Word of Mouth: Rumor Dissemination in Social Networks Jan Kostka - - PowerPoint PPT Presentation

▶
word of mouth rumor dissemination in social networks
SMART_READER_LITE
LIVE PREVIEW

Word of Mouth: Rumor Dissemination in Social Networks Jan Kostka - - PowerPoint PPT Presentation

Oct 11, 2023 19 likes •163 views

Word of Mouth: Rumor Dissemination in Social Networks Jan Kostka Yvonne Anne Oswald Roger Wattenhofer D istributed C omputing 1 G roup Introduction social networks everywhere: facebook, co-authors, email .... => effects dissemination of

slide-1
SLIDE 1

1

Word of Mouth: Rumor Dissemination in Social Networks

Jan Kostka Yvonne Anne Oswald Roger Wattenhofer

Distributed Computing Group

slide-2
SLIDE 2

2 Yvonne Anne Oswald @ SIROCCO 2008

Introduction

social networks everywhere: facebook, co-authors, email .... => effects dissemination of information => influences decisions

Windows or Mac?

slide-3
SLIDE 3

3 Yvonne Anne Oswald @ SIROCCO 2008

Introduction

Windows or Mac?

  • viral marketing
  • competing theses, theories
  • virus vs immunisation

social networks everywhere: facebook, co-authors, email .... => effects dissemination of information => influences decisions

GOAL: select optimal initiator set, to convince as many nodes as possible

slide-4
SLIDE 4

4 Yvonne Anne Oswald @ SIROCCO 2008

Related Work

1 rumour

  • epidemics, physical processes:

sophisticated propagation models + simulation

  • Kempe et al. [KDD03] :

selecting optimal initiators is NP-hard greedy hill climbing algorithm: (1-1/e)-approximation 2 competing rumours

  • Bharati et al.[WINE07], Carnes et al.[ICEC07]

2nd player: selecting optimal initiators is NP-hard hill climbing works as well

1st player What about the 1st player?

slide-5
SLIDE 5

5 Yvonne Anne Oswald @ SIROCCO 2008

Basic Model

  • strategy: select set of nodes to initiate the rumour
  • rumour propagation:

accept first rumour encountered forward rumour to all adjacent nodes variations: more players, payoff definition, propagation model (cascade, threshold, …), weighted or directed edges, …

Payoff: # convinced nodes

Example

1st player: 4 nodes 2nd player: 3 nodes

slide-6
SLIDE 6

6 Yvonne Anne Oswald @ SIROCCO 2008

Warm-Up: 1 vs 1

  • Complete graph
  • Trees
  • Grid
  • Bottleneck
slide-7
SLIDE 7

7 Yvonne Anne Oswald @ SIROCCO 2008

He who laughs last, laughs best?

Intuition: 1st player has more choice ⇒ better chance to win There are networks where the 1st player always loses!!! (computational power irrelevant)

slide-8
SLIDE 8

8 Yvonne Anne Oswald @ SIROCCO 2008

How hard is it to compute the optimal strategy?

Centroid Problem 1st player: how do I choose the optimal starting set? (knowing how many nodes the second player can select) Medianoid Problem 2nd player: how do I choose my optimal starting set? (knowing the nodes selected by the 1st player) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

slide-9
SLIDE 9

9 Yvonne Anne Oswald @ SIROCCO 2008

NP-hardness of Medianoid Problem

  • Theorem. The (r|p)-medianoid problem is NP-hard.

Proof: Reduce Dominating Set (DS) problem to (r|1)-medianoid problem. 1st player chooses x1 2nd player selects Yr

payoff 2nd player: # nodes closer to Yr than to Xp.

Yr = DS

Idea: show that ∃ Yr s.t. 2nd player wins at least |V|+r nodes <=> ∃ DS with r nodes

slide-10
SLIDE 10

10 Yvonne Anne Oswald @ SIROCCO 2008

NP-hardness of Medianoid Problem

  • Theorem. The (r|p)-medianoid problem is NP-hard.

Proof: Reduce Dominating Set (DS) problem to (r|1)-medianoid problem. 1st player chooses x1 2nd player selects Yr

payoff 2nd player: # nodes closer to Yr than to Xp .

Yr != DS

slide-11
SLIDE 11

11 Yvonne Anne Oswald @ SIROCCO 2008

“<=” V’ solution to VC problem, let Xp = V’ 2nd player wins ≤ 2 “=>“ 2nd player wins ≤ 2 if xi on every diamond, we are ok if no xi on diamond, contradiction

NP-hardness of Centroid Problem

  • Theorem. The (r|p)-centroid problem is NP-hard.

Proof: Reduce Vertex Cover (VC) problem to (1|p)-centroid problem. Given graph G(V,E), replace each edge with “diamond structure”

Idea: show that ∃ Xp s.t. less than 2 nodes closer to Y1(Xp) <=> ∃ VC with p nodes

1st player chooses Xp 2nd player selects Y1(Xp)

slide-12
SLIDE 12

12 Yvonne Anne Oswald @ SIROCCO 2008

NP-hardness of Approximating the Centroid Problem

  • Theorem. Computing an α-approximation of the (r|p)-centroid

problem is NP-hard. Proof: Reduce Vertex Cover (VC) problem to (1|p)-centroid problem. Given graph G(V,E), replace each edge with “clique structure”

Y_1(X_p) := node same idea, a little bit more complicated 4α-2

slide-13
SLIDE 13

13 Yvonne Anne Oswald @ SIROCCO 2008

More findings…

  • relationship Condorcet vertex – centroid
  • characterize weaknesses of heuristics for centroid
  • small radius
  • high degrees
  • midpoint of spanning tree
  • (not in paper) simulation of strategies in random graphs:

Kleinberg, Watts, Epstein model

slide-14
SLIDE 14

14 Yvonne Anne Oswald @ SIROCCO 2008

The End!

Thank Thank you! you!

Questions? Comments?