Fact-checking Effect on Viral Hoaxes:
A Model of Misinformation Spread in Social Networks
Marcella Tambuscio1
∗
, Giancarlo Ruffo1
, Alessandro Flammini2
, Filippo Menczer2
1
Computer Science Department, University of Turin, Italy
2
School of Informatics and Computing, Indiana University, Bloomington, USA
ABSTRACT
The Internet and online social networks have greatly facilitated
and accelerated information diffusion processes, but at
the same time they provide fertile ground for the spread of
misinformation, rumors and hoaxes. The goal of this work
is to introduce a simple modeling framework to study the
diffusion of hoaxes and in particular how the availability of
debunking information may contain their diffusion. As traditionally
done in the mathematical modeling of information
diffusion processes, we regard hoaxes as viruses: users can
become infected if they are exposed to them, and turn into
spreaders as a consequence. Upon verification, users can also
turn into non-believers and spread the same attitude with a
mechanism analogous to that of the hoax-spreaders. Both
believers and non-believers, as time passes, can return to a
susceptible state. Our model is characterized by four parameters:
spreading rate, gullibility, probability to verify a
hoax, and that to forget one’s current belief. Simulations on
homogeneous, heterogeneous, and real networks for a wide
range of parameters values reveal a threshold for the factchecking
probability that guarantees the complete removal
of the hoax from the network. Via a mean field approximation,
we establish that the threshold value does not depend
on the spreading rate but only on the gullibility and forgetting
probability. Our approach allows to quantitatively
gauge the minimal reaction necessary to eradicate a hoax.
Categories and Subject Descriptors
[Human-centered computing]: Collaborative and social
computing—Social networks; [Computing methodologies]:
Modeling and simulation
Keywords
Misinformation spread; fact-checking; viral hoaxes; epidemiology;
information diffusion models
∗Contact author. Email: tambuscio@di.unito.it
Copyright is held by the International World Wide Web Conference Committee
(IW3C2). IW3C2 reserves the right to provide a hyperlink to the
author’s site if the Material is used in electronic media.
WWW 2015 Companion, May 18–22, 2015, Florence, Italy.
ACM 978-1-4503-3473-0/15/05.
http://dx.doi.org/10.1145/2740908.2742572.
1. INTRODUCTION
In a hyper-connected world where information spreads increasingly
fast, online social networks and new media play
a crucial role in the diffusion of misinformation, i.e., false
claims that are mostly spread unintentionally. Few people
seem to check the reliability of news before sharing them
with their friends and potentially with millions of others.
This is mainly due to the fact that the Internet, in particular
social networking services, provide a complete decentralization
of information on a large scale: every user is
potentially a news source, and often it is not trivial to establish
the truth. In the last few years there has been a
growing interest in this topic, with different approaches and
techniques. Several publications have focused on studying
characteristics of rumor propagation, analyzing features [5,
11, 18, 7] and proposing diffusion models [4, 12, 21, 20, 1].
Great effort has been devoted to the creation of effective
classifiers to detect false content or fake accounts, highlighting
recurrent patterns [10, 6, 14, 18]. On the other hand,
several theories have been proposed to limit the diffusion of
hoaxes, identifying the most influential users or working on
prejudices and personal beliefs, from a psychological point
of view [3, 9].
Here we focus on analyzing how long a hoax survives in a
network, i.e., the duration of time in which there are users
who believe it true. Intuitively, hoaxes are very similar to
common viruses: people, as nodes in the social networks,
can become “infected” and believe fake news upon coming
into contact with other infected nodes, or “recover” with
a simple fact-checking action. The virus spread problem
has been studied extensively and many epidemic models
were proposed since the 1920s. Later scientists realized that
those mathematical models could describe a large range of
other phenomena like social contagion, information spread,
and computer virus attacks [7, 21, 13]. We consider compartmental
epidemic models, like SIR (Susceptible-InfectedRecovered)
and SIS (Susceptible-Infected-Susceptible) [2], in
which nodes are characterized by different behaviors represented
by states, and the dynamical evolution of the system
is ruled by transition rates among the states.
We propose a stochastic epidemic model to describe the
simultaneous diffusion of a hoax and its relative debunking:
it can be seen as a SIS model in which the Infected status
is split in two sub-compartments, believers (B) and fact
checkers (F), and the transition I → S can be interpreted as a
forgetting process. Moreover we have the transition B → F
with a fixed probability pverify that indicates the fraction
of infected users that check the reliability of the informa-
977
tion received, revealing the hoax. Empirical observations
suggest that some hoaxes seem to become endemic in social
networks, since periodically same false news re-emerge
and “infect” other users, even many years after their first
appearance. We are interested in comparing our model with
SIS for what concerns the absence of an epidemic threshold
(infection can become endemic) in scale-free networks [17].
Furthermore we analyze the role of a fact-checking activity:
can we find a threshold for pverify, i.e., a value that assures
the complete removal of the hoax from the network?
2. THE MODEL
We want to simulate the spread of a hoax and its debunking
at the same time, assuming that some users believe the
fake news and some others do not, because they decide to
verify the information or because they already know it is
not true. Therefore we build upon a model for the competitive
spread of two rumors [19] to describe the competition
among believers and fact-checkers. We extend the model by
introducing verifying and forgetting processes.
We consider a network represented by a graph G = (V, E).
Each node i is associated to a triple of binary indicators,
representing its state at time t, that can assume one of three
possible values:
∀i ∈ V si(t) = [sB
i (t), sF
i (t), sS
i (t)] =



[1, 0, 0]
[0, 1, 0]
[0, 0, 1]
(1)
corresponding to the three possible behaviors for agent i:
• sB
i (t)=1 (Believer): i believes the fact is true,
• sF
i (t)=1 (Fact checker): i believes the fact is false,
• sS
i (t)=1 (Susceptible): i is neutral.
We represent the following three phenomena:
• spreading [S → B, S → F]: each agent modifies
with some probability its state considering the points
of view (states) of its neighbors;
• verifying [B → F]: each agent can fact-check the
hoax with a fixed probability pverify;
• forgetting [B → S, F → S]: each agent, regardless of
belief state, forgets the news with a fixed probability
pforget.
Let pi(t) be the probability mass function of i at time t:
pi(t) = [pB
i (t), pF
i (t), pS
i (t)] (2)
representing the probability of assuming each of the possible
behaviors (B, F, S) at time t. The dynamics of the system
are given by a random realization for pi(t + 1):
si(t + 1) = MultiRealize[pi(t + 1)]. (3)
The probability mass function pi(t + 1) is defined by:
pB
i (t + 1) = fisS
i (t) + (1 − pforget − pverify)sB
i (t)
pF
i (t + 1) = gisS
i (t) + pverifysB
i (t) + (1 − pforget)sF
i (t)
pS
i (t + 1) = pforget(sB
i (t) + sF
i (t)) + (1 − fi − gi)sS
i (t)
(4)
Figure 1: States and transitions of the model
where pforget, pverify are constant probabilities and fi, gi
are the “spreading” functions that provide the network effect,
describing how the hoax (fi) and its debunking (gi)
disseminate among the immediate neighborhood of a vertex.
These “spreading” functions are given by:
fi(t) = β
nB
i (t) · (1 + α)
nB
i (t) · (1 + α) + nF
i (t) · (1 − α)
gi(t) = β
nF
i (t) · (1 − α)
nB
i (t) · (1 + α) + nF
i (t) · (1 − α)
(5)
where β ∈ [0, 1] is a constant parameter for the spreading
rate, α ∈ [0, 1) is a constant parameter for the credibility of
the hoax (or agents gullibility), and nB
i , nF
i are the number
of neighbors of i that are believers or fact checkers at time t,
respectively. Notice that fi(t) + gi(t) = β, i.e., the infection
rate, as in the SIS model: indeed, if we consider the two
states Believer and Fact checker as a unique state Infected,
we recover the SIS model exactly, where pforget is the recovery
probability usually denoted by µ. In summary, we have
four parameters: the spreading rate β, the gullibility α, the
probability pverify to fact-check a hoax, and the probability
pforget to forget one’s current belief. We consider here
values of pforget and pverify such that pforget + pverify < 1.
The model is illustrated in Figure 1.
3. RESULTS
To explore whether the network topology plays a role in
the persistence of infections, as it happens in SIR/SIS cases
[17, 16], we tested our model on different types of networks:
random, scale-free, and a real social network from Facebook.
In the rest of the paper we will denote with B∞, F∞ and
S∞ the densities of believers, fact checkers and susceptible
nodes in the infinite-time (equilibrium) limit.
3.1 Scale-free and random networks
Let us consider Barabasi-Albert (BA) and Erdos-Renyi
(ER) networks with the same size (N = 1000) and same
mean degree ( k = 6). To understand the influence of
fact-checking activity, in these simulations we fix the values
of the spreading rate (β = 0.5) and forgetting probability
(pforget = 0.1), varying only pverify and the gullibility
parameter α. In Figure 2 we show results obtained with
978
BA network
BELIEVER FACT CHECKER SUSCEPTIBLE
ER network
Figure 2: Model behavior in BA and ER networks
with N = 1000 and k = 6. Each line represents
number of nodes of same compartment, averaged
on 30 iterations, fixing β = 0.5 and pforget = 0.1.
α = 0.3 and α = 0.8, pverify = 0.05 in BA and ER networks.
The obtained behaviors are similar to those of known
epidemic models. We find that:
• S∞ does not depend on topology, pverify or spreading
rate β, as we will prove with mean-field equations;
• α and pverify rule the dynamics of believers and fact
checkers, determining the victory of one of the two
behaviors;
• increased fact-checking (pverify) has the power to remove
the hoax (B∞ = 0), however the total infection
(intended as the sum B∞ + F∞) remains active in the
network.
Let us focus on the last point: even with an high gullibility
(α ≈ 1), it seems always possible to find a value of pverify
such that B∞ = 0. Let us build a phase diagram in which
we vary only α ∈ [0, 1) and pverify ∈ [0, 0.3] (the other parameters
remain fixed as before). For each point (α, pverify)
in the diagram we indicate the value of B∞, using a color
palette to show the density of believers. Figure 4 illustrates
two phase diagrams for BA and ER networks. They confirm
that a relatively small fact-checking activity can cancel the
hoax, even when users tend to believe it with high probability.
We will derive analytically a theoretical threshold for
pverify to guarantee the disappearance of fake content.
We tested the model also on real networks, and results
confirm the behaviors we obtained with the synthetic networks.
In Figure 3 we show results of simulations on a real
Facebook network with N ≈ 4000 [8] and BA/ER networks
of the same size, with fixed parameters as in Figure 2. When
the hoax is more credible, or the agents are more gullible,
the population of believers is larger, as expected.
FB network BA network ER network
Figure 3: Comparison of simulation results on different
network types with same size. Parameters:
β = 0.5, pforget = 0.1, pverify = 0.05, α = 0.3 (top) and
0.8 (bottom).
BA network ER network
Figure 4: Phase diagram showing B∞, varying α
(hoax credibility or agent gullibility) and pverify.
Each point is the averaged result on 30 simulations.
3.2 Comparison with SIS/SIR
As explained earlier, our model can be interpreted as an
SIS model in which the Infected status is split in two subcompartments,
therefore we can investigate analogies between
the two models. In Figure 2 and Figure 4 it is immediately
evident that the model has basically the same
behavior in BA and ER networks. In particular, even when
the hoax is removed, its debunking keeps spreading in the
network: this means that the “infection” (believers and fact
checkers) is still active. This is not surprising: a classic result
in epidemic theory [16] says that there is no difference of
behavior in heterogeneous and homogeneous networks when
the reproduction number
R0 =
β · k
pforget
is greater than 1, as in our configuration (R0 = 30 1).
But when R0 < 1, traditional SIR and SIS models perform
differently depending on the network topology: in a random
network, the virus expires; in a scale-free networks, under
the right assumptions, the epidemic threshold goes to zero
and the infection can reach an endemic level — although
with a very small number of infected individuals. In our
model we confirm the absence of an epidemic threshold in
scale-free networks, as shown by the example in Figure 5.
We set the parameters so that R0 = 0.85 < 1. While only
979
BA network ER network
Figure 5: As in SIS/SIR models, under the right
assumptions, our model has a different behavior in
BA or ER networks. In this example the parameter
values are β = 0.1, pforget = 0.7, α = 0.8, and
pverify = 0.05. Considering the infection as the sum
of believers and fact checkers, we observe two different
behaviors: in the ER network the infection
is removed, while in the BA network the infection
reaches an endemic level (with only fact checkers).
susceptible users survive in ER networks, a few fact checkers
survive in BA networks.
3.3 Analytical results
In previous sections we show several simulation results:
now we want to prove some of these with a mathematical
analysis of the model. We derive mean-field equations for
our model, in the hypothesis that all vertices have the same
number of neighbors k and these neighbors are chosen randomly,
therefore the values pi(t) do not depend on i. With
some calculations, we obtain:
pS
(∞) =
pforget
β + pforget
(6)
pB
(∞) =



0
β(2α · pforget − (1 − α) · pverify)
2α(β + pforget)(pforget + pverify)
(7)
pF
(∞) = 1 − pB
(∞) − pS
(∞). (8)
We can read these three values as S∞, B∞ and F∞; they
match with the simulations results in the previous section.
First, the density of susceptible individuals at the infinitetime
limit depends only on the spreading rate β and the
forgetting probability: Eq. 6 fits very well with numerical
results. Second, from Eq. 7 we can obtain a sufficient condition
for the hoax to be removed from the network:
pB
(∞) =
β(2α · pforget − (1 − α) · pverify)
2α(β + pforget)(pforget + pverify)
> 0
⇒ pverify <
2α
1 − α
· pforget
and therefore
pverify ≥
2α
1 − α
· pforget ⇒ pB
(∞) = 0. (9)
The threshold is plotted in Figure 6 for some value of pforget.
This result is consistent with simulations and phase diagrams
illustrated in Figure 4.
Figure 6: Analytical values of pverify, as a function
of α (with pforget = 0.1), such that the hoax is completely
removed from the network.
4. CONCLUSIONS
In this work we proposed a stochastic epidemic model
to describe the propagation of a hoax in a social network.
This model can be interpreted as a SIS model in which the
compartment I of infected users has two sub-compartments:
believers and fact-checkers. We implemented and tested
the model on heterogeneous (scale-free), homogeneous (random),
and real networks, varying parameters and topology.
We focused on analyzing the crucial role of fact-checking
activity, ruled by a verifying probability. Analytically, we
found a threshold for this probability, a sufficient condition
that assures the hoax will be removed. This is interesting
because it provides an idea of how many fact-checkers would
be sufficient to guarantee the complete removal of fake news.
We analyzed the results of several simulations, discussing
the role of each parameter in the dynamic evolution of the
system, and confirming similarities of behaviors with traditional
SIS and SIR epidemic models. On the basis of the
results presented here, work on this point is continuing to
explore if there are regions of the network in which it is
easier that the hoax infection becomes endemic.
Future work could also involve some extensions to the
model: we could insert a “memory effect” or delay the debunking
diffusion, as realistically the two propagations may
not be simultaneous. Moreover, this model does not take
into account the heterogeneity of agents — all have the same
gullibility and verification probability, for example. In real
life, people may have different tendencies to believe claims
that are consistent with their world views and selectively
discard factual evidence that is not consistent [15]. Additionally
we began to explore how the dynamics respond to
a periodical re-injection of the hoax: a spiky behavior appears,
with periodic bursts of spreading. Recent work on
detection of rumors and fake content identified such bursts
as features with high predictive power [6, 10], so they deserve
further study. We are studying a more sophisticated
980
model in which we consider different gullibility values in two
communities (skeptic and gullible), observing the role of segregation
level among the groups, and the polarization of the
hoax in the gullible group, a fact that has been empirically
observed in [11]. Finally, the model need to be validated
with empirical data, analyzing the diffusion of real hoaxes
in social networks.
Acknowledgments
We would like to thank Prof. James Kauffman (University
of Virginia) who has been the target of a misinformation
phenomenon some years ago. He debunked the fake story
that involved him creating a letter (http://drlauraletter.
com), but occasionally that rumor starts spreading again
through the social networks without any apparent control.
He provided us with data about his experience, helping us
get some insights at a very early stage of our study.
This work was supported in part by NSF (grant CCF-
1101743), DoD (grant W911NF-12-1-0037), and the James
S. McDonnell Foundation (grant 220020274).
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APPENDIX
A. MEAN-FIELD ANALYSIS
We can approximate the infinite-time behavior of the system
with mean-field theory. First we simplify Eq. 4 substituting
si(t) with pi(t) and considering that pi(t + 1) = pi(t)
when t → ∞. Then, in the hypothesis that all vertices
have the same number of neighbors k and these neighbors
are chosen randomly, we substitute nB
i (t) with pB
(∞) and
nF
i (t) with pF
(∞) in the spreading functions, Eq.5. We obtain
equations only in terms of pB
(∞), pF
(∞) and pS
(∞):
p
B
(∞) =f · p
S
(∞) + (1 − pforget − pverify) · p
B
(∞) (10)
p
F
(∞) =g · p
S
(∞) + pverify · p
B
(∞) + (1 − pforget) · p
F
(∞) (11)
p
S
(∞) =pforget · (p
B
(∞) + p
F
(∞)) + (1 − f − g) · p
S
(∞). (12)
Moreover, we know that pS
i (t) = 1−pB
i (t)−pF
i (t), therefore
we can substitute
pS
(∞) = 1 − pB
(∞) − pF
(∞) = 1 − β
981
in 12 and trivially obtain the solution for the susceptible
rate at stationary state:
pS
(∞) =
pforget
β + pforget
. (13)
Similarly, we can re-write:
pF
(∞) =1 − pB
(∞) − pS
(∞)
=1 − pB
(∞) −
pforget
β + pforget
(14)
and, substituting it in 10, we obtain an equation only in
terms of pB
(∞):
pB
(∞) = f·
pforget
β + pforget
+(1−pforget−pverify)·pB
(∞) (15)
where
f =
β · pB
(∞) · (1 + α)
pB(∞) · (1 + α) + (1 − pB(∞) −
pforget
β + pforget
) · (1 − α)
.
We can observe that pB
(∞) = 0 is solution of 15, i.e. the
situation in which the hoax is defeated. To find the rate
of believers at the stationary state when the hoax survives,
with a little algebra on 15, it is easy to obtain
pB
(∞) =
β(2α · pforget − (1 − α) · pverify)
2α(β + pforget)(pforget + pverify)
. (16)
For fact-checkers we also have two solutions (hoax survives
or not) and we can trivially find them substituting the values
for believers and susceptible users in
pF
(∞) = 1 − pB
(∞) − pS
(∞).
982