•  Voter Dynamics •  Opinion Formation as Bayesian Learning

•  Model •  Simulations

•  The Role of Priors for Innovation •  Model •  Simulations

Arnim Bleier, Haiko Lietz and Markus Strohmaier Contact: arnim.bleier@gesis.org

ChASM, 23.07.2014

Consensus Formation in Social Networks through Bayesian Iterated Learning

Agenda: Background

Research

Voter Dynamics

p(xi

= k | {xj

}j2Fo(i),↵) /

n

ik

+ ↵

K

n

i.

+ ↵

X i

Fo(i)

nik

No recovery for extinct states, nor introduction of new states.

*) Valid for degree-regular networks only.

*

F. Palombia, S. Toti: Stochastic Dynamics of the Multi–State Voter Model over a Network based on Interacting Cliques and Zealot Candidates, 2014

Normalized frequency of voter i observing state k.

Opinion Formation as Bayesian Learning

X i

Fo(i)

θi Dirichlet prior in form of pseudo counts before the states of neighbors are observed. nik

p(xi

= k | {xj

}j2Fo(i),↵) /

n

ik

+ ↵

K

n

i.

+ ↵

No recovery for extinct states, nor introduction of new states. no

R

T. Griffiths, M. Kalish: Language evolution by iterated learning with Bayesian agents, 2007

Effects of the prior on the evolution of opinions in a fully connected network.

= 1 = 2.5 Prior density for different values of and two different states.

Each panel shows the evolution of the proportion of voters being in state one in a single simulation.

=.1

0

25

50

75 none all none none all all

Simulations

Simulations =.1 = 1 = 2.5 Prior density for

different values of and two different states.

Each panel shows the evolution in the probability distribution of voters being in one of the two states, i.e. p(X = 1).

Effects of the prior on the evolution of opinions in a fully connected network. none all none none all all

0

10

20

30

The Role of Priors for Innovation

X i

Fo(i)

θi

Dirichlet Process prior probability of voting for a new state.

nik

p(xi

= k | {xj

}j2Fo(i),↵) /

n

ik

+ ↵

K

n

i.

+ ↵

p(xi

= k | {xj

}j2Fo(i),↵) /

8><

>:

n

ik

n

i.

+ ↵

if xi

= k

↵

n

i.

+ ↵

if xi

= k

new

No recovery for extinct states nor Introduction of new states.

Allowing for an infinite number of possible states, of which only a finite number is realized by the voters.

R. M. Neal: Markov Chain Sampling Methods for Dirichlet Process Mixture Models, 2000

Simulations

1

10

100

1000

100 200 300 400

K

iterations

- α = .01

- α = .02

Network: Politicians twitter follower network BTW13: nodes 856, 11136 reciprocal edges, average degree 26 and clustering coefficient 0.4. Left: Number of distinct states over iterations for α = .01 and α = .02 and different initializations. Right: Empirical distribution of the number of present states (K) for different settings of α.

10

20

30

%

α = .02

10

20

30

%

α = .01

25

50

75

1 2 3 4 5 6 7 8 9 10

%

K

α = .001

steps