Papers in Physics, vol. 6, art. 060005 (2014)

Received: 3 August 2014, Accepted: 3 August 2014
Edited by: G. Martinez Mekler
Licence: Creative Commons Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.060005

www.papersinphysics.org

ISSN 1852-4249

Reply to the Commentary on “Critical phenomena in the spreading of
opinion consensus and disagreement”

A. Chacoma,1 D. H. Zanette1, 2∗

1. Taking into account the reviewer’s concern
(see Ref. [1]), we have replaced the right panel of
Fig. 1 in our main article (see Ref. [2]) by a plot
showing the evolution of an entire 200-agent array.
We hope that this dissipates the possible confusion
pointed out by the reviewer. The caption and the
main text have been modified accordingly.

2. Indeed, the equivalence between the one-
dimensional (1D) versions of the voter model
with nearest-neighbor interactions and of diffusion-
limited binary annihilation (A + A → 0) has been
recognized since the first studies of coarsening pro-
cesses [3]. The boundaries separating domains with
different opinions in the 1D voter model move as
random walkers, which annihilate with each other
when they meet during their motion. In view that,
in the case of a linear array with pD = pC = 1, our
model reduces to two mutually intercalated voter
systems, the scaling laws of binary annihilation also
apply to our results. It is well known, for instance,
that the number of particles a(t) in 1D diffusion-
limited annihilation decays with time as a ∼ t−1/2
[4]. This implies that, in a finite system, the time
needed for complete annihilation of an initial (even)
number of particles, a(0), goes as T ∼ a(0)2. In our

∗E-mail: zanette@cab.cnea.gov.ar

1 Instituto Balseiro and Centro Atómico Bariloche, 8400
San Carlos de Bariloche, Ŕıo Negro, Argentina.

2 Consejo Nacional de Investigaciones Cient́ıficas y
Técnicas, Argentina.

model, in turn, the initial number of boundaries
between opinion domains can be seen to behave as
b(0) ∼ n+(0)n−(0)N, where n±(0) is the initial
fraction of agents with each opinion, and N is the
system size. The above result indicates that the
time needed for all the boundaries to disappear is
T ∼ n2+(0)n2−(0)N2. In other words, as illustrated
by the results shown in the lower panel of Fig. 2, the
product N−2T depends only on the initial concen-
tration of each opinion. A similar argument makes
it possible to show that the probability of reaching
consensus Pcons depends on n±(0), but it is inde-
pendent of the system size, as shown in the upper
panel of the same figure.

On the other hand, the possible connection
between the case pD, pC 6= 1 and branching-
annihilation random walks, is less clear. As re-
marked by the reviewer, this connection should,
in turn, establish a link with the universality class
of directed percolation. However, the facts that
our model exhibits multiple absorbing states and
that there is no phase where fluctuations persist at
asymptotically long times (as well as the absence
—in the 1D case, where the connection is expected
to hold— of nontrivial critical exponents) do not
seem to suggest a relation to that universality class
[5]. The point, nevertheless, is worth considering
in future work.

3. Certainly, as acknowledged in the paper’s fi-
nal section, the most important direction in which
our model should be extended is to consider more

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Papers in Physics, vol. 6, art. 060005 (2014) / A. Chacoma et al.

complex topologies, in particular, those that rep-
resent real-life social systems. The one- and two-
dimensional arrays studied in the paper are just a
convenient —and probably the simplest— way of
defining the groups of agents that participate in
the opinion dynamics. It must be realized, how-
ever, that the existence of an underlying network
of social contacts (either ordered or not) is not nec-
essary to specify the structure of groups relevant to
our class of models. In fact, the most general defini-
tion of the group structure is to provide a list of all
the groups present in the population, enumerating
the agents that belong to each group. This pro-
cedure encompasses all the possible partitions into
groups of any given population —even those that
cannot be represented by means of an underlying
network [6]— and thus allows for the consideration
of any degree of complexity compatible with the
population size. The active group G and the ref-
erence group G′ involved in each interaction event
can then be chosen —for instance, at random—
from the list that specifies the group structure.

Note that, from this perspective, a network —
whose topology is entirely defined by the list of all
its links— is nothing but a structure formed by a
set of two-agent groups. In this sense, the notion of
group structure generalizes that of network, intro-
ducing a kind of higher-degree connection between
population members [6, 7].

[1] F Bagnoli, Commentary on “Critical phenom-
ena in the spreading of opinion consensus and
disagreement”, Pap. Phys. 6, 060004 (2014).

[2] A Chacoma, D H Zanette, Critical phenomena
in the spreading of opinion consensus and dis-
agreement, Pap. Phys. 6, 060003 (2014).

[3] S Redner, A guide to first-passage processes,
Cambridge University Press, Cambridge (2001).

[4] A S Mikhailov, A Y Loskutov, Foundations
of synergetics II. Chaos and noise, Springer,
Berlin (1996).

[5] H Hinrichsen, Nonequilibrium critical phenom-
ena and phase transitions into absorbing states,
Adv.Phys. 49, 815(2000).

[6] D H Zanette, Beyond networks: Opinion for-
mation in triplet-based populations, Phil. Trans.
R. Soc. A 367, 3311 (2009).

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

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