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arXiv:0706.4058v1 [physics.soc-ph] 27 Jun 2007

Opinion Formation in Laggard Societies

Peter Klimek

1

, Renaud Lambiotte

2

, Stefan Thurner

1,3∗

1

Complex Systems Research Group; HNO; Medical University of Vienna; W¨ahringer G¨urtel 18-20; A-1090; Austria

2

GRAPES; Universit´e de Li`ege; Sart-Tilman; B-4000 Li`ege; Belgium

3

Santa Fe Institute; 1399 Hyde Park Road; Santa Fe; NM 87501; USA

We introduce a statistical p hysics model for opinion dynamics on random networks where agents

adopt the opinion held by the majority of their direct neighbors only if the fraction of these neighbors

exceeds a certain threshold, p

u

. We ﬁnd a transition from total ﬁnal consensus to a mixed ph ase

where opinions coexist amongst the agents. The relevant parameters are the relative sizes in the

initial opinion distribution within the population and the connectivity of the underlying network.

As the order parameter we deﬁ ne the asymptotic state of opinions. In the phase diagram we ﬁnd

regions of t otal consensus and a mixed phase. As the ’laggard parameter’ p

u

increases the regions of

consensus shrink. In addition we introduce rewiring of the underlying network during the opinion

formation process and discuss the resulting consequences in the phase diagram.

PACS numbers: 89.75.Fb, 87.23.Ge, 05.90.+m

INTRODUCTION

Many decisions of human beings are often strongly in-

ﬂuenced by their social surroundings , e.g. the opinion

of friends, colleagues or the neighborhood. Only a few

ty pes of decisions in few individuals emerge from abso-

lute norms and ﬁr m convictions which are independent

of the opinion of others. Much more common is the situ-

ation where some sort of social pressure lea ds individuals

to conform to a group, and take decisions which minimize

conﬂict within their nearest neighborhood. For example,

if a la rge fraction of my friends votes for one party, this

is likely to inﬂuence my opinion on whom to vote for ;

if I observe my peers realizing huge proﬁts by invest-

ing in some s tock this might have an inﬂuence on my

portfolio as well; and if the fra c tion of physicist friends

(coauthors) publishing paper s on networks exceeds a cer-

tain threshold, I will have to reconsider and do the same;

the social pressure would otherwise be just unbearable.

Lately, the study of opinion formation within societies

has become an issue of more quantitative scientiﬁc in-

terest. In ﬁrst attempts agents were considered as s ites

on a lattice, and opinion dynamics was incorporated by

the so-called voter model (VM) [1] (only two neighbors

inﬂuence e ach other at one timestep), the majority rule

(MR) [2, 3] (each member of a group of odd size adopts

the state of the local majority), o r the Axelrod model [4]

(where two neighbors inﬂuence themselves on possibly

more than one topic with the objective to become more

similar in their se ts of opinio ns). Imposing regular lattice

structure on social environments is convenient however,

most observed structures of r e al-world networks belong

to one of three classes: Erd¨os-Renyi (ER) [5], scale-free

[6] or small-world netwo rks [7]. This has been accounted

for the VM [8, 9, 10] as well as for the MR on diﬀerent

topologies [11, 12, 13]. For a review of further eﬀorts in

this directions see [14, 15] and citations therein. Another

FIG. 1: Update process for two diﬀerent conﬁgurations of

neighbors and an update threshold of p

u

= 0.8. The node in

the center gets updated. (a) Three out of ﬁve neighbors are

in a diﬀerent state, so the threshold is not exceeded and the

node stays unchanged. (b) Four out of ﬁve neighbors are in a

diﬀerent state; 4/5 ≥ 0.8 thus the node adopts the state.

approach to model social interaction was developed out

of the notion of catalytic sets [16], leading to an una-

nimity rule (UR) model [17] on arbitrary networks in an

irreversible formulation.

As a realistic model for many real world situations here

we present a reversible generalization to the UR and MR

models introducing an a rbitrary thresho ld governing up-

dates (’laggard’ parameter). The UR and MR are e x-

tremal cases of the model. In [1 8] the idea of a threshold

was intro duced in the context of investigating the origin

of global cascades in ER netwo rks of ’early-adopters’. In

contrast to this work, where updates were only allowed

in one direction, i.e. irreversible, the following model is

fully reversible in the sense that two opinions compete

against each other in a fully sy mmetric way.

2

THE MODEL

Each individual i is represente d as a node in a net-

work. The state of the node represents its opinion on

some subject. For simplicity we restrict ourselves on bi-

nary opinions, ye s/no, 0/1, Bush/Mother Theresa, etc.

Linked nodes are in contact with e ach other, i.e. they

’see’ or know each others opinion. The opinion fo rma-

tion process of node i is a three-step process (see Fig.1):

Suppose i is initially in state ’0’(’1’).

• Check the state of all nodes connected to i.

• If the fraction of state ’1’(’0’)-nodes of i’s neighbors

exceeds a threshold p

u

, i adopts opinion ’1’(’0’).

• Otherwise i remains in state ’0’(’1’).

As a substrate network we chose random graphs [5],

i.e. N nodes are randomly linked with L links (self-

interactions are forbidden), the averag e connectivity be-

ing

¯

k = L/N . The update threshold necessary for a

node’s change o f opinion, p

u

has to be higher than 0.5 in

order to be meaningful in the above sense. The update is

carried out asynchronously. In a network containing N

nodes, at time t, there are A

0

t

nodes with opinion ’0’ and

A

1

t

nodes with opinion ’1’. The relative number of nodes

are a

0/1

t

= A

0/1

t

/N . One time step is associated with

applying the update pro cedure N times, i.e. each node

gets up dated once pe r timestep on average. As time goe s

to inﬁnity, the relative population of nodes with opinion

0/1 will be denoted by a

0/1

∞

.

ANALYTICAL AND NUMERICAL RESULTS

To derive a master equation for the evolution of this

system we calculate opinion-transition probabilities via

combinatorial considera tio ns in an iterative fa shion, mo-

tivated by [16]. A master equation for a

0

t

is found ex-

plicitly, the situation for a

1

t

is completely analogous. At

t = 0, we have a fraction of a

0

0

nodes in state ’0’. The

probability that at time t one node belonging to a

0

t

will

ﬂip its opinion to ’1’ is denoted by p

0→1

t

. This probability

is nothing but the sum over all combinations where more

than a fraction of p

u

of the neighbor s are in state ’1’,

weighted by the probabilities for the neighboring nodes

to be either from a

0

t

or a

1

t

=

1 − a

0

t

,

p

0→1

t

=

¯

k

X

i=⌈

¯

kp

u

⌉

¯

k

i

1 − a

0

t

i

a

0

t

¯

k−i

, (1)

where ⌈.⌉ denotes the ceiling function, i.e. the near-

est integer being greater or equal. The same consid-

eration leads to an expression for the opp osite tran-

sition p

1→0

t

, where 1 and 0 are exchanged in Eq.(1).

The probability for a node to be switched from ’0’ to

’1’, ∆

0→1

0

, is the product of the transitio n probability,

p

0→1

t

, and the probability to be originally in the frac-

tion a

0

0

, i.e. ∆

0→1

0

= p

0→1

0

a

0

0

. The same reasoning gives

∆

1→0

0

= p

1→0

0

1 − a

0

0

and provides the master equation

for the ﬁrst time step (i.e. updating each node once on

average),

a

0

1

= a

0

0

+ ∆

1→0

0

− ∆

0→1

0

. (2)

Let us now examine some specia l cases.

The low connectivity limit

For suﬃciently low connectivities

¯

k there are no more

possible updates in the network after the ﬁrst iteration.

For a given update threshold p

u

, this is the ca se if a

change in opinion requires all neighboring nodes to have

the same state. To see this more c learly, consider the case

of a network with constant k = 2 (1D circle). Choos e a

node whose state is e.g. ’0’. There a re four possible

conﬁgurations of neighbors: both b e ing in state ’0’, one

being in ’0’ and the other in ’1’ and both being in ’1’.

Irrespective of p

u

, only the latter conﬁguration a llows

an update. For all o ther cases at least one neighbor in

state ’0’ must be updated to ’1’, i.e. has to have two

neighboring nodes in ’1’. But this is not possible, since

at least one neighbor will always be in ’0’. The same

holds for higher values of

¯

k, as long as every neighbor

has to hold the same opinion to a llow an update, i.e.

we eﬀectively use an unanimity rule. For the special case

k = 2 the ﬁnal popula tio n in state ’0’ is given by a

0

∞

= a

0

1

.

Inserting this in Eq.(2) yields

a

0

∞

= 3(a

0

0

)

2

− 2(a

0

0

)

3

. (3)

A compar ison between the theoretical prediction of

Eq.(3) and the simulatio n of this system (on a regular

1D circ le network with N = 10

4

) is see n in Fig.2(a).

Higher connectivities

For higher connectivities there are much more conﬁg-

urations allowing for potential updates, the evolution

does not stop after one single iteration. At the sec-

ond time step the update probability is given by the

product of the transition probability at t = 1 and the

probability to be initially in the respective state, re-

duced by the probability to already have undergone this

transition during the ﬁrst time step. We have, for ex-

ample, ∆

1→0

1

=

p

1→0

1

− p

1→0

0

1 − a

0

0

. For arbitrary

times t this is straight forwardly seen to be ∆

1→0

t

=

p

1→0

t

− p

1→0

t−1

1 − a

0

0

, and the master equation be -

comes a

0

t+1

= a

0

t

+ ∆

1→0

t

− ∆

0→1

t

. Inserting for a

0

t

in

a recursive way yields the master equation

a

0

t+1

= a

0

0

+ p

1→0

t

1 − a

0

0

− p

0→1

t

a

0

0

. (4)

3

FIG. 2: Asymptotic population sizes of the ’0’-state fraction, a

0

∞

, as a function of its initial size, a

0

0

, for N = 10

4

, p

u

= 0.8. (a)

k = 2 for all nodes (1D circle), (b) ER graph with

¯

k = 9000 and (c) ER graph with

¯

k = 10.

Again, theoretical predictions of Eq.(4) agree perfectly

with numerical ﬁndings, Fig.2(b). Three regimes can be

distinguished: two of them correspond to a network in

full conse ns us. Between these there is a mixed phase

where no consensus can be reached.

High connectivity limit. For the fully connected net-

work the as ymptotic population sizes can easily be de-

rived: if a

0

0

> p

u

or a

0

0

< 1 − p

u

consensus is reached.

For 1 − p

u

< a

0

0

< p

u

the system is frustrated and no up-

date will take place, giving rise to a diagram like Fig.2(b).

Compared to Fig.2(a) a sharp transition betwe e n the con-

sensus phases and the mixed phase has app e ared. We

now try to understand the origin of this transition.

Intermediate regime. The transition between the

smooth solution for the ﬁnal populations as a function of

a

0

0

and the sharp one for higher connectivities becomes

discontinuous when the possibility for an individual node

to ge t updated in a later timestep ceases to play a neg-

ligible role. Systems with small update probabilities will

then be driven towards the consensus states. However,

if the initial populations are too far from the consensus

states they will not be reached. For p

u

= 0.8 the sharp

transition arises for values of

¯

k around 10. Fig.2(c) shows

simulation data for ER graphs with N = 10

4

nodes and

¯

k = 10 with p

u

= 0.8 . Here we already ﬁnd two regimes

with consensus and an almost linear regime in-between.

The analy tical curve obtained from numerical summa-

tions of Eq.(4) resembles the qualitative behavior of the

simulations up to ﬁnite-size deviations. The dynamics of

the sys tem is shown in the phase diagram, Fig.3(a). It

illustrates the size of the respective regimes a nd their de-

pendence on the parameters a

0

0

and connectedness

¯

k/N.

The order parameter is a

0

∞

. Along the dotted lines a

smooth trans itio n takes place, solid lines indicate dis-

continuous transitions from the consensus phase to the

mixed phase. The change from smooth to sharp appears

at

¯

k/N ≈ 0.01. For larger p

u

the regions of consensus

shrink toward the left and right margins of the ﬁg ure.

So far we assumed static networks. However, this is

far from being realistic, as social ties ﬂuctuate. We thus

allow links to get randomly rewired with the rewirement

process taking place on a larger time scale than the opin-

ion update, since otherwise the new connection would not

lead to state changes. Let us assume that the number of

rewired links per rewirement-timestep is ﬁxed to L

′

, so

that it becomes natural to deﬁne a social temperature,

T = L

′

/L. T quantiﬁes the individual’s urge to recon-

sider a topic with new acquaintances, or equivalently, the

ﬂuctuation of ties in their social surrounding.

The evolution of opinions in a network at T 6= 0 is

as follows: We ﬁx a network and perfo rm the same dy-

namics as for T = 0, until the system has converged and

no further upda tes occur. Then per tur b the system by

a rewirement step a nd randomly rew ire L

′

links among

the N nodes (N and L are kept constant over time), in-

crease the time-unit for the rewirement steps by one and

let the system rela x into a (converged) opinion conﬁg-

uration. Iterate this procedure. Note that this process

can be viewed as a dynamical map of the curves shown

in Figs.2(a)-(c). With this view it becomes intuitively

clear that consensus will be reached for a wider range

of parameters, where the time to arrive there crucially

depends on the value of

¯

k.

To incorpo rate the temperature eﬀect in the ma ster

equation we introduce the second timescale and denote

the population in state ’0’ as a

0

t,

¯

t

. Her e t is the time for

the update proce ss as before and

¯

t is the time step on the

temper ature time scale, i.e. counts the number of rewire-

ment s teps. We use a

0

0

≡ a

0

0,0

. a

0

∞,0

can be obtained

from a

0

∞,0

= lim

t→∞

a

0

t,0

+ ∆

1→0

t,0

− ∆

0→1

t,0

for high

¯

k,

and from E q.(2) fo r low

¯

k, when we only observe up-

dates during the ﬁrst itera tio n. This evolution is nothing

but a dynamical map. The probabilities to ﬁnd a con-

ﬁguration of neighbors a llowing an update are no longer

given only by ∆

0→1

t,0

and ∆

1→0

t,0

, instea d we have to count

the ones constituted by a rewiring, which happens with

probability T . That is why we can consider this kind

of evolution as a dynamical map of the former proc ess,

with a

0

∞,0

as the initial population for the ﬁr st rewire-

4

FIG. 3: Phase diagram for a

0

∞

as a function of initial frac-

tion size a

0

0

and connectedness,

¯

k/N . Simulations where per-

formed with ER graphs with N = 10

3

and p

u

= 0.8. Two

symmetrical regions of consensus and a mixed phase in be-

tween are observed. The dotted line indicates a smooth tran-

sition, the solid line a discontinuous one. Inset: Detail for

small

¯

k. Arrows mark the change from smooth to sharp tran-

sitions, positioned at

¯

k/N ∼ 0.01. (b) Phase diagram for

a

0

∞,∞

. Technically adjacency matrices with N = 10

4

were

generated and checked by Monte-Carlo simulations whether

they allow an update at ﬁxed a

0

0

and

¯

k.

ment step evolving to a

0

∞,1

, and so on. The transition

probabilities are now given by T ∆

1→0

t,

¯

t

and T ∆

1→0

t,

¯

t

, since

only new co nﬁgurations ca n give rise to an update. We

thus assume the master equation for a system at T 6= 0

after the ﬁrst rewiring to be

a

0

∞,

¯

t+1

= lim

t→∞

a

0

t,

¯

t

+ T

∆

1→0

t,

¯

t

− ∆

0→1

t,

¯

t

. (5)

Furthermore, one expects the exis tence of a critical value

k

c

, below which the intermediate regime (mixed state)

will disappear. This will oc c ur whenever there is no

chance that a conﬁguration of neighbors can be found

leading to an update. The value for k

c

can be easily

estimated: Say we have a node in state ’1’ and ask if

an update to state ’0’ is possible under the given circum-

stances. For a given

¯

k this requires that there are at least

⌈

¯

kp

u

⌉ neighbors in state ’0’ present in the set A

0

0

. If

¯

k is

above the cr itical value k

c

it occurs that even if all nodes

FIG. 4: Half-life time τ (

¯

t until half th e population reached

consensus) vs. relative number of neighbors for T =

0.25, 0.5, 1. Inset: Same in log-log scale. Scaling around the

pole k

c

/N ∼ 0.61 with an exponent γ ≈ 7.4 is suggested. 10

3

initial populations with a

0

0

= 0.5 and N = 10

2

were averaged.

from A

0

0

were neighbors of the node in state ’1’, there a re

still too many other neighboring nodes (which are then

necessarily in state ’1’) to ex c e ed the upda te threshold.

This means tha t we can not have updates if ⌈

¯

kp

u

⌉ > A

0

0

,

and we get

k

c

=

a

0

0

N

p

u

. (6)

For p

u

= 0.8 and a

0

0

= 0.5, k

c

≈ 0.61 N . We next con-

sider the time-to-convergence in the system. To this end

we measure the half-life time τ, of initial populations at

a

0

0

= 0.5 for diﬀerent connectivities

¯

k, see Fig.4. The

ﬁgure suggests that the observed scaling of τ could be of

power-law type, with a pole at k

c

/N , i.e. τ ∝

k

c

−

¯

k

N

−γ

.

The estimated critical exponent γ ≈ 7.4 seems to be in-

dependent of temperature. Note, that the estimate is

taken r ather far from the pole at k

c

, which suggests to

interpret the actual numbers with some ca re.

The phase diagram for the T 6= 0 system is shown

in Fig.3(b). There are still thr e e regimes, which are ar-

ranged in a diﬀerent manner than before. Consensus is

found for a much wider range of or der par ameters; the

mixed phase is found for high connectivities, i.e.

¯

k > k

c

.

The value of k

c

at a

0

0

= 0.5, as found in Fig.3(b), is 0.63,

slightly above the prediction of 0.61. This mismatch is

because we used networks with inho mogeneous degr ee

distributions (Poisson). Whether a network allows for an

upda te or not is solely determined by the node with the

lowest degree k, which explains why we can still observe

upda tes when the average degree

¯

k is near to but already

above k

c

. Systems in the mixed phase are frustrated. k

c

is linear in a

0

0

which we conﬁrm by ﬁnding a straight line

separating the frustrated drom the consensus phase, see

5

Fig.3(b). For larger p

u

the regions of consensus shrink.

CONCLUSION

Summarizing we presented a model bridg ing the gap

between existing MR and UR models. Opinion dynamics

happ e ns on static random networks where agents adopt

the opinion held by the majority of their direct neighbo rs

only if the fraction of neighbors exceeds a pre-spec iﬁed

laggard-threshold, p

u

. The la rger this par ameter the

more stimulus the agent needs to adapt his opinion to

the one o f his direct neighborhood. This system shows

two phases, full c onsensus and a mixed phase where opin-

ions coexist. We studied the corresponding phase di-

agram as a function of the initial opinion distribution

and the connectivity of the underlying networks. As the

laggard-parameter p

u

increases the regions of full co n-

sensus shrink. We introduced rewiring of the underlying

network during the opinion formatio n process and discuss

the resulting consequences for the phase diagram. This

social temperature introduced here diﬀers from the usual

temper ature of statistical mechanics. It acc ounts for link

ﬂuctuations and not for the ﬂuctuations of the sta te of

the nodes. For T > 0, the s ystem can escape the frozen

state a

0/1

∞

6= 1, and globa l consensus can be obtained. In

the c ase of usual temperature (opinions of nodes switch

randomly) [12], a diﬀerent behavior is expected. For low

temper ature, the system a lso can escape the frozen sta te,

however for higher values of T the system undergoes a

transition from an ordered to an unordered phase, where

a

∞

= 1/2. Even though laggards sometimes enjoy a

bad reputation as being slow and backward-oriented, so-

cieties of laggar ds are shown to have remarkable levels of

versatility as long as they are not forced to interact too

much.

Supported by Austrian Science Fund FWF Projects

P17621 and P19132 and COST P10 action.

∗

Electronic address: thurner@univie.ac.at

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