Papers in Physics, vol. 1, art. 010004 (2009)

Received: 7 October 2009, Accepted: 7 October 2009
Edited by: M. C. Barbosa
Licence: Creative Commons Attribution 3.0
DOI: 10.4279/PIP.010004

www.papersinphysics.org

ISSN 1852-4249

Answer to the Commentary on “A note on the consensus time of
mean-field majority-rule dynamics”

Damián H. Zanette1∗

In his Commentary [1], H. Fort begins by point-
ing out two aspects of the majority-rule (MR)
model that, as presented in the main paper [2], may
need some clarification.

It was suggested in the paper that the equiva-
lence of the mean-field MR dynamics and a ran-
dom walk can be formulated in terms of the evo-
lution of N+, the number of agents with opinion
+1. Specifically, in order to write down a master
equation for the process, one should find the tran-
sition probability that, in a single evolution step,
N+ changes to any other N ′+. It can be readily
seen that such event requires that the size G of the
group of agents chosen to reach consensus at that
step satisfies G > 2|∆|, with ∆ = N ′+ − N+. If
∆ is positive (respectively, negative) the number
of agents with opinion −1 (respectively, +1) in the
group must be exactly equal to |∆|. Summing the
probabilities for all the possible values of G yields
the transition probability for a given ∆.

Of course, one could cut off and renormalize the
probability distribution for the group size, pG, in
such a way that G can be at most equal to the
population size N . The two regimes in the size
dependence of the consensus time would certainly
still exist –although explicitly working out analyt-

∗E-mail: zanette@cab.cnea.gov.ar

1 Consejo Nacional de Investigaciones Cient́ıficas y
Técnicas, Centro Atómico Bariloche and Instituto Bal-
seiro, 8400 San Carlos de Bariloche, Ŕıo Negro, Ar-
gentina.

ical results such as Eq. (7) of the main paper may
become trickier. However, it is somehow artificial
to conceive that, in a social process driven by events
which involve agent groups, probabilities depend
on the total population size. It sounds more nat-
ural to just allow any group size and, in the case
that G ≥ N , admit that the population falls into
an absorbing, frozen state. Also, from an opera-
tional viewpoint, a cut-off probability distribution
with G ≤ N would be difficult to implement if the
population size varies with time.

The Commentary also addresses a series of gen-
eralizations of the MR dynamics –some of which
have already been considered in the literature–
that certainly add realism to the model. Hetero-
geneity among agents is crucial to more realisti-
cally approach any population-based complex sys-
tem. In the context of the MR model, inflexible
and unsettled agents as well as “contrarians” have
been considered in a series of papers by S. Galam
and coworkers (and recently reviewed by Castel-
lano et al. [3]). Also, we have discussed the ef-
fects of several forms of heterogeneity on synchro-
nization dynamics [4, 5], which bear close simi-
larities with opinion formation. Overall, the ex-
pected consequence of heterogeneity is that full
consensus is never reached, but a dynamical state –
where the degree of consensus fluctuates with time
around a well-defined average– is asymptotically
approached.

Spatially-distributed populations, where agent
groups are localized in space, have been taken into

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Papers in Physics, vol. 1, art. 010004 (2009) / D. H. Zanette

account since the first formulations of the MR dy-
namics [6, 7]. It has been shown that dimensional-
ity has nontrivial consequences in the attainment of
consensus. For a fixed group size G, the consensus
time grows with the population size N as a power
whose exponent depends on the dimension [3]. In
one-dimensional arrays, the final state is sensibly
dependent on the initial condition, while in higher
dimensions, the dynamics coincides with diffusive
coarsening.

Either random or deterministic time-dependent
effects, such as stochastic fluctuations in the rules
that govern single consensus events or population
dynamics, are certainly worth considering. One
may ask, for instance, whether sufficiently frequent
“births” or “arrivals” of dissenters are able to over-
come consensus attainment. To my knowledge,
population dynamics has not been addressed in the
context of MR model. On the other hand, random-
ness in each consensus event has already been taken
into account [3].

Finally, global opinion-formation factors such as
mass media, advertising, and propaganda play the
role of external fields in the spin-like dynamics of
the MR model, as also discussed by S. Galam [3].

[1] H Fort, Commentary on “A note on the con-
sensus time of mean-field majority-rule dy-
namics”, Pap. Phys. 1, 010003 (2009).

[2] D H Zanette, A note on the consensus time of
mean-field majority-rule dynamics, Pap. Phys.
1, 010002 (2009).

[3] C Castellano, S Fortunato, V Loreto, Statis-
tical physics of social dynamics, Rev. Mod.
Phys. 81, 591 (2009).

[4] G H Paissan, D H Zanette, Synchronization of
phase oscillators with heterogeneous coupling:
A solvable case, Physica D 237, 818 (2008).

[5] D H Zanette, Interplay of noise and coupling
in heterogeneous ensembles of phase oscilla-
tors, Eur. Phys. J. B 69, 269 (2009).

[6] P L Krapivsky, S Redner, Dynamics of major-
ity rule in two-state interacting spin systems,
Phys. Rev. Lett. 90, 238701 (2003).

[7] C J Tessone, R Toral, P Amengual, H S Wio,
M San Miguel, Neighborhood models of minor-
ity opinion spreading, Eur. Phys. J. B 39, 535
(2004).

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