Brief Announcement: On the Voting Time of the Deterministic Majority - - PowerPoint PPT Presentation

▶

brief announcement on the voting time of the

Brief Announcement: On the Voting Time of the Deterministic Majority - - PowerPoint PPT Presentation

Oct 20, 2023 244 likes •449 views

Brief Announcement: On the Voting Time of the Deterministic Majority Process Dominik Kaaser 1 Frederik Mallmann-Trenn 2 , 3 Emanuele Natale 4 1 University of Salzburg 2 cole Normale Suprieure 4 Sapienza Universit di Roma 3 Simon Fraser

slide-1

SLIDE 1

Brief Announcement: On the Voting Time of the Deterministic Majority Process

Dominik Kaaser 1 Frederik Mallmann-Trenn 2,3 Emanuele Natale 4

1University of Salzburg 2École Normale Supérieure 4Sapienza Università di Roma 3Simon Fraser University

October 7, 2015

slide-2

SLIDE 2

Deterministic Binary Majority Voting Process

◮ given a graph G = (V, E)

slide-3

SLIDE 3

Deterministic Binary Majority Voting Process

◮ given a graph G = (V, E) ◮ initial opinion assignment f0 : V → {0, 1}

slide-4

SLIDE 4

Deterministic Binary Majority Voting Process

◮ given a graph G = (V, E) ◮ initial opinion assignment f0 : V → {0, 1}

f0

slide-5

SLIDE 5

Deterministic Binary Majority Voting Process

◮ given a graph G = (V, E) ◮ initial opinion assignment f0 : V → {0, 1} ◮ process runs in discrete rounds

f0

slide-6

SLIDE 6

Deterministic Binary Majority Voting Process

◮ given a graph G = (V, E) ◮ initial opinion assignment f0 : V → {0, 1} ◮ process runs in discrete rounds ◮ every node adopts the majority opinion

f0

slide-7

SLIDE 7

Deterministic Binary Majority Voting Process

◮ given a graph G = (V, E) ◮ initial opinion assignment f0 : V → {0, 1} ◮ process runs in discrete rounds ◮ every node adopts the majority opinion

f1

slide-8

SLIDE 8

Deterministic Binary Majority Voting Process

◮ given a graph G = (V, E) ◮ initial opinion assignment f0 : V → {0, 1} ◮ process runs in discrete rounds ◮ every node adopts the majority opinion

f2

slide-9

SLIDE 9

Deterministic Binary Majority Voting Process

◮ given a graph G = (V, E) ◮ initial opinion assignment f0 : V → {0, 1} ◮ process runs in discrete rounds ◮ every node adopts the majority opinion

f3

slide-10

SLIDE 10

Deterministic Binary Majority Voting Process

◮ given a graph G = (V, E) ◮ initial opinion assignment f0 : V → {0, 1} ◮ process runs in discrete rounds ◮ every node adopts the majority opinion

f4

slide-11

SLIDE 11

Deterministic Binary Majority Voting Process

◮ given a graph G = (V, E) ◮ initial opinion assignment f0 : V → {0, 1} ◮ process runs in discrete rounds ◮ every node adopts the majority opinion

feven

slide-12

SLIDE 12

Deterministic Binary Majority Voting Process

◮ given a graph G = (V, E) ◮ initial opinion assignment f0 : V → {0, 1} ◮ process runs in discrete rounds ◮ every node adopts the majority opinion

fodd

slide-13

SLIDE 13

Known Results

◮ Goles and Olivos 1980, and Poljak and Sůra 1983: The process always converges to a

two-periodic state.

slide-14

SLIDE 14

Known Results

◮ Goles and Olivos 1980, and Poljak and Sůra 1983: The process always converges to a

two-periodic state.

◮ Winkler 2008: The process converges after at most O (|E|) rounds.

slide-15

SLIDE 15

Known Results

◮ Goles and Olivos 1980, and Poljak and Sůra 1983: The process always converges to a

two-periodic state.

◮ Winkler 2008: The process converges after at most O (|E|) rounds. ◮ Frischknecht, Keller, Wattenhofer 2013: These bounds are tight.

slide-16

SLIDE 16

Our Contribution

Bounds on the Voting Time

The voting time of the majority process is at most 1 + min{ |E| − |Vodd|/2 , |E|/2 + |Veven|/4 + 7/4 · |V | } .

slide-17

SLIDE 17

Our Contribution

Bounds on the Voting Time

The voting time of the majority process is at most 1 + min{ |E| − |Vodd|/2 , |E|/2 + |Veven|/4 + 7/4 · |V | } .

Exploiting Symmetries

The voting time is bounded by the voting time in G∆ obtained by contracting its families. A family is a set of nodes which share the same neighborhood.

slide-18

SLIDE 18

Our Contribution

Bounds on the Voting Time

The voting time of the majority process is at most 1 + min{ |E| − |Vodd|/2 , |E|/2 + |Veven|/4 + 7/4 · |V | } .

Exploiting Symmetries

The voting time is bounded by the voting time in G∆ obtained by contracting its families. A family is a set of nodes which share the same neighborhood.

NP Hardness

For a given simple graph G and an integer k, computing whether there exists an initial opinion assignment for which the voting time of G is at least k is NP-complete.

slide-19

SLIDE 19

Applications

Community Detection

Compute convergence time for distributed community detection based on label propagation clustering algorithms (Raghavan, Albert, Kumara 2007)

slide-20

SLIDE 20

Thank You for Your Attention — Questions Welcome!

Introduction Related Work Our Contribution