Erds Rnyi Graphs Grant Schoenebeck, Fang-Yi Yu Interacting Particle - PowerPoint PPT Presentation

Consensus of Interacting Particle Systems on Erdős– Rényi Graphs Grant Schoenebeck, Fang-Yi Yu

Interacting Particle Systems • A perfect toy model of opinion dynamics – Agents on a graph G with opinions/types – Opinions update locally • Phenomena of interest – Convergence – Consensus

Interacting Particle Systems • A perfect toy model of opinion dynamics – Agents on a graph G with opinions/types – Opinions update locally • Phenomena of interest – Convergence – Consensus

Goal The < dynamic > converge to consensus quickly in < graphs >

Outline • What is our model of <dynamic> ? • The < dynamic > reaches consensus quickly in complete graph? • The < dynamic > reaches consensus quickly in 𝐻 𝑜,𝑞 ?

Voter model • Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set {0,1} • Given an initial configuration 𝑌 0 :V ↦ {0,1}

Voter model • Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set {0,1} • Given an initial configuration 𝑌 0 :V ↦ {0,1} • At round t, • A node v is picked uniformly at random

Voter model • Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set {0,1} • Given an initial configuration 𝑌 0 :V ↦ {0,1} • At round t, • A node v is picked uniformly at random

Voter model • Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set {0,1} • Given an initial configuration 𝑌 0 :V ↦ {0,1} • At round t, • A node v is picked uniformly at random • 𝑌 𝑢 𝑤 updates to a random neighbor’s opinion

Voter model [Aldous 13] • Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set {0,1} • Given an initial configuration 𝑌 0 :V ↦ {0,1} • At round t, • A node v is picked uniformly at random • 𝑌 𝑢 𝑤 updates to a random neighbor’s opinion

Iterative majority • Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set {0,1} • Given an initial configuration 𝑌 0 :V ↦ {0,1}

Iterative majority • Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set {0,1} • Given an initial configuration 𝑌 0 :V ↦ {0,1} • At round t, • A node v is picked uniformly at random

Iterative majority [Mossel et al 14] • Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set {0,1} • Given an initial configuration 𝑌 0 :V ↦ {0,1} • At round t, • A node v is picked uniformly at random • 𝑌 𝑢 𝑤 = 1 if 1 is the majority opinion in its neighborhood. 𝑌 𝑢 𝑤 = 0 otherwise

Iterative majority [Mossel et al 14] • Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set {0,1} • Given an initial configuration 𝑌 0 :V ↦ {0,1} • At round t, • A node v is picked uniformly at random • 𝑌 𝑢 𝑤 = 1 if 1 is the majority opinion in its neighborhood. 𝑌 𝑢 𝑤 = 0 otherwise

Iterative 3-majority • Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set {0,1} • Given an initial configuration 𝑌 0 :V ↦ {0,1}

Iterative 3-majority • Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set {0,1} • Given an initial configuration 𝑌 0 :V ↦ {0,1} • At round t, • A node v is picked uniformly at random

Iterative 3-majority • Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set {0,1} • Given an initial configuration 𝑌 0 :V ↦ {0,1} • At round t, • A node v is picked uniformly at random • Collects the opinion of 3 randomly chosen neighbors

Iterative 3-majority [Doerr et al 11] • Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set {0,1} • Given an initial configuration 𝑌 0 :V ↦ {0,1} • At round t, • A node v is picked uniformly at random • Collects the opinion of 3 randomly chosen neighbors • Updates 𝑌 𝑢 𝑤 to the opinion of the majority of those 3 opinions.

Common Property • Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set 𝑠 𝑌 𝑢−1 𝑤 = 1 7 {0,1} • Given an initial configuration 𝑌 0 :V ↦ {0,1} • At round t, • A node v is picked uniformly at random The update of opinion only depends on the fraction of opinions amongst its neighbors

Node Dynamic (𝐻, 𝑔, 𝑌 𝟏 ) • Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set 𝑠 𝑌 𝑢−1 𝑤 = 1 7 {0,1} , an update function 𝒈 • Given an initial configuration 𝑌 0 :V ↦ {0,1} • At round t, • A node v is picked uniformly at random • 𝒀 𝒖 𝒘 = 1 w.p. 𝒈 𝒔 𝒀 𝒖−𝟐 𝒘 ; = 0 otherwise

Node Dynamic (𝐻, 𝑔, 𝑌 𝟏 ) • Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set Voter Majority 3-Majority 1 {0,1} , an update function 𝑔 • Given an initial configuration 0.8 𝑌 0 :V ↦ {0,1} • At round t, 0.6 • A node v is picked uniformly at random • 𝑌 𝑢 𝑤 = 1 w.p. 𝑔 𝑠 𝑌 𝑢−1 𝑤 ; 0.4 = 0 otherwise 0.2 0 0 0.2 0.4 0.6 0.8 1

Outline • What is our model of <dynamic> ? • The < dynamic > reaches consensus quickly in complete graph? Which are similar to iterative majority, 3-majority

A Warm-up Theorem • Given a node dynamic (𝐿 𝑜 , 𝑔, 𝑌 𝟏 ) over the complete graph. If the update function f is “rich get richer”, then the maximum expected consensus time 𝑃(𝑜 2 ) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1

A Warm-up Theorem • Given a node dynamic (𝐿 𝑜 , 𝑔, 𝑌 𝟏 ) over the complete graph. If the update function f is “rich get richer”, then the maximum expected consensus time 𝑃(𝑜 2 )

Hitting Time • (𝑌 0 , 𝑌 1 , . . . ) is a discrete time-homogeneous Markov chain with finite state space Ω and transition kernel 𝑄 . • Hitting time for 𝐵 ⊂ Ω : 𝜐 𝐵 = min{𝑢 ≥ 0 ∶ 𝑌 𝑢 ∈ 𝐵} .

A Warm-up Theorem • Given a node dynamic (𝐿 𝑜 , 𝑔, 𝑌 𝟏 ) over the complete graph. If the update function f is “like majority”, then the maximum expected hitting time for consensus configuration is small

More about Hitting Time • Expected hitting time and potential function 𝜐 𝐵 Expected hitting time for 𝐵 ⊂ Ω

More about Hitting Time • Expected hitting time and potential function 𝜐 𝐵 Expected hitting time for 𝐵 ⊂ Ω 𝜔 Potential function for 𝜐 𝐵

More about Hitting Time • Expected hitting time and potential function 𝜐 𝐵 Expected hitting time for 𝐵 ⊂ Ω 𝜔 Potential function for 𝜐 𝐵 ∀𝑦 ∈ Ω, 𝜐 𝐵 x ≤ 𝜔 𝑦

A Conventional Approach for the Theorem • Expected hitting time and potential function 𝜐 𝐵 Expected hitting time for 𝐵 ⊂ Ω 𝜔 Potential function for 𝜐 𝐵 ∀𝑦 ∈ Ω, 𝐹[𝜐 𝐵 x ] ≤ 𝜔 𝑦 • Guess a function 𝜔 (only depends on the number of 1 )

Outline • What is our model of <dynamic> ? • The < dynamic > reaches consensus quickly in complete graph? • The < dynamic > reaches consensus quickly in 𝐻 𝑜,𝑞 ?

The Main Theorem • Given a node dynamic (𝐻, 𝑔, 𝑌 𝟏 ) over 𝐻 ∼ 𝐻 𝑜,𝑞 where 𝑞 = Ω(1) , and f be “ smooth rich get richer”, the maximum expected consensus time is 𝑃(𝑜 log 𝑜) with high probability. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1

The Conventional Approach • Expected hitting time and potential function 𝜐 𝐵 Expected hitting time for 𝐵 ⊂ Ω 𝜔 Potential function for 𝜐 𝐵 ∀𝑦 ∈ Ω, 𝐹[𝜐 𝐵 x ] ≤ 𝜔 𝑦 • Guess a function 𝜔 (only depends on the number of 1 s)

Observation 1 • Expected hitting time and potential function 𝜐 𝐵 Expected hitting time for 𝐵 ⊂ Ω 𝜔 Potential function for 𝜐 𝐵 ∀𝑦 ∈ Ω, 𝐹[𝜐 𝐵 x ] ≤ 𝜔 𝑦 • Guess a function 𝜔 (only depends on the number of 1 s)

𝜔 𝑦 111 𝑦 101 𝑦 111 𝜔 𝑦 001 𝑦 001 𝑦 011 𝑦 100 𝑦 110 𝑦 000 𝑦 010 𝜔 𝑦 000

Reduce to One Dimension 𝜚 3 𝑦 101 𝑦 111 𝜚 1 𝑦 001 𝑦 011 𝑦 100 𝑦 110 𝜚 0 𝑦 000 𝑦 010

Observation 2 • Expected hitting time and potential function 𝜐 𝐵 Expected hitting time for 𝐵 ⊂ Ω 𝜔 Potential function for 𝜐 𝐵 ∀𝑦 ∈ Ω, 𝜐 𝐵 x ≤ 𝜔 𝑦 • Guess a function 𝜔 (only depends on the number of 1 s)

Observation 2 • Expected hitting time and potential function 𝜐 𝐵 Expected hitting time for 𝐵 ⊂ Ω 𝜔 Potential function for 𝜐 𝐵 ∀𝑦 ∈ Ω, 𝐹[𝜐 𝐵 x ] ≤ 𝜔 𝑦 • Construct a function 𝜔 (only depends on the number of 1 s)

Observation 2 • Expected hitting time and potential function 𝜐 𝐵 Hitting time for 𝐵 ⊂ Ω 𝜔 Potential function for 𝜐 𝐵 A system of linear inequalities with variable 𝜔 𝑦 𝑦∈Ω ∀𝑦 ∈ Ω, 𝜐 𝐵 x ≤ 𝜔 𝑦 • Construct a function 𝜔 (only depends on the number of 1 s)

Proof Outline • Control the system of linear inequalities • Construct 𝜚 𝑙 𝑙∈[𝑜] iteratively satisfying the system of linear inequalities.

𝜚 3 𝑦 101 𝑦 111 𝜚 1 𝑦 001 𝑦 011 𝑦 100 𝑦 110 𝜚 0 𝑦 000 𝑦 010

Reduce to one dimensional Number of 1 = n/2 Number of 1 = 0 Number of 1 = 𝑜

Reduce to birth-death process Number of 1 = n/2 𝑞 + (𝑦) 𝑞 + (𝑦) Number of 1 = 0 Number of 1 = 𝑜 𝑞 − (𝑦) 𝑞 − (𝑦)