consensus formation in the deffuant modelhirscher/talks/stockholm.pdf · higher dimensions on the...
TRANSCRIPT
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
Univariate opinions � Critical value for Z Timo Hirscher - Consensus in the De�uant model 11/26
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.
Univariate opinions � Critical value for Z Timo Hirscher - Consensus in the De�uant model 12/26
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.
Higher dimensions � On the in�nite percolation cluster Timo Hirscher - Consensus in the De�uant model 15/26
bbbbb
bbbbb
bbbbb
bbbbb
u v
bbbbb
bbbbb
bbbbb
bbbbb
u ve
Copy 1 Copy 2
Higher dimensions � On the in�nite percolation cluster Timo Hirscher - Consensus in the De�uant model 16/26
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).
Multivariate opinions � Achievable opinion values Timo Hirscher - Consensus in the De�uant model 20/26
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).
Multivariate opinions � Achievable opinion values Timo Hirscher - Consensus in the De�uant model 21/26
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).
Multivariate opinions � Achievable opinion values Timo Hirscher - Consensus in the De�uant model 22/26
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.
Multivariate opinions � Critical value for Z Timo Hirscher - Consensus in the De�uant model 24/26
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) 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.
Multivariate opinions � Critical value for Z Timo Hirscher - Consensus in the De�uant model 25/26
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