Roland M ¨uhlenbernd

Convention & Innovation in Social Networks

Roland M ¨uhlenbernd

November 7, 2014

O VERVIEW

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Signaling Games, Learning & Innovation

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Basic Simulation Results in a Scale-Free Network

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Theories from Sociolinguistics

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Analysis and Conclusion

I NTRODUCTION

The signaling game (Lewis 1969)

I

is a game-theoretical model of communication

I

that is mainly approached for questions of the emergence of semantic meaning

→

the signaling game is a reasonable model to approach questions of the

of semantic meaning

Source: David Lewis (1969):Convention. A Philosophical Study.Harvard University Press.

Signaling Game : SG

S, R

T, M, A, Pr, U

N S

R

1 0

R

1 0

S R

0 1

R

0 1

.5 .5

t

t

m

m

m

m

a

a

a

a

a

a

a

a

I

T

t

I

M

m

I

A

a

a

I

Pr(t

Pr(t

I

U(t

a

1 if i

j 0 else a

a

t

1 0

t

0 1

P URE STRATEGIES

Pure strategies are contingency plans, players act according to.

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sender strategy:

T

M

I

receiver strategy:

M

A

:

tS m₂

σ₂

:

m₂ tS

m1

σ₃

:

tS m₂

σ₄

:

m2 tS

m1

ρ₁

:

m2 aS

ρ₂

:

aS m2

aL

ρ₃

:

m2 aS

ρ₄

:

aS m2

aL

S IGNALING S YSTEMS ...

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are combinations of pure strategies. The Lewis game has two: L

and L

L

: t

t

m

m

a

a

L

:

t

t

m

m

a

a

I

are strict Nash equilibria of the EU-table:

ρ₁ ρ₂ ρ₃ ρ₄

σ₁

1 0 .5 .5

σ₂

0 1 .5 .5

σ₃

.5 .5 .5 .5

.5 .5 .5 .5

I

associate messages to states in an unique way

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are evolutionary stable states

R EINFORCEMENT L EARNING

Agents play the signaling gameSG=h(S,R),T,M,A,Pr,C,Ui

I as probabilistic strategiess:T→∆(M),r:M→∆(A)

I communicative outcome shifts probabilities in appropriate directions

I successful communication viaht∈T,m∈M,a∈Aimakes the choicess(m|t)andr(a|m)more probable in subsequent rounds

S IGNALING G AMES & R EINFORCEMENT L EARNING

I Initially unbiased agents that play the signaling game

repeatedly and update via reinforcement learning might finally end up in a signaling system

I BUT: with the standard accountRoth-Erevreinforcement learning, signaling systems evolve with guarantee only for the simplest variant of a signaling game in two players setups

I for richer games and larger populations the emergence of signaling systems becomes more and more improbable

I by applyingBush-Mostellerreinforcement learning with i)lateral inhibition, ii)negative reinforcementand iii)innovation, signaling systems emerge for large societies that interact with rich games

E XPERIMENTS ON S CALE -F REE N ETWORKS

Settings:

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network: 500 agents on a scale-free network (Holme-Kim algorithm with m

2, p

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game type: 3

9 game

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update: Bush-Mosteller reinforcement learning with negative reinforcement, lateral inhibition and innovation

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break condition: after 100,000 simulation steps

E MERGENCE OF L OCAL C ONVENTIONS

Figure: Regional conventions after 100,000 simulation steps.

E MERGENCE OF M ULTIPLE C ONVENTIONS

Figure: The number of regional conventions over the first 20,000 simulation steps for all 10 simulation runs.

B IRTH AND D EATH OF L OCAL C ONVENTIONS

Figure: The number of learners for 6 specific different signaling conventions for the first 10,000 simulation steps.

R ESUME Basic results:

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multiple regional conventions (10-30) coexist in the population

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long-term stability: the number of different conventions remain stable within an interval

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short-term reactivity: the number of different conventions oscillates inside this interval

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semantic change: new conventions emerge and spread, others get extinct

Do particular network properties support semantic change or,

on the contrary, semantic stability?

T HE T HRESHOLD P ROBLEM

The threshold problem: how can a new linguistic variant spread and reach a particular threshold of speakers that enables to replace a concurrent old variant?

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to reach such a threshold is rather improbable considering the facts that

1. the new variant is expected to be initially used by a minority (generally starts with one speaker)

2. the old variants are expected to be wide-spread linguistic conventions that serves for perfect communication

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sociolinguists expect that new variants mostly do not disseminate but remain in small social groups, often with short durability

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What enables new variants in rare cases to spread and

establish a new linguistic convention?

T HE W EAK -T IES T HEORY

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Milroy & Milroy (1985) expect particular environmental patterns of the social network structure to be source and engine for language change

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weak-ties theory:

I new (innovative) variants emerge most often among edges that constituteweak tiesin the social network

I new variants disseminate viacentral nodes

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assumption: exactly the combination of weak ties and central nodes supports new variants to overcome the threshold problem

Source: James Milroy & Lesley Milroy (1985): Linguistic change, social network and speaker innovation.Journal of Linguistics21(02), 339–384.

D EFINITION OF W EAK -T IES

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Easley & Kleinberg (2010) show that the strength of ties generally increases with the overlap of neighbors

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the neighborhood overlap NO is defined as:

NO(hv

v

N(v

N(v

N(v

N(v

I

where

v

v

is a tie between nodes v

and v

, and N(v

is the set of neighbors of v

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to keep things easy, I will use neighborhood overlap as the measure for tie strength

Source: David Easley & Jon Kleinberg (2010):Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press.

D ATA A NALYSIS

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Are there particular correlations between network properties and behavioral features?

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Do the data support the weak-ties theory?

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Do we find further correlations that might give us further

insights in the dynamics of semantic change?

S TATIC N ETWORK P ROPERTIES Degree Centrality

DC(v) =

|V| −

1 Closeness Centrality

CC(v) =

1

w∈V

SP(v, w) Betweenness Centrality

BC(v) =

u,w∈V|

SP(u, w)

u,w∈V|

SP(u, w)|

1 3 6 150

1 3 6

150 ₁₁₁₅¹₂⁶⁸

1 3 6 10

9 15

1 2 6 11

8 15

1 3 6 150

1 3 6 150

S TATIC N ETWORK P ROPERTIES

Tie Strength

TS(v) =

w∈N(v)

N(v)∩N(w) N(v)∪N(w)

|

N(v)|

Clustering

CL(v) = 2

triangles(v)

|N(v)|(|N(v)| −

1)

0.31

0.31 ^0.171₃

0 0

0.171

3 0.31

0.31

D YNAMIC F EATURE

I innovation skill INV(v): the proportion of simulation steps at which agentvswitched to a new strategy, which no neighbor has actually learned

I impact IMP(v): the proportion of simulation steps at which a neighbor of agentvswitched to agentv’s strategy

I loyalty LOY(v): the proportion of simulation steps agentv played her favorite strategy (most often played strategy)

I interiority INT(v): the proportion of simulation steps for which agentvhas exclusively neighbors using the same strategy

I mutual intelligibility MI(v): the averageMIvalue (expected utility of signaling systems) of agentvto her neighborhood at a given simulation step, averaged over all simulation steps

C ORRELATION OF F EATURES

Static Environmental Features:

I Tie Strength (TS) I Degree Centrality (DC) I Closeness Centrality (CC) I Betweeness Centrality (BC) I Individual Clustering (CL)

Dynamic (Behavioral) Features:

I Innovation Skill (INV) I Impact (IMP) I Loyalty (LOY) I Interiority (INT)

I Mutual Intelligibility (MI)

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−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8

TS DC CC BC CL INV IMP LOY INT MI 1

TS DC CC BC CL INV IMP LOY INT MI

Figure: The correlations for all different pairs of features

R ESULTS

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data support the weak-tie theory

1. INVhas a high negative correlation withTS

2. IMPreveals a high positive correlation with all three centrality propertiesDC,CCandBC

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INV has a high negative correlation with LOY, MI and INT

convention

2. innovative agents are not very intelligible to neighbors 3. innovative agents are rather positioned at the border of a

convention region

O UTLOOK

I Experiments on different network structures (e.g. on non-scale-free small-world networks innovative agents are much harder to characterize)

I More fine-grained analysis of the data (e.g. using regression models to find out, if there are non-trivial interactions like non-linear dependencies between features)

I Exteded analysis of the data (e.g. measure closeness vitality, individual force of innovation, the number of known messages or the growth magnitudes of an agent’s newly innovated signaling system)