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 Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 1 / 18

Outline Description of the model 1 Trivialization for Finite Population 2 Trivialization for Infinite Population 3 Future work and conclusion 4 Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 2 / 18

Description of the Model A pairwise interaction model is specified by 1 An initial distribution D 0 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 D 0 with support in [0 , 1] . 2 An interaction function f : [0 , 1] 2 → [0 , 1] 2 . t ) ∈ [0 , 1] n denote the opinions of the n agents at Let X t := ( X 1 t , ..., X n time t . iid Assume X i ∼ D 0 , 0 t , X j At every step, pick a random pair ( X i t ) and set t +1 , X j t , X j ( X i t +1 ) = f ( X i t ) X k t +1 = X k for any k � = i, j. t Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 3 / 18

Bounded Confidence Model No interaction Attraction effect if opinion dissimilarity > τ . if opinion dissimilarity ≤ τ , 0 1 0 1 0 1 0 1 Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 4 / 18

Bounded Confidence Model No interaction Attraction effect if opinion dissimilarity > τ . if opinion dissimilarity ≤ τ , 0 1 0 1 0 1 0 1 Initial distribution: D 0 = U [0 , 1] Interaction function: x + λ 2 ( y − x ) , y + λ � � � 2 ( x − y ) if | x − y | ≤ τ, f τ,BC ( x, y ) := ( x, y ) if | x − y | > τ, where τ, λ ∈ (0 , 1) . Deffuant et al. (2000); G´ omez-Serrano et al. (2012); Hegselmann and Krause (2002) Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 4 / 18

Attraction-Repulsion Model Repulsion effect Attraction effect if opinion dissimilarity > τ . if opinion dissimilarity ≤ τ , 0 1 0 1 0 1 0 1 Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 5 / 18

Attraction-Repulsion Model Repulsion effect Attraction effect if opinion dissimilarity > τ . if opinion dissimilarity ≤ τ , 0 1 0 1 0 1 0 1 Initial distribution: D 0 = U [0 , 1] Interaction function: � � x + λ 2 ( y − x ) , y + λ � 2 ( x − y ) if | x − y | ≤ τ, f τ ( x, y ) := ( 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 Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 6 / 18

Attraction-Repulsion Model: n = 100 , λ = µ = 1 2 Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 6 / 18

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 T n the set of trivialized configurations. Consensus Polarization 0 1 0 1 Definition (Trivialization) We say that the process trivializes if for any ε > 0 , there exist Y ∈ T n and t 0 ∈ N such that for any t ≥ t 0 � X t − Y � ∞ < ε. Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 7 / 18

Outline Description of the model 1 Trivialization for Finite Population 2 Trivialization for Infinite Population 3 Future work and conclusion 4 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] , C A f < 1 and C R 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 ′ | ≤ C A f | x − y | (attraction); if | x − y | > τ , then x ′ , y ′ ∈ [0 , 1] \ [ x, y ] and | x ′ − y ′ | ≥ C R f | x − y | (repulsion). Then, the process trivializes with probability 1 . Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 9 / 18

Proof Outline Let A ε := { x ∈ [0 , 1] n : min y ∈T n � x − y � ∞ < ε } be the set of states ε -close to a trivialized configuration, for ε > 0 . A ε absorbing, for ε small enough. Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 10 / 18

Proof Outline Let A ε := { x ∈ [0 , 1] n : min y ∈T n � x − y � ∞ < ε } be the set of states ε -close to a trivialized configuration, for ε > 0 . A ε absorbing, for ε small enough. Let V ε ( m ) := { x ∈ A C ε : P m ( x, A ε ) > m − 1 } be the set of “promising” states at m steps, for m > 0 . For any m , the set V ε ( m ) is uniformly transient, i.e. ∞ � P t ( x, V ε ( m )) < M m for x ∈ [0 , 1] n . E x [ # visits in V ε ( m )] = t =0 Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 10 / 18

Proof Outline Let A ε := { x ∈ [0 , 1] n : min y ∈T n � x − y � ∞ < ε } be the set of states ε -close to a trivialized configuration, for ε > 0 . A ε absorbing, for ε small enough. Let V ε ( m ) := { x ∈ A C ε : P m ( x, A ε ) > m − 1 } be the set of “promising” states at m steps, for m > 0 . For any m , the set V ε ( m ) is uniformly transient, i.e. ∞ � P t ( x, V ε ( m )) < M m for x ∈ [0 , 1] n . E x [ # visits in V ε ( m )] = t =0 There exists M < ∞ such that V ε ( M ) = A C ε . A C = ⇒ ε uniformly transient, i.e. � ∞ t =0 P t ( x, A C for x ∈ [0 , 1] n ε ) < M ⇒ lim t →∞ P t ( x, A ε ) = 1 . = Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 10 / 18

Outline Description of the model 1 Trivialization for Finite Population 2 Trivialization for Infinite Population 3 Future work and conclusion 4 Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 11 / 18

Probability of Polarization as n → ∞ , λ = µ = 1 2 Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 12 / 18

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. Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 12 / 18

x x/(1-μ) (x-μ)/(1-μ) x (x-λ/2y)/(1-λ/2) x y 0 1 0 1 0 1 x x/(1-μ) (x-μ)/(1-μ) x (x-λ/2y)/(1-λ/2) x y 0 1 0 1 0 1 x x/(1-μ) (x-μ)/(1-μ) x (x-λ/2y)/(1-λ/2) x y 0 1 0 1 0 1 Trivialization for Infinite Population Let f ( n ) ( x ) be the empirical distribution of the n points at time t . t For any λ, µ there exists τ C such that � 1 2 δ ( x ) + 1 2 δ ( x − 1) if τ < τ C , n →∞ f ( n ) t →∞ lim lim ( x ) = t δ ( x − 1 2 ) if τ > τ C . Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 13 / 18

Trivialization for Infinite Population Let f ( n ) ( x ) be the empirical distribution of the n points at time t . t For any λ, µ there exists τ C such that � 1 2 δ ( x ) + 1 2 δ ( x − 1) if τ < τ C , n →∞ f ( n ) t →∞ lim lim ( x ) = t δ ( x − 1 2 ) if τ > τ C . For any t ≥ 0 , lim n →∞ f ( n ) = f t in distribution t f 0 = U ([0 , 1]) x x/(1-μ) (x-μ)/(1-μ) x (x-λ/2y)/(1-λ/2) x y � x +(1 − λ � x − λ � ∂f t ( x ) 1 2 ) τ 2 y = f t f t ( y ) dy 0 1 0 1 1 − λ 0 1 1 − λ ∂t x − (1 − λ 2 ) τ 2 2 � x − µ x x/(1-μ) (x-μ)/(1-μ) x (x-λ/2y)/(1-λ/2) x y x − µ 1 1 − µ − τ � � + f t f t ( y ) dy 1 − µ 1 − µ 0 1 0 1 0 0 1 � � 1 1 x � + f t f t ( y ) dy − f t ( x ) . x x/(1-μ) (x-μ)/(1-μ) x (x-λ/2y)/(1-λ/2) x y 1 − µ 1 − µ x 1 − µ + τ 0 1 0 1 0 1 Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 13 / 18

Outline Description of the model 1 Trivialization for Finite Population 2 Trivialization for Infinite Population 3 Future work and conclusion 4 Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 14 / 18

Can we find a “potential”? t ) 2 is non-increasing For Bounded Confidence Model : S ( X t ) = � n i =1 ( X i along any sample path (G´ omez-Serrano et al. (2012)). Elisabetta Cornacchia, Neta Singer, Emmanuel Abbe Polarization in Attraction-Repulsion Models 15 / 18