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 𝐻𝑜,𝑞?

Voter model

• Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set

{0,1}

• Given an initial configuration

𝑌0:V ↦ {0,1}

Voter model

• Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set

{0,1}

• Given an initial configuration

𝑌0: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

𝑌0: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

𝑌0:V ↦ {0,1}

• At round t,
• A node v is picked uniformly at random
• 𝑌𝑢 𝑤

updates to a random neighbor’s

• pinion

Voter model [Aldous 13]

• Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set

{0,1}

• Given an initial configuration

𝑌0:V ↦ {0,1}

• At round t,
• A node v is picked uniformly at random
• 𝑌𝑢 𝑤

updates to a random neighbor’s

• pinion

Iterative majority

• Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set

{0,1}

• Given an initial configuration

𝑌0:V ↦ {0,1}

Iterative majority

• Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set

{0,1}

• Given an initial configuration

𝑌0: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

𝑌0:V ↦ {0,1}

• At round t,
• A node v is picked uniformly at random
• 𝑌𝑢 𝑤 = 1 if 1 is the majority opinion in its

neighborhood. 𝑌𝑢 𝑤 = 0 otherwise

Iterative majority [Mossel et al 14]

• Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set

{0,1}

• Given an initial configuration

𝑌0:V ↦ {0,1}

• At round t,
• A node v is picked uniformly at random
• 𝑌𝑢 𝑤 = 1 if 1 is the majority opinion in its

neighborhood. 𝑌𝑢 𝑤 = 0 otherwise

Iterative 3-majority

• Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set

{0,1}

• Given an initial configuration

𝑌0:V ↦ {0,1}

Iterative 3-majority

• Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set

{0,1}

• Given an initial configuration

𝑌0: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

𝑌0: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

𝑌0:V ↦ {0,1}

• At round t,
• A node v is picked uniformly at random
• Collects the opinion of 3 randomly chosen

neighbors

• Updates 𝑌𝑢 𝑤 to the opinion of the

majority of those 3 opinions.

Common Property

• Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set

{0,1}

• Given an initial configuration

𝑌0: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 𝑠𝑌𝑢−1 𝑤 = 1 7

Node Dynamic (𝐻, 𝑔, 𝑌𝟏)

• Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set

{0,1}, an update function 𝒈

• Given an initial configuration

𝑌0:V ↦ {0,1}

• At round t,
• A node v is picked uniformly at random
• 𝒀𝒖 𝒘 = 1 w.p. 𝒈 𝒔𝒀𝒖−𝟐 𝒘

; = 0 otherwise

𝑠𝑌𝑢−1 𝑤 = 1 7

Node Dynamic (𝐻, 𝑔, 𝑌𝟏)

• Fixed a graph 𝐻 = (𝑊, 𝐹) opinion set

{0,1}, an update function 𝑔

• Given an initial configuration

𝑌0:V ↦ {0,1}

• At round t,
• A node v is picked uniformly at random
• 𝑌𝑢 𝑤 = 1 w.p. 𝑔 𝑠𝑌𝑢−1 𝑤

; = 0 otherwise

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Voter Majority 3-Majority

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 (𝐿𝑜, 𝑔, 𝑌𝟏) over the complete graph. If

the update function f is “rich get richer”, then the maximum expected consensus time 𝑃(𝑜2)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1

A Warm-up Theorem

• Given a node dynamic (𝐿𝑜, 𝑔, 𝑌𝟏) over the complete graph. If

the update function f is “rich get richer”, then the maximum expected consensus time 𝑃(𝑜2)

Hitting Time

• (𝑌0, 𝑌1, . . . ) is a discrete time-homogeneous Markov chain

with finite state space Ω and transition kernel 𝑄.

• Hitting time for 𝐵 ⊂ Ω: 𝜐𝐵 = min{𝑢 ≥ 0 ∶ 𝑌𝑢 ∈ 𝐵}.

A Warm-up Theorem

• Given a node dynamic (𝐿𝑜, 𝑔, 𝑌𝟏) 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

𝜐𝐵 Expected hitting time for 𝐵 ⊂ Ω

More about Hitting Time

• Expected hitting time and potential function

𝜐𝐵 Expected hitting time for 𝐵 ⊂ Ω 𝜔 Potential function for 𝜐𝐵

More about Hitting Time

• Expected hitting time and potential function

𝜐𝐵 Expected hitting time for 𝐵 ⊂ Ω 𝜔 Potential function for 𝜐𝐵 ∀𝑦 ∈ Ω, 𝜐𝐵 x ≤ 𝜔 𝑦

A Conventional Approach for the Theorem

• Expected hitting time and potential function
• Guess a function 𝜔 (only depends on the number of 1)

𝜐𝐵 Expected hitting time for 𝐵 ⊂ Ω 𝜔 Potential function for 𝜐𝐵 ∀𝑦 ∈ Ω, 𝐹[𝜐𝐵 x ] ≤ 𝜔 𝑦

Outline

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

The Main Theorem

• Given a node dynamic (𝐻, 𝑔, 𝑌𝟏) over 𝐻 ∼ 𝐻𝑜,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.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1

The Conventional Approach

• Expected hitting time and potential function
• Guess a function 𝜔 (only depends on the number of 1s)

𝜐𝐵 Expected hitting time for 𝐵 ⊂ Ω 𝜔 Potential function for 𝜐𝐵 ∀𝑦 ∈ Ω, 𝐹[𝜐𝐵 x ] ≤ 𝜔 𝑦

• Expected hitting time and potential function
• Guess a function 𝜔 (only depends on the number of 1s)

Observation 1

𝜐𝐵 Expected hitting time for 𝐵 ⊂ Ω 𝜔 Potential function for 𝜐𝐵 ∀𝑦 ∈ Ω, 𝐹[𝜐𝐵 x ] ≤ 𝜔 𝑦

𝑦001 𝑦011 𝑦101 𝑦111 𝑦000 𝑦010 𝑦100 𝑦110 𝜔 𝑦001 𝜔 𝑦111 𝜔 𝑦000

Reduce to One Dimension

𝑦001 𝑦011 𝑦101 𝑦111 𝑦000 𝑦010 𝑦100 𝑦110 𝜚 1 𝜚 0 𝜚 3

Observation 2

• Expected hitting time and potential function
• Guess a function 𝜔 (only depends on the number of 1s)

𝜐𝐵 Expected hitting time for 𝐵 ⊂ Ω 𝜔 Potential function for 𝜐𝐵 ∀𝑦 ∈ Ω, 𝜐𝐵 x ≤ 𝜔 𝑦

Observation 2

• Expected hitting time and potential function
• Construct a function 𝜔 (only depends on the number of 1s)

𝜐𝐵 Expected hitting time for 𝐵 ⊂ Ω 𝜔 Potential function for 𝜐𝐵 ∀𝑦 ∈ Ω, 𝐹[𝜐𝐵 x ] ≤ 𝜔 𝑦

Observation 2

• Expected hitting time and potential function
• Construct a function 𝜔 (only depends on the number of 1s)

𝜐𝐵 Hitting time for 𝐵 ⊂ Ω 𝜔 Potential function for 𝜐𝐵 ∀𝑦 ∈ Ω, 𝜐𝐵 x ≤ 𝜔 𝑦

A system of linear inequalities with variable 𝜔 𝑦

𝑦∈Ω

Proof Outline

• Control the system of linear inequalities
• Construct 𝜚 𝑙

𝑙∈[𝑜] iteratively satisfying the system of

linear inequalities.

𝑦001 𝑦011 𝑦101 𝑦111 𝑦000 𝑦010 𝑦100 𝑦110 𝜚 1 𝜚 0 𝜚 3

Reduce to one dimensional

Number of 1 = n/2 Number of 1 = 0 Number of 1 = 𝑜

Reduce to birth-death process

𝑞+(𝑦) 𝑞−(𝑦) 𝑞−(𝑦) 𝑞+(𝑦) Number of 1 = n/2 Number of 1 = 0 Number of 1 = 𝑜

Proof Outline

• Control the system

– Drift: {𝑞+ 𝑦 − 𝑞−(𝑦)}𝑦∈Ω – Non-laziness: {𝑞+ 𝑦 }𝑦∈Ω

• Construct 𝜚 𝑙

𝑙∈[𝑜] iteratively satisfying the system of

linear inequalities.

Future Work

• Does iterative majority reach consensus fast in dense Erdős–

Rényi random graphs?

• Does iterative majority reach consensus fast in sparse Erdős–

Rényi random graphs? Or expander+?