Polarization in Attraction-Repulsion Models Elisabetta Cornacchia, - - PowerPoint PPT Presentation

Polarization in Attraction-Repulsion Models

Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe

ISIT 2020

20-26 June 2020

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Outline

1

Description of the model

2

Trivialization for Finite Population

3

Trivialization for Infinite Population

4

Future work and conclusion

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Description of the Model

A pairwise interaction model is specified by

1 An initial distribution D0 with support in [0, 1]. 2 An interaction function f : [0, 1]2 → [0, 1]2. Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 3 / 18

Description of the Model

A pairwise interaction model is specified by

1 An initial distribution D0 with support in [0, 1]. 2 An interaction function f : [0, 1]2 → [0, 1]2.

Let Xt := (X1

t , ..., Xn t ) ∈ [0, 1]n denote the opinions of the n agents at

time t. Assume Xi

iid

∼ D0, At every step, pick a random pair (Xi

t, Xj t ) and set

(Xi

t+1, Xj t+1) = f(Xi t, Xj t )

Xk

t+1 = Xk t

for any k = i, j.

Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 3 / 18

Bounded Confidence Model

Attraction effect if opinion dissimilarity ≤ τ,

1 1

No interaction if opinion dissimilarity > τ.

1 1

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Bounded Confidence Model

Attraction effect if opinion dissimilarity ≤ τ,

1 1

No interaction if opinion dissimilarity > τ.

1 1

Initial distribution: D0 = U[0, 1] Interaction function: fτ,BC(x, y) := x + λ

2(y − x), y + λ 2(x − y)

• if |x − y| ≤ τ,

(x, y) if |x − y| > τ, where τ, λ ∈ (0, 1).

Deffuant et al. (2000); G´

• mez-Serrano et al. (2012); Hegselmann and Krause (2002)

Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 4 / 18

Attraction-Repulsion Model

Attraction effect if opinion dissimilarity ≤ τ,

1 1

Repulsion effect if opinion dissimilarity > τ.

1 1

Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 5 / 18

Attraction-Repulsion Model

Attraction effect if opinion dissimilarity ≤ τ,

1 1

Repulsion effect if opinion dissimilarity > τ.

1 1

Initial distribution: D0 = U[0, 1] Interaction function: fτ(x, y) := x + λ

2(y − x), y + λ 2(x − y)

• if |x − y| ≤ τ,

(x − µx, y + µ(1 − y)) if |x − y| > τ, where τ, λ, µ ∈ (0, 1).

Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 5 / 18

Attraction-Repulsion Model: n = 100, λ = µ = 1

2

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Attraction-Repulsion Model: n = 100, λ = µ = 1

2

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Trivialization

Definition (Trivialized configuration)

Y := (Y 1, ..., Y n) ∈ [0, 1]n is a trivialized configuration if for any i, j ∈ [n], |Y i − Y j| ∈ {0, 1}. Denote by Tn the set of trivialized configurations. Consensus Polarization

1 1

Definition (Trivialization)

We say that the process trivializes if for any ε > 0, there exist Y ∈ Tn and t0 ∈ N such that for any t ≥ t0 Xt − Y ∞ < ε.

Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 7 / 18

Outline

1

Description of the model

2

Trivialization for Finite Population

3

Trivialization for Infinite Population

4

Future work and conclusion

Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 8 / 18

Conditions for Trivialization

Theorem

Denote (x′, y′) = f(x, y) and assume that f satisfies the following attraction-repulsion condition: there exist τ ∈ [0, 1], CA

f < 1 and CR f > 1

such that for any x, y ∈ [0, 1] (assume x < y) if |x − y| < τ, then x′, y′ ∈ [x, y] and |x′ − y′| ≤ CA

f |x − y|

(attraction); if |x − y| > τ, then x′, y′ ∈ [0, 1] \ [x, y] and |x′ − y′| ≥ CR

f |x − y|

(repulsion). Then, the process trivializes with probability 1.

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Proof Outline

Let Aε := {x ∈ [0, 1]n : miny∈Tn x − y∞ < ε} be the set of states ε-close to a trivialized configuration, for ε > 0. Aε absorbing, for ε small enough.

Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 10 / 18

Proof Outline

Let Aε := {x ∈ [0, 1]n : miny∈Tn x − y∞ < ε} be the set of states ε-close to a trivialized configuration, for ε > 0. Aε absorbing, for ε small enough. Let Vε(m) := {x ∈ AC

ε : P m(x, Aε) > m−1} be the set of “promising”

states at m steps, for m > 0. For any m, the set Vε(m) is uniformly transient, i.e. Ex [# visits in Vε(m)] =

∞

• t=0

P t(x, Vε(m)) < Mm for x ∈ [0, 1]n.

Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 10 / 18

Proof Outline

Let Aε := {x ∈ [0, 1]n : miny∈Tn x − y∞ < ε} be the set of states ε-close to a trivialized configuration, for ε > 0. Aε absorbing, for ε small enough. Let Vε(m) := {x ∈ AC

ε : P m(x, Aε) > m−1} be the set of “promising”

states at m steps, for m > 0. For any m, the set Vε(m) is uniformly transient, i.e. Ex [# visits in Vε(m)] =

∞

• t=0

P t(x, Vε(m)) < Mm for x ∈ [0, 1]n. There exists M < ∞ such that Vε(M) = AC

ε .

= ⇒ AC

ε uniformly transient, i.e.

∞

t=0 P t(x, AC ε ) < M

for x ∈ [0, 1]n = ⇒ limt→∞ P t(x, Aε) = 1.

Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 10 / 18

Outline

1

Description of the model

2

Trivialization for Finite Population

3

Trivialization for Infinite Population

4

Future work and conclusion

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Probability of Polarization as n → ∞, λ = µ = 1

2

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Probability of Polarization as n → ∞, λ = µ = 1

2

n = 2 n = 4 n = 6 n = 20 n = 100 As the population size increases, the probability of polarizing depending on τ tends to a step function.

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Trivialization for Infinite Population

Let f(n)

t

(x) be the empirical distribution of the n points at time t. For any λ, µ there exists τC such that lim

t→∞ lim n→∞ f(n) t

(x) = 1

2δ(x) + 1 2δ(x − 1)

if τ < τC, δ(x − 1

2)

if τ > τC.

1 1 1 x/(1-μ) x (x-μ)/(1-μ) x (x-λ/2y)/(1-λ/2) x y 1 1 1 x/(1-μ) x (x-μ)/(1-μ) x (x-λ/2y)/(1-λ/2) x y 1 1 1 x/(1-μ) x (x-μ)/(1-μ) x (x-λ/2y)/(1-λ/2) x y

Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 13 / 18

Trivialization for Infinite Population

Let f(n)

t

(x) be the empirical distribution of the n points at time t. For any λ, µ there exists τC such that lim

t→∞ lim n→∞ f(n) t

(x) = 1

2δ(x) + 1 2δ(x − 1)

if τ < τC, δ(x − 1

2)

if τ > τC. For any t ≥ 0, limn→∞ f(n)

t

= ft in distribution

f0 = U([0, 1]) ∂ft(x) ∂t = 1 1 − λ

2

x+(1− λ

2 )τ x−(1− λ 2 )τ

ft

• x − λ

2 y

1 − λ

2

• ft(y)dy

+ 1 1 − µ ft x − µ 1 − µ

x−µ 1−µ −τ

ft(y)dy + 1 1 − µ ft

• x

1 − µ 1

x 1−µ +τ

ft(y)dy − ft(x).

1 1 1 x/(1-μ) x (x-μ)/(1-μ) x (x-λ/2y)/(1-λ/2) x y 1 1 1 x/(1-μ) x (x-μ)/(1-μ) x (x-λ/2y)/(1-λ/2) x y 1 1 1 x/(1-μ) x (x-μ)/(1-μ) x (x-λ/2y)/(1-λ/2) x y

Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 13 / 18

Outline

1

Description of the model

2

Trivialization for Finite Population

3

Trivialization for Infinite Population

4

Future work and conclusion

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Can we find a “potential”?

For Bounded Confidence Model: S(Xt) = n

i=1(Xi t)2 is non-increasing

along any sample path (G´

• mez-Serrano et al. (2012)).

Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 15 / 18

Can we find a “potential”?

For Bounded Confidence Model: S(Xt) = n

i=1(Xi t)2 is non-increasing

along any sample path (G´

• mez-Serrano et al. (2012)).

For Attraction-Repulsion Model: Find a function h : [0, 1]n → R such that {h(Xt)}t∈N is a non-negative super-martingale (or a bounded sub-martingale); for any ε > 0, |h(Xt+1) − h(Xt)| < ε = ⇒ minY ∈Tn Xt − Y ∞ < δ(ε), where δ(ε) is such that limε→0+ δ(ε) = 0.

Theorem (Doob’s martingale convergence theorem)

Let (Mt)t≥0 be a non-negative super-martingale (or a bounded sub-martingale). Then, the limit limt→∞ Mt = M∞ exists almost surely. Example (for n = 3): h(Xt) := n

i,j=1 ||Xi t − Xj t | − τ|.

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Future Work

For Attraction-Repulsion model: probability of polarization or agreement depending on n and τ; expected mixing time depending on τ, n, λ and µ. Possible extensions: random interaction function; network structures → individuals tend to interact more often with people with similar opinions; 2D model.

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2D Model

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2D Model Thank you.

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Bibliography

Deffuant, G., Neau, D., Amblard, F., and Weisbuch, G. (2000). Mixing beliefs among interacting agents. Adv. Complex Syst., 3(1–4):87–98. G´

• mez-Serrano, J., Graham, C., and Le Boudec, J.-Y. (2012). The

bounded confidence model of opinion dynamics. Mathematical Models and Methods in Applied Sciences, 22(02):1150007. Hegselmann, R. and Krause, U. (2002). Opinion dynamics and bounded confidence: Models, analysis and simulation. Journal of Artificial Societies and Social Simulation, 5:1–24.

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