Papers in Physics, vol. 1, art. 010003 (2009)

Received: 2 October 2009, Accepted: 6 October 2009
Edited by: M. C. Barbosa
Licence: Creative Commons Attribution 3.0
DOI: 10.4279/PIP.010003

www.papersinphysics.org

ISSN 1852-4249

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

Hugo Fort,1∗

In this Commentary, I review the article by D.
H. Zanette on the consensus time of mean-field
majority-rule dynamics [1]. The paper identifies
two different regimes for the mean field (MF) ver-
sion of the majority-rule (MR) opinion dynamics,
characterized by different dependences on the pop-
ulation size N . In one of them, corresponding to
gradual persuasion, the typical known logarithmic
dependence is observed. The novelty appears in
the alternative regime, associated with very drastic
events, which is governed by a power law. In this
Commentary, I point out a couple of minor points
that, in my opinion, deserve further clarification. I
also make some general remarks and briefly discuss
some features which can be incorporated in order
to use the model in more realistic contexts.

The author addresses the problem of the depen-
dence of the consensus time with the population
size in a mean field (MF) version of the majority-
rule (MR) opinion dynamics. The size of the group
of agents selected at each evolution step, G, is
drawn from a probability distribution pG, which
for large G decays as a power law.

The main result is that, for MFMR, the consen-
sus time S exhibits two distinct regimes, charac-
terized by different dependences on the population
size N . If the exponent of pG is larger than 2, S has

∗E-mail: hugo@fisica.edu.uy

1 Instituto de F́ısica, Facultad de Ciencias, Universidad
de la República, Iguá 4225, CP 11.400 Montevideo,
Uruguay.

a dependence of the kind N log N which is already
known from analytical results for constant G. On
the other hand, if the exponent of the distribution
of group sizes is less or equal than 2, the dependence
of S on N is also given by a power law. It is interest-
ing that the two regimes are related to two different
mechanisms of consensus attainment: gradual per-
suasion versus drastic large-G events which involve
the whole population at a single evolution step.

Some points that would need further clarification
are:

a. The equivalence of the MFMR to a random walk
under the action of a force field, mentioned at
the end of page 1, is an interesting issue. For
those who are not experts on this area I think
that it deserves a more detailed explanation; in-
deed, the walk has variable step length since only
the agents with the minority opinion flip, and
the direction of the force seems to exhibit a sort
of persistence.

b. What is the rationale for considering values of
G larger than the population size? In fact, the
maximum physically possible G is N . Perhaps it
is for technical reasons or difficulties implement-
ing the constraint that G cannot be greater than
N ?

Some considerations, a little beyond the scope of
the majority-rule opinion model, on features con-
cerning realistic situations which are not included
in this model:

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Papers in Physics, vol. 1, art. 010003 (2009) / H. Fort

1. Individual heterogeneity. There are always in-
dividuals who are not susceptible to the MR
for their convictions, or by necessity, and resist
the majority opinion. In addition, there are in-
dividuals who can change their mind not only
by following herd behaviour but also through
other mechanisms like learning from experi-
ence, etc.

2. Space. The spatial structure might have a rel-
evant effect on the MR opinion dynamics. Has
this been studied? In general, it turns out
that spatial correlation may introduce impor-
tant differences in agent based models.

3. Chance. The application of the MR is com-
pletely deterministic. The effects of introduc-
ing a stochastic component is something worth
exploring. For example, in the form that some
of the individuals in the minority of G do not
flip or some in the majority suffer spontaneous
flips.

4. Population dynamics. For long times it seems
unavoidable to consider births and deaths, this
turnover population seems to be a dynamic
source for the opinion formation. Maybe this
could be implemented by a noise term like the
one mentioned above.

5. Opinion Formation Factors. Mass media sig-
nals, advertising and propaganda operating
over the individuals play an important role in
opinion formation and can have an important
impact. It seems that this could be modeled
by external fields.

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

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