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A process of rumour scotching on finite populations

Abstract and Figures

Rumour spreading is a ubiquitous phenomenon in social and technological networks. Traditional models consider that the rumour is propagated by pairwise interactions between spreaders and ignorants. Only spreaders are active and may become stiflers after contacting spreaders or stiflers. Here we propose a competition-like model in which spreaders try to transmit an information, while stiflers are also active and try to scotch it. We study the influence of transmission/scotching rates and initial conditions on the qualitative behaviour of the process. An analytical treatment based on the theory of convergence of density-dependent Markov chains is developed to analyse how the final proportion of ignorants behaves asymptotically in a finite homogeneously mixing population. We perform Monte Carlo simulations in random graphs and scale-free networks and verify that the results obtained for homogeneously mixing populations can be approximated for random graphs, but are not suitable for scale-free networks. Furthermore, regarding the process on a heterogeneous mixing population, we obtain a set of differential equations that describes the time evolution of the probability that an individual is in each state. Our model can also be applied for studying systems in which informed agents try to stop the rumour propagation, or for describing related susceptible–infected–recovered systems. In addition, our results can be considered to develop optimal information dissemination strategies and approaches to control rumour propagation.
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Research
Cite this article: de Arruda GF, Lebensztayn
E, Rodrigues FA, Rodríguez PM. 2015 A process
of rumour scotching on nite populations.
R. Soc. open sci. 2: 150240.
http://dx.doi.org/10.1098/rsos.150240
Received: 1 June 2015
Accepted: 18 August 2015
Subject Category:
Mathematics
Subject Areas:
applied mathematics/mathematical
modelling/statistical physics
Keywords:
rumour process, asymptotic behaviour,
density-dependent Markov Chain, Monte Carlo
simulation, epidemic model, stochastic model
Author for correspondence:
Pablo Martín Rodríguez
e-mail: pablor@icmc.usp.br
A process of rumour
scotching on nite
populations
Guilherme Ferraz de Arruda1, Elcio Lebensztayn2,
Francisco A. Rodrigues1and Pablo Martín Rodríguez1
1Departamento de Matemática Aplicada e Estatística, Instituto de Ciências
Matemáticas e de Computação,Universidade de São Paulo - Campus de São Carlos,
Caixa Postal 668, São Carlos, São Paulo 13560-970, Brazil
2Instituto de Matemática, Estatística e Computação Cientíca, UniversidadeEstadual
de Campinas - UNICAMP, Rua Sérgio Buarquede Holanda 651, Campinas,
São Paulo 13083-859, Brazil
Rumour spreading is a ubiquitous phenomenon in social
and technological networks. Traditional models consider that
the rumour is propagated by pairwise interactions between
spreaders and ignorants. Only spreaders are active and may
become stiflers after contacting spreaders or stiflers. Here we
propose a competition-like model in which spreaders try to
transmit an information, while stiflers are also active and try
to scotch it. We study the influence of transmission/scotching
rates and initial conditions on the qualitative behaviour of
the process. An analytical treatment based on the theory of
convergence of density-dependent Markov chains is developed
to analyse how the final proportion of ignorants behaves
asymptotically in a finite homogeneously mixing population.
We perform Monte Carlo simulations in random graphs and
scale-free networks and verify that the results obtained for
homogeneously mixing populations can be approximated for
random graphs, but are not suitable for scale-free networks.
Furthermore, regarding the process on a heterogeneous mixing
population, we obtain a set of differential equations that
describes the time evolution of the probability that an
individual is in each state. Our model can also be applied
for studying systems in which informed agents try to stop
the rumour propagation, or for describing related susceptible–
infected–recovered systems. In addition, our results can be
considered to develop optimal information dissemination
strategies and approaches to control rumour propagation.
1. Introduction
Spreading phenomena is ubiquitous in nature and technology [1].
Diseases propagate from person to person [2], viruses contaminate
computers worldwide and innovation spreads from place to
2015 The Authors. Published by the Royal Society under the terms of the Creative Commons
Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted
use, provided the original author and source are credited.
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place [3,4]. In the last decades, the analysis of the phenomenon of information transmission from a
mathematical and physical point of view has attracted the attention of many researchers [1,510]. The
expression ‘information transmission’ is often used to refer to the spreading of news or rumours in
a population or the diffusion of a virus through the Internet. These random phenomena have similar
properties and are often modelled by mathematical models [57].
In this paper, we propose and analyse a process of rumour scotching on finite populations. An
interacting particle system is considered to represent the spreading of the rumour by agents on a given
graph, representing a finite population of size n. We assume that each agent, or node of the graph, may
be in any of the three states belonging to the set {0, 1, 2}, where 0 stands for ignorant, 1 for spreader and
2 for stifler. Finally, the model is formulated by considering that a spreader tells the rumour to any of its
(nearest) ignorant neighbours at rate λand that a spreader becomes a stifler owing to the action of its
(nearest neighbour) stifler nodes at rate α.
When the considered graph is the complete graph, representing a finite homogeneously mixing
population, we obtain limit theorems regarding the proportion of ignorants at the end of the process.
That is, when there are not more spreaders in the population. In addition, we study the model in random
graphs and scale-free networks through Monte Carlo simulations. The computational approach allows
us to verify that the results obtained for homogeneously mixing populations can be approximated for
random graphs, but are not suitable for scale-free networks. Finally, we provide an analytical framework
to understand the behaviour of the process on a heterogeneous mixing population. More precisely, we
obtain a set of differential equations describing the time evolution of the probability that an individual
is in each state. We show that there is a remarkable matching between these analytical results and those
obtained from computer simulations.
We point out that a removal mechanism different from the one considered in the usual models is
considered here. We assume that stifler nodes can scotch the rumour propagation. Our model is inspired
by the stochastic process discussed in [11]. In such work, the author assumes that the propagation of a
rumour starts from one individual, who after an exponential time learns that the rumour is false and
then starts to scotch the propagation by the individuals previously informed. When the population is
homogeneously mixed, Bordenave [11] showed that the scaling limit of this process is the well-known
birth-and-assassination process, introduced in the probabilistic literature by Aldous & Krebs [12]asa
variant of the branching process [13]. In order to introduce a more realistic model we consider two
modifications. We suppose that each stifler tries to stop the rumour diffusion by all the spreaders that
he/she meets along the way. It is assumed that the rumour starts with general initial conditions.
Our model can be applied to describe the spreading of information through social networks. In this
case, a person propagates a piece of information to another one and then possibly becomes a stifler. That
event may occur if, for instance, such person discovers that the piece of information is false and then tries
to scotch the spreading. The same dynamics can model the spreading of data in a network. A computer
can try to scotch the diffusion of a file after discovering that it contains a virus.
These dynamics are related to the well-known Williams–Bjerknes (WB) tumour growth model [14],
which is studied on infinite regular graphs like hypercubic lattices and trees (see for instance [1517]).
The same model on the complete graph is studied by Kortchemski [18] in the context of a predator–
prey susceptible–infected–recovered (SIR) model. As a description of a rumour dynamic on graphs with
a finite number of vertices, including random graphs and scale-free networks, this model has not been
addressed yet. In this way, here we apply the theory of convergence of density-dependent Markov chains
and use computational simulations to study the asymptotic behaviour of rumour scotching on finite
populations.
Our results can contribute to the analysis of optimal information dissemination strategies [19]aswell
as the statistical inference of rumour processes [20]. In addition, given the competition-like structure of
the process, it may be applied as a toy model of marketing policies. In such a situation, the first spreader
may represent the first individual to try a new product and his/her neighbours can imitate him/her at
rate λ. On the other hand, stiflers may represent individuals who know that the product is low quality
and therefore, they can persuade other users to dismiss the product at rate α. We refer the reader to [3,4],
for a review of related models and results in this direction.
2. Previous works on rumour spreading
The most popular models to describe the spreading of news or rumours are based on the stochastic
or deterministic version of the classical SIR, SIS (susceptible–infected–susceptible) and SI (susceptible–
infected) epidemic models [1,21]. In these models, it is assumed that an infection (or information) spreads
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through a population subdivided into three classes (or compartments), i.e. susceptible, infective and
removed individuals. In the case of rumour dynamics, these states are referred as ignorant, spreader and
stifler, respectively.
The first stochastic rumour models are due to Daley & Kendall (DK) [22,23] and to Maki &
Thompson (MT) [24]. Both models were proposed to describe the diffusion of a rumour through a closed
homogeneously mixing population of size n, i.e. a population described by a complete graph. Initially,
it is assumed that there is one spreader and n1 are in the ignorant state. The evolution of the DK
rumour model can be described by using a continuous time Markov chain, denoting the number of
nodes in the ignorant, spreader and stifler states at time tby X(t), Y(t)andZ(t), respectively. Thus, the
stochastic process {(X(t), Y(t))}t0is described by the Markov chain with transitions and corresponding
rates given by
transition rate
(1, 1) XY,
(0, 2) Y
2,
(0, 1) Y(nXY).
This means that if the process is in state (X,Y)attimet, then the probability that it will be in state
(X1, Y+1) at time t+his XYh +o(h), where o(h) is a function such that limh0o(h)/h=0. In this
model, it is assumed that individuals interact by pairwise contacts and the three possible transitions
correspond to spreader–ignorant, spreader–spreader and spreader–stifler interactions. In the first
transition, the spreader tells the rumour to an ignorant, who becomes a spreader. The two other
transitions indicate the transformation of the spreader(s) into stifler(s) because of its contact with a subject
who already knew the rumour.
MT formulated a simplification of the DK model by considering that the rumour is propagated by
directed contact between the spreaders and other individuals. In addition, when a spreader icontacts
another spreader j,onlyibecomes a stifler. Thus, in this case, the continuous-time Markov chain to be
considered is the stochastic process {(X(t), Y(t))}t0that evolves according to the following transitions
and rates:
transition rate
(1, 1) XY,
(0, 1) Y(nX).
The first references about these models [2224] are the most cited works about stochastic rumour
processes in homogeneously mixing populations and have triggered numerous significant research in
this area. Basically, generalizations of these models can be obtained in two different ways. The first
generalizations are related to the dynamic of the spreading process and the second ones to the structure
of the population. In the former, there are many rigorous results involving the analysis of the remaining
proportion of ignorant individuals when there are no more spreaders on the population [25,26]. Note that
this is one way to measure the range of the rumour. After the first rigorous results, namely limit theorems
for this fraction of ignorant individuals [25,26], many authors introduced modifications in the dynamic of
the basic models in order to make them more realistic. Recent papers have suggested generalizations that
allow various contact interactions, the possibility of forgetting the rumour [27], long-memory spreaders
[28] or a new class of uninterested individuals [29]. Related processes can be found for instance in [30,31].
However, all these models maintain the assumption that the population is homogeneously mixing.
On the other hand, recent results have analysed how the topology of the considered population affects
the diffusion process. In this direction, Coletti et al. [32] studied a rumour process when the population is
represented by the d-dimensional hypercubic lattice and Comets et al. [33] modelled the transmission of
information of a message on the Erd˝os–Rényi (ER) random graph. Related studies can be found in [3438]
and references therein. In the previous works, authors dealt with different probabilistic techniques to
get the desired results. Such techniques allow extending our understanding of a rumour process in a
more structured population, namely, represented by lattices and random graphs. Unfortunately, when
one deals with the analysis of these dynamics in real-world networks, such as online social networks or
the Internet [39], whose topology is very heterogeneous, it is difficult to apply the same mathematical
arguments and a different approach is required. In this direction, general rumour models are studied
in [40,41] where the population is represented by a random graph or a complex network and important
results are obtained by means of approximations of the original process and computational simulations.
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3. Homogeneously mixing populations
The model proposed here assumes that spreaders propagate the rumour to their direct neighbours, as in
the original MT model [24]. However, differently from this model, stifler nodes try to scotch the rumour
propagation. Indeed, we assume that a spreader tells the rumour to an ignorant at rate λand a spreader
becomes a stifler at rate αowing to the action of a stifler.
Let us formalize the stochastic process of interest. Consider a population of fixed size n.Asusual,we
denote the number of nodes in the ignorant, spreader and stifler state at time tby Xn(t), Yn(t)andZn(t),
respectively. We assume that xn
0,yn
0and zn
0are the respective initial proportions of these individuals in
the population and suppose that the following limits exist:
x0:=lim
n→∞ xn
0>0,
y0:=lim
n→∞ yn
0,
and z0:=lim
n→∞ zn
0>0.
(3.1)
Our rumour model is the continuous-time Markov chain V(n)(t)={(Xn(t), Yn(t))}t0with transitions and
rates are given by
transition rate
(1, 1) λXY,
(0, 1) αY(nXY).
This means that if the process is in state (X,Y)attimetthen the probabilities that it will be in states
(X1, Y+1) or (X,Y1) at time t+hare, respectively, λXYh +o(h)andαY(nXY)h+o(h). Note
that while the first transition corresponds to an interaction between a spreader and an ignorant, the
second one represents the interaction between a stifler and a spreader. When ngoes to infinity, the entire
trajectories of this Markov chain, rescaled by n, have as a limit the set of differential equations given by
x(t)=−λx(t)y(t),
y(t)=λx(t)y(t)αy(t)z(t),
z(t)=αy(t)z(t)
and x(0) =x0,y(0) =y0,z(0) =z0.
(3.2)
The solutions rely on the initial conditions, as the stifler class is an absorbing state. Figure 1 shows
this dependency. In figure 1a, the initial conditions are fixed and two parameters αand λare evaluated,
showing that an increase in the values of αreduces the maximum fraction of spreader nodes. In figure 1b,
the rates are fixed and the initial conditions are varied, which shows that the time evolution of the system
changes, evidencing the dependency on the initial conditions.
We solved the system of equations (3.2) numerically for every pair of parameters, λand α,eachone
starting from 0.05 and incrementing them with steps of 0.05 until reaching the unity. Figure 2apresents
the results in terms of the fraction of ignorants at the end of the process. The higher the probability α,
the higher the fraction of the ignorants for low values of λ. On the other hand, the fraction of ignorants
is lower when the parameter λis increased, even when α1.
The analysis of equations (3.2) allows us to obtain some information about the remaining proportion
of ignorants at the end of the process. However, this procedure refers to the limit of the process and it does
not say anything about the relation between such value and the size of the population. In order to study
such relation, we consider the theory of density-dependent Markov chains, from which we can obtain
not only information of the remaining proportion of ignorants, but also acquire a better understanding
of the magnitude of the random fluctuations around this limiting value. This approach has already been
used for rumour models, see for instance [28,29]. In the rest of the paper, we denote the ratio α/λ by ρ.
Let τ(n)=inf{t:Yn(t)=0}be the absorption time of the process. More specifically, τ(n)is the first time
at which the number of spreaders in the population vanishes. Our purpose is to study the behaviour of
the random variable Xn(τ(n))/n,fornlarge enough, by stating a weak law of large numbers and a central
limit theorem.
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(b)
(a)
a= 0.05, l=0.05, ignorants
a= 0.05, l=0.05, spreaders
a= 0.05, l=0.05, stiflers
a= 0.10, l=0.05, ignorants
a= 0.10, l=0.05, spreaders
a= 0.10, l=0.05, stiflers
x0= 0.98, y0=z0=0.01, ignorants
x0= 0.98, y0=z0=0.01, spreaders
x0= 0.98, y0=z0=0.01, stiflers
x0= 0.8, y0=z0=0.1, ignorants
x0= 0.8, y0=z0=0.1, spreaders
x0= 0.8, y0=z0=0.1, stiflers
50 100 150 200 250 300 3500
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
time
% of population
0.2
0
0.4
0.6
0.8
1.0
1.2
1.4
1.6
% of population
Figure 1. Time evolution of the rumour model (equation (3.2)) according to (a) the variation of the parameters αand λfor the xed
initial condition x0=0.98, y0=z0=0.01 and (b) the variation of the initial condition for the xed parameters α=0.05, λ=0.05.
1.0
0.8
0.6
0.4
0.2
1.0
0.8
0.6
0.4
0.2
0
0.2 0.4 0.6 0.8 1.0
a
l
Figure 2. Fraction of ignorant individuals for the theoretical model,obtained by the numerical evaluation of the system of equations (3.2)
for x0=0.98, y0=0.01 and z0=0.01.
Themainideaistodene,bymeansofarandomtimechange,anewprocess{˜
V(n)(t)}t0, with the
same transitions as {V(n)(t)}t0, so that they terminate at the same point. The transformation is done in
such a way that {˜
V(n)(t)}t0is a density-dependent Markov chain for which we can apply well-known
convergence results (see for instance [4244]).
The first step in this direction is to define
θn(t)=t
0
Yn(s)ds,
for 0 tτ(n). Notice that θnis a strictly increasing, continuous and piecewise linear function. In this
way, we can define its inverse by
Γn(s)=inf{t:θn(t)>s}, (3.3)
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1.0
0.5
–0.5
–1.0
0
f(x)
y0
x0
f(x)
y0
x0
f(x)
x0
f(x)
x0
f (x)
x
0 0.5 1.0
x
0 0.5 1.0
x
0 0.5 1.0
x
0 0.5 1.0
Figure 3. Four dierent cases for the function f(x) given by equation (3.8). (a)ρ<x0/z0and y0>0, (b)ρ>x0/z0and y0>0,
(c)ρ<x0/z0and y0=0and(d)ρ>x0/z0and y0=0.
for 0 s
0Yn(u)du. Then it is not difficult to see that the process defined as
˜
Vn(t):=Vn(Γn(t)) (3.4)
has the same transitions as {Vn(t)}t0. As a consequence, if we define ˜τn=inf{t:˜
Yn(t)=0}we get that
Vn(τn)=˜
Vn(˜τn). This implies that it is enough to study ˜
Xn(˜τ(n))/n. The gain of the previous comparison
relies on the fact that {˜
Vn(t)}t0is a continuous-time Markov chain with initial state (xn
0n,yn
0n)and
transitions and rates given by
transition rate
0=(1, 1) λX,
1=(0, 1) α(nXY).
In particular, the rates of the process can be written as
nβli˜
X
n,˜
Y
n,
where β0(x,y)=λxand β1(x,y)=α(1 xy). Processes defined as above are called density dependent as
the rates depend on the density of the process (i.e. normed by n). Then {˜
Vn(t)}t0is a density-dependent
Markov chain with possible transitions in the set {0,1}. By applying convergence results of [44], we
obtain an approximation of this process, as the population size becomes larger, by a system of differential
equations. Similar arguments have been applied for stochastic rumour and epidemic models [28,29,45]
and we include them for the sake of completeness. We use the notation used in [44] except for the
Gaussian process that we would rather denote by V=(Vx,Vy). Here ϕ(x,y)=y,and
τ=inf{t:y(t)0}=−1
λlog x
x0,
where Xrepresents the limiting fraction of ignorant individuals of the process, which is defined later. It
is known that the limit behaviour of the density-dependent Markov chain {˜
Vn(t)}t0can be determined
by the drift function F(x,y)=l0β0(x,y)+l1β1(x,y).
In other words,
F(x,y)=(λx,(λ+α)x+αyα) (3.5)
and the limiting system of ordinary differential equations is given by
x(t)=−λx(t),
y(t)=(λ+α)x(t)+αy(t)α
and x(0) =x0,y(0) =y0.
(3.6)
The solution of (3.6) is
x(t)=x0exp(λt)
and y(t)=f(x(t)), (3.7)
where f:(0,x0]Ris given by
f(x)=1(1 x0y0)x0
xρ
x. (3.8)
Figure 3 shows the behaviour of f(x) for four possible relations between ρand the initial conditions.
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According to theorem 11.2.1 of [44] we have that, on a suitable probability space, ˜
Vn(t)/nconverges to
(x(t), y(t)) given by (3.7), almost surely uniformly on bounded time intervals. Then the following results
can be obtained as a consequence of theorem 11.4.1 of [44].
3.1. Law of large numbers
If xdenotes the root of f(x)=0in(0,x0], then
lim
n→∞
Xn(τn)
n=x(3.9)
in probability. This means that, for nlarge enough, it is a high probability the process dies out leaving
approximately a proportion xof remaining ignorant nodes of the population. In order to prove the
limit of equation (3.9), note that y0>0and
ϕ(v(τ)) ·F(v(τ)) =y(τ)=(λ+α)xα<0 (3.10)
imply that y(τε)>0andy(τ+ε)<0for0<ε<τ
. Therefore, the almost surely convergence of
˜
Yn(t)/nto y(t) uniformly on bounded intervals implies that
lim
n→∞ ˜τ(n)=τa.s. (3.11)
When y0=0andx0z0, this result is also valid because y(0) >0 and (3.10) still holds. On the other
hand, if y0=0andx0ρz0, then y(t)<0forallt>0, and again the almost sure convergence of ˜
Yn(t)/n
to y(t) uniformly on bounded intervals yields that limn→∞ ˜τ(n)=0=τalmost surely. Therefore, as
˜
Xn(t)/nconverges to x(t) almost surely, we obtain the law of large numbers from (3.11) and the fact
that X(n)(τ(n))=˜
X(n)(˜τ(n)).
3.2. Central limit theorem
Furthermore, we can describe the distribution of the random fluctuations around the limiting value x.
More precisely, by assuming that y0>0, or that y0=0andρ<x0/z0, we obtain the following central
limit theorem:
nXn(τ(n))
nxN(0, σ2)asn→∞, (3.12)
where denotes convergence in distribution and N(0, σ2) is the Gaussian distribution with mean zero
and variance σ2:=σ2(α,λ,x0,y0,z0) given by
xz[x0x(1 z0x)+z0ρ2z(x0x)]
x0z0[ρx(ρ+1)]2, (3.13)
where z:=1x. Indeed, from theorem 11.4.1 of [44] we have that if, y0>0ory0=0and
x0z0, then n(n1˜
Xn(˜τ(n))x)
converges in distribution as n→∞to
Vx(τ)+x
(1 +δ)xδ
Vy(τ). (3.14)
The resulting normal distribution has mean zero, so, to complete the proof of central limit theorem, we
need to calculate the corresponding variance. To compute the covariance matrix Cov(V(τ), V(τ)), we
use Eq. (2.21) from [44, ch. 10] which translates to
Cov(V(t), V(t)) =t
0
Φ(t,s)G(x(s), y(s))[Φ(t,s)]Tds. (3.15)
In our case,
G(x,y)=λxλx
λx(λα)xαy+α
and
Φ(t,s)=eλ(ts)0
eα(ts)eλ(ts)eα(ts),
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thus we obtain that Cov(V(τ), V(τ)) is given by
xx2
x0
x(xx0)
x0
x(xx0)
x0
2x1+(x1)2
z0x2
x0
.
We get the closed formula (3.13) for the asymptotic variance by using last expression and properties
of variance.
As mentioned previously, Kortchemski [18] deals with this model on the complete graph in the context
of epidemic spreading. More precisely, the case X(0) =nand Y(0) =Z(0) =1 is considered in a population
of size n+2. Interesting results related to limit theorems and phase transitions are obtained. The results
stated here concerning the asymptotic behaviour of the rumour process are proved under a different
initial configuration and have a different convergence scale. We observe that the case considered in [18]
is, using our notation, x0=1andy0=z0=0 (see equation (3.1)). Therefore, our work complements the
results by Kortchemski [18].
4. Heterogeneously mixing populations
As an interacting particle system, our model can be formulated in a finite graph (or network) Gas a
continuous-time Markov process (ηt)t0on the state space {0, 1, 2}V,whereV:={1, 2, ...,n}is the set of
nodes. A state of the process is a vector η=(η(i):iV), where η(i)∈{0, 1, 2}and0,1,2representthe
ignorant, spreader and stifler states, respectively. The rumour is spread at rate λand a spreader becomes
astieratrateαafter contacting stiflers. We assume that the state of the process at time tis ηand let
iV.Then
P(ηt+h(i)=1|ηt(i)=0) =λhN1(i)+o(h)
and
P(ηt+h(i)=2|ηt(i)=1) =αhN2(i)+o(h),
where N(i):=N(η,i) is the number of neighbours of ithat are in state ,for=1, 2 and for the
configuration η. In the previous section, we present a rigorous analysis of our rumour model on a
complete graph with nvertices. Our results in such a case are related to the asymptotic behaviour of
the random variables
X(n)(t)=
n
i=1
I{ηt(i)=0}
and
Y(n)(t)=
n
i=1
I{ηt(i)=1},
where IAdenotes the indicator random variable of the event A. This mean-field approximation assumes
that the possible contacts between each pair of individuals occur with the same probability. This
assumption enables an analytical treatment, but does not represent the organization of real-world
networks, whose topology is very heterogeneous [39,4648]. In this case, we use a different approach that
allows us to describe the evolution of each node. Such formulation assumes the independence among
the state of the nodes. More precisely, we are interested in the behaviour of the probabilities
xi(t):=P(ηt(i)=0),
yi(t):=P(ηt(i)=1)
and zi(t):=P(ηt(i)=2),
(4.1)
for all i=1, 2, ...,n. We describe our process in terms of a collection of independent Poisson processes
Nλ
iand Nα
iwith intensities λand α, respectively, for i=1, 2, ...,n. We associate the processes Nλ
iand
Nα
ito the node iand we say that at each time of Nλ
i(Nα
i), if iis in state 1 (2) then it chooses a nearest
neighbour jat random and tries to transmit (scotch) the information provided jis in state 0 (1). In this
way, we obtain a realization of our process (ηt)t0.
In order to study the evolution of the functions (4.1), we fix a node iand analyse the behaviour of
its different transition probabilities on a small-time window. More precisely, consider a small enough
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positive number hand note that
P(ηt+h(i)=0) =P(ηt+h(i)=0|ηt(i)=0)P(ηt(i)=0), (4.2)
where the first factor of the right-hand side of last expression is given by
P(ηt+h(i)=0|ηt(i)=0) =1P(ηt+h(i)=1|ηt(i)=0) P(ηt+h(i)=2|ηt(i)=0)
=1P(ηt+h(i)=1|ηt(i)=0) +o(h). (4.3)
The o(h) term appears in the above equation, because the occurrence of a transition from state 0 to
state 2 in a time interval of size himplies the existence of at least two marks of a Poisson process at the
same time interval.
To develop (4.3), for a node j,letBji(h) denote the intersection of the events: (i) {Nj(t,t+h)=1};
(ii) {jtransmit the information to iin (t,t+h)}; (iii) {ηj(t)=1}; and (iv) {ηj(s)=1, for t<st+h}.Also
let Aji =1ifiis a direct neighbour of jin the network (equals 0 other case) and ki=jAij is the degree
of the node i.
We observe that the event (i) only takes into account the Poisson process with rate λ, and that the
probability of a contact between nodes jand i, which is related to (ii), is given by Aji/kj.
Consequently, we obtain
P(ηt+h(i)=1|ηt(i)=0) =P(ηt+h(i)=1|ηt(i)=0, n
j=1Bji(h))P(Bji (h)|ηt(i)=0) +o(h),
=
n
j=1
Aji
kj
(λh+o(h))P(ηt(j)=1) +o(h). (4.4)
Thus, we obtain
P(ηt+h(i)=0) =
1
n
j=1
Aji
kj
(λh+o(h))P(ηt(j)=1) +o(h)
P(ηt(i)=0)
or
P(ηt+h(i)=0) P(ηt(i)=0) =−
n
j=1
Aji
kj
(λh+o(h))P(ηt(j)=1) +o(h)
P(ηt(i)=0).
Finally, as x
i(t)=limh0(xi(t+h)xi(t))/hwe conclude x
i(t)=−λxi(t)n
j=1(Aji/kj)yj(t). Same arguments
allow us to obtain the equations for yi(t)andzi(t). In this way, we have the following set of dynamical
equations:
x
i(t)=−λxi(t)
n
j=1
Pjiyj(t),
y
i(t)=λxi(t)
n
j=1
Pjiyj(t)αyi(t)
n
j=1
Pjizj(t),
z
i(t)=αyi(t)
n
j=1
Pjizj(t)
and xi(0) =x0,yi(0) =y0,zi(0) =z0,
(4.5)
for all i=1, 2, ...,n,andPji :=Aji /kj. We observe that when the network considered is a complete graph
of nvertices, the system of equations (4.5) matches with the homogeneous approach (see the system of
equations (3.2)).
Observe that our formalism assumes that the network is fixed and static during the whole
spreading process. Such formalism is similar to the so-called quenched mean field (QMF) for epidemic
spreading [49,50]. In this manner, for a fixed network we have one set of equations that describes its
behaviour. Such approach contrasts with the heterogeneous mean field (HMF), applied to epidemic
spreading [51,52] and the MT in [40,52]. Regarding the HMF, only the degree distribution is considered
and all nodes with degree kare considered statistically equivalent. Such formalism neglects specific
structures of the network (e.g. the number of triangles), as many different networks can have the same
degree distribution.
In order to verify the influence of network structure on the dynamical behaviour of the models, we
consider random graphs of ER and scale-free networks of Barabási and Albert (BA). Random graphs
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50 100 150
1.0
0.8
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
0
130
120
110
100
90
80
70
iyizi
t
50 100 150
t
50 100 150
t
Figure 4. Time evolution of the nodal probabilities considering our model for an ER network with n=104nodes and k≈100. We
consider the spreading rate λ=0.2 and stiing rate α=0.1. Each curve represents the probability that a node is in one of the three
states (ignorant, spreader or stier) and the colour represents the degree of the node i. The initial conditions are x0=0.98, y0=0.01
and z0=0.01.
1.0
0.8
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
1.0
0.8
0.6
0.4
0.2
050 100 150 0
t
50 100 150
t
50 100 150
t
1000
800
600
400
200
1200
yizi
x
i
Figure 5. Time evolution of the nodal probabilities considering our model for an BA network with n=104nodes and k≈100. The
spreading rate as λ=0.2, while the stiing rate is α=0.1. Each curve represents the probability that a node is in one of the three
states (ignorant, spreader or stier) and the colour represents the degree of the node i. The initial conditions are x0=0.98, y0=0.01
and z0=0.01.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
g= 2.2
g= 2.4
g= 2.6
g= 2.8
ER
·xÒ
0.2 0.4 0.6 0.8 1.0
l
Figure 6. Phase diagram of the nal fraction of ignorants as a function of λfor α=0.5, x×λ. The initial conditions are
x0=0.98, y0=0.01 and z0=0.01. All the networks have n=103nodes and k≈10.
are created by a Bernoulli process, connecting each pair of vertices with the same probability p.The
degree distribution of random graphs follows a Poisson distribution for large values of nand small
p, as a consequence of the law of rare events [53]. On the other hand, the BA model generates scale-
free networks by taking into account the network growth and preferential attachment rules [54]. The
networks generated by this model present degree distribution following a power-law, P(k)kγ, with
γ=3. In random graphs most of the nodes have similar degrees, whereas scale-free networks are
characterized by a very heterogeneous structure.
Figures 4and 5show the time evolution of the nodal probabilities, considering ER and BA networks,
respectively. These results are obtained by solving numerically the system of equations (4.5). Both
networks have n=104nodes and k≈100. The spreading rate is λ=0.2 and the stifling rate is α=0.1.
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1.0
0.8
0.6
0.4
0.2
00 20 40 60 80 100 120
1.0
0.8
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
0
130
120
110
100
90
80
70
x
iyizi
t
0 20406080100120
t
0 20 40 60 80 100 120
t
Figure 7. Time evolution of the nodal probabilities considering the MT model in an ER network with n=104nodes and k≈100.
The spreading rate is λ=0.2 and the stiing rate is α=0.1. Each curve represents the probability that a node is in one of the three
states (ignorant, spreader or stier) and the colour represents the degree of the node i. The initial conditions are x0=0.98, y0=0.01
and z0=0.01.
The colour of each curve denotes the degree of each node i. Comparing figures 4and 5,wecanseethat
the variance of xi,yiand ziin BA networks is higher than in ER networks. Moreover, in both networks,
higher degree nodes tend to turn into a stifler earlier than lower degree ones.
In addition to the homogeneous versus heterogeneous comparison performed before, we can also
compare different levels of heterogeneity. As many real networks rely on power-law degree distributions,
P(k)kγ, with 2 <γ <3[55] we use the configuration model [56] to generate such networks without
degree correlations. More precisely, we use the algorithm proposed in [57]. Figure 6 shows the phase
diagram of the final fraction of ignorants as a function of λfor α=0.5. Here we use five networks
with n=103nodes and k≈10, four of them are power-law degree distributions, P(k)kγwith
γ=2.2, γ=2.4, γ=2.6 and γ=2.8 and one ER. Besides the initial conditions are x0=0.98, y0=0.01
and z0=0.01. This experiment is based on the numerical solution of the ODE set of equations (4.5)
for a sufficiently large value of time, where the number of spreaders is negligible. We observe that the
higher the γthe lower the final fraction of ignorants, suggesting that the spreading is favoured by such
structural feature. Interestingly, the ER network showed the lowest final fraction of ignorants. Regarding
scale-free networks, such results suggest that hubs on our model present a similar behaviour to hubs on
the MT model, suggesting that on a first moment, hubs favour the spreading; however, once it becomes
a stifler it acts efficiently, stifling its neighbours. Again, similarly to MT model, the homogeneity seems
to favour the spreading which contrasts with the epidemic spreading processes, which are favoured
by heterogeneity.
We compare the behaviour of our model, described by equation (4.5), with the MT model [24]inER
and BA networks. The time evolution of this model is given by
x
i(t)=−λxi(t)
n
j=1
Pjiyj(t),
y
i(t)=λxi(t)
n
j=1
Pjiyj(t)αyi(t)
n
j=1
Pij(xj(t)+zj(t)),
z
i(t)=αyi(t)
n
j=1
Pij(xj(t)+zj(t))
and xi(0) =x0,yi(0) =y0,zi(0) =z0,
(4.6)
where, as before, xi,yiand ziare the micro-state variables, quantifying the probability that the node iis an
ignorant, a spreader or a stifler at time t, respectively, for i=1, 2, ...,n. Note xi(t)+yi(t)+zi(t)=1, i,t.
Figures 7and 8show the time evolution of the nodal probabilities, by numerically solving
equation (4.6). Similarly to our model, the variances in BA networks are higher than in ER networks.
Besides, the hubs and leaves of the BA networks present a completely different behaviour, as can be
seen in figure 8b. Moreover, the nodes having higher degrees also tend to become stifler earlier than low
degree nodes.
We consider the same initial conditions for both rumour models, i.e. x0=0.98, y0=0.01 and z0=0.01.
It is worth emphasizing that the initial conditions in figures 7and 8are not usual in the MT model,
as most of the works on this model considers the initial fraction of stiflers as zero [1]. However, our
model needs an initial non-zero fraction of stiflers, otherwise there is no manner to contain the rumour
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1200
1000
800
600
400
200
0 20 40 60 80 100 120
t
0 20406080100120
t
0 20406080100120
t
1.0
0.8
0.6
0.4
0.2
0
xi
1.0
0.8
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
0
yizi
Figure 8. Time evolution of the nodal probabilities considering the MT model in an BA network with n=104nodes and k≈100.
The spreading rate is λ=0.2 and the stiing rate is α=0.1. Each curve represents the probability that a node is in one of the three
states (ignorant, spreader or stier) and the colour represents the degree of the node i. The initial conditions are x0=0.98, y0=0.01
and z0=0.01.
propagation, so we assumed the same initial condition in order to perform a comparative analysis of both
models, as important differences emerge. The main feature that emerges from the comparison between
figures 5and 4with 8and 7is the peak of the probability of a node being a spreader. In our model it tends
to be higher than in the MT process. Such a feature evinces the differences between two formulations.
In the MT model, the spreaders lose the interest in the rumour propagation owing to the contact with
individuals who have already known the rumour, whereas in our model spreaders are convinced only
by stifler vertices to stop spreading the information.
As mentioned before, the hubs on our model present a similar behaviour as on the MT model, having
a large number of edges it spreads and is stifled very efficiently. Aside from this similarity on the MT such
phenomenon happens at a faster rate, as an individual can lose interest on the rumour just by contacting
twice to one of its neighbour individuals (on the first contact spreads the information, on the second it
becomes a stifler, subject to the rates of the process). On the other hand, in our model stiflers are active
and depend only on the probability of finding a spreader. In this manner, at the beginning of the process
yiand ziare low, implying that z
i(t) is also low and the dominant term of y
i(t)isλxi(t)n
j=1Pjiyj(t). Then,
when the fraction of spreaders increase it also increases the z
i(t). Such a process seems to be faster on
the MT model than ours. Consequently, the final fraction of ignorants and the time to achieve the steady
state are different.
In addition, it is noteworthy that the MT model allows an individual to spread the information to a
neighbour, then lost interest by contacting the same individual, which seems to be different from real-
world situations. Such feature is absent in our model, however, in our model the individuals do not lose
interest in the information, they are convinced to stop the spreading.
5. MonteCarlosimulation
The analytical methods presented in §§3 and 4 assume that there is no correlation between the states.
However, it is not true on most real cases, due to triangles, assortativity, community structure, among
other features. This is also the assumption made on many epidemic [4952] and rumour spreading
models [40,52]. In [58], the authors compared the accuracy of some mean field approaches, considering
many different dynamical processes, and showed that some approaches present relative accuracy on
disassortative networks even when the mean degree is low. Although some approximations are still
valid, the Monte Carlo simulations mimic the process itself in a computational manner assuming only
the pattern of connections and the contact relationships. As it does not assume the absence of correlations,
those simulations are expected to be more similar to the real process. In this manner, the analytical and
numerical methods exposed in §4 and the Monte Carlo simulations are complementary. On the one hand,
the ODE system give us insights about the process, allowing us to threaten it mathematically, on the other
hand, the Monte Carlo simulations make less assumptions.
In this way, we perform extensive numerical simulations to verify how our rigorous results obtained
for homogeneously mixing populations can be considered as approximations for random graphs and
scale-free networks. The rumour spreading simulation is based on the contact between two individuals.
At each time step each spreader makes a trial to spread the rumour to one of its neighbours and each
stifler makes a trial to stop the spreading. If the spreader contacts an ignorant, it spreads the rumour with
probability λ. Similarly, if the stifler contacts an spreader that spreader becomes a stifler with probability
α. The updates are performed in a sequential asynchronous fashion. For the simulation procedure, it is
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300
200
100
0
n= 10 000
n= 5000
0.005 0.010 0.015 0.020
PDF (x)
x
Figure 9. Distribution of the fraction of ignorants obtained from 1000 simulations in a complete graph varying the number of nodes.
The bars are obtained experimentally, while the tted Gaussian are based on the theoretical values obtained from equations (3.2), (3.8)
and (3.12).
300 250
200
150
100
50
0
250
200
150
100
50
0
ER ·kÒ=10 BA ·kÒ=10
BA ·kÒ=50
BA ·kÒ= 1000
BA ·kÒ= 5000
BA ·kÒ= 8000
ER ·kÒ= 100
theoretical theoretical
0.005 0.010 0.015 0.020 0.025 0.030 0.02 0.04 0.06 0.08
xx
PDF (x)
(b)(a)
Figure 10. Distribution of the fraction of ignorants considering 1000 Monte Carlo simulations of the rumour scotching model in networks
with n=104nodes generated from the (a)ERand(b) BA network models. The simulations consider λ=0.5, α=0.5 and initial
conditions x0=0.98, y0=0.01 and z0=0.01. Theoretical curves, obtained by equations (3.2), (3.8) and (3.12), are in red.
important to randomize the state of the initial conditions, especially for the heterogeneous networks.
In order to overcome statistical fluctuations in our simulations, every model is simulated 50 times with
random initial conditions.
5.1. Complete graph
The results are quantified as a function of the fraction of ignorant nodes, as when the time tends
to infinity, the proportion of spreaders tends to zero and the fraction of ignorants and stiflers has
complementary information about the population. Figure 9 compares the distribution of the fraction
of ignorants obtained by Monte Carlo simulations with the central limit theorem by fitting a Gaussian
distribution according to the theoretical values obtained from equations (3.2), (3.8) and (3.12). Complete
graphs of two different sizes are considered to show the dependency on the number of nodes n. Note
that equations (3.8) and (3.12) assert that only the variance depends on the network size, i.e. σ21/n.
Thus, the numerical simulations agree remarkably with the theoretical results.
5.2. Complex networks
In order to verify the behaviour of the rumour scotching model on complex networks, we evaluate
networks generated by random graphs of the ER and scale-free networks of BA. Figure 10 shows the
distribution of the final fraction of ignorants considering 1000 Monte Carlo simulations of the rumour
scotching model in networks with n=104vertices generated from the ER and BA models. The theoretical
results for the homogeneously mixing populations, obtained from equations (3.2), (3.8) and (3.12), are
also shown. In ER networks, the distribution converges to the theoretical results as the network becomes
denser. In this way, even in sparse networks, k=100, the results are close to the mean-field predictions.
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1.0
0.8
0.6
0.4
0.2
1.0
0.8
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
1.0
0.8
0.6
0.4
0.2
1.0
0.8
0.6
0.4
0.2
0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
l
1.0
0.8
0.6
0.4
0.2
1.0
0.8
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
1.0
0.8
0.6
0.4
0.2
1.0
0.8
0.6
0.4
0.2
l
aa a a
0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
(b)(a) (c)
(g)(h)
(d)
(e)(f)
Figure 11. Fraction of ignorants (given by colour intensities) according to the rates αand λfor dierent initial conditions considering ER, from (ad), and BA network models, from (eh). Networks with n=104and k≈8are
considered. Every point is as an average over 50 simulations. (a)x0=0.98, y0=0.005 and z0=0.015, (b)x0=0.98, y0=0.015 and z0=0.005, (c)x0=0.98, y0=0.01 and z0=0.01, (d)x0=0.9, y0=0.05 and z0=0.05,
(e)x0=0.98, y0=0.005 and z0=0.015, (f)x0=0.98, y0=0.015 and z0=0.005, (g)x0=0.98, y0=0.01 and z0=0.01, and (h)x0=0.9, y0=0.05 and z0=0.05.
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1.0
0.8
0.6
0.4
0.2
0
t
50 100 150
t
0 50 100 150
·xÒ
·yÒ
·zÒ·xÒ
·yÒ
·zÒ
f
(b)(a)
Figure 12. Comparison of the Monte Carlo simulations and the solution of the nodal time evolution dierential equations,
equations (4.5). The continuous curves are the numerical solution of the dierential equations (4.5), while the symbols are the Monte
Carlo simulations with its respective standard deviation. Every point is as an average over 50 simulations. In (a)anERnetwork
while in (b) a BA network. Both with n=104nodes and k≈100. Moreover, the initial conditions are x0=0.98, y0=0.01 and
z0=0.01.
On the other hand, the convergence of scale-free networks to the theoretical results does not occur even
for k=8000 because of their high level of heterogeneity.
The system of equations (3.2) that describes the evolution of rumour dynamics on homogeneous
populations can characterize the same dynamics in random regular networks if we consider λ=kλand
α=kα. In this case, the probabilities of spreading and scotching the rumour depend on the number of
connections, but the solution of the system of equations does not change. As random networks present
an exponential decay near the mean degree, their dynamical behaviour is similar to the mean-field
predictions. On the other hand, this approximation is not accurate for scale-free networks, because they
do not present a typical degree and the second-moment of their degree distribution diverges for 23
as n→∞. Therefore, the homogeneous mixing assumption is suitable only for ER networks.
Figure 11 shows the Monte Carlo simulation results as a function of the parameters αand λfor
different initial conditions. The simulation considers every pair of parameters, λand α,startingfrom
λ=α=0.05 and incrementing them with steps of 0.05 until reaching the unity. In the rumour spreading
dynamics, the role played by the stiflers is completely different from the recovered individuals in
epidemic spreading. Note that stifler and recovered are absorbing states. However, in the disease
spreading, the recovered individuals do not participate in the dynamics and are completely excluded
from the interactions, whereas in our model, stiflers are active and try to scotch the rumour to the
spreaders.
The number of connections of the initial propagators influences the spread of disease [21,59], but
does not impact the rumour dynamics [60]. We investigate if the number of connections of the initial
set of spreaders and stiflers affects the evolution of the rumour process with scotching in BA scale-free
networks. In a first configuration, the initial state of the hubs is set as spreaders and stiflers are distributed
uniformly in the remaining of the network. In another case, stiflers are the main hubs and spreaders
are distributed uniformly. In both cases, we verify that the final fraction of ignorants is the same as in
completely uniform distribution of spreader and stifler states (figure 11eh). Therefore, we infer that the
degree of the initial spreaders and stiflers does not influence the final fraction of ignorants.
Figure 12 shows numerical solutions of equation (4.5) and the Monte Carlo simulations for ER and BA
networks. Regarding the simulations, figure 12a,bcorrespond to the average behaviour of the variables
shown in figures 4and 5. We can see that the maximum fraction of spreaders occurring in BA networks
is lower than in ER networks. This happens because most of the vertices in BA networks are lowly
connected (owing to the power-law degree distribution). Moreover, we can see that the variance decays
over time, which is a consequence of the presence of an absorbing state. In addition, we also find that for
sparser networks the matching is less accurate (results not shown).
6. Conclusion
The modelling of rumour-like mechanisms is fundamental to understanding many phenomena in society
and online communities, such as viral marketing or social unrest. Many works have investigated the
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dynamics of rumour propagation in complete graphs (e.g. [22]) and complex structures (e.g. [40]). The
models considered so far assume that spreaders try to propagate the information, whereas stiflers are not
active. Here, we propose a new model in which stiflers try to scotch the rumour to the spreader agents.
We develop an analytical treatment to determine how the fraction of ignorants behaves asymptotically
in finite populations by taking into account the homogeneous mixing assumption. We perform Monte
Carlo simulations of the stochastic model on ER random graphs and BA scale-free networks. The
results obtained for homogeneously mixing populations can be used to approximate the case of random
networks, but are not suitable for scale-free networks, owing to their highly heterogeneous organization.
The influence of the number of connections of the initial spreaders and stiflers is also addressed. We
verify that the choice of hubs as spreaders or stiflers has no influence on the final fraction of ignorants.
The study performed here can be extended by considering additional network models, such as small-
world or spatial networks. The influence of network properties, such as assortativity and community
organization can also be analysed in our model. In addition, strategies to maximize the range of the
rumour when the scotching is present can also be developed. The influence of the fraction of stiflers on
the final fraction of ignorant vertices is another property that deserves to be investigated.
Authors’ contributions. F.A.R. and P.M.R. conceived and designed the theoretical model. P.M.R. and E.L. developed the
analysis of density-dependent Markov chains to study the model on homogeneously mixing populations. G.F.A. and
F.A.R. performed the computational simulations of the process. All the authors contributed to the theoretical analysis
of the model on heterogeneous mixing populations. All authors contributed in writing and reviewing the manuscript.
All authors gave final approval for publication.
Competing interests. We have no competing interests.
Funding. P.M.R. acknowledges FAPESP (grant no. 2013/03898-8) and CNPq (grant no. 479313/2012-1) for financial
support. F.A.R. acknowledges CNPq (grant no. 305940/2010-4), FAPESP (grant nos. 2011/50761-2 and 2013/26416-
9) and NAP eScience - PRP - USP for financial support. G.F.A. acknowledges FAPESP for the sponsorship provided
(grant no. 2012/25219-2). E.L. acknowledges CNPq (grant no. 303872/2012-8), FAPESP (grant no. 2012/22673-4) and
FAEPEX - UNICAMP for financial support.
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