Consensus Formation in Social Networks through Bayesian Iterated Learning

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arnimbleier 
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Transcript of Consensus Formation in Social Networks through Bayesian Iterated Learning
• 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: [email protected]
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 degreeregular 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
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25
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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
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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
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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