Signaling Conventions in Small-World Networks Introduction & Overview

(1)

Signaling Conventions in Small-World Networks

Introduction & Overview Lewis (1969) invented signaling games to show that meaning conventions can arise simply from regularities in communicative behavior. Since then Lewisean signaling games have become a standard model for the evolution of semantic meaning (cf. Steels, 1997; Nowak and Krakauer, 1999; Skyrms, 2010). This paper contributes to the theory of signaling games by showing how language evolution depends on the social structure of a population, thus paving the way for a more thorough understanding of language variability and language contact phenomena in terms of evolutionary game theory. Previous studies have mainly looked at imitation-based dynamics, to show that (i) multiple language conventions can coevolve in suitably structured populations (e.g.Wagner, 2009; Zollman, 2005), (ii) which network properties are most conducive for the formation of uniform conventions (cf. Ke, Gong, and Wang, 2008), or (iii) which nodes in the network are responsible for the initiation of language change (cf. Fagyal et al., 2010). Our simulations built not on imitation, but on usage-based learning dynamics from evolutionary game theory, so as to relate more directly to previous theoretical work on conventionalization in signaling games.

Experimental Set-up. A signaling game is a game played between a senderSand a receiverR.

We looked only at the simplest non-trivial variant of this game, as studied originally by Lewis.

Each time the game is played a statet∈T ={t₁, t₂}is selected randomly with equal probability.

S observes the selected state, but R doesn’t. S then selects a message m ∈ M = {m₁, m2}, and R responds with a choice of action a ∈ A = {a₁, a₂}. Success of a round of a play is measured by a utility function, which we assume to be the same for both players, that rewards successful communication: U(ti, tj) = 1 ifi=jand 0 otherwise. Although messages are initially meaningless in this game, meaningfulness arises from regularities in behavior. There are two patterns of sender and receiver behavior in which messages get a unique meaning from use. We will refer to these as languages:

L₁: t₁ t2

m1

m₂ a1

a₂ L₂: t₁ t₂

m₁ m₂

a₁ a₂ Classical evolutionary game theory assumes a homogenous population of agents and studies which evolutionary processes lead to the selection of a single unique language for the whole population under which initial conditions. In this paper we focus instead on more fine-grained agent-based evolutionary dynamics to study also when different parts of the population adopt different languages. In our simulations agents repeatedly play a Lewis game with those agents they are connected with in their social network, and adapt their behavior based on learning from previous interactions, either by less rational reinforcement learning (RL) (e.g. Skyrms 2010) or more rational combination of belief learning and best-response dynamics (BR) (Nachbar 2008).

Following Wagner (2009) we modeled a structured populations using so-called β-graphs (Watts & Strogatz 1998) which exhibit small-world properties argued to persist in social networks, namely a high clustering coefficient, paired with a low characteristic path length. For our analysis, we created 10 such β-graphs with 300 nodes, k = 6 and β ∈ {.08, .09, .1}. These parameter choices ensured the small-worldliness of our networks that we had to keep small for obtaining enough data points at manageable computation costs. For each network, we ran 20 trials each with either only BR- or only RL-agents. Agents played the Lewis game. Commu- nication happened randomly between neighbors on the network, and each agent’s behavior was updated separately after each round of communication the agent was involved in. We recorded strategies of agents in suitably 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. The latter was to ensure a compromise between a short running time and sufficient time for learning, but also because we were interested in the results of learning after a realistic time-span, not in limit behavior. Figure 1 depicts a resulting network.

(2)

Data Analysis & Results. In order to determine which local network properties best characterize where, on average, learning would be most likely successful, we looked at what we will call language regions. A language region is a maximal subset of agents that have acquired the same language that forms a connected subgraphs. Most of the time, two big language regions formed, one for each signaling convention. We also found that each connected language region of a given type had always a higher average clustering and transitivity value than the expected average value for a connected subgraph with the same sizen(= number of nodes), whereas the density value didn’t exhibit such a divergence (see Figure 2).¹ We may conclude from this that local cliquishness supports the evolution of a local language, whereas density doesn’t.

We also investigated the relationship between meaning evolution and social network structure at a local level. Given our data, a rather straightforward cross-classification based on whether an agent is globally and/or locally well-connected turned out to have high explanatory value.

Using suggestive terminology, we speak of two types of agents,family menandglobetrotters. The former have tight local connections, with less global connections; the latter show the opposite pattern plus a high degree of connectivity. In order to capture these notions more adequately, we look at suitable notions from social network theory (Jackson 2008): betweenness centrality (BC), closeness centrality (CC), degree centrality (DC) and individual clustering (CL). Family men and globetrotters are thus characterized as follows:

BC CC DC CL

family man low low - high globetrotter high high high low

A certainly surprising result of our experiments was that the learning dynamics did not have much impact on the local network properties that characterize regional learning success. Phrased more strikingly, we could conclude that an agent’s location in the network was more influential to his behavioral adaptation than his rationality. Still, there were, of course, notable differences between learning dynamics. The most obvious difference is that BR-learners settle into conventions much faster than RL-learners (see Figure 3). The slower RL-dynamics moreover showed a very interesting connection between the temporal development of meaning formation and network structure (see Figure 3 on the right): there seem to be three phases of conventionalization which affect different network roles. In phase 1 (ca. 0-50) the first agents to adopt a convention, called founding fathershave a much higher degree of connectivity as the agents of phase 2 (ca. 50-100), calledstabilizers, who stabilize the language region around founding fathers. By comparing both groups, stabilizers are classical family men, whereas founding fathers are high-connected family men with global influence. The last agents to adopt a convention, (after ca. 100 rounds) show more and more the mark of globetrotters. This suggests the interpretation that a convention is usually sparked by influential family men, while it takes a locally well-connected set of real family men to fix a meaning convention, so that it can also affect the globetrotters.

References. Fagyal et al. (2010), “Centers and peripheries: Network roles in language change”, Lingua ? Jackson (2008), Social and Economic Networks ? Ke, Gong & Wang (2008),

“Language change and social networks”,Communications in Computational Physics ? Lewis (1969), Convention ? Nachbar (2008), “Learning and Evolution in Games: Belief Learning”, The New Palgrave Dictionary of Economics ? Nowak & Krakauer (1999), “The evolution of language”, PNAS ? Skyrms (2010), Signals: Evolution, Learning & Information ? Steels (1997), “The synthetic modeling of language origins”,Evolution of Communication ? Wagner (2009) “Communication and structured correlation”,Erkenntnis ? Watts & Strogatz (1998),

“Collective Dynamics of ‘Small-World’ Networks”, Nature ? Zollman (2005), “Talking to neighbors: The evolution of regional meaning”, Philosophy of Science

1Given the (sub)-graphG: average clustering depicts the average CL value (see Jackson (2008)) over all nodes inG, transitivity depicts the fraction of all possible triangles inGthat are in fact inGand density depicts fraction of maximal edges inG.

(3)

Figure 1: Small-world network after a simulation with 90% learners and 10% non-learners.

Density

n

0 50 100 150 200 250 300

0 0.2 0.4 0.6 0.8 1

language region (RL-dynamics) language region (BR-dynamics)

Clustering

n

0 50 100 150 200 250 300

0.4 0.5 0.6 0.7 0.8 0.9 1

Transitivity

n

0 50 100 150 200 250 300

0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 2: Comparing observed density, clustering and transitivity of language regions with expected values from randomly chosen subgraphs (solid lines, subgraph size along the x-axis).

% of agents BR-learners

0%

20%

40%

60%

80%

100%

1 2 3 4 5 6 7 8 9 10 11

% of agents (CL) RL-learners ^{BC (CC/DC)}

0 25 50 75 100 125 150 175 200 225 250 275 300 0%

20%

40%

60%

80%

100%

(.4) (.42) (.44) (.46) (.48) (.5)

0 .005 .01 .015 .02 .025

(.185) (.19) (.195) (.2) (.205) (.21) DC×10

CC BC CL

Figure 3: Temporal development of the proportion of agents having settled into their final language for BR-dynamics (left) and RL-dynamics (right). The right picture also plots the average values for CL, BC, CC and DC for those RL-learners who have settled into their final language during the specified interval of rounds.