1. • 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

2. Voter Dynamics
p(xi = k | {xj}j2Fo(i), ↵) /
nik + ↵
K
ni. + ↵
Xi
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.

3. Opinion Formation as Bayesian Learning
Xi
Fo(i)
θi Dirichlet prior in form of pseudo counts
before the states of neighbors are observed.nik
p(xi = k | {xj}j2Fo(i), ↵) /
nik + ↵
K
ni. + ↵
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

4. Effects of the prior
on the evolution of
opinions in a fully
connected network.
= 1 = 2.5Prior 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 noneall all
Simulations

5. Simulations
=.1 = 1 = 2.5Prior 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 noneall all
0
10
20
30

6. The Role of Priors for Innovation
Xi
Fo(i)
θi
Dirichlet Process prior
probability of voting for a
new state.
nik
p(xi = k | {xj}j2Fo(i), ↵) /
nik + ↵
K
ni. + ↵
p(xi = k | {xj}j2Fo(i), ↵) /
8
><
>:
nik
ni. + ↵
if xi = k
↵
ni. + ↵
if xi = knew
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

7. 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