ETH Zurich – Distributed Computing – www.disco.ethz.ch Silvio Frischknecht, Barbara Keller, Roger Wattenhofer Convergence in (Social) Influence Networks
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ETH Zurich – Distributed Computing – www.disco.ethz.ch
Silvio Frischknecht, Barbara Keller, Roger Wattenhofer
Convergence in (Social) Influence Networks
Simple World
2 Opinions:
Opinion changes: Whatever the majority of
my friends think
b
b
b
b
b
What Can Happen?
and/or
Goles and Olivios 1980
Easy Lower Bound: Ω(n)
Easy Lower Bound: Ω(n)
Easy Lower Bound: Ω(n)
Easy Lower Bound: Ω(n)
Easy Lower Bound: Ω(n)
Upper Bound:
v
)( 2nO
v
Upper Bound: )( 2nO
v
Upper Bound: )( 2nO
Good edge: Friend takes advised opinion on next day
Bad edge: Friend does not take the proposed opinion
v
Upper Bound: )( 2nO
Good edge: Friend takes advised opinion on next day
Bad edge: Friend does not take the proposed opinion
v
t t+1 t+2
v g
b
g: Nr. of good edges b: Nr. of bad edges
case g > b
Upper Bound: )( 2nO
Good edge: Friend takes advised opinion on next day
Bad edge: Friend does not take the proposed opinion
v
t t+1 t+2
v g
b
g: Nr. of good edges b: Nr. of bad edges
case g > b
Upper Bound: )( 2nO
Good edge: Friend takes advised opinion on next day
Bad edge: Friend does not take the proposed opinion
v
t t+1 t+2
v g
b
g: Nr. of good edges b: Nr. of bad edges
case g > b
Upper Bound: )( 2nO
v
t t+1 t+2
v g
b
case b > g
g: Nr. of good edges b: Nr. of bad edges
Upper Bound: )( 2nO
v
t t+1 t+2
v g
b
case b > g
g: Nr. of good edges b: Nr. of bad edges
Upper Bound: )( 2nO
v
t t+1 t+2
v g
b
case b > g
g: Nr. of good edges b: Nr. of bad edges
Upper Bound: )( 2nO
v
t t+1 t+2
v g
b
case b > g
g: Nr. of good edges b: Nr. of bad edges
Upper Bound: )( 2nO
b
g
Tight Bound?
Lower bound Upper bound
vs. 2nn
Let`s Vote
vs. 2nn
n
n2
2
log
Simpler Example: nn
Simpler Example: nn
A Transistor
A Transistor
B
C
E
B
C
E
B C
E
E
C B
B C
E
B E
C
B B C
E E
C
B C
E
B B B E E E
C C C
B E
C B B
C C
E E
Other Results
Iterative model: Adversary picks nodes instead of synchronous rounds:
1 Step = 1 node change its opinion
Convergence Time: θ(n²)
Iterative Model
Benevolent algorithm: θ(n)
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