immediate neighbour. For
each agent, the probability of generating an existing conven-
tion is determined by the sum of the number of times that it
has, itself, previously generated that convention added to the
sum of the number of times it has received that convention
from an immediately neighbouring agent ( provided the
agent has already learnt that convention). Thus, in this
model, agents tend not merely to generate what they have
generated before; but also to generate what they have
‘heard’ (and learned) from neighbouring agents.
As the simulation progresses, agents will invent conven-
tions and pass them on to each other. Thus, initially the
number of conventions used by the agents (i.e. the complex-
ity of the language) will gradually increase. However, the
number of conventions is limited by restrictions in cultural
transmission. Two versions of information transmissions are
implemented. In the horizontal transmission version, conven-
tions are passed among immortal but forgetful peers. Each
time an agent picks a convention (new or old), then, for
each of the M convention tokens that it currently stores,
there is a probability that this token is forgotten. In the verti-
cal transmission version, peers eventually ‘die-off’ (their
convention repertoire disappearing with them) and are
replaced by new peers who are initially ‘blank slates’.
So far, we have not distinguished between Easy conventions
(which can be learned from another agent by minimal
exposure—these correspond to lexical items) and Hard conven-
tions (which require multiple exposures—these correspond to
structural properties of the language). To get started, we make
the simplest of distinctions between them: Easy conventions
can be learned by an agent from a single exposure. Once a con-
vention has been generated by a neighbour, an agent can
immediately generate that convention. Hard conventions can
only be learned from two exposures: only when an agent has
encountered two examples of the exact same convention from
its neighbours (whether from the same or different neighbour),
will this convention be seated at a new table (representing that
convention in the agent).
(b) Networks
Agents are represented as nodes in a non-directed graph
(one in which edges have no orientation) and links between
neighbouring agents are represented by edges between
nodes. Networks are characterized by three parameters: n
is the number of nodes (i.e. the population size), kis the
mean nodal degree (i.e. the number of links (neighbours)
that an agent can communicate with, averaged across
agents) and Cis the clustering coefficient (i.e. a measure of
the degree to which nodes (agents) in a graph tend to cluster
together).
The structure of our networks is inspired by real social
networks, based on recent work finding quantitative relations
between city size and the structure of human interaction net-
works from mobile communication records in Portugal and
the UK [28]. Mobile phone communication has been argued
to be a reliable proxy for the strength of individual-based
social interactions [29]. The results in [28] revealed that
the number of average contacts per mobile phone (nodal
degree, k) grows superlinearly with city population size,
according to the well-defined scaling relation: kn
b
21
,
where nis population size. These results fit prior theoretical
work suggesting that superlinear scaling stems from the
nature of human interactions [30]. Interestingly, the prob-
ability that an individual’s contacts are also connected
with each other—i.e. the clustering coefficient C—remained
constant (C0.25) across city sizes [28].
(c) Network sampling
Twenty-five networks were sampled using the method devel-
oped in [31], which consists of a graph Hamiltonian that
allows the creation of random networks close to specified
nodal degree and clustering coefficient values. Sampling con-
verges to networks with desired specified connectivity (details
on the algorithm and implementation can be found in [31]).
For sampling, values of kand Cwere set so that they matched
real social networks described in [28]. For each value of popu-
lation size (n¼30, 50, 100, 200 and 500), five networks were
sampled using a target value of kso that kn
b
21
,where
b
was set to a constant value of 1.677 for all population sizes
n, yielding target mean degree kof 10, 14.1, 22.6, 36.1 and
67.1 for population sizes n¼30, 50, 100, 200 and 500, respect-
ively. Note that, as nincreases, so does the number of
neighbours that an agent has on average. The value of
b
was set to be the minimum so that an agent has (at least)
10 neighbouring agents for the smallest population size (n¼
30). The target value of Cwas set to a constant of 0.25
across all sampling—i.e. the invariable value in real social
networks, regardless of population size [28].
Twenty networks were sampled each run (one for each
population size n), from which five were selected that had
parameters close to the target values. Results are shown in
table 1. All simulations were implemented using R [32].
(d) Implementation
A single run of our simulation is composed of many iter-
ations. On a given iteration, each agent ‘utters’ one
convention to one of its neighbours, who is randomly
picked from the set of all its neighbours in the graph. The
convention produced by the agent can be either part of its
repertoire (conventions that have been previously generated
or learned by the agent) or invented anew. Conventions are
divided into two types: Easy and Hard to learn. Each time
an agent ‘invents’ a new convention, that convention is
randomly defined to belong to one of these two categories
with probability 0.5.
We use an extension of the Chinese restaurant stochastic
sampling process to model an agent’s selection of a convention
to generate. The probability of choosing a given convention, c,
is proportional to the number of ctokens that it has previously
generated or heard from its neighbours. More precisely,
the probability of selecting an already used convention is
defined as
Pðconvention ¼cÞ¼ tc
Mþ1,ð2:1Þ
where t
c
is the number of tokens of convention cthat are
part of the agent’s repertoire and Mis the number of
convention tokens that the agent has stored in memory, thus
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Pt
c
¼M. The probability of inventing a convention anew is
defined as
Pðconvention ¼anewÞ¼ 1
Mþ1:ð2:2Þ
The value of Mincreases over subsequent iterations. How-
ever, conventions are eventually lost, either by token forgetting
(Poisson forgetting in the horizontal transmission version) or
by death of the agent (vertical transmission version). Poisson
forgetting is defined at the level of tokens. Each time an
agent picks a convention (new or old), then, for each of the
Mconvention tokens that it currently stores, there is a prob-
ability pthat this token is ‘forgotten’. This would imply that,
on average Mptokens are forgotten each time a convention
is updated. Given that each time a new convention token is
generated, 1 new token is added to M, then M will be in bal-
ance when, on average, Mp¼1. Forgetfulness of tokens
captures the idea that cognitive constraints affect the cultural
evolution of language [33].
In the vertical transmission version of the model, each
time an agent conveys a convention, there is a probability p
that an agent ‘dies off’—i.e. all the ‘tokens’ in their ‘restau-
rant’ would disappear. That location in the network would
still exist, but is completely cleared, and the ‘dead agent’ is
just replaced by a ‘blank slate’ new agent at the same location
in the network (like being born into the social network).
Agents can learn conventions from neighbours. The learned
convention becomes part of the agent’s repertoire and can be
sampled during its own production. In the current simulations,
Easy conventions are defined as those that are learned from only
a single exposure, whereas Hard conventions require at least
two exposures to be learned.
When an agent uses a convention, to ‘communicate’ with its
neighbours, what is the probability that this communication will
be successful? We take ‘successful’ communication to imply
only that the ‘receiving’ agent also knows that same
convention. We are interested in determining the number of
Easy and Hard conventions that are successfully used at the
population level. Thus, a convention is considered ‘successful’
when it has been learned or generated by one of the agent’s
neighbours at some point across iterations. Additionally, to get
a better sense of successful communication, we measured the
proportion of neighbours that share each agent’s conventions.
For vertical and horizontal transmission, five separate
runs of 1000 iterations were carried out across a range of the
parameters n(population size) and p(probability of forgetting
or dying off). At the end of each run,three measures were taken
and compared as a function of population size: (i) the (absolute
and relative) number of Easy and Hard successful conventions
that remained part of the agents’ memory (tables in the restau-
rant), (ii) the (absolute and relative) number of Easy and Hard
conventions that remained part of the memory of at least 10%
of the agents in the population, and (iii) the mean proportion
of neighbours sharing an agent’s conventions—that is, for
each Easy and Hard convention and for each agent, the pro-
portion of neighbouring agents who had that convention as
part of their repertoire was counted. This quantity was averaged
over all conventions-agents.
(e) Results
Absolute and relative values of Hard and Easy conventions
after 1000 iterations are shown in figure 1, reflecting a
general trend towards an increasing frequency of Easy
conventions compared to Hard conventions as the popu-
lation size increases, in both the vertical and horizontal
transmission cases. When the population is small, Hard con-
ventions represent a sizable proportion of the total number
of conventions. As population size increases and the overall
number of conventions grows, the absolute and relative
number of Hard conventions decreases. Both the absolute
and relative patterns remain the same across the different
conditions, suggesting a robust effect of population size on
the proportion of Hard versus Easy to learn conventions.
The predictions of the model are, we stress, qualitative: pre-
dicting a cross-over between the prevalence of Easy and
Hard conventions as population size increases. The popu-
lation size at which the cross-over occurs depends on
parameters, such as the difference between the number of
learning trials required for Easy and Hard items (see
electronic supplementary material, appendix).
3. Discussion
The results suggest that the differential effects of population
size on structural complexity and vocabulary size can be
accommodated within a parsimonious model of cultural trans-
mission constrained by one cognitive constraint: Ease of
Learning. Linguistic innovations that are easy to learn tend
to increase in number as a linguistic community grows,
because the number of potential innovators increases, and
innovations can spread more rapidly. By contrast, small
linguistic communities favour linguistic innovations that are
hard to learn because they require multiple interactions
between individual speakers. It is likely, of course, that
many additional forces have shaped the relative development
of different aspects of linguistic complexity [2]. One factor that
may partly underlie the Easy/Hard distinction considered
Table 1. Graph connectivity properties: mean connectivity values averaged across the five graphs selected for each value of population size, n¼30, 50, 100,
200 and 500 (s.d., standard deviations).
population size
n
mean
b
in
k5n
b21
mean nodal
degree k
nodal degree,
s.d.
mean clustering
coefficient, C
clustering
coefficient, s.d.
30 1.676 9.9 0.08 0.251 0.007
50 1.685 14.8 0.2 0.256 0.002
100 1.684 23.4 0.089 0.242 0.001
200 1.681 36.9 0.9 0.250 0.001
500 1.655 62.4 2.2 0.246 0.009
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here concerns the degree to which properties of language can
be learned independently. Perhaps an additional reason that
learning a lexical item is relatively easy is that word meanings
can, to a considerable degree, be learned independently of one
another. By contrast, structural aspects of language may inter-
lock in more complex ways, making the propagation of such
linguistic innovations more difficult.
More broadly, it is interesting to speculate whether other
aspects of cultural evolution may be subject to the pressures
described here. For example, perhaps an increase in commu-
nity size might be associated with a reduction in the
prevalence of complex dances, music, rituals, myths or reli-
gious beliefs, but an increase in the prevalence of simpler
variants (we leave aside skills relevant to survival, such as
tool use, whose diffusion will depend on objective measures
of efficacy, as well as direct person-to-person contact
[34– 36]). Of course, such effects may, to some extent, be
counteracted by the ability of people to self-assemble into
10–1
1
10
102
103
no. active conventions
population size
0
10
20
30
40
50
60
70
80
90
100
% of active conventions
population size
10–1
1
10
102
103
30 50 100 200 500
30 50 100 200 500
conventions shared by 10% or more
population size
0
10
20
30
40
50
60
70
80
90
100
30 50 100 200 500
30 50 100 200 500
% conventions shared by 10% or more
population size
0
0.2
0.4
0.6
0.8
1.0
30 50 100 200 500
proportion of neighbours sharing
conventions
p
o
p
ulation size
EASY; horizontal; p= 1/200
HARD; horizontal; p= 1/200
EASY; vertical; p= 1/500
HARD; vertical; p= 1/500
EASY; vertical; p= 1/200
HARD; vertical; p= 1/200
EASY; horizontal; p= 1/500
HARD; horizontal; p= 1/500
(a)(b)
(c)(d)
(e)
Figure 1. Panels (aand b) display the results corresponding to the average number of successful conventions per agent—that is, conventions in the agent’s
repertoire that can be understood by at least one of its neighbours. Panels (cand d) display the results corresponding to the average number of conventions
that are shared by at least 10% of the population. Left panels display absolute numbers (aand c), and right panels display relative proportions (band d)of
conventions after 1000 iterations, obtained for increasing values of population size (displayed in the x-axis). Panel (e) displays the mean proportion of neighbours
that share an agent’s convention, averaged across all convention-agents. Blue lines correspond to Easy conventions, and red lines correspond to Hard conventions.
Dashed lines correspond to results of the horizontal transmission version (circles correspond to the agent’s probability of Poisson forgetting p¼1/500, while squares
correspond to a probability of p¼1/200). Solid lines correspond to results of the vertical transmission model (circles correspond to the agent’s probability of dying-
off p¼1/500, while squares correspond to p¼1/200).
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5

small specialist groups whether face-to-face or virtual, and
formal (educational institutions) or informal (salons, discus-
sions groups, artistic movements), to innovate and propagate
cultural forms of high complexity. In the absence of the abil-
ity for people to self-organize in this way, our simulations
raise the possibility that language and culture might
become unrelentingly simpler, at the structural level, as
human societies become increasingly interconnected.
Data accessibility. All source code, data and results are available from:
https://github.com/mhchristiansen/lang-paradox.
Authors’ contributions. F.R., N.C. and M.H.C. conceived and designed
the study. F.R. conducted the simulations and analyses and wrote
the first draft of the paper. N.C. and M.H.C. edited the paper. All
authors gave final approval for publication.
Competing interests. The authors have no competing interests.
Funding. N.C. was supported by ERC grant no. 295917-RATION-
ALITY, the ESRC Network for Integrated Behavioural Science
(grant no. ES/K002201/1), the Leverhulme Trust (grant no.
RP2012-V-022) and Research Councils UK Grant EP/K039830/1.
Acknowledgements. We thank Daniel Nettle and an anonymous reviewer
for valuable comments on this work.
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