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arXiv:1011.1124v2 [nlin.AO] 5 Nov 2010
EPJ manuscript No.
(will be inserted by the editor)
Opinion formation and cyclic dominance in adaptive networks
G¨ uven Demirel1, Roshan Prizak2, P. Nitish Reddy2, and Thilo Gross1
1Max-Planck-Institute for the Physics of Complex Systems – N¨ othnitzer Straße 38, 01187 Dresden, Germany
2Department of Electrical Engineering, Indian Institute of Technology – Bombay, Powai, 400076, Mumbai, India
Received: November 8, 2010
Abstract. The Rock-Paper-Scissors(RPS) game is a paradigmatic model for cyclic dominance in biological
systems. Here we consider this game in the social context of competition between opinions in a networked
society. In our model, every agent has an opinion which is drawn from the three choices: rock, paper or
scissors. In every timestep a link is selected randomly and the game is played between the nodes connected
by the link. The loser either adopts the opinion of the winner or rewires the link. These rules define an
adaptive network on which the agent’s opinions coevolve with the network topology of social contacts. We
show analytically and numerically that nonequilibrium phase transitions occur as a function of the rewiring
strength. The transitions separate four distinct phases which differ in the observed dynamics of opinions
and topology. In particular, there is one phase where the population settles to an arbitrary consensus
opinion. We present a detailed analysis of the corresponding transitions revealing an apparently paradoxial
behavior. The system approaches consensus states where they are unstable, whereas other dynamics prevail
when the consensus states are stable.
1 Introduction
Networks have been used as a metaphor for describing
complex systems from a vast range of fields [1,2,3,4,5,6].
Many previous studies considered the dynamics of net-
works, a line of research in which the network itself is
treated as a dynamical system. A prominent example is
the preferential attachment mechanism for network growth
leading to scale-free topologies [7]. Other works focused on
the dynamics on networks, where each node carries a dy-
namical state, the time evolution of which is coupled to
other states according to the network topology. Examples
include for instance the synchronization of phase oscilla-
tors [8,9] and epidemics [10,11] on complex networks.
Until ten years ago the dynamics of and on networks
were studied separately in models, whereas it is clear that
both processes typically take place simultaneously in the
real world [12,13]. If one considers both types of dynamics
in the same model, an adaptive network is formed [12,13].
In the past decade adaptive networks have been studied
for instance in the context of epidemiology [14,15,16,17,
18], opinion formation [19,20,21,22,23,24], neuroscience
[25,26,27,28,29,30,31], and emergence of cooperation [32,
33,34,35,36,37,38,39]. They have been shown to exhibit
dynamical phenomena such as robust self-organization to
critical behavior [25], formation of complex topologies[40],
complex system-level dynamics [41], and emergence of lead-
ership [34]. A comprehensive collection of works can be
found in [42].
Correspondence to: guven@pks.mpg.de
An area that has received considerable attention in
adaptive networks research is the behavior of simple para-
digmatic models of opinion formation. These include con-
tinuous models such as Deffuant-type compromise models
[43] and discrete models such as voter-type models [19,20,
21,22,23,24].
The original voter model [44] considers the spread-
ing of opinions across a static network. The nodes, rep-
resenting agents in this network, change their opinion dy-
namically according to a rule capturing social adjustment,
the alignment of opinions with neighboring nodes. A sec-
ond fundamental mechanism not considered in the voter
model is social segregation, i.e. preferential linking be-
tween agents holding similar opinions and rejection of con-
nections to agents of different opinion. Coupling social
segregation with social adjustment leads to an adaptive
network. Adaptive voter models [21,22,23] in which the
network topology coevolves with the state of the nodes are
especially appealing targets for research, because many re-
sults that are found in this simple model appear to hold
also in significantly more complex systems in which many
different opinions interact by a pairwise-symmetric com-
petition [19,23].
For motivating the model considered in the present
paper, let us speculate a bit about opinion formation in
the real world. In many questions of broad importance
such as religious belief or mitigation of climate change,
it is evident that there is multitude of different opinions.
If pairwise comparisons between opinions are made, there
will certainly be some pairings in which the two opinions
under consideration are almost equally attractive, whereas
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2 G¨ uven Demirel et al.: Opinion formation and cyclic dominance in adaptive networks
in other pairings one opinion is clearly much more attrac-
tive than the alternative. Previous results [19,23] seem to
suggest that even such complex situation can in princi-
ple be understood by considering fundamental motifs of
relationships between opinions: Those opinions that are
clearly inferior to the opinions held by the majority will
certainly vanish exponentially. The competition between
the remaining opinions, which are almost equally attrac-
tive, may then be adequately be captured by the voter-like
models. However, one building block is still missing – the
motif of cyclic dominance. This motif depicts a cycle of
opinions in which every opinion appears superior in com-
parison with the next one in the cycle, but inferior when
compared to the previous one.
We believe that cyclic dominance in opinion dynamics
can appear when topics of broad interest are discussed in
the general public. Although a globally advantageous so-
lution may indeed exist, the agents participating in the
opinion formation process are typically non-specialists,
which may have only partial knowledge of the situation.
For instance in the discussion of public security, it is often
quoted that a) additional security measures are necessary
to stop an increase in crime, b) closed-circuit cameras are
much cheaper than additional police patrols and c) cam-
eras have no quantifyable effect on crime, so one might as
well save the expense for the camera. Together, these (ad-
mittedly partial) arguments define a cycle of dominance
between the three options “do-nothing”, “more-police”,
and “more-cameras”.
The most simple model of cyclic dominance is known
as the rock-paper-scissors(RPS) game. A single RPS game
is played by two agents, each choosing between the three
pure strategies: rock (R), paper(P), and scissors (S). An
agent who has chosen R wins against an agent who has
chosen S, an agent who has chosen S wins against an agent
who has chosen P, and finally an agent who has chosen P
wins against an agent who has chosen R. Thus none of the
options is globally advantageous (Fig. 1).
Paper
Scissors
Rock
Fig. 1. The cyclic dominance between strategies Rock(R), Pa-
per(P), and Scissors(S). Every option wins against the option
its arrow is pointing to, but loses against the third option.
The RPS game has been studied in different contexts
such as bacterial competition [46,47,48], mating strategies
[49,50], learning in social interaction systems [51,52], and
emergence of cooperation [53,54]. It has been shown that
the cyclic dominance functions as the source of biological
diversity and species coexistence in many biological sys-
tems [46,47,48,49,50,55]. For instance, the three-morph
mating system in side-blotched lizards displays sustained
oscillations, a dynamic form of species coexistence, due
to the cyclic dominance relationship between the three
morphs [49].
The effects of network topology on the RPS game dy-
namics has been the subject of several previous investi-
gations [56,57,58,59,60,61]. In well-mixed systems, cyclic
dominance can be insufficient for ensuring coexistence [56,
62], whereas on regular lattices, it is maintained through
spiral chaos [59,60,61]. Further, in populations of mobile
agents on regular lattices, a non-equilibrium phase tran-
sition from coexistence to exclusion is observed as dif-
fusion of agents is increased [59,61]. When studied on
degree-regularsmall-world networks with annealed and/or
quenched randomness, the RPS game displays non-equilibrium
phase transitions between stable coexistence, oscillations,
and uniformity [57,58].
Here we consider the RPS game on an adaptive social
network. Our model captures the cyclic dominance rela-
tionship between three opinions and the processes of social
adjustment and social segregation. We start by defining
the adaptive RPS game in Section 2. Then, we develop
a low-dimensional analytical approximation in Section 3.
Analytical results from the low-dimensional model and nu-
merical results from agent-based simulations are discussed
and compared in Section 4. We finish by the summary and
discussion of results in Section 5.
2 Adaptive RPS game
We consider a network of N nodes and L links, where
nodes correspond to agents and links represent social in-
teractions. On the network, each node has an internal state
representing its opinion (R, P, or S). The network is ini-
tialized as an Erd˝ os-Renyi random graph and agents are
assigned states randomly with equal probability. The sys-
tem is then left to evolve according to the following rules:
In an update step, a link is chosen at random. If the se-
lected link connects agents in the same state (inert link)
then nothing happens. If the link connects agents in dif-
ferent states (active link) then an interaction takes place,
from which one agent emerges as the winner and the other
as the loser according to the rules of the RPS game. With
probability p, the loser cuts its connection to the winner
and establishes a new connection (rewires) to a randomly
chosen agent of its own state. Otherwise (with probability
¯ p = 1 − p), the loser adopts the opinion of the winner.
We note that both rewiring and adoption events con-
serve the total number of nodes and links in the sys-
tem. Therefore the mean degree ?k? = 2L/N is time-
independent and can be treated as a parameter.
Throughout this paper we measure time in terms of
time steps, corresponding to N/2 update steps, such that
the expected number of games in which an agent partic-
ipates per time step is one. We remark that by selecting
links, highly connected agents participate in more games.
This link update rule has been chosen as it presents a
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G¨ uven Demirel et al.: Opinion formation and cyclic dominance in adaptive networks3
compromise between the direct and reverse update rules
commonly used in the voter model [24].
3 Low-dimensional model
Previous studies showed that the dynamics of discrete
adaptive networks can be captured by low-dimensional
approximations, describing the global densities of certain
subgraphs in the network [14,16,21,22,23,39,63,64]. In
the following we consider the densities of nodes of a certain
opinion A ∈ {R,P,S}, denoted by [A], and the densities of
links between nodes with given opinions A and B, denoted
by [AB]. These densities are understood to be normalized
to the total number of nodes N, such that
[R] + [P] + [S] = 1(1)
and
[RR] + [PP] + [SS] + [RS] + [PR] + [SP] = ?k?/2. (2)
Following the procedure in [14], we derive the equa-
tions of motion
d
dt[R] = ¯ p([RS] − [PR]),
d
dt[P] = ¯ p([PR] − [SP]),
d
dt[S] = ¯ p([SP] − [RS]),
?
?
d
dt[SS] = ¯ p
d
dt[RS] = ¯ p
−[RS]2
d
dt[SP] = ¯ p
−[SP]2
d
dt[PR] = ¯ p
−[PR]2
d
dt[RR] = ¯ p
d
dt[PP] = ¯ p
[RS] +[RS]2
[S]
−2[PR][RR]
[R]
−2[SP][PP]
−2[RS][SS]
[S]
+[RP][PS]
?
?
?
+ p[PR],
[PR] +[PR]2
[R][P]
+ p[SP],
?
?
[SP] +[SP]2
[P]
+ p[RS]
−[RS] +2[RS][SS]
−[RS][PR]
?
−[SP][RS]
?
−[PR][SP]
[S][P]
[S]
−[SP] +2[SP][PP]
[R]
?
− p[RS],
+[RS][PR]
[P][R]
[P]
−[PR] +2[PR][RR]
[S]
?
− p[SP],
+[SP][RS]
[R][S]
[R][P]
?
− p[PR].
(3)
The first three equations describe change in the den-
sity of agents holding a given opinion. For instance the
first equation captures the change in the density of agents
holding opinion R, [R], which depends on the gain from
agents of opinion S adopting opinion R at the rate p[RS]
and the loss from nodes of opinion R adopting opinion P at
the rate p[PR]. The next three equations in equation (3)
describe the densities of inert links. Every adoption or
rewiring event creates at least one inert link. In addition,
more inert links can be created in adoption events if the
adopting node has multiple neighbors holding the opinion
that is adopted. This creation of additional inert links is
captured by the quadratic terms in the equations. Finally,
inert links can be destroyed if one of connected agents
adopts the opinion of a third agent. The corresponding
rates then scale both with the density of inert links of
a given type, say [RR], and the rate at which a given
node of the respective type adopts a different opinion,
here p[PR]/[P]. A factor of two arises because changing
the opinion of either of the two nodes on a inert link is
sufficient to turn the inert link into an active link. The
last three equations in equation (3) describe the dynamics
of active links, which are affected by the same processes
as the inert links.
0
1
p=0.2
[P]
p=0.3
0 600
0
1
p=0.4
time
[P]
0600
p=0.8
time
[S]
[P]
[R]
Fig. 2. (Color online) Transient and long term dynamics of the
RPS system. The timeseries show the density of players holding
the opinion P in four different phases: stationary phase (top-
left, p = 0.2), oscillatory phase (top-right, p = 0.3), consensus
phase (bottom-left, p = 0.4), and fragmented phase (bottom-
right, p = 0.8) The insets show the corresponding long-term
behavior in the ternary phase space spanned by the opinion
densities R, P, and S. Parameters: ?k? = 4, N = 106.
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4 G¨ uven Demirel et al.: Opinion formation and cyclic dominance in adaptive networks
4 Results
In the following we analyze the dynamics of the RPS game
as a function of the parameter p, which can be interpreted
as the relative frequency of social segregation as response
to discomforting interactions. In representative simulation
runs of the agent based model (Figure 2), four different
types of dynamical behavior are evident: First, at small
values of p we observe a stationary phase. Although the
network remains dynamic on the microscopic level of indi-
vidual nodes and links, the macroscopic densities of opin-
ions, [R], [P], and [S], approach a state of stationary co-
existence in the stationary phase. Second, at higher values
of p there is an oscillatory phase where the three opinions
show oscillatory behavior as they go through a stable cy-
cle of succession (Figure 3). Third, if p is increased further
then oscillatory behavior is only observed transiently while
the system goes through oscillations of increasing ampli-
tude. The system then hits the absorbing boundary where
all agents are of the same opinion. In this consensus phase
all links are inert and the dynamics freezes. Fourth, for
high values of p we observe a fragmented phase in which
rewiring rapidly drives the system to an absorbing state in
which the network consists of disconnected communities
in which a local consensus is reached. These results are
summarized in Table 1
Table 1. Dynamical Phases
Phase
Micro
Dynamics
Macro
Dynamics
stationary
(mixed)
oscillatory
(mixed)
stationary
(polarized)
stationary
(mixed)
Network
Stationaryongoing connected
Oscillatoryongoing connected
Consensusfrozen connected
Fragmented frozenfragmented
We locate the transitions between the four phases nu-
merically by simulation of the agent-based model and an-
alytically by computation of bifurcations in the low-di-
mensional approximation. For illustration, a comparison
of the expected maximum and minimum values of [P] in
the long-term dynamics is shown in Figure 4. Although the
diagrams in Figure 4 are similar, there are notable differ-
ences: In the analytical model a) the oscillatory phase is
absent and b) the location of the transition points seems
to be relatively poorly approximated. We argue that the
analytical approximation is nevertheless a valuable tool.
Regarding a) it will become apparent below that the ana-
lytical model, despite the absence of the oscillatory phase,
provides a conceptual framework for understanding the
transitions leading to this phase. Regarding b) we note
that the discrepancies arise mainly from the relatively low
value of ?k?, which we chose specifically to accentuate the
differences. The phase diagram in Figure 5 shows that for
higher ?k? the numerical results approach the analytical
5080
1/6
1/3
1/2
time
[P],[R],[S]
[P]
[R]
[S]
Fig. 3. (Color Online) Cyclic succession in the oscillatory
phase. Shown are time series of the three variables [R], [P], and
[S] in the long-term dynamics. Parameters: ?k? = 4, N = 106,
p = 0.3.
prediction. It is remarkable that for networks with high
mean degree ?k? the oscillatory and fragmented phase oc-
cupy only a small portion of the parameter space, such
that the system is with high probability in the stationary
phase if p < 0.5 and in the consensus phase otherwise.
For considering the transitions between phases more
closely, we start by computing the location of the steady
state in which the system resides in the stationary phase
[P] = [R] = [S] =1
[RS] = [SP] = [PR] =1
3,
9
?
?
?k? −
?k?
2+
1
1−p
?
?
,
[PP] = [RR] = [SS] =1
9
1
1−p
.
(4)
The corresponding state is stationary irrespective of
p, but is dynamically unstable for p > 0.5. Computation
of the eigenvalues of the system’s Jacobian matrix reveals
that the destabilization at p = 0.5 occurs in a Hopf bi-
furcation. Generically, this type of bifurcation can appear
in two different forms [65]. In a supercritical Hopf bifur-
cation, a stable limit cycle emerges as the steady state
loses stability, whereas in a subcritical Hopf bifurcation,
an unstable limit cycle contracts around the stable steady
state and vanishes as the state is destabilized. For distin-
guishing between the two forms of Hopf bifurcation one
computes the first Lyapunov coefficient in the bifurcation
point. This coefficient is negative in the supercritical case
and positive in the subcritical case.
In the present system the corresponding Lyapunov co-
efficient is exactly zero, which corresponds to a degenerate
situation in which the steady forms a center of a dense set
of cycles with neutral stability.
The degeneracy of the Hopf bifurcation in the analyt-
ical model is certainly an artefact of the approximation.
This problem could be fixed by considering higher order
moments in the expansion, which should turn the degener-
ate bifurcation either into a supercritical or a subcritical
Hopf bifurcation. However, already considering network
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G¨ uven Demirel et al.: Opinion formation and cyclic dominance in adaptive networks5
0
1/3
2/3
1
[P]
0 0.250.5
p
0.751
0
1/3
2/3
1
[P]
Fig. 4. (Color Online) Comparison of numerical results (top)
with the analytical approximation (bottom). Shown are the
maximum (purple-filled circles) and minimum (blue-empty cir-
cles) values of [P] in the long-term dynamics. Both the sta-
tionary phase (low p) and the fragmented phase (high p) are
characterized by maxima and minima close to [P] = 1/3. By
contrast in the consensus phase some timeseries approach an
all-P state ([P] = 1) while others approach an all-R or all-S
state ([P] = 0). An oscillatory phase, where [P] oscillates in a
finite range, is only observed in the numerical model. Param-
eters: ?k? = 4, N = 106.
moments of third order increases the number of dynamical
equations in the model to 30 and introduces some addi-
tional complications [66]. In the present paper we therefore
do not present the third order expansion, but note that
the numerical results strongly suggest a supercritical Hopf
bifurcation, which explains the onset of oscillations.
We note that the location of the Hopf bifurcation chan-
ges at low mean degree. It occurs at p = (15?k? − 14 −
√6)/(15?k?+ 2− 2√6) if (14+√6)/15 ≤ ?k? ≤ 2. In this
range the Hopf bifurcation is supercritical. However, the
band of oscillations is very narrow; they appear only for
(15?k? − 14 −√6)/(15?k? + 2 − 2√6) ≤ p ≤ 1 − 1/?k?
(Figure 5).
0 25
0
0.25
0.5
0.75
1
<k>
p
0
0.25
0.5
0.75
1
p
consensus
stationary
fragmented
consensus
stationary
fragmented
oscillatory
Fig. 5. (Color online) Phase diagram of the adaptive RPS
game from numerical simulations (top) and the analytical ap-
proximation (bottom). Depending on the values of the mean
degree ?k? and the frequency of rewiring p we find four distinct
phases (labeled, see also Table 1). The oscillatory phase is ab-
sent in the analytical approximation. The numerical results are
based on networks with N = 106.
Let us now move on to the transition to the consen-
sus phase. In this phase the system approaches one of
three consensus states in which all agents on the network
hold the same opinion. Consequently, the density of active
links is zero ([RS] = [SP] = [PR] = 0) and the dynamics
freezes. Any finite system encountering such an absorbing
state must therefore remain there for all time. In small
systems there is a significant probability that a consensus
state is encountered “by accident” regardless of the pa-
rameter values. However, our numerical results show that
in sufficiently large networks the consensus states are only
observed in a certain parameter range. For understanding
the transitions delimiting this range we therefore consider
the thermodynamic limit, N → ∞, captured by the ana-
lytical model.
In the limit of infinite network size, the consensus states
are stationary but not absorbing. Because of the normal-
ization, there can be a finite number of active links, even
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6 G¨ uven Demirel et al.: Opinion formation and cyclic dominance in adaptive networks
if [RS] = [SP] = [PR] = 0. Consider a system resting in
one of the consensus states, say [P] = 1, in which a finite
number of nodes of state S and a finite number of SP-
links remain. If a game is played on one of the SP-links,
the agent holding opinion P will lose and react either by
rewiring the SP-link or by adopting S. Thus two scenar-
ios are possible, if p is sufficiently large then the removal
of SP-links by rewiring events dominates over creation of
SP-links by adoption events and the number of SP-links
declines exponentially. In this case [P] = 1 is stable. How-
ever, if creation of SP-links by adoption is faster than re-
moval by rewiring then [SP] and [S] grow exponentially.
In this case the [P] = 1 state is a saddle point which has an
unstable manifold on which the system can depart toward
[S] = 1.
Intuition suggests that consensus dynamics should be
observed only if the consensus states are stable, however,
almost the opposite is the case. By stability analysis of the
consensus states in the analytical model we find that the
consensus state is unstable for p < p∗= 1−1/?k?, showing
that consensus states are unstable in the consensus phase.
For understanding why these states are observed despite
their instability, we return to the example from the previ-
ous paragraph. In the parameter range under considera-
tion the [P] = 1 state is susceptible to the invasion of play-
ers holding opinion S. The system will therefore depart
exponentially from the [P] = 1 state and eventually ap-
proach [S] = 1. However, for reasons of symmetry [S] = 1
must be susceptible to the invasion of R, and [R] = 1 must
be susceptible to the invasion of P. Thus a limit cycle is
closed, which connects all three consensus states. Such a
cycle connecting multiple saddle-points is called a hetero-
clinic loop [65]. Because the approach and departure from
a consensus state are exponential, a round-trip on the het-
eroclinic loop takes infinite time and an observer studying
the system on the loop at an arbitrary time will find it in
one of the saddle points (here, the consensus states) with
probability one.
For finding the transition marking the onset of the con-
sensus phase we have to ask for the value of p at which not
the consensus states, but the heteroclinic loop connecting
them becomes dynamically stable. This analysis is compli-
cated by the general difficulties in the stability analysis of
limit cycles and the specific degeneracy of our analytical
approximation described above. Instead, we therefore set-
tle for a plausibility argument. Consider that the stability
of any limit cycle can only change in bifurcations. One
such bifurcation is the transcritical bifurcation of cycles,
in which two cycles meet and interchange their stability.
In the numerical results we have already observed that
a stable limit cycle emerges from what we identified as
a Hopf bifurcation. When p is increased this cycle grows
until it coincides with the heteroclinic loop. At this point
the heteroclinic loop becomes stable while the limit cycle
is destabilized and leaves the physical space. In the ana-
lytical model we do not observe an oscillatory phase, but
the dense set of cycles created in the degenerate Hopf bi-
furcation extends to the heteroclinic loop where it should
likewise induce a change in stability.
Finally, we consider the transition from the consensus
phase to the fragmented phase. In the analytical model
this transition occurs at p = p∗, which we identified at
the parameter value at which the consensus states become
stable. For understanding why the stabilization of the con-
sensus states marks the end of the consensus phase, note
that the consensus states are not the only absorbing states
in the system. To qualify as absorbing, a given state has
to satisfy [RS] = [SP] = [PR] = 0, whereas the other
variables, [R], [P], [S], [RR], [PP], [SS], can assume ev-
ery set of values that is consistent with the normalization.
The absorbing states thus include the consensus states and
a much larger mass of fragmented states in which different
opinions survive in disconnected network components. If
p < p∗all of these states are dynamically unstable. Be-
cause only the consensus states profit from the heteroclinic
mechanism, all other do not appear in the long-term dy-
namics. By contrast, for p > p∗all absorbing states are dy-
namically stable. The consensus states are now only three
points on a huge manifold of stable states and fragmented
behavior is observed with high probability.
In numerical simulations of networks in the fragmented
phase we observed that active links are removed so rapidly
that only few adoption events occur before the absorbing
state is reached. Therefore, the distribution of opinions in
the fragmented absorbing state closely mirrors the initial
distribution.
5 Summary and Discussion
In the present paper we have proposed a model for the
cyclic dominance of three opinions spreading across an
adaptive network. An agents i in this network respond to
discomforting interactions with another agent j either by
social adjustment, adopting j’s opinion, or by social seg-
regation, breaking the connection to j and establishing a
new connection to a third agent k who shares the opin-
ion of i. We showed that the dynamics of this systems
depends on the mean degree of the agents and the rela-
tive frequency of social segregation. Depending on the val-
ues of these parameters there are four distinct dynamical
phases, which we characterized as stationary, oscillatory,
consensus and fragmented.
We note that our model combines features that were
previously observed in models of cyclic dominance on non-
adaptive networks [49,56,57,58] and in voter-like mod-
els on adaptive networks [21,22,23]. Specifically, behav-
ior closely reminiscent of the stationary, oscillatory and
consensus phasea was observed in models of cyclic domi-
nance with different degrees of disorder [57] whereas the
fragmentation transition, leading to the fragmented phase,
was observed in adaptive voter models [21,23]. One can
therefore suspect that the transitions connecting the sta-
tionary, oscillatory, and consensus phase could be under-
stood in terms of the disorder that is generated, in our
model, intrinsically by rewiring of network connections.
By contrast, the fragmentation transition is a genuine
adaptive network effect that cannot be observed in non-
adaptive networks. In contrast to previous voter-like mod-
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G¨ uven Demirel et al.: Opinion formation and cyclic dominance in adaptive networks7
els, the present system can exhibit sustained long-term
dynamics, whereas the voter-like models always reach an
absorbing state.
Our analysis revealed that global consensus is con-
nected to the presence of a heteroclinic loop. A closely re-
lated homoclinic mechanism was recently shown to lead to
a state of full cooperation in a snowdrift game on adaptive
networks [39]. A similar mechanism was also suspected to
be at work in the model proposed in [58]. This evidence
suggests that homoclinic and heteroclinic structures could
also play a role in other adaptive networks. A more de-
tailed analysis thus appears a promising target for future
mathematical research.
In the introduction we argued that the motif of cyclic
dominance could appear in opinion formation processes
on questions of broad importance where a population of
non-specialists tries to reach a conclusion based on a set
of partial arguments (for instance provided by the me-
dia). A sensible behavior in this case were to maintain an
ongoing discussion until new evidence becomes available
that highlights one of the opinions as globally advanta-
geous. In our model we find such an ongoing discussion
in the stationary phase. In networks of high connectiv-
ity, this phase is observed when social adjustment is more
frequent than social segregation. However, if segregation
dominates then the system most likely ends up in the con-
sensus phase, where all agents agree on one opinion. Which
opinion is thus selected is arbitrary, because no opinion is
globally advantageous based on the available information.
Since this unreasonale response is caused by a collective
mechanism we are tempted to call it “swarm stupidity”.
If the consensus process, described above, occurred
in a real system, the agents would probably be unaware
that the decision was made arbitrarily. Because of the
cyclic approach to consensus, the consensus decision, say
R, has been proven to be superior to the opinion S, which
was previously held by the majority of the population.
Some agents may remember that earlier S itself replaced
P, which was clearly inferior to R (cf. Figure 3). Thus look-
ing back from the arbitrary consensus state it may appear
that this state was reached by making steady progress
from P to S to R.
Although we are not aware of sociological studies con-
cerning cyclic dominance of opinions, one may speculate
on the impact of the internet on opinion formation pro-
cesses. Personal acquaintances, friends, and family have
a high subjective value so the contact to them is likely
to be maintained even if they hold a different opinion in
certain matters. By contrast it is very easy to “rewire”
links to online sources of information. It therefore seems
natural to assume that an increasing importance of online
communication corresponds to an increased frequency of
segregation in our model and should therefore promote
the “swarm stupidity” behavior.
We believe that in the future the present model may
be a building block for investigations focusing on the more
realistic scenarios, where there is a complex networks of
pairwise dominance relationships between a larger number
of opinions.
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