Article

Opinion Formation in Laggard Societies

Abstract and Figures

We introduce a statistical physics 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 phase 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 define the asymptotic state of opinions. In the phase diagram we find regions of total 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. Comment: 5 pages, eps figs
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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 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: Eros-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
01
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
01
t
=
¯
k
X
i=
¯
kp
u
¯
k
i
1 a
0
t
i
a
0
t
¯
ki
, (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
10
t
, where 1 and 0 are exchanged in Eq.(1).
The probability for a node to be switched from ’0’ to
’1’,
01
0
, is the product of the transitio n probability,
p
01
t
, and the probability to be originally in the frac-
tion a
0
0
, i.e.
01
0
= p
01
0
a
0
0
. The same reasoning gives
10
0
= p
10
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
+
10
0
01
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,
10
1
=
p
10
1
p
10
0
1 a
0
0
. For arbitrary
times t this is straight forwardly seen to be
10
t
=
p
10
t
p
10
t1
1 a
0
0
, and the master equation be -
comes a
0
t+1
= a
0
t
+
10
t
01
t
. Inserting for a
0
t
in
a recursive way yields the master equation
a
0
t+1
= a
0
0
+ p
10
t
1 a
0
0
p
01
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
+
10
t,0
01
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
01
t,0
and
10
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
10
t,
¯
t
and T
10
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
10
t,
¯
t
01
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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... al.[177] andKlimek et. al.[178]. In the former an unanimity rule is implemented, i.e., an agent can change its state if and only if all its neighbours sharing the same opinion, otherwise, its state remains unchanged. ...
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