![Properties for Four Types of Opinion Distributions: Uða; bÞ is the Uniform Distribution on the Interval [a, b], Betaða; bÞ is the Beta Distribution on the Interval [0, 1] with a,b> 0, PLðx 0 ; gÞ is the Power-Law (Pareto) Distribution on ½x 0 ; 1Þ with x 0 ; g > 0, and](https://www.researchgate.net/profile/Yilun_Shang/publication/259537135/figure/tbl1/AS:667876117712903@1536245483294/Properties-for-Four-Types-of-Opinion-Distributions-Uda-bTH-is-the-Uniform-Distribution_Q320.jpg)
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Deffuant Model with General Opinion
Distributions: First Impression and
Critical Confidence Bound
YILUN SHANG
SUTD-MIT International Design Center, Singapore University of Technology and Design, 20 Dover
Drive, 138682, Singapore
Received 13 May 2013; Revised 13 July 2013; accepted 22 July 2013
In the Deffuant model for social influence, pairs of adjacent agents interact at a constant rate and mix up their
opinions (represented by continuous variables) only if the distance between opinions is short according to a thresh-
old. We derive a critical threshold for the Deffuant model on Z, above which the opinions converge toward the aver-
age value of the initial opinion distribution with probability one, provided the initial distribution has a finite
second order moment. We demonstrate our theoretical results by performing extensive numerical simulations on
some continuous probability distributions including uniform, Beta, power-law and normal distributions. Noticed
is a clear differentiation of convergence rate that unimodal opinions (regardless of being biased or not) achieve con-
sensus much faster than even or polarized opinions. Hereby, the emergence of a single mainstream view is a promi-
nent feature giving rise to fast consensus in public opinion formation and social contagious behavior. Finally, we
discuss the Deffuant model on an infinite Cayley tree, through which general network architectures might be
factored in. V
C2013 Wiley Periodicals, Inc. Complexity 19: 38–49, 2013
Key Words: Deffuant model; social dynamics; consensus; phase transition; Monte Carlo simulation
1. INTRODUCTION
People in everyday life meet and discuss their opin-
ions toward something; they influence one another
and as a consequence may adapt their opinions
toward other people’s opinion. In the last decade, research
that focuses on public opinion formation in social net-
works has gained lots of momentum and varied mathe-
matical opinion dynamics models have been developed.
Most established ones include the voter model [1], the
majority rule model [2], the social impact model [3], the
Sznajd model [4], the Deffuant model [5], and the HK
model [6]. We refer the reader to the comprehensive sur-
vey [7] for this prolific field.
In this article, we study a continuous opinion dynam-
ics, the Deffuant model [5], where opinion adjustments
only proceed when two opinions differ by less in
Correspondence to: Yilun Shang; SUTD-MIT International
Design Center, Singapore University of Technology and
Design, 20 Dover Drive, Singapore 138682.
E-mail: shylmath@hotmail.com
38 COMPLEXITY Q2013 Wiley Periodicals, Inc., Vol. 19 No. 2
DOI 10.1002/cplx.21465
Published online 30 September 2013 in Wiley Online Library
(wileyonlinelibrary.com)
magnitude than a given threshold, that is, discussion
under bounded confidence [6,8]. Formally, consider a
graph G5(V,E) with node set Vrepresenting the individu-
als in a population and edge set Ecapturing the potential
social interaction amongst individuals. Initially, nodes are
assigned i.i.d. opinions according to a continuous random
variable X. (Typically, XUð0;1Þis uniformly distributed
on [0, 1].) Each pair of nodes fu,vg僆Emeets at the times
of a unit rate Poisson process, independent for different
pairs. Denote by X
t
(u) the opinion value of node u2Vat
time t. Thus X
0
(u) has the same distribution as X. The
above collection of Poisson processes dictates the meeting
time of the nodes. When at sometime tthe Poisson event
occurs at edge fu,vgsuch that the premeeting states of
the two nodes are Xt2ðuÞ(i.e., Xt2ðuÞ:5lim s"tXsðuÞ) and
Xt2ðvÞ, we set
XtðuÞ5Xt2ðuÞ1lðXt2ðvÞ2Xt2ðuÞÞ if jXt2ðuÞ2Xt2ðvÞj d;
Xt2ðuÞotherwise;
(
(1)
and
XtðvÞ5Xt2ðvÞ1lðXt2ðuÞ2Xt2ðvÞÞ if jXt2ðuÞ2Xt2ðvÞj d;
Xt2ðvÞotherwise;
(
(2)
where l2ð0;1=2is referred to as the convergence parame-
ter and d2Ris the so-called confidence bound [6]. The
updating rule (1) and (2) roughly means that the opinions
of the interacting individuals shift toward each other by a
relative amount lwhen they meet and find their opinion
difference is small enough. A value of l51=2 means that
the two discussing individuals will meet halfway.
Results on the Deffuant dynamics are mostly presented
under the assumption that XUð0;1Þ, namely the initial
opinions are uniformly and randomly chosen in the range
[0, 1]. In this scenario, it is shown through Monte Carlo
simulations that [9,10], for d>dc51=2, all individuals
eventually share the same opinion 1=2 on a variety of net-
works, be them complete graphs, regular lattices, random
graphs or even scale-free networks. When dbecomes
smaller than 1=2, the complete consensus can not be
achieved and numerical simulations again unravel that
[5,11] the number of clusters in the final configuration
can be approximated by 1=(2d). The parameter lonly
influences the speed of convergence [5].
One issue of interest concerns the unequal initial opin-
ion distribution: what would happen if the initial opinions
were unevenly distributed or had preference=bias? Laslier
[12] conjectured that the opinions will tend to average ini-
tial opinion. Through extensive simulations on a directed
Barab
asi-Albert network, Jacobmeier [13] found that the
final consensus value is always around the mean of the
initial opinions (which is dubbed as ‘‘first impression’’),
leading to the conclusion that for a communicative social
community ‘‘the first impression guides the opinion form-
ing’’ [13]. Nonuniformly distributed opinions are simulated
recently in [14]. The authors argue that initial opinion
plays a key role in the collective opinion evolution and
the final opinions converge more easily when the initial
ones are closer.
This sort of first impression effect is commonplace in
the real life. In this article, we aim to provide rigorous
analytical results to support this first impression observa-
tion. More specifically, we consider the Deffuant model on
the real line Z. As long as the initial distribution has finite
second moment, namely EðX2Þ<1, we show that there
exists a critical confidence bound d
c
such that, when
d>dc,
Plim
t!1 XtðuÞ5EX0ðuÞ5EX51 (3)
for all u2Z. Although, there has been a plethora of
numerical results on the Deffuant model (and its variants)
in sociophysics, the mathematical analyses are done only
very recently by Lanchier [15] and H€
aggstr€
om [16] inde-
pendently using different methods. They considered the
one-dimensional Deffuant model on Zwith uniform initial
opinion distribution in the interval [0, 1]. A consequent
critical confidence bound d
c
51=2 was obtained. Here, we
follow the line of H€
aggstr€
om’s work by first establishing a
pairwise average procedure (called sharing a drink (SAD)
[16]) on Zand then deriving the critical value d
c
with the
help of SAD procedure and e-flat points concept. Our
main result is summarized in Theorem 1 (see section 3
below).
Taking some concrete continuous distributions such as
uniform, Beta, power-law, and normal distributions as the
initial opinion distribution, we provide the exact expres-
sions of critical confidence bound d
c
for them (see Table 1
below). In section 4, extensive simulations are performed
to illustrate the availability of our theoretical results. Inter-
estingly, we observe that either even or polarized opinions
in the initial configuration may delay the process of form-
ing collective opinion, which agrees with our lay intuition
that widely divergent topics and controversial issues are
difficult (take much longer time!) to seek consensus on. In
the meantime, unimodal opinions (either unbiased or
biased) are prone to achieve consensus fast, indicating
that the emergence of a single mainstream view is critical
to guiding consensus formation efficiently.
Finally, in the Discussion section, we consider the
applicability and limitations of our methodology by treat-
ing the Deffuant model on an infinite Cayley tree, which
is hoped to bridge the gap between path and more com-
plex and realistic networks.
COMPLEXITY 39Q2013 Wiley Periodicals, Inc.
DOI 10.1002/cplx
2. PRELIMINARIES
2.1. SAD Procedure
In this section, we recall the methodology of
H€
aggstr€
om [16], which is applicable to our general situa-
tion. First define
Y0ðuÞ51 for u50;
0 for u2Znf0g:
((4)
A discrete-time process fYiðuÞgu2Zfor i0 can be
defined iteratively as follows. Given a sequence u1;u2;2
Zand l2ð0;1=2, we obtain the configuration fYiðuÞgu2Z
by letting
YiðuÞ5
Yi21ðuÞ1lðYi21ðu11Þ2Yi21ðuÞÞ for u5ui;
Yi21ðuÞ1lðYi21ðu21Þ2Yi21ðuÞÞ for u5ui11;
Yi21ðuÞfor u2Znfui;ui11g:
8
>
>
<
>
>
:(5)
This process is called SAD, which can be viewed as a
liquid-exchanging procedure for glasses located at each
site u2Z. Initially, only the glass at 0 is full, namely Y
0
(0)
51, while all others are empty, namely, Y
0
(u)50 for
u6¼ 0. At each subsequent time step i, we pick two adja-
cent glasses at u
i
and u
i
11, and pouring liquids from the
glass with higher level to that with lower level by a relative
amount l. The following are a couple of basic properties
regarding SAD [16].
Lemma 1. (monotonicity)
Suppose that fYiðuÞgu2Zis obtained via a SAD such that
uj6¼ 21for all 1ji.Then Yið0ÞYið1ÞYið2Þ.
TABLE 1
Properties for Four Types of Opinion Distributions: Uða;bÞis the
Uniform Distribution on the Interval [a, b], Betaða;bÞis the Beta
Distribution on the Interval [0, 1] with a,b>0, PLðx0;gÞis the
Power-Law (Pareto) Distribution on ½x0;1Þ with x0;g>0, and
Nð^
l;^
r2Þis the Normal Distribution on R
Opinion
Distribution
First
Impression
Second
Order
Moment
Critical
Confidence
Bound
XEXEðX2Þd
c
Uða;bÞa1b
2
a21ab1b2
3
b2a
2
Betaða;bÞa
a1b
a2ða11Þ1abðb11Þ
ða1bÞ2ða1b11Þ
max fa;bg
a1b
PLðx0;gÞx0g
g21
x2
0g
g22(when g>2) 1
Nð^
l;^
r2Þ^
l^
l21^
r21
They all have finite second order moments satisfying the assump-
tion of Theorem 1.
Lemma 2. (unimodality)
Suppose that fYiðuÞgu2Zis obtained via a SAD. Then
there exists a v 2Zsuch that Yiðv22ÞYiðv21Þ
YiðvÞYiðv11ÞYiðv12Þ.In other words, the con-
figuration fYiðuÞgu2Zis unimodal and v is the mode.
Lemma 3. (mode height)
For any u 2Z;sup i0YiðuÞ51=ðjuj11Þ.
Now fix any time t>0 and consider the Deffuant
model on Z. It is easy to see that there exists a finite
interval ½ua;ubZcontaining 0 such that the Poisson
events on the two edges fua21;uagand fub;ub11g
have not happened up to time t(recall that there is a
‘‘Poisson clock’’ on each edge of Z) [16]. Denote by N
the number of opinion adjustments occur in [u
a
,u
b
]up
to time t. We arrange the times of these events in the
chronological order as
0:5sN11<sN<sN21<<s1t:
For i51;;N, we set u
i
be the left end node of the
edge fui;ui11gfor which u
i
and u
i
11 adjust opinions
at time s
i
. Given the sequence u1;;uNand
l2ð0;1=2, we obtain a SAD procedure fYiðuÞgu2Z. The
following proposition is instrumental in understanding
the Deffuant dynamics by establishing a link between
fXtðuÞgu2Zand fYiðuÞgu2Z.
Proposition 1.
For i50;1;;N,
Xtð0Þ5X
u2Z
YiðuÞXsi11ðuÞ:
In particular;Xtð0Þ5X
u2Z
YNðuÞX0ðuÞ:5X
u2Z
YtðuÞX0ðuÞ:
In other words, the SAD procedure is linked to the Def-
fuant model in a simple and linear way. The opinion at
the origin of any time tcan be expressed as a weighted
combination of initial opinions across the real line with
weights being the above designed SAD. Proposition 1
can be proved straightforwardly by induction over i
exactly as in [[16], Lemma 3.1] based on the above con-
struction, since the initial opinion distribution plays no
role in the decomposition.
2.2. Flat Points
Another key ingredient toward the solution of Deffuant
dynamics is the notion of flat points proposed in [16]. A
counterpart in the proof of Lanchier is the set Xj[15].
Given e>0 and the initial configuration fX0ðvÞgv2Zwith
finite mean (i.e., EjXj<1), u2Zis said to be an e-flat
point to the right if for all n0,
40 C O M P L E X I T Y Q2013 Wiley Periodicals, Inc.
DOI 10.1002/cplx
1
n11X
u1n
v5u
X0ðvÞ2½EX2e;EX1e:
Similarly, u2Zis said to be an e-flat point to the left if
for all n0,
1
n11X
u
v5u2n
X0ðvÞ2½EX2e;EX1e;
and u2Zis said to be two-sidedly e-flat if for all n;m0,
1
n1m11X
u1m
v5u2n
X0ðvÞ2½EX2e;EX1e:
When uis e-flat to the right, the Kolmogorov strong
law of large numbers implies
P lim
n!1
1
n11X
u1n
v5u
X0ðvÞ5EX
!
51:
Reasoning as in [16], Lemma 4.2] and using the transla-
tion invariance of the configuration fX0ðvÞgv2Z, we have
Lemma 4.
For e>0and u 2Z;Pðuise2flat to the right Þ5
Pðuise2flat to the left Þ>0.
By considering three independent events A15fu21ise
flat to the leftg;A25fX0ðuÞ2½EX2e;EX1eg and A35
fu11iseflat to the rightg, we further have the following
result (see [16, Lemma 4.3])
Lemma 5.
For e>0and u 2Z;Pðu is two 2sidedly e2flatÞ>0.
3. DEFFUANT MODEL: CONSENSUS FORMATION AND
CRITICAL VALUE
In this section, we establish the following main result
concerning the first impression and critical confidence
bound for the Deffuant model.
Theorem 1.
Consider the Deffuant model on Zwith parameters l2
ð0;1=2and d 2R.Suppose that EðX2Þ<1and define
dc5inf fd:PðjX2EXj>dÞ50g:(6)
If dc<1then d
c
is the critical confidence bound in
the following sense. If d <dc, then with probability 1
the limiting value X1ðuÞ5lim t!1 XtðuÞexists and
fjX1ðuÞ2X1ðu11Þjg 2 f0g[½d;1Þ for every u 2Z.If
d>dc, then with probability 1, X1ðuÞ5
lim t!1 XtðuÞ5EX for every u 2Z.
If dc51then for any d 2R,with probability 1 the
limiting value X1ðuÞ5lim t!1 XtðuÞexists and
fjX1ðuÞ2X1ðu11Þjg 2 f0g[½d;1Þ for every u 2Z.
This theorem means that d
c
(either finite or infinite) is
the critical confidence bound such that when d<dcthe
limiting configuration is piecewise constant interrupted
by jumps of size at least d; when d>dcthe complete
consensus is formed and the first impression (i.e., the
mean of the initial distribution) is confirmed. We men-
tion that dc51corresponds to the situation where the
initial opinion distribution Xhas an infinite support. It
is intuitively plausible that consensus can not be
achieved for any d2Rsince there can be an edge fv,v
11gsuch that jX0ðvÞ2X0ðv11Þj >d.
In the following, we show Theorem 1 in two regimes
d<dcand d>dc, respectively. Since the proof is by
and large similar as in [16], we focus on the differences
and only sketch the similarities.
3.1. Subcritical Regime: d<d
c
In this section, we consider the Deffuant model with
d<dc, where d
c
is defined as in (6).
First we assume that dc<1. Take d5ðdc2dÞ=2>0. For
u2Z, define the following events
BðuÞ5fjXtðuÞ2Xtðu11Þj >d;for all t0g;
C1ðuÞ5fu21isdflat to the leftg,
C2ðuÞ5fX0ðuÞ>EX1dc2dg;C3ðuÞ5fX0ðuÞ<EX2dc1dg,
and C4ðuÞ5fu11isdflat to the rightg.
Proposition 2.
Under the assumption of Theorem 1, if d <dc,then for any
u2ZPðBðuÞÞ >0.
Proof
For u2Z, define two events CðuÞ5C1ðuÞ\C2ðuÞ\C4ðuÞand
C0ðuÞ5C1ðuÞ\C3ðuÞ\C4ðuÞ. It follows from the independ-
ence that PðCðuÞÞ5PðC1ðuÞÞPðC2ðuÞÞPðC4ðuÞÞ and similarly,
PðC0ðuÞÞ5PðC1ðuÞÞPðC3ðuÞÞPðC4ðuÞÞ. By the definition (6) we
have either (i) PðC2ðuÞÞ >0 or (ii) P ðC3ðuÞÞ >0.
If (i) holds, using Lemma 4 we know that P ðCðuÞÞ >0.
It suffices to show
CðuÞBðuÞ:(7)
Assume that CðuÞholds. Let T<1be the first time
that opinion adjustment occurs across either of the
edges fu21;ugor fu;u11g. Therefore, X
t
(u)5X
0
(u) for
any t<T. We will show that such a Tdoes not exist
at all.
COMPLEXITY 41Q2013 Wiley Periodicals, Inc.
DOI 10.1002/cplx
Indeed, on one hand there must exist some t0<Tsuch
that either Xt0ðu21Þor Xt0ðu11Þexceeds
ðEX1dc2dÞ2d5EX1d. Moreover, for any t<Twe
obtain from Proposition 1 (by replacing 0 with u11
due to translation invariance)
Xtðu11Þ5X
v2Z
YtðvÞX0ðvÞ;
and YtðvÞ50 for all vu. By Lemma 1 we have
Ytðu11ÞYtðu12ÞYtðu1NÞ>05Ytðu1N11Þ5
for some 1 N<1. Set ck5kðYtðu1kÞ2Ytðu1k11ÞÞ 0
for k51;;N. Calculating as in [16, Eqs. (19) and (20)]
gives PN
n51cn51 and
Xtðu11Þ5X
N
n51
cn
1
nX
n
k51
X0ðu1kÞ
!
:(8)
Since the event C4ðuÞholds, we see from (8) that
Xtðu11Þ2½EX2d;EX1d:(9)
Analogously, we can show that Xtðu21Þ2½EX2d;EX1d.
This fact contradicts with the existence of such a t0<T
above. Therefore, we must have T51.
Furthermore, noting that C2ðuÞholds and using (9) we
obtain
jXtðuÞ2Xtðu11Þj >EX1dc2d2ðEX1dÞ5dc22d5d
for all t0. Hence, BðuÞholds, and (7) is established.
If (ii) holds, using Lemma 4 likewise we know that
PðC0ðuÞÞ >0. It suffices to prove
C0ðuÞBðuÞ:(10)
This can be shown analogously as in case (i), which
concludes the proof.
Notice that the above proof essentially indicates that,
for all u2Z;CðuÞ[C0ðuÞBðuÞand PðBðuÞÞ PðCðuÞ
[C0ðuÞÞ >0. By the ergodicity ([17, p. 340 Theorem
(1.3)]) of the indicator processes fIBðuÞgu2Zand
fICðuÞ[C0ðuÞgu2Z, we can obtain the following corollary (c.f.
[[16], Lemma 5.2])
Corollary 1.
With probability 1, there are infinitely many nodes u to the
left (and right) of 0 such that BðuÞhappens. The same
thing holds for CðuÞ[C0ðuÞ.
Proposition 3.
Under the assumption of Theorem 1, if d <dc,then with
probability 1 the limiting value X1ðuÞ5lim t!1 XtðuÞexists
and fjX1ðuÞ2X1ðu11Þjg 2 f0g[½d;1Þ for every u 2Z.
Proof
Given the initial opinion configuration fX0ðuÞgu2Z, let u
1
be a node such that Cðu121Þ[C0ðu121Þhappens, and let
u25min fu>u1:CðuÞ[C0ðuÞhappensg. Since CðuÞ[
C0ðuÞBðuÞfor every u2Z, the opinions in the interval
fu1;u111;;u2gwill not be affected by nodes outside
and vice versa. From Corollary 1, we know that each u2Z
must sit in some such interval. Hence, we only need to
show the proposition for every u2fu1;u111;;u2g.
The rest of the proof essentially follows from Theorem
5.3 [16]. We outline the argument as follows. Define the
energy of the interval fu1;u111;;u2gat time tas
Wt5X
u2fu1;u111;;u2g
XtðuÞ20:
If the nodes uand u11 in the interval exchange
opinions, W
t
drops by an amount of 2lð12lÞj
Xt2ðuÞ2Xt2ðu11Þj, and W
t
is always decreasing with
respect to time t. This fact together with the conditional
version of the Borel-Cantelli Lemma [17, p. 240, Corol-
lary (3.2)] can be used to show
lim
t!1 max fjXtðuÞ2Xtðu11ÞjIfjXtðuÞ2Xtðu11Þjdg:
u2fu1;u111;;u221gg50:(11)
For any edge fu,u11gin the interval
fu1;u111;;u2g, a single opinion adjustment can only
increase jXtðuÞ2Xtðu11Þj by at most ld. Exploiting (11)
we can see that either jXtðuÞ2Xtðu11Þj >dfor all large
enough tor lim t!1 jXtðuÞ2Xtðu11Þj50. Finally, this
together with the fact that the quantity Pu2fu1;u111;;u2g
XtuðÞremains unchanged over time can be used to
show the existence of the limit lim t!1 XtðuÞ.
If dc51, it is easy to check that all the arguments in
this section still hold true by replacing d
c
with d1efor
any e>0. Hence, the subcritical part of Theorem 1 is
completed.
3.2. Supercritical Regime: d>d
c
To understand the behavior of the Deffuant model in
the regime d>dc(with dc<1) the following two lemmas
are critical.
Lemma 6.
Suppose that the assumption of Theorem 1 holds. Fix d 2R.
With probability 1, for any u 2Z,either jXtðuÞ2Xtðu11Þj >d
for all large enough t or lim t!1 jXtðuÞ2Xtðu11Þj50.
Proof
For each u2Z, similarly as in Proposition 3, we define the
energy at node uas WtðuÞ5XtðuÞ2. We further define a
continuous-time step function W†
tðuÞby W†
0ðuÞ50 and
42 C O M P L E X I T Y Q2013 Wiley Periodicals, Inc.
DOI 10.1002/cplx
W†
tðuÞjumps an amount of 2lð12lÞjXt2ðuÞ2Xt2ðu11Þj2at
time twhen opinion adjustment occurs on the edge fu,u
11g[16]. We can show as in Lemma 6.2 [16] that for any
u2Zand t0,
EWtðuÞ1EW†
tðuÞ5EðX2Þ<1:
Using this fact and the conditional Borel-Cantelli
Lemma, we can finish the proof as in Proposition 3
(see the proof of Proposition 6.1 [16]).
Using the powerful tool linking SAD and the Deffuant
model (Proposition 1), we have the following result,
which can be shown verbatim follows the proof of
Lemma 6.3 [16]. The proof entails combining an argu-
ment similar as Proposition 2 and a discussion for the
location of the mode on Z(see Lemma 2).
Lemma 7
Given an initial configuration fX0ðuÞgu2Zand e>0. If u 2
Zis two-sidedly e-flat, then XtðuÞ2½EX26e;EX16efor all
t0.
Proposition 4
Under the assumption of Theorem 1, if d >dc,then with
probability 1 X1ðuÞ5lim t!1 XtðuÞ5EX for every u 2Z.
Proof
Take an e>0 satisfying d>dc16e. We first show that with
probability 1
lim
t!1 jXtðuÞ2Xtðu11Þj50 (12)
for any u2Z(see Proposition 6.4 [16]). By Lemma 6, it
suffices to show that for each u,
PðjXtðuÞ2Xtðu11Þj >dfor all large enough tÞ50:(13)
Suppose that the probability in (13) is positive. Then
the event in (13) happens for infinitely many vwith
probability 1 by using the ergodicity theorem. Arguing
as in Proposition 3, we obtain that the limit X1ðuÞ
exists and fjX1ðuÞ2X1ðu11Þjg 2 f0g[½d;1Þ for every
u2Z. Lemma 5 implies that, with probability 1, there
exists a node wwhich is two-sidedly e-flat. From
Lemma 7, we know that X1ðwÞ2½EX26e;EX16e.By
Lemma 6, X1ðw11Þmust either exceed X1ðwÞby at
least d, or fall below X1ðwÞby at least d, or equal
X1ðwÞ. In other words, by the choice of e, we have
either (i) X1ðw11Þ>EX1dc, or (ii) X1ðw11Þ<EX2dc,
or (iii) X1ðw11Þ5X1ðwÞ. By the definition (6) and the
fact that min u2ZX0ðuÞX1ðw11Þmax u2ZX0ðuÞ, the
cases (i) and (ii) are impossible. Accordingly, we can
show X1ðwÞ5X1ðuÞiteratively for all u2Z. This, how-
ever, contradicts the infinitely many v(13) is established
and (12) holds.
Next, for the node wobtained above, we have XtðwÞ2
½EX26e;EX16efor all t0. For any u2Z, we obtain
with probability 1 that XtðuÞ2½EX27e;EX17efor large
enough tby invoking (12) (since there are only finitely
many edges between uand w). Taking e!0 completes
the proof.
4. SIMULATION STUDY
4.1. Methodology
In order to demonstrate and deepen our theoretical
results, we carry out the simulations on rings of different
sizes n, where each node is connected to its two neighbors
on either side. A real life parallel can be seen with resi-
dents in modern large apartment buildings where people
tend to build walls of privacy in an intellectual=emotional
sense and only know their neighbors live right next door
[18]. We propose several initial opinion distributions
which are modeled by continuous probability distribu-
tions. To be specific, we consider the following four
classes of probability distributions (see Figure 1 for their
density curves):
Uða;bÞ: uniform distribution on the interval [a, b],
whose probability density function is
fUða;bÞðxÞ51
b2aIfaxbg. We interpret it as even
opinions.
Betaða;bÞ: beta distribution on the interval [0, 1]
with parameters a>0andb>0. Its probability
density function is fBetaða;bÞðxÞ5xa21ð12xÞb21
Bða;bÞIf0x1g,
where B is the beta function. We will consider three
different pairs of parameters, namely, (a,b)5
(0.3,0.3) representing polarized opinions; (a,b)5
(3,3) representing unbiasedunimodalopinions;and
(a,b)5(2,5) representing biased unimodal opinions.
PLðx0;gÞ: power-law (Pareto) distribution on ½x0;1Þ
with parameter x0>0andg>0. Its probability den-
sity function is fPLðx0;gÞðxÞ5gxg
0x2g21Ifxx0g.Weinter-
pret it as divergent biased opinions.
Nð^
l;^
r2Þ: normal (or Gaussian) distribution on R,
whose probability density function is
fNð^
l;^
r2ÞðxÞ51
ffiffiffiffiffiffiffiffi
2p^
r2
pexp 2ðx2^
lÞ2
2^
r2
. It represents divergent
unbiased unimodal opinions.
Table 1 summarizes some important properties regard-
ing our theoretical result (Theorem 1) for these
distributions.
4.2. First Impression Effect
In this section, the simulations have been made to dis-
play the time evolution of opinions among a population
of n5500 individuals on a ring; see Figure 2. We deal
with six typical initial opinion distributions, namely
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DOI 10.1002/cplx
Uð0;2Þ;Betað0:3;0:3Þ;Betað3;3Þ;Betað2;5Þ;PLð1;2:5Þ, and
N(0,1), as described above. The parameters we used in the
simulations are summarized in Table 2.
The straightforward method for illustrating the evolu-
tion of opinions is to consider continuous-time evolution
by monitoring each Poisson jumps. However, the conver-
gence in our situation is much slower than that in a fully
mixed population [6,19] and the machine incurs out-of-
memory error due to the overwhelming computation.
Since our main goal here is to confirm the first impression
effect, we plot in Figure 2 the opinion evolutions by com-
pressing (and discretizing) time axis. Specifically, each
time unit in Figure 2(a–f ) corresponds to 50,000 times of
Poisson events. In the insets of Figure 2(a–f), we perform
independent simulations with each time unit correspond-
ing to 1,000 times of Poisson events, respectively.
We observe from Figure 2 the following. First, the opin-
ions converge to the average EXof initial opinions for all
the six situations. For the first four cases in Figure 2(a–d),
the first impression effect predicted in Theorem 1 is con-
firmed since we take d>dc(c.f. Table 2). For the last two
cases in Figure 2(e–f), we see that final consensus are also
reached at the average EX. We performed a number of
tests by using different confidence bound d(taking d>dc
when dc<1, and taking dlarge enough when dc51)
and different size nof nodes varying from 500 to 1000,
and they confirm the first impression effect.
Second, scattered opinion distributions (such as U(0,
2)) and B(0.3, 0.3) converge much slower than unimodal
opinion distributions (such as B(3, 3), B(2, 5) and N(0, 1)),
especially at early times. For example, in Figure 2(b) the
opinion discrepancy is around [0.3, 0.7] at t10 while it
is around [0.4, 0.6] in Figure 2(c). These differences can
also be seen clearly from the insets: there are no evident
convergence until time t560 for Figure 2(b) inset, but
the convergence can be discerned as early as t30 for
Figure 2(c,d) insets. For opinion distributions with diver-
gent supports [such as PL(1, 2.5) and N(0, 1)] the consen-
sus becomes sensitively contingent on d. When dis large,
fast consensus can be expected (as is shown in Figure
2(e,f), where we choose dalmost equal to the maximal
initial opinion difference). The reason why for PL(1, 2.5)
we need a much larger dthan other cases (c.f. Table 2) is
that power-law distribution has a heavy tail, and ‘‘outliers’’
are more likely to appear. For example, if we take d510
in Figure 2(e), we probably can not get consensus since
there is an opinion at around 20 and the only second larg-
est opinion is at around 10. The rate of consensus for dif-
ferent opinion distributions will be further studied in
section 4.4 by rescaling opinions on the same range and
fixing both land d.
4.3. Critical Confidence Bound
To determine the critical confidence bound d
c
,we
resort to Monte Carlo simulations. We fix l50:5 as above
since it does not affect the final configuration. Given the
number nof nodes, confidence bound dand the initial
opinion distribution X, we conduct the Deffuant model
algorithm on 1000 samples. The Deffuant algorithm pro-
ceeds until no node changes its opinion by more than
0.0001 for 100,000 times of consecutive Poisson events.
Let P
c
be the fraction of samples which reach a complete
FIGURE 1
Depiction of probability density functions of initial opinions studied in
the simulations. (a) uniform distribution Uða;bÞwith a50,b52
(green solid line), power-law distribution PLðx0;gÞwith
x051;g52:5 (red dashed line), and normal distribution Nð^
l;^
r2Þ
with ^
l50;^
r51 (blue dotted line); (b) beta distributions Betaða;bÞ
with ða;bÞ5ð0:3;0:3Þ(blue dotted line), ða;bÞ5ð3;3Þ(green solid
line), and ða;bÞ5ð2;5Þ(red dashed line).
44 C O M P L E X I T Y Q2013 Wiley Periodicals, Inc.
DOI 10.1002/cplx
FIGURE 2
Evolution of opinions with n5500 and l50:5. Each opinion is represented by a hollow diamond. In the main pictures, one time unit corresponds to
50,000 times of Poisson events; while in the insets one time unit corresponds to 1000 times of Poisson events. In each subfigure, the inset displays
another independent simulation with respect to the main picture.
COMPLEXITY 45Q2013 Wiley Periodicals, Inc.
DOI 10.1002/cplx
consensus. In Figure 3, we plot P
c
as a function of confi-
dence bound dfor four initial opinion distributions U(0,
2), Beta(0.3, 0.3), Beta(3, 3), and Beta(2, 5).
From Figure 3(a–d), we clearly observe that P
c
increases for d<dcand then saturates to 1 for d>dcin
each of the four cases. By examining the growth of P
c
against different population size n, we may conclude that
P
c
will converge to a step function in the limit n!1,
implying a critical confidence bound d
c
. This agrees with
our Theorem 1. Our results are compatible with an early
study of the critical bound [9], where individuals sit on
the sites of a square lattice and random graphs. Finally,
we remark that d
c
can not be finite for those with diver-
gent opinion supports [such as PL(1, 2.5) and N(0, 1)],
TABLE 2
The Parameters Used in the Simulations Corresponding to Figure 2
XEXd
c
dn lFig. no.
U(0, 2) 1 1 1.2 500 0.5 2(a)
Beta(0.3, 0.3) 0.5 0.5 0.6 500 0.5 2(b)
Beta(3, 3) 0.5 0.5 0.6 500 0.5 2(c)
Beta(2, 5) 0.286 0.714 0.8 500 0.5 2(d)
PL(1, 2.5) 1.67 120 500 0.5 2(e)
N(0, 1) 0 14 500 0.5 2(f)
FIGURE 3
Fraction of samples with complete opinion consensus as a function of the confidence bound. Three different numbers of individuals located on rings
are studied: n 51000, 5000, and 10,000. The critical confidence bound d
c
is indicated by a vertical dotted line in each subfigure (c.f. Table 2).
46 C O M P L E X I T Y Q2013 Wiley Periodicals, Inc.
DOI 10.1002/cplx
since it is always possible (i.e., with positive probability)
to find two nodes with distance arbitrarily large for large
enough n.
4.4. Comparison of Consensus Rate
In this section, we explore the rate of consensus in the
Deffuant model by considering four initial opinion distri-
butions U(0, 1), Beta(0.3, 0.3), Beta(3, 3), and Beta(2, 5).
Opinions with these distributions lie in the same range [0,
1]. Set DðtÞ5max u;vfXtðuÞ2XtðvÞg and
T5TX5min ft:DðtÞ<0:0001g:
We refer to Tas the time of reaching consensus. Simu-
lations are done for a population of n51000 individuals
on a ring and l50:5. Figure 4 represents the rate of con-
sensus versus confidence bound d. We observe the follow-
ing. First, for each given d, the times of reaching
consensus are arranged decreasingly as
TBetað0:3;0:3Þ>TUð0;1Þ>TBetað2;5Þ>TBetað3;3Þ:
This relation suggests that
polarized opinions even opinions
unimodal ðbut biasedÞopinions
unimoal and unbiased opinions;
where ‘‘’’ means ‘‘converges slower than’’. This agrees
with the observation in [14]. Furthermore, we see that the
difference between TBetað3;3Þand TBetað2;5Þis relatively small.
Note that there is absent of mainstream view in an even
opinion configuration and there are (at least) two main-
stream views in a polarized opinion configuration. Hence,
we may conclude that unimodality (the emergence of a
single mainstream view) is a prominent feature which
contributes to fast consensus of opinions.
Second, for each of these distributions convergence
becomes faster for larger d. This is intuitively clear since
when dis rising, more nodes are able to adjust their opin-
ions with each other.
Third, the convergence time Tis more sensitive to
U(0, 1) and Beta(0.3, 0.3) than Beta(3, 3). For example, T
reduces by almost 25% for Beta(0.3, 0.3) but around 15%
for Beta(3, 3) when dincreases all the way from 0.5 to 1.
This can be explained as follows. Even and polarized
opinions [represented by, e.g., U(0, 1) and Beta(0.3, 0.3)]
are essentially scattered, and the increase of dplays a
more important role in compromising the opinions since
the unimodal opinions [represented by, e.g., Beta(3, 3)
and Beta(2, 5)] are aggregated to some extent at the
outset.
5. DISCUSSION
We have shown that the Deffuant model on Zwith ini-
tial opinion distribution Xexhibits a critical confidence
bound d
c
provided the second order moment EX<1.
When d>dc, the opinions will converge eventually with
probability one to the initial mean value EX, a first
impression phenomenon. When d<dc, the limit configu-
ration is piecewise constant interrupted by jumps of size
at least d. It would be very interesting to better
FIGURE 4
Time of reaching complete consensus for different initial opinion
distributions as a function of confidence bound d. The results for
Beta(2, 5) starts from d50.8 since we only consider the super-
critical regime d>dc.
FIGURE 5
Depiction of an infinite Cayley tree CTrwith r53. The root node
is labeled 0. Each node uhas a unique father node u
f
. For any
node v,T(v) is the subtree with root v.
COMPLEXITY 47Q2013 Wiley Periodicals, Inc.
DOI 10.1002/cplx
understand the first impression effect of the Deffuant
model on general networks. An intermediate step toward
this goal can be an infinite Cayley tree (see e.g., [20,21]).
To formalize the question precisely we need the following
definitions.
For r2N, let CTrbe a labeled r-regular infinite tree
without leaf nodes; see Figure 5. The root node is labeled
0. Denote by T(u) the subtree with root u2CTr. Hence,
Tð0Þ5CTr. For any 0 6¼ u2CTr, there is a unique father
node, denoted by u
f
. Thus each edge in CTrhas a unique
representation like fu;ufg. Analogous to (5), given a
sequence u1;u2;2CTrand l2ð0;1=2, the associated
SAD process fYiðuÞgu2CTrcan be defined as
YiðuÞ5
Yi21ðuÞ1lðYi21ðufÞ2Yi21ðuÞÞ for u5ui;
Yi21ðuÞ1lðYi21ðuiÞ2Yi21ðuÞÞ for u5uf
i;
Yi21ðuÞfor u2CTrnfui;uf
ig:
If fu1$ukg:5fu1;u2;;ukgis a path between nodes
u
1
and u
k
on CTr, and Z:CTr!R, then we write
Zðu1ÞZðukÞ
meaning that Zðu1ÞZðu2ÞZðukÞ. Since the path
between u
1
and u
k
is unique, the above notations are
well-defined.
Note that when r3, the monotonicity (c.f. Lemma 1)
no longer holds, since liquids can pass through different
branches along the tree and much more complicated phe-
nomenon may happen. However, we are able to establish
the following ‘‘constrained’’ monotonicity result.
Proposition 5
Suppose that fYiðuÞgu2CTris obtained via a SAD such that
uj2f0$vg[TðvÞfor all 1ji.Then
Yið0ÞYiðvfÞmax u2TðvÞYiðuÞ:(14)
Proof
Assume that (14) holds for i5k– 1, and we need to
show it holds for i5k. It suffices to prove the following
three cases:
a. If uk2f0$vf
fg;Ykuf
k
f
Ykuf
k
Ykuk
ðÞ
YkwðÞ, where wf5uk;
b. If uk5vf;Ykuf
k
f
Ykuf
k
Ykuk
ðÞmax u2TvðÞ
YkuðÞ;
c. If uk5v;Ykuf
k
f
Ykuf
k
max u2TvðÞ
YkuðÞ.
To see (a) holds, we note that
Ykuf
k
f
5Yk21uf
k
f
12lðÞYk21uf
k
1lYk21uk
ðÞ5Ykuf
k
;
Ykuf
k
5Yk21uf
k
1lYk21uk
ðÞ2Yk21uf
k
Yk21uf
k
112lðÞYk21uk
ðÞ2Yk21uf
k
5Ykuk
ðÞ;
and
YkwðÞ5Yk21wðÞ12lðÞYk21uk
ðÞ1lYk21uf
k
5Ykuk
ðÞ:
To see (b) holds, we note that the first two inequalities
can be shown similarly as in case (a). The last inequal-
ity holds since
max
u2TðvÞYkuðÞ5max
u2TðvÞYk21uðÞ12lðÞYk21uk
ðÞ1lYk21uf
k
5Ykuk
ðÞ:
To see (c) holds, we note that the first inequality can
be shown similarly as above. Since Yk21uf
k
Yk21uk
ðÞ,
we have Ykuf
k
Ykuk
ðÞ. Moreover, we obtain
max
u2TðvÞnfukgYkuðÞ5max
u2TðvÞnfukgYk21uðÞ12lðÞYk21uf
k
1lYk21uf
k
f
5Ykuf
k
Therefore, the second inequality also holds.
Let ‘u
ðÞ
be the distance of uto the root 0. We have the follow-
ing result similarly as Lemma 3 (see [16, Theorem 2.3]).
Proposition 6
For any u 2CTr;sup i0YiuðÞ51=‘uðÞ11ðÞ.
It is an intriguing open problem to get a suitable char-
acterization of ‘‘flat points’’ on CTrso that the critical
value of confidence bound can be established. We are
currently working on the related issues and will release
the results elsewhere.
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