This document summarizes research on opinion formation and consensus building in social networks using Bayesian learning models. It presents three key models: 1) a voter dynamics model where individuals update their states based on neighbors, 2) an opinion formation model where individuals treat neighbors' states as evidence in a Bayesian updating process, and 3) a model allowing for innovation where a Dirichlet process prior enables the introduction of new states. Simulations show the impact of different prior probabilities on consensus formation and the role of priors in allowing for innovation and the emergence of new ideas.

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