=Paper= {{Paper |id=Vol-1283/paper6 |storemode=property |title=Meaning, Evolution, and the Structure of Society |pdfUrl=https://ceur-ws.org/Vol-1283/paper_6.pdf |volume=Vol-1283 |dblpUrl=https://dblp.org/rec/conf/ecsi/MuhlenberndF14 }} ==Meaning, Evolution, and the Structure of Society== https://ceur-ws.org/Vol-1283/paper_6.pdf

Meaning, Evolution, and the Structure of Society

Roland Mühlenbernd and Michael Franke

Seminar für Sprachwissenschaft, Universität Tübingen

Abstract. Leading models of language evolution seek to explain key properties
of language as emerging from repeated interactions of language-using agents.
This paper will explore some of the consequences that integrating a more realistic
social interaction structure into established models of language evolution in terms
of evolutionary game theory has and reflects on prospective applications.

Leading models of language evolution seek to explain key properties of language as
emerging from repeated goal-oriented interactions of language-using agents [26, 32, 4,
2, 31]. Models of meaning evolution delineate exact conditions under which semantic
meaningfulness can emerge, by imitation of others or by rationally responding to the
behavior of one’s environment. It is obvious that the psychology of language users plays
a pivotal role here: among other things, the particulars of agents’ perception and mem-
ory, their disposition to adapt their behavior, including the extent to which they make
rational choices will heavily influence the way linguistic behavior evolves over time.
Consequently, there has been a lot of research into the impact of different agent-based
learning dynamics [18, 17, 3, 21, 8, 13]. Moreover, the sociology of language-using
populations plays a role in determining the time course and outcome of evolutionary
processes. Many evolutionary models make explicit assumptions about the interaction
patterns within a population of language users, such as who interacts with whom [25].
Different interaction structures may give different predictions about uniformity or di-
versity of language [37, 34, 23]: if everybody in a large population were to interact with
everybody else equally, language uniformity is to be expected; but given a tendency to
interact mostly with nearby kinsmen, we expect dialects and regional variations. Fur-
ther obvious consequences of different social interaction structures are the readiness
and speed with which new innovations may spread in a population [16, 6], or the social
standing of the innovators [6, 24].
The aim of this paper is to highlight where and how psychological and sociological
constraints interact and influence the outcome of evolutionary processes relevant for
the evolution of meaning. Our goal is to communicate ideas, not mathematical detail.
Section 1 introduces basic notions of evolutionary game theory and formal network the-
ory. Section 2 demonstrates how restrictions on social interaction patterns influence the
equilibrium outcomes for a simple game that has been used frequently in the literature
to study structural iconicity in language [5]. We show that more realistic interaction
structures make it more likely that iconic language use evolves. Section 3 recaps the
conditions under which simple signaling game models of meaning evolution give rise to
regional variability, and Section 4 zooms in on the global and local network properties
that characterize, among others, language boundaries or origins of conventionalization.
t1 m1 a1 t1 m1 a1 t1 m1 a1

t2 m2 a2 t2 m2 a2 t2 m2 a2
(a) strategy profile s1 (b) strategy profile s2 (c) strategy profile s3

Fig. 1. Different strategy profiles for a signaling game with 2 states/messages/actions.

1 Background: Evolutionary Games & Networks

Signaling Games. We will restrict our attention to signaling games [20]. A sender has
private information which a receiver lacks. The sender sends a message, which has no
pre-established meaning. The receiver then takes an action in response to observing the
sender’s message. If the receiver’s action matches the sender’s private information, the
game is a success; if not, it’s a failure. For illustration, we consider signaling games
with a set of two states T = {t1 , t2 }, a set of two messages M = {m1 , m2 } and a set of two
receiver actions A = {a1 , a2 }. With probability p state t1 occurs, and with probability
1 − p it is t2 . The utilities for sender and receiver of one round of play are given by
functions US ,R : T × M × A → R as follows:

1 if i = k

UR (ti , mj , ak ) =  US (ti , mj , ak ) = UR (ti , mj , ak ) − j

0 otherwise


where 0 ≤ 1 ≤ 2 < 1 are costs of sending messages m1 and m2 respectively. When
the state probabilities are equal p = .5 and the message costs are zero 1 = 2 = 0,
then we call the game a Lewis game. We also consider the case where p > .5 and
0 ≤ 1 < 2 and call it a Horn game. This case has been used frequently to study
the structural iconicity principle a.k.a. Horn’s division of pragmatic labor [11] from
an evolutionary perspective, namely the observation that most often languages encode
frequent or stereotypical meanings with short and economic forms [27, 19, 15, 7, 23].
A pure sender strategy is a function s : T → M from states to messages; a pure
receiver strategy is a function r : M → A from messages to acts. A pair of pure strate-
gies is a pure strategy profile. There are 4 pure sender and 4 pure receiver strategies for
the signaling games we look at here and consequently 16 pure strategy profiles. Next to
pure strategies, there are also mixed strategies. These are probability distributions over
pure strategies. So, a mixed sender (receiver) strategy s̃ (r̃) is a function assigning a
probability to each pure sender (receiver) strategy. In an evolutionary context, we inter-
pret these as giving the frequency with which each pure strategy occurs in a population
of agents. A mixed strategy profile is then a pair of mixed sender and receiver strate-
gies that describes the current population state, i.e., the frequencies of pure sender and
receiver strategies occurring in the population at a given time.

Static Solutions. For the Lewis game, the two pure strategy profiles s1 and s2 in Fig-
ure 1 are of particular interest. In each of these, all of the agents play the same strategy
in either sender or receiver role. Lewis called profiles like these, where agents succeed
to communicate perfectly, signaling systems. Although messages are initially meaning-
less, they become meaningful if a population of agents uses them in the way described
by a signaling system. E.g., in strategy profile s1 from Figure 1(a), message m1 de-
notes state t1 . The population states in s1 and s2 are evolutionarily stable states (esss)
[22, 35]. When, as we assume here, agents hold both a sender and receiver strategy
independently, the esss of the game are just the strict Nash equilibria [30].

Dynamic Solutions. Knowing which population states are esss is important, but we may
also wish to know whether and how a population which is not in an ess develops if every
agent adapts her behavior in some way or another. The most common evolutionary
dynamic studied for that purpose is the replicator dynamic of [33]. The general idea
behind the replicator dynamic is that the frequency s̃t (s) of pure sender strategy s at time
t in the population changes proportional to how successful s fares against the averaged
receiver behavior r̃t at time t. If s does better than the average over all pure sender
strategies, then the frequency of s increases proportionally to how much better it does
than average; if s does worse, then its frequency decreases proportionally to how much
worse it does in comparison with other sender strategies. Similarly, for the receiver. A
fixed point of the replicator dynamic is a population state that does not change under the
replicator dynamic. An attractor is a fixed point such that population states close to it
converge towards that fixed point. For each attractor, we call its basin of attraction the
region of population states that converge to it. To simplify parlor, we restrict attention
to relevant attractors that have a non-negligible basin of attraction.
The behavior of signaling games under the replicator dynamic is well-studied [12,
31, 13]. For populations where sender and receiver role are treated separately, we obtain
the following picture. The two signaling systems s1 and s2 in Figure 1 are the only
relevant attractors of the replicator dynamic for the Lewis game and have equally sized
basins of attraction. For the Horn game, there are three relevant attractors (Figure 1):
strategy s1 is called the Horn state since it captures the structural iconicity principle
mentioned above according to which the more efficient form is linked with the more
frequent meaning; strategy s2 is called the Anti-Horn state, since it operates exactly the
other way around; and strategy s3 is called the Smolensky state [11, 5, 19]. The Horn
state is the more efficient way of using language than the Anti-Horn state, although both
achieve perfect communication throughout. The Smolensky state is worse than Horn or
Anti-Horn, as communication is achieved in only p times cases. This is reflected in the
relative sizes of the basins of attraction. To estimate these by numerical simulations,
we generated 1 million randomly chosen initial population states, ran the replicator
dynamic for a Horn game (p = .75, 1 = 0, 2 = .1) and recorded the number of times
the system converged to one of the three states. 54.5% or trials converged to the Horn
state, 33.0% to the Anti-Horn state, and 11.9% to the Smolensky state.
The bigger size of the basin of attraction of the Horn state could be taken as a partial
explanation for the relative prevalence of the structural iconicity principle. However, al-
though Horn and Anti-Horn states support full communication, the Smolensky state is
communicatively inefficient but still attracts almost 12% of initial population states for
the chosen parameter values. Interestingly, this changes when we consider communica-
tion in structured populations, which is what we will work towards next.
Network Games. The replicator dynamic describes how the frequencies of various
strategies in a population of agents develop over time. This is a macro-level perspec-
tive, because there is no mention in the formulation of the replicator dynamic of what
each individual agent is doing. But we can also link the replicator dynamic to a micro-
level perspective. It can be shown that the replicator dynamic describes the most likely
path of strategy distributions in a virtually infinite and homogeneous population when
every agent updates her behavior by conditional imitation [9, 29]. A population is ho-
mogenous if every agent repeatedly interacts with everybody else equally. An agent who
updates her behavior by conditional imitation plays a constant pure strategy for a while,
but, every now and then, checks how well all neighbors fare with the strategy that they
play. If some set X of neighbors do better than the agent, he adopts the pure strategy
of some x ∈ X, where x is chosen from X with a probability that is proportional to the
relative success of x’s strategy if compared with that of the other elements in X.
The conditional imitation rule adopts a micro-level, agent-based perspective. The
micro-level perspective makes it possible to dispense with the assumption that the pop-
ulation is virtually infinite and homogeneous. Instead, we can fix an explicit interaction
structure for a population by defining which agents can interact with one another. At
the same time, we can also look at different update rules, i.e. different ways of changing
behavior over time. We refer to a fixed interaction structure and a fixed update rule as a
network game, when the interaction structure of the agents is given by a social network
of relations. Such a network is formally represented as an undirected graph G = hN, Ei
where N = {1, . . . , n} is the set of nodes, representing the agents, and E ⊆ N × N
is an irreflexive and symmetric ordering on N. Agents i and j interact if and only if
hi, ji ∈ E. In the following, we look at some relevant graph structures whose interaction
with various update rules we will explore thereafter.

Network Types. The most trivial graph structure is a completely connected network:
every agent interacts with everybody else equally; the population is homogeneous. In
a k-ring network all of the agents are ordered and connected in a circular way to their
k-nearest neighbors. In a grid network the agents are arranged on an n × m toroid struc-
ture, where every agent interacts with the 8 nearest neighbors. Rings and grids are easy
to implement and facilitate proofs, but they are quite unrealistic models of human in-
teraction patterns. We therefore also consider two more complicated graph types that
are created with a random component. The first are so-called β-graphs. A β-graph is
obtained by considering a k-ring network and subsequently, for each node, rewiring
its k/2 left neighbors to a random vertex n with probability β [36]. β-graphs have so-
called small-world properties: a short characteristic-path length and a high clustering
coefficient, features that also show, for instance, in the analysis of human friendship
networks [14]. Nodes in a β-graph almost all have the same number of connections,
but it seems more realistic to assume that most agents interact with a smaller number
of agents, but there are also agents who interact with many. If the frequency of agents
with ever larger numbers of connections follows a power-law distribution, the network
is said to be scale-free. We consider a special kind of scale-free networks here, which
is both scale-free and has small-world properties [1]. These are constructed by a simple
parameterized preferential attachement algorithm [10].
Update Rules. We will here only consider general classes of update rules, namely the
imitation class and the learning class. The imitation class comprises update rules where
agents would adopt other agents’ behavior under certain conditions. The conditional
imitation update rule that we linked to the replicator dynamic above is, of course, a first
example. Moreover, we consider the imitate-the-best update rule used by [37] and [34].
This update rule is the special case of the conditional imitation where agents adopt the
strategy of the neighbor who is doing best with probability 1 if this neighbor is doing
better than the agent himself.
When imitation is conditional on success of neighboring strategies, imitation-based
update rules are not as innocuous and resource efficient as they may seem at first. Payoff-
dependent imitation requires keeping track of the neighbors’ average success. It may
be easier to keep track just of one’s own. For that reason, we also look at two spec-
imen from the learning class of update dynamics. Learning-based dynamics assume
that agents try to optimize their behavior by keeping track of the past interactions that
they were engaged in. Whereas imitation-based dynamics have agents change their en-
tire pure strategy when they imitate, learning-based update rules often induce only local
and gradual changes to the agents’ behavioral strategies. A behavioral sender strategy σ
is a function from T to probability distributions over M. Similarly, a behavioral receiver
strategy ρ is a function from M to probability distributions over A.
Reinforcement learning (RL) is the most popular learning dynamic when it comes
to signaling games [31]. Here, agents are more likely to repeat a certain choice the more
successful is was in the past. This can be pictured as a process of updating Pólya urns
[28]. An urn models a behavioral strategy, in the sense that the probability of making a
particular decision is proportional to the number of balls in the urn that correspond to
that action choice. By adding or removing balls from an urn after each encounter, an
agent’s behavior is gradually adjusted. For signaling games, the sender has an urn Ωt
for each state t ∈ T , which contains balls for different messages m ∈ M. The number
of balls of type m in urn Ωt is m(Ωt ), the overall number of balls in urn Ωt is |Ωt |. If the
sender is faced with a state t she draws a ball from urn Ωt and sends message m, if the
ball is of type m. The resulting behavioral strategy is: σ(m|t) = m(Ωt )/|Ωt |. (Similarly, for
the receiver.) The learning dynamics are realized by changing the urn content depen-
dent on the communicative success. If communication via t, m and a is successful, the
number of balls in urn Ωt is increased by α ∈ N balls of type m and reduced by γ ∈ N
balls of type m0 , m. Similarly, for the receiver. In this way successful communicative
behavior is more probable to reappear in subsequent rounds. In our experiments, all
urns were initially filled with 100 balls and we set α = 10 and γ = 4.

2 Network Structure Shifts the Basin of Attraction

We have seen above that there are three relevant attractor states for the replicator dy-
namic in the Horn game (Figure 1), with differently sized basins of attraction. Let’s see
what happens when we look at various network games instead. To isolate the impact of
interaction structure, we look at a network game with the conditional imitation update
rule. This is because, as explained above, for huge and completely connected networks,
average behavior of conditional imitation imitates that of the replicator dynamic. Com-
Smolensky Smolensky

45.9% 50.1%
36.5% 41.0%
16.0% 6.5%

Anti-Horn Anti-Horn

Horn Horn
(a) Homogeneous population (b) Heterogeneous population

Fig. 2. Comparing the basins of attraction for the conditional imitation rule by numerical simu-
lation for different network topologies. Figure 2(a) shows the results for a completely connected
network, while 2(b) shows the results for a small-world network, both for 100 agents.

paring completely connected networks to other types of interaction structure, we get a
good glimpse at the consequences of the assumption of homogeneity that is hidden in
the elegant formulation of the replicator dynamic.
We fix a Horn game with p = .75, 1 = 0 and 2 = .1, as before, and compare,
by numerical simulation, several trials on both homogeneous and heterogeneous pop-
ulations. Figure 2 shows the averaged outcomes of several trials of a discrete traver-
sal of the strategy space. Each dot in the graph corresponds to an initial state of the
sender population. (The picture for the receiver would be largely parallel.) Since pop-
ulation states for this game are probability distributions over four strategies, it suffices
to describe these states by three numbers, leaving implicit the frequency of strategies
{t1 , t2 } → m2 and {m1 , m2 } → a2 . The color of the dots represents the most frequent
sender strategy to which the initial strategy eventually evolved. Depicted are trials over
100 runs. Figure 2(a) visualizes the basins of attraction for a homogeneous population
of 100 agents with a completely connected network as interaction structure. Notice that
these results are for a very small population only, but still meet the predictions of the
replicator dynamic in 93.2% of the trials. Figure 2(b) shows the results for a heteroge-
neous population of 100 agents located on a β-graph (k = 4; β = .1) as the underlying
interaction structure. The pictures in Figure 2 show clearly that conditional imitation on
small-world networks leads to fewer Smolensky outcomes.
Table 1 compares basins of attraction (BoA), not only for complete and small-world
networks, but also for scale-free network (with m = 3 and p = .8) and grid topologies.
They all differ not only in this distribution, but also in the correspondence to the basin of
attraction of the replicator dynamic (RD-correspondence). These results show clearly
that social factors of interaction have non-trivial effects on the results of evolutionary
processes of language games.
Table 1. Basins of attraction and replicator dynamic correspondence for different topologies

BoA Horn BoA Anti-Horn BoA Smolensky RD-corresp.
complete network 45.9% 36.5% 16.0% 93.2%
scale-free 49.7% 38.3% 10.5% 91.0%
β-graph 50.1% 41.0% 6.5% 89.1%
grid topology 50.1% 40.8% 6.7% 88.2%

L1 L2 other Horn Anti-Horn other
(a) Lewis game (b) Horn game

Fig. 3. Exemplary distribution of players with different strategies. Figure 3(a) shows the result of
a Lewis game, Figure 3(b) of a Horn game with p = .6 and 1 = .1, 2 = .2.

3 Emergence of Regional Meaning

[37] analyzed the Lewis game played on 100 × 100 grid under the imitate the best rule.
The main finding was that all agents learned a signaling system and that the population
was divided in multiple language regions: regions of agents playing either the one or the
other signaling system. In other words, heterogeneous interaction structures can lead to
the emergence of regional meaning.
[23] applied reinforcement learning to a Horn-game, played on a 30 × 30 grid net-
work. Multiple language regions emerged in ca. 80% of all simulation runs and, on
average, the number of Anti-Horn players amounted to ca. 3% (30 agents) of all agents.
Furthermore, language regions were always separated by border players, who never
learned a language, but rather switched between different strategies continuously. Ex-
emplary distributions for Lewis and Horn game are depicted in Figure 3.
[23] also extended the grid structure to a social map [25]: agents communicate not
only with their immediate neighbors on the toroid structure, but with all agents of the
population, whereby the probability of communicating with another agent was inversely
proportional to the Manhattan distance. A parameter degree of locality γ determines the
probability percentage
1 100%

0.8 80%

60% Lewis game
0.6
Horn game 1
0.4 40% Horn game 2

0.2 20%

0 γ 0% γ
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
(a) Prob. of neighborhood communication (b) Percentage of trials with multiple language re-
as a function of degree of locality γ gions as a function of degree of locality γ

Fig. 4. The probability of neighborhood communication increases with degree of locality γ (4(a)).
The emergence of a society of multiple language regions is supported by high γ-values (4(b)).

relation between distance and probability. While for γ = 0 each agent in the network
is chosen as interlocutor with the same probability, by increasing γ, the probability of
choosing a close agent increases. Figure 4(a) depicts the probability to communicate
with a direct neighbor dependent on γ for a 30 × 30 social map. By increasing γ the
social map structure approximates neighborhood communication on a grid network.
We can then examine how the emergence of multiple language regions depends on the
degree of locality γ in three different games: the Lewis game, Horn game 1 (p = .6,
1 = .1, 2 = .2) and Horn game 2 (p = .7, 1 = .1, 2 = .2). Since for γ = 0 the social
map corresponds to a complete network, in any trial for any game only one signaling
system emerges, as expected. For γ = 8 the social map equals a grid structure and all
trials of the Lewis game, ca. 80% of the trials of Horn game 1 and only ca. 10% of
Horn game 2 result in a society of multiple language regions. Figure 4(b) depicts how
the percentage increases with γ for 0 ≤ γ ≤ 8.
In sum, the emergence of variability depends strongly on the network topology, es-
pecially the level of connectedness. Moreover, [34] showed for β-graphs that the num-
ber and kind of language regions depends on global network properties like clustering
coefficient and average path length. In the next section we extend this line of research
by looking at the evolution of signaling conventions, not only on β-graphs, but also
scale-free networks. Adding to previous work, we investigate the network properties
associated with regions of agents that have successfully learned a language.

4 Structure of Regional Meaning
Deepening our understanding of synthetic evolutionary processes in structured popula-
tions us useful for a more thorough understanding of the sociological factors of linguis-
tic variability. Most previous work has focused on studying which network structures
are conducive to innovation and its spread [16, 6], less has been done to investigate the
structure of the sub-regions of agents that have successfully learned a language. To-
wards this goal, we modeled structured populations as β-graph and scale-free networks
and created appropriate 10 networks of each type with 300 nodes. For each network, 20
simulation runs were conducted. Agents played the Lewis game with randomly chosen
neighbors on the network, and each agent’s behavior was updated via reinforcement
learning, separately after each round of play. We recorded strategies of agents in suit-
ably chosen regular intervals. Each trial ran until at least 90% of agents had acquired a
language, or each network connection had been used 3000 times in either direction.
In order to determine which global network properties best characterize where, on
average, learning would be most likely successful, we looked at what we will call lan-
guage regions. A language region is a maximal subset of agents that have acquired the
same language that forms a connected subgraph. Despite using reinforcemant learning
instead of imitation dynamics, our data confirmed Wagner’s [34] results that in small-
world networks like ours the number of language regions is small while the size of
language regions is relatively big. On β-graphs multiple language regions emerged for
each simulation run, whereas on scale-free networks more than one language region
emerged in ca. 90% of all runs. However, most of the time, only two big language
regions formed on scale-free networks and β-graphs as well, one for each signaling
convention.
To analyze the properties of language regions, we looked at relevant properties of
(sub-)graphs: density, average clustering and transitivity.1 Our simulation data show
that each connected language region of a given type had always a higher average clus-
tering value than the expected value for a random connected subgraph of the same size,
whereas the density value didn’t exhibit such a divergence (Figure 5). So, a regular
local clique structure supports locally coherent language, but mere number of connec-
tions does not. For β-graphs this conclusion is also supported by the fact, that language
regions always had a higher transitivity value than expected from random sampling.
Interestingly this doesn’t hold for scale-free networks. This divergence results from the
nature of both network types: in a diffuse network like β-graphs transitivity and cluster-
ing roughly correspond to each other, whereas scale-free networks can have high indi-
vidual clustering because of the cliquishness of sub-communities, but unions of those
communities are connected by only few hubs and therefore the more global transitivity
value stays low, or doesn’t exceed the expected average value.

5 Conclusion & Outlook
We presented cursorily data from a number of studies that all support a very general
conclusion: the social structure in a society matters to the evolution of conventional
meaning. Not only can different interaction structures in network games lead to shifts
in the basins of attraction of different strategies under identical dynamics, constraints
on social interaction patterns can also lead to the emergence of regional variety. Using
notions from formal network theory, this paper tried to dig deeper into the relation
between social structure and evolutionary dynamics of meaning by investigation where
languages are most likely situated in the population.
1
For the formal definitions, we refer to [14].
β-graph scale-free
density density
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 n 0 n
0 50 100 150 200 250 300 0 50 100 150 200 250 300
clustering clustering
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
n n
0 50 100 150 200 250 300 0 50 100 150 200 250 300
transitivity transitivity
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
n n
0 50 100 150 200 250 300 0 50 100 150 200 250 300

Fig. 5. Comparing observed density, clustering and transitivity of a language region (o) with
expected values from randomly chosen subgraphs (solid lines, subgraph size n along the x-axis).

The models and results we presented explore the impact of social factors on lan-
guage evolution on a very high level of abstraction. But it is nonetheless clear that this
approach promises to lend itself to a number of concrete linguistic applications. We
therefore consider the presented material here, as a first step into new terrain. The most
obvious future applications of this line of work are language contact phenomena and
the sociolinguistics of innovation and innovation-adoption. But also, for example, by
looking at scale-free networks with a few super-agents connected to (almost) all other
agents, the impact of modern media on language change can be formally studied. More-
over, once we move beyond static interaction structures and also take generational mod-
els of language evolution into account, it becomes possible to study the way changes in
social interaction structures could have interacted with, i.e., causing and being caused
by, language use.
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