Consensus formation in the Deuant model

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1 Consensus formation in the Deuant model Timo Hirscher Chalmers University of Technology Seminarium i matematisk statistik, KTH October 29, 2014

2 Outline The seminar is structured as follows: Introduction: Description of the model Limiting behavior Essential concepts Deuant model with univariate opinions Phase transition for the model on Z Partial results for Z d, d 2 as well as the innite percolation cluster on Z d, d 2 Deuant model on Z with multivariate opinions Outline Timo Hirscher - Consensus in the Deuant model 2/26

3 Introduction The Deuant model Simple connected graph G = (V, E) represents interrelations between the individuals {η t (v)} v V denotes the opinion prole at time t 0, the initial conguration {η 0 (v)} v V being i.i.d. ν Model parameters: condence bound θ 0 and willingness to compromise µ (0, 1 2 ] Update step If at time t edge u, v is chosen and the current values are η t (u) =: a and η t (v) =: b, the update rule reads { a + µ(b a) if a b θ, η t (u) = a otherwise, { b + µ(a b) if a b θ, η t (v) = b otherwise. Introduction The model Timo Hirscher - Consensus in the Deuant model 3/26

4 Introduction Limiting behavior Scenarios, the conguration can approach: (i) No consensus There will be nally blocked edges, i.e. e = u, v s.t. for all times t large enough. η t (u) η t (v) > θ, (ii) Weak consensus Every pair of neighbors {u, v} will nally concur, i.e. lim t η t(u) η t (v) = 0 almost surely. (iii) Strong consensus The opinion value at every vertex converges to a common limit as t. Introduction The model Timo Hirscher - Consensus in the Deuant model 4/26

5 Introduction Energy For a convex function E : R R 0, dene the energy at a given vertex v V at time t to be W t (v) := E ( η t (v) ). Note that, due to convexity of E, an update step can only decrease the sum of energies of the two involved vertices. Introduction Essential concepts Timo Hirscher - Consensus in the Deuant model 5/26

6 Introduction Share a drink Fix v V, start with the initial conguration {ξ 0 (u)} u V = δ v and perform updates like in the Deuant model, with the same µ but ignoring the condence bound θ. The outcome after nitely many updates {ξ n (u)} u V is called SAD-prole. The opinion value η t (v) is a convex combination of the initial opinions {η 0 (u)} u V. If the SAD-procedure starting from δ v is mimicking the updates in the Deuant model backwards in time, the contribution of a vertex u is given by ξ n (u). Introduction Essential concepts Timo Hirscher - Consensus in the Deuant model 6/26

7 The Deuant model with univariate opinions Univariate opinions Timo Hirscher - Consensus in the Deuant model 7/26

8 Pairwise long-term behavior Lemma For the Deuant model on Z with bounded i.i.d initial opinions and threshold parameter θ, the following holds a.s. for every two neighbors u, v Z: Either η t (u) η t (v) > θ for all suciently large t, i.e. the edge u, v is nally blocked, or lim η t(u) η t (v) = 0, t i.e. the two neighbors will nally concur. Univariate opinions on Z Timo Hirscher - Consensus in the Deuant model 8/26

9 Flatness Consider the line graph Z as underlying network for the model. v Z is called ε-at to the right in the initial conguration {η 0 (u)} u Z if for all n 0: v+n 1 η 0 (u) [E η 0 ε, E η 0 + ε]. n + 1 u=v It is called ε-at to the left if the above condition is met with the sum running from v n to v instead. v is called two-sidedly ε-at if for all m, n 0: 1 m + n + 1 v+n u=v m η 0 (u) [E η 0 ε, E η 0 + ε]. Univariate opinions on Z Timo Hirscher - Consensus in the Deuant model 9/26

10 Theorem (Lanchier, Häggström) Consider the Deuant model on the graph (Z, E), where E = { v, v + 1, v Z} with ν = unif([0, 1]) and xed µ (0, 1 2 ]. Then the critical value is θ c = 1 2 : (a) If θ > 1 2, the model converges almost surely to strong consensus, i.e. with probability 1 we have: lim t η t(v) = 1 2 for all v Z. (b) If θ < 1 2 however, the integers a.s. split into nite clusters; no global consensus is approached. Univariate opinions Critical value for Z Timo Hirscher - Consensus in the Deuant model 10/26

11 Crucial properties of the initial distribution If ν, the distribution of η 0, has a nite expectation, dene its radius by R := inf{r 0, P(η 0 [E η 0 r, E η 0 + r]) = 1}. If the initial distribution is bounded, let h denote the largest gap in its support. supp(ν) R a E η 0 h b Univariate opinions Critical value for Z Timo Hirscher - Consensus in the Deuant model 11/26

12 Theorem Consider the Deuant model on Z with i.i.d. initial opinions. (a) If the initial distribution ν is bounded, there is a phase transition from a.s. no consensus to a.s. strong consensus at θ c = max{r, h}. The limit value in the supercritical regime is E η 0. (b) Suppose ν is unbounded but its expected value exists, either in the strong sense, i.e. E η 0 R, or the weak sense, i.e. E η 0 {, + }. Then for any θ (0, ), the Deuant model will a.s. approach no consensus in the long run. Univariate opinions Critical value for Z Timo Hirscher - Consensus in the Deuant model 12/26

13 Limiting behavior on Z d, d 2 Theorem (a) If the initial values are distributed uniformly on [0, 1] and θ > 3 4, the conguration will a.s. approach weak consensus, i.e. for all u, v P ( lim t η t (u) η t (v) = 0 ) = 1. (b) For general initial distributions on [0, 1], this threshold is non-trivial if the support is not {0, 1}. Higher dimensions On the full grid Z d Timo Hirscher - Consensus in the Deuant model 13/26

14 Bond percolation on Z d In i.i.d. bond percolation on the grid Z d every edge is independently chosen to be open with probability p [0, 1]. For d 2 there exists a critical probability p c (0, 1), s.t. for subcritical percolation, i.e. p < p c, one a.s. has only nite clusters and for supercritical percolation, i.e. p > p c, there a.s. exist a (unique) innite cluster. Let us consider the Deuant model on the random subgraph of supercritical i.i.d. bond percolation on Z d which is independent of the initial conguration and Poisson events. Higher dimensions On the innite percolation cluster Timo Hirscher - Consensus in the Deuant model 14/26

15 Limiting behavior on the innite percolation cluster Theorem Consider the Deuant model on the innite cluster of supercritical bond percolation with parameter p < 1 and i.i.d. bounded initial opinion values. Then the conguration can not approach strong consensus on the innite cluster for θ < R. Higher dimensions On the innite percolation cluster Timo Hirscher - Consensus in the Deuant model 15/26

16 u v u v e Copy 1 Copy 2 Higher dimensions On the innite percolation cluster Timo Hirscher - Consensus in the Deuant model 16/26

17 The Deuant model on Z with multivariate opinions Multivariate opinions Timo Hirscher - Consensus in the Deuant model 17/26

18 Crucial change for multivariate opinions If we extend the model to vector-valued opinions replacing the absolute value by the Euclidean distance there is a non-trivial change: By compromising, two opinions can get closer to a third one that was further than θ away from both. η(u) η(v) η(w) Multivariate opinions Timo Hirscher - Consensus in the Deuant model 18/26

19 Finite congurations Consider a nite section {1,..., n} of the line graph Z, a nite sequence (e i ) N i=1 of edges e i { 1, 2,..., n 1, n } and some values x 1,..., x n in supp(ν). Call such a triplet a nite conguration. To run the dynamics of the Deuant model with parameter θ on this setting will mean that we set η 0 (v) = x v for all v {1,..., n}, and then update those values interpreting (e i ) N i=1 as the locations of the rst N Poisson events on 1, 2,..., n 1, n. Multivariate opinions Achievable opinion values Timo Hirscher - Consensus in the Deuant model 19/26

20 Simultaneously achievable opinion values For θ > 0 and initial distribution ν, let D θ (ν) denote the set of vectors in R k which the opinion values of nite congurations can collectively approach, if the dynamics are run with condence bound θ. More precisely, x D θ (ν) if and only if for all r > 0, there exists a nite conguration such that running the dynamics with respect to θ will bring all its opinion values inside B(x, r). Multivariate opinions Achievable opinion values Timo Hirscher - Consensus in the Deuant model 20/26

21 Properties of D θ (ν) Lemma (a) D θ (ν) is closed and increases with θ. (b) supp(ν) D θ (ν) conv(supp(ν)) B[E η 0, R] for all θ > 0, where conv(a) denotes the convex hull, A the closure of a set A. (c) The connected components of D θ (ν) are convex and at distance at least θ from one another. If D θ (ν) is connected, then D θ (ν) = conv(supp(ν)). (d) For R <, the set-valued mapping θ D θ (ν) is piecewise constant. (e) If D θ (ν) is connected and E η 0 nite, then E η 0 D θ (ν). Multivariate opinions Achievable opinion values Timo Hirscher - Consensus in the Deuant model 21/26

22 Relation to the support of η t For θ > 0 and t 0, let the support of the distribution of η t be denoted by supp θ (η t ). Theorem If ϑ D ϑ (ν) has no jump in [θ ε, θ + ε] for xed θ and some ε > 0, the following equality holds true for all t > 0: supp θ (η t ) = D θ (ν). Multivariate opinions Achievable opinion values Timo Hirscher - Consensus in the Deuant model 22/26

23 Adapted denition of the largest gap Given an initial distribution ν, dene the length of the largest gap in its support as h := inf{θ > 0, D θ (ν) is connected}. Note that this is consistent with the univariate case. Multivariate opinions Critical value for Z Timo Hirscher - Consensus in the Deuant model 23/26

24 Limiting behaviour for the model on Z with multivariate opinions Theorem Consider the Deuant model on Z with an initial distribution on (R k,. ). (a) If the initial distribution is bounded, i.e. R = inf { r > 0, P ( η 0 B[E η 0, r] ) = 1 } <, there is a phase transition from a.s. no consensus to a.s. strong consensus at θ c = max{r, h}. The limit value in the supercritical regime is E η 0. Multivariate opinions Critical value for Z Timo Hirscher - Consensus in the Deuant model 24/26

25 Limiting behaviour for the model on Z with multivariate opinions Theorem Consider the Deuant model on Z with an initial distribution on (R k,. ). (a) Bounded initial distribution (b) Let η 0 = (η (1) 0,..., η(k) 0 ) be the random initial opinion vector. If at least one of the coordinates η (i) 0 has an unbounded marginal distribution, whose expected value exists (regardless of whether nite, + or ), then the limiting behavior will a.s. be no consensus, irrespectively of θ. Multivariate opinions Critical value for Z Timo Hirscher - Consensus in the Deuant model 25/26

26 Literature Deuant, G., Neau, D., Amblard, F. and Weisbuch, G., Mixing beliefs among interacting agents, Advances in Complex Systems, Vol. 3, pp , Häggström, O., A pairwise averaging procedure with application to consensus formation in the Deuant model, Acta Applicandae Mathematicae, Vol. 119 (1), pp , Lanchier, N., The critical value of the Deuant model equals one half, Latin American Journal of Probability and Mathematical Statistics, Vol. 9 (2), pp , Bibliography Timo Hirscher - Consensus in the Deuant model 26/26

27 Appendix Ergodicity Theorem (ergodic theorem for Z d -actions) Let ξ denote a Z d -stationary random element, (B n ) n N an increasing sequence of boxes and f be a bounded function. For n one gets 1 B n z B n f(t z ξ) E[f(ξ) ξ 1 I] a.s., where T z is the translation x x z and I the σ-algebra of Z d -invariant events. Appendix Timo Hirscher - Consensus in the Deuant model 27/26

28 Appendix General metrics Denition Consider a metric ρ on R k. (i) Call ρ locally dominated by the Euclidean distance, if there exist γ, c > 0 such that for x, y R k with x y 2 γ: ρ(x, y) c x y 2. (ii) Let ρ be called weakly convex if for all x, y, z R k : ρ(x, αy+(1 α) z) max{ρ(x, y), ρ(x, z)} for all α [0, 1]. (iii) ρ is called sensitive to coordinate i, if there exists a function ϕ : [0, ) [0, ) such that lim s ϕ(s) = and for any two vectors x, y R k with x i y i > s, it holds that ρ(x, y) > ϕ(s). Appendix Timo Hirscher - Consensus in the Deuant model 28/26

Stochastic domination in space-time for the supercritical contact process

Stochastic domination in space-time for the supercritical contact process Stein Andreas Bethuelsen joint work with Rob van den Berg (CWI and VU Amsterdam) Workshop on Genealogies of Interacting Particle