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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 Cientíca, 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 [1,5–10]. 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 [5–7].

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 [15–17]).

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 n−1 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))}t≥0is described by the Markov chain with transitions and corresponding

rates given by

transition rate

(−1, 1) XY,

(0, −2) Y

2,

(0, −1) Y(n−X−Y).

This means that if the process is in state (X,Y)attimet, then the probability that it will be in state

(X−1, Y+1) at time t+his XYh +o(h), where o(h) is a function such that limh→0o(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))}t≥0that evolves according to the following transitions

and rates:

transition rate

(−1, 1) XY,

(0, −1) Y(n−X).

The ﬁrst references about these models [22–24] 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 [34–38]

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))}t≥0with transitions and

rates are given by

transition rate

(−1, 1) λXY,

(0, −1) αY(n−X−Y).

This means that if the process is in state (X,Y)attimetthen the probabilities that it will be in states

(X−1, Y+1) or (X,Y−1) at time t+hare, respectively, λXYh +o(h)andαY(n−X−Y)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 α≈1.

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.

Themainideaistodeﬁne,bymeansofarandomtimechange,anewprocess{˜

V(n)(t)}t≥0, with the

same transitions as {V(n)(t)}t≥0, so that they terminate at the same point. The transformation is done in

such a way that {˜

V(n)(t)}t≥0is a density-dependent Markov chain for which we can apply well-known

convergence results (see for instance [42–44]).

The ﬁrst step in this direction is to deﬁne

θn(t)=t

0

Yn(s)ds,

for 0 ≤t≤τ(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 dierent 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)}t≥0. 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)}t≥0is 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(n−X−Y).

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 −x−y). 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)}t≥0is 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 X∞represents 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)}t≥0can 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+αy−α) (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 −x0−y0)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 x∞denotes 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 x∞of 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)x∞−α<0 (3.10)

imply that y(τ∞−ε)>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)=τ∞a.s. (3.11)

When y0=0andx0>ρz0, this result is also valid because y(0) >0 and (3.10) still holds. On the other

hand, if y0=0andx0≤ρz0, 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=τ∞almost 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))

n−x∞⇒N(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

x∞z∞[x0x∞(1 −z0−x∞)+z0ρ2z∞(x0−x∞)]

x0z0[ρ−x∞(ρ+1)]2, (3.13)

where z∞:=1−x∞. Indeed, from theorem 11.4.1 of [44] we have that if, y0>0ory0=0and

x0>ρz0, then √n(n−1˜

Xn(˜τ(n))−x∞)

converges in distribution as n→∞to

Vx(τ∞)+x∞

(1 +δ)x∞−δ

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)=λx−λx

−λx(λ−α)x−αy+α

and

Φ(t,s)=e−λ(t−s)0

eα(t−s)−e−λ(t−s)eα(t−s),

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thus we obtain that Cov(V(τ∞), V(τ∞)) is given by

⎛

⎜

⎜

⎜

⎝

x∞−x2

∞

x0

x∞(x∞−x0)

x0

x∞(x∞−x0)

x0

2x∞−1+(x∞−1)2

z0−x2

∞

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)t≥0on 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):i∈V), 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

astiﬂeratrateαafter contacting stiﬂers. We assume that the state of the process at time tis ηand let

i∈V.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 [39,46–48]. 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)t≥0.

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) =1−P(ηt+h(i)=1|ηt(i)=0) −P(ηt+h(i)=2|ηt(i)=0)

=1−P(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<s≤t+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)=limh→0(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

x

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 stiing rate α=0.1. Each curve represents the probability that a node is in one of the three

states (ignorant, spreader or stier) 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 stiing rate is α=0.1. Each curve represents the probability that a node is in one of the three

states (ignorant, spreader or stier) 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

·x•Ò

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)∼k−γ, 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 stiing rate is α=0.1. Each curve represents the probability that a node is in one of the three

states (ignorant, spreader or stier) 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)∼k−γ, 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)∼k−γwith

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 stiing rate is α=0.1. Each curve represents the probability that a node is in one of the three

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 [49–52] 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

x•x•

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. σ2∝1/√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 dierent initial conditions considering ER, from (a–d), and BA network models, from (e–h). 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 dierential equations,

equations (4.5). The continuous curves are the numerical solution of the dierential 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 2<γ ≤3

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 11e–h). 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

16

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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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