consensus formation in the deffuant modelhirscher/talks/stockholm.pdf · higher dimensions on the...

Consensus formation

in the De�uant model

Timo Hirscher

Chalmers University of Technology

Seminarium i matematisk

statistik, KTH

October 29, 2014

Outline

The seminar is structured as follows:

• Introduction:• Description of the model• Limiting behavior• Essential concepts

• De�uant model with univariate opinions• Phase transition for the model on Z• Partial results for Zd, d ≥ 2 as well as

the in�nite percolation cluster on Zd, d ≥ 2

• De�uant model on Z with multivariate opinions

Outline Timo Hirscher - Consensus in the De�uant model 2/26

IntroductionThe De�uant model

• Simple connected graph G = (V,E) represents interrelationsbetween the individuals

• {ηt(v)}v∈V denotes the opinion pro�le at time t ≥ 0, theinitial con�guration {η0(v)}v∈V being i.i.d. ν

• Model parameters: con�dence bound θ ≥ 0 and willingness to

compromise µ ∈ (0, 12 ]

Update step

If at time t edge 〈u, v〉 is chosen and the current values are

ηt−(u) =: a and ηt−(v) =: b, the update rule reads

ηt(u) =

{a+ µ(b− a) if |a− b| ≤ θ,a otherwise,

ηt(v) =

{b+ µ(a− b) if |a− b| ≤ θ,b otherwise.

Introduction � The model Timo Hirscher - Consensus in the De�uant model 3/26

IntroductionLimiting behavior

Scenarios, the con�guration can approach:

(i) No consensus

There will be �nally blocked edges, i.e. e = 〈u, v〉 s.t.

|ηt(u)− ηt(v)| > θ,

for all times t large enough.

(ii) Weak consensus

Every pair of neighbors {u, v} will �nally concur, i.e.

limt→∞|ηt(u)− ηt(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 De�uant model 4/26

IntroductionEnergy

For a convex function E : R→ R≥0, de�ne the energy at a given

vertex v ∈ V at time t to be Wt(v) := E(ηt(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 De�uant model 5/26

IntroductionShare a drink

Fix v ∈ V , start with the initial con�guration {ξ0(u)}u∈V = δv and

perform updates like in the De�uant model, with the same µ but

ignoring the con�dence bound θ. The outcome after �nitely many

updates {ξn(u)}u∈V is called SAD-pro�le.

The opinion value ηt(v) is a convex combination of the initial

opinions {η0(u)}u∈V . If the SAD-procedure starting from δv is

mimicking the updates in the De�uant model backwards in time,

the contribution of a vertex u is given by ξn(u).

Introduction � Essential concepts Timo Hirscher - Consensus in the De�uant model 6/26

The De�uant model with

univariate opinions

Univariate opinions Timo Hirscher - Consensus in the De�uant model 7/26

Pairwise long-term behavior

Lemma

For the De�uant 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 |ηt(u)− ηt(v)| > θ for all su�ciently large t, i.e. the edge

〈u, v〉 is �nally blocked, or

limt→∞|ηt(u)− ηt(v)| = 0,

i.e. the two neighbors will �nally concur.

Univariate opinions � on Z Timo Hirscher - Consensus in the De�uant model 8/26

Flatness

Consider the line graph Z as underlying network for the model.

v ∈ Z is called ε-�at to the right in the initial con�guration

{η0(u)}u∈Z if for all n ≥ 0:

1

n+ 1

v+n∑u=v

η0(u) ∈ [E η0 − ε,E η0 + ε] .

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

η0(u) ∈ [E η0 − ε,E η0 + ε] .

Univariate opinions � on Z Timo Hirscher - Consensus in the De�uant model 9/26

Theorem (Lanchier, Häggström)

Consider the De�uant model on the graph (Z, E), whereE = {〈v, v + 1〉, v ∈ Z} with ν = unif([0, 1]) and �xed µ ∈ (0, 12 ].Then the critical value is θc =

12 :

(a) If θ > 12 , the model converges almost surely to strong

consensus, i.e. with probability 1 we have:

limt→∞

ηt(v) =12 for all v ∈ Z.

(b) If θ < 12 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 De�uant model 10/26

Crucial properties of the initial distribution

If ν, the distribution of η0, has a �nite expectation, de�ne its radius

by

R := inf{r ≥ 0, P(η0 ∈ [E η0 − r,E η0 + r]) = 1}.

If the initial distribution is bounded, let h denote the largest gap in

its support.

-supp(ν)

a bE η0 -�

-�

h

R


Theorem

Consider the De�uant 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

θc = max{R, h}.

The limit value in the supercritical regime is E η0.(b) Suppose ν is unbounded but its expected value exists, either in

the strong sense, i.e. E η0 ∈ R, or the weak sense, i.e.

E η0 ∈ {−∞,+∞}.Then for any θ ∈ (0,∞), the De�uant model will a.s. approach

no consensus in the long run.


Limiting behavior on Zd, d ≥ 2

Theorem

(a) If the initial values are distributed uniformly on [0, 1] andθ > 3

4 , the con�guration will a.s. approach weak consensus, i.e.

for all 〈u, v〉

P(limt→∞|ηt(u)− ηt(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 Zd Timo Hirscher - Consensus in the De�uant model 13/26

Bond percolation on Zd

In i.i.d. bond percolation on the grid Zd every edge is independently

chosen to be open with probability p ∈ [0, 1].

For d ≥ 2 there exists a critical probability pc ∈ (0, 1), s.t. forsubcritical percolation, i.e. p < pc, one a.s. has only �nite clusters

and for supercritical percolation, i.e. p > pc, there a.s. exist a

(unique) in�nite cluster.

Let us consider the De�uant model on the random subgraph of

supercritical i.i.d. bond percolation on Zd which is independent of

the initial con�guration and Poisson events.

Higher dimensions � On the in�nite percolation cluster Timo Hirscher - Consensus in the De�uant model 14/26

Limiting behavior on the in�nite percolation cluster

Theorem

Consider the De�uant model on the in�nite cluster of supercritical

bond percolation with parameter p < 1 and i.i.d. bounded initial

opinion values.

Then the con�guration can not approach strong consensus on the

in�nite cluster for θ < R.


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The De�uant model on Zwith multivariate opinions

Multivariate opinions Timo Hirscher - Consensus in the De�uant model 17/26

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(w)

η(u)

η(v)

Multivariate opinions Timo Hirscher - Consensus in the De�uant model 18/26

Finite con�gurations

Consider a �nite section {1, . . . , n} of the line graph Z, a �nite

sequence (ei)Ni=1 of edges ei ∈ {〈1, 2〉, . . . , 〈n− 1, n〉} and some

values x1, . . . , xn in supp(ν). Call such a triplet a �nite

con�guration.

To run the dynamics of the De�uant model with parameter θ on

this setting will mean that we set η0(v) = xv for all v ∈ {1, . . . , n},and then update those values interpreting (ei)

Ni=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 De�uant model 19/26

Simultaneously achievable opinion values

For θ > 0 and initial distribution ν, let Dθ(ν) denote the set of

vectors in Rk which the opinion values of �nite con�gurations can

collectively approach, if the dynamics are run with con�dence

bound θ.

More precisely, x ∈ Dθ(ν) if and only if for all r > 0, there exists a

�nite con�guration such that running the dynamics with respect to

θ will bring all its opinion values inside B(x, r).


Properties of Dθ(ν)

Lemma

(a) Dθ(ν) is closed and increases with θ.

(b) supp(ν) ⊆ Dθ(ν) ⊆ conv(supp(ν)) ⊆ B[E η0, R] for all θ > 0,where conv(A) denotes the convex hull, A the closure of a set

A.

(c) The connected components of Dθ(ν) are convex and at

distance at least θ from one another. If Dθ(ν) is connected,then Dθ(ν) = conv(supp(ν)).

(d) For R <∞, the set-valued mapping θ 7→ Dθ(ν) is piecewiseconstant.

(e) If Dθ(ν) is connected and E η0 �nite, then E η0 ∈ Dθ(ν).


Relation to the support of ηt

For θ > 0 and t ≥ 0, let the support of the distribution of ηt bedenoted by suppθ(ηt).

Theorem

If ϑ 7→ Dϑ(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:

suppθ(ηt) = Dθ(ν).


Adapted de�nition of the largest gap

Given an initial distribution ν, de�ne the length of the largest gap

in its support as

h := inf{θ > 0, Dθ(ν) is connected}.

Note that this is consistent with the univariate case.

Multivariate opinions � Critical value for Z Timo Hirscher - Consensus in the De�uant model 23/26

Limiting behaviour for the model on Z with multivariate

opinions

Theorem

Consider the De�uant model on Z with an initial distribution on

(Rk, ‖ . ‖).(a) If the initial distribution is bounded, i.e.

R = inf{r > 0, P

(η0 ∈ B[E η0, r]

)= 1}<∞,

there is a phase transition from a.s. no consensus to a.s. strong

consensus at

θc = max{R, h}.

The limit value in the supercritical regime is E η0.


Limiting behaviour for the model on Z with multivariate

opinions

Theorem

Consider the De�uant model on Z with an initial distribution on

(Rk, ‖ . ‖).(a) Bounded initial distribution

(b) Let η0 = (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.


Literature

De�uant, G., Neau, D., Amblard, F. and Weisbuch, G., Mixing

beliefs among interacting agents, Advances in Complex

Systems, Vol. 3, pp. 87-98, 2000.

Häggström, O., A pairwise averaging procedure with

application to consensus formation in the De�uant model, Acta

Applicandae Mathematicae, Vol. 119 (1), pp. 185-201, 2012.

Lanchier, N., The critical value of the De�uant model equals

one half, Latin American Journal of Probability and

Mathematical Statistics, Vol. 9 (2), pp. 383-402, 2012.

Bibliography Timo Hirscher - Consensus in the De�uant model 26/26

AppendixErgodicity

Theorem (ergodic theorem for Zd-actions)

Let ξ denote a Zd-stationary random element, (Bn)n∈N an

increasing sequence of boxes and f be a bounded function.

For n→∞ one gets

1

|Bn|∑z∈Bn

f(Tzξ)→ E[f(ξ) | ξ−1I] a.s.,

where Tz is the translation x 7→ x− z and I the σ-algebra of

Zd-invariant events.

Appendix Timo Hirscher - Consensus in the De�uant model 27/26

AppendixGeneral metrics

De�nition

Consider a metric ρ on Rk.(i) Call ρ locally dominated by the Euclidean distance, if there

exist γ, c > 0 such that for x, y ∈ Rk 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 ∈ Rk:

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 lims→∞ ϕ(s) =∞ and for any

two vectors x, y ∈ Rk with |xi − yi| > s, it holds thatρ(x, y) > ϕ(s).

Appendix Timo Hirscher - Consensus in the De�uant model 28/26

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