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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