Papers in Physics, vol. 1, art. 010002 (2009)

Received: 6 July 2009, Accepted: 2 September 2009
Edited by: M. C. Barbosa
Reviewed by: H. Fort (Universidad de la República, Uruguay)
Licence: Creative Commons Attribution 3.0
DOI: 10.4279/PIP.010002

www.papersinphysics.org

ISSN 1852-4249

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

Damián H. Zanette1∗

In this work, it is pointed out that in the mean-field version of majority-rule opinion dy-
namics, the dependence of the consensus time on the population size exhibits two regimes.
This is determined by the size distribution of the groups that, at each evolution step,
gather to reach agreement. When the group size distribution has a finite mean value,
the previously known logarithmic dependence on the population size holds. On the other
hand, when the mean group size diverges, the consensus time and the population size
are related through a power law. Numerical simulations validate this semi-quantitative
analytical prediction.

Much attention has been recently paid, in the
context of statistical physics, to models of so-
cial processes where ordered states emerge spon-
taneously out of disordered initial conditions (ho-
mogeneity from heterogeneity, dominance from di-
versity, consensus from disagreement, etc.) [1].
Not unexpectedly, many of them are adaptations
of well-known models for coarsening in interacting
spin systems, whose dynamical rules are reinter-
preted in the framework of social-like phenomena.
The voter model [2, 3] and the majority rule model
[4, 5] are paradigmatic examples. In the latter,
consensus in a large population is reached by ac-
cumulative agreement events, each of them involv-
ing just a group of agents. The present note is
aimed at briefly revisiting previous results on the
time needed to reach consensus in majority-rule dy-
namics, stressing the role of the size distribution of
the involved groups. It is found that the growth of
the consensus time with the population size shows

∗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.

distinct behaviors depending on whether the mean
value of the group size distribution is finite or not.

Consider a population of N agents where, at any
given time, each agent has one of two possible opin-
ions, labeled +1 and −1. At each evolution step,
a group of G agents (G odd) is selected from the
population, and all of them adopt the opinion of
the majority. Namely, if i is one of the agents in
the selected group, its opinion si changes as

si → sign
∑
j

sj, (1)

where the sum runs over the agents in the group. Of
course, only the agents, not the majority, effectively
change their opinion. In the mean-field version of
this model, the G agents selected at each step are
drawn at random from the entire population.

It is not difficult to realize that the mean-field
majority-rule (MFMR) dynamics is equivalent to a
random walk under the action of a force field. For a
finite-size population, this random walk is moreover
subject to absorbing boundary conditions. Think,
for instance, of the number N+ of agents with opin-
ion +1. As time elapses, N+ changes randomly,
with transition probabilities that depend on N+
itself, until it reaches one of the extreme values,

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Papers in Physics, vol. 1, art. 010002 (2009) / D. H. Zanette

N+ = 0 or N. At this point, all the agents have
the same opinion, the population has reached full
consensus, and the dynamics freezes.

In view of this overall behavior, a relevant quan-
tity to characterize MFMR dynamics in finite pop-
ulations is the consensus time, i.e. the time needed
to reach full consensus from a given initial condi-
tion. In particular, one is interested in determining
how the consensus time depends on the population
size N. The exact solution for three-agent groups
(G = 3) [5] shows that the average number of steps
needed to reach consensus, Sc, depends on N as

Sc ∝ N log N, (2)

for large N. The proportionality factor depends
in turn on the initial unbalance between the two
opinions all over the population. The analogy of
MFMR dynamics with random walks suggests that
this result should also hold for other values of the
group size G, as long as G is smaller than N. This
can be easily verified by solving a rate equation
for the evolution of N+ [1]. Numerical results and
semi-quantitative arguments [6] show that Eq. (2)
is still valid if, instead of being constant, the value
of G is uniformly distributed over a finite interval.

What would happen, however, if, at each step, G
is drawn from a probability distribution pG that al-
lows for values larger than the population size? If,
at a given step, the chosen group size G is equal to
or largen than N, full consensus will be instantly at-
tained and the evolution will cease. In the random-
walk analogy, this step would correspond to a single
long jump taking the walker to one of the bound-
aries. Is it possible that, for certain forms of the
distribution pG, these single large-G events could
dominate the attainment of consensus? If it is so,
how is the N-dependence of the consensus time
modified?

To give an answer to these questions, assume that
G is drawn from a distribution which, for large G,
decays as

pG ∼ G−γ, (3)

with γ > 1. Tuning the exponent γ of this power-
law distribution, large values of G may become suf-
ficiently frequent as to control consensus dynamics.

The probability that at the S-th step the selected
group size is G ≥ N, while in all preceding steps
G < N, reads

PS =

(
N−1∑

G=Gmin

pG

)S−1 ∞∑
G=N

pG, (4)

where Gmin is the minimal value of G allowed for
by the distribution pG. The average waiting time
(in evolution steps) for an event with G ≥ N is thus

Sw =
∞∑
S=1

SPS =

(
∞∑

G=N

pG

)−1
∝ Nγ−1, (5)

where the last relation holds for large N when pG
verifies Eq. (3).

Compare now Eqs. (2) and (5). For γ > 2 (re-
spectively, γ ≤ 2) and asymptotically large popu-
lation sizes, one has Sw � Sc (respectively, Sw �
Sc). This suggests that above the critical expo-
nent γcrit = 2, the attainment of consensus will
be driven by the asymptotic random-walk features
that lead to Eq. (2). For smaller exponents, on the
other hand, consensus will be reached by the occur-
rence of a large-G event, in which all the population
is entrained at a single evolution step. Note that
γcrit stands at the boundary between the domain
for which the mean group size is finite (γ > γcrit)
and the domain where it diverges (γ < γcrit).

In order to validate this analysis, numerical sim-
ulations of MFMR dynamics have been performed
for population sizes ranging from 102 to 105. The
probability distribution for the group size G has
been introduced as follows. First, define G = 2g+1.
Choosing g = 1, 2, 3, . . . ensures that the group size
is odd and G ≥ 3. Then, take for g the probability
distribution

pg =
1

ζ(γ)
g−γ, (6)

where ζ(z) is the Riemann zeta function. With this
choice, pG satisfies Eq. (3). The average waiting
time for a large-G event, given by Eq. (5), can be
exactly given as

Sw =
ζ(γ)

ζ(γ, 1 + N/2)
, (7)

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Papers in Physics, vol. 1, art. 010002 (2009) / D. H. Zanette

where ζ(z,a) is the generalized Riemann (or Hur-
witz [7]) zeta function. In the numerical simula-
tions, both opinions were equally represented in the
initial condition. The total number of steps needed
to reach full consensus, S, was recorded and aver-
aged over series of 102 to 106 realizations (depend-
ing on the population size N).

Figure 1: Numerical results for the number of steps
needed to reach consensus, S, normalized by the
population size N, as a function of N, for three
values of the exponent γ. The straight dotted lines
emphasize the validity of Eq. (2) for γ = 2.5 and 3.
For γ = 2 the line is horizontal, suggesting S ∝ N.

The two upper data sets in Fig. 1 show the ratio
S/N for two values of the exponent γ > γcrit. Since
the horizontal scale is logarithmic, a linear depen-
dence in this graph corresponds to the proportion-
ality given by Eq. (2). Dotted straight lines illus-
trate this dependence. For these values of γ, there-
fore, the relation between the consensus time and
the population size coincides with that of the case
of constant G. For the lowest data set, which corre-
sponds to γ = γcrit, the relation ceases to hold. The
horizontal dotted line suggests that now S ∝ N, as
predicted for γ = 2 by Eq. (5).

The log-log plot of Fig. 2 shows the number of
steps to full consensus as a function of the popu-
lation size for three exponents γ ≤ γcrit. The dot-
ted straight line has unitary slope, representing the
proportionality between S and N for γ = 2. For
lower exponents, the full curves are the graphic rep-
resentation of Sw as given by Eq. (5). The excellent

Figure 2: Number of steps needed to reach consen-
sus as a function of the population size, for three
values of the exponent γ. The slope of the straight
dotted line equals one. Full curves correspond to
the function Sw given in Eq. (7).

agreement between Sw and the numerical results
for S demonstrates that, for these values of γ, the
consensus time in actual realizations of the MFMR
process is in fact dominated by large-G events.

Figure 3: Fraction of realizations where consensus
is attained through a large-G event as a function of
the population size, for several values of the expo-
nent γ.

A further characterization of the two regimes of
consensus attainment is given by the fraction of
realizations where consensus is reached through a

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Papers in Physics, vol. 1, art. 010002 (2009) / D. H. Zanette

large-G event. This is shown in Fig. 3 as a function
of the population size. For γ < γcrit, consensus is
the result of a step involving the whole population
in practically all realizations. As N grows, the fre-
quency of such realizations increases as well. The
opposite behavior is observed for γ > γcrit. For the
critical exponent, meanwhile, the fraction of large-
G realizations is practically independent of N, and
fluctuates slightly around 0.57.

In summary, it has been shown here that in
majority-rule opinion dynamics, the dependence of
the consensus time on the population size exhibits
two distinct regimes. If the size distribution of the
groups of agents selected at each evolution step de-
cays fast enough, one reobtains the logarithmic an-
alytical result for constant group sizes. If, on the
other hand, the distribution of group sizes decays
slowly, as a power law with a sufficiently small ex-
ponent, the dependence of the consensus time on
the population size is also given by a power law.
The two regimes are related to two different mecha-
nisms of consensus attainment: in the second case,
in particular, consensus is reached during events
which involve the whole population at a single evo-
lution step. The logarithmic regime occurs when
the mean group size is finite, while in the power-law
regime the mean value of the distribution of group
sizes diverges. In connection with the random-walk
analogy of majority-rule dynamics, this is reminis-
cent of the contrasting features of standard and
anomalous diffusion [8].

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