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The spreading of misinformation online
Michela Del Vicario
a
, Alessandro Bessi
b
, Fabiana Zollo
a
, Fabio Petroni
c
, Antonio Scala
a,d
, Guido Caldarelli
a,d
,
H. Eugene Stanley
e
, and Walter Quattrociocchi
a,1
a
Laboratory of Computational Social Science, Networks Department, IMT Alti Studi Lucca, 55100 Lucca, Italy;
b
IUSS Institute for Advanced Study, 27100
Pavia, Italy;
c
Sapienza University, 00185 Rome, Italy;
d
ISC-CNR Uos “Sapienza,”00185 Rome, Italy; and
e
Boston University, Boston, MA 02115
Edited by Matjaz Perc, University of Maribor, Maribor, Slovenia, and accepted by the Editorial Board December 4, 2015 (received for review September
1, 2015)
The wide availability of user-provided content in online social media
facilitates the aggregation of people around common interests,
worldviews, and narratives. However, the World Wide Web (WWW)
also allows for the rapid dissemination of unsubstantiated rumors
and conspiracy theories that often elicit rapid, large, but naive social
responses such as the recent case of Jade Helm 15––where a simple
military exercise turned out to be perceived as the beginning of a
new civil war in the United States. In this work, we address the
determinants governing misinformation spreading through a thor-
ough quantitative analysis. In particular, we focus on how Facebook
users consume information related to two distinct narratives: scien-
tific and conspiracy news. We find that, although consumers of
scientific and conspiracy stories present similar consumption pat-
terns with respect to content, cascade dynamics differ. Selective
exposure to content is the primary driver of content diffusion and
generates the formation of homogeneous clusters, i.e., “echo cham-
bers.”Indeed, homogeneity appears to be the primary driver for the
diffusion of contents and each echo chamber has its own cascade
dynamics. Finally, we introduce a data-driven percolation model
mimicking rumor spreading and we show that homogeneity and
polarization are the main determinants for predicting cascades’size.
misinformation
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virality
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Facebook
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rumor spreading
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cascades
The massive diffusion of sociotechnical systems and micro-
blogging platforms on the World Wide Web (WWW) creates a
direct path from producers to consumers of content, i.e., allows
disintermediation, and changes the way users become informed,
debate, and form their opinions (1–5). This disintermediated envi-
ronment can foster confusion about causation, and thus encourage
speculation, rumors, and mistrust (6). In 2011 a blogger claimed
that global warming was a fraud designed to diminish liberty and
weaken democracy (7). Misinformation about the Ebola epidemic
has caused confusion among healthcare workers (8). Jade Helm 15,
a simple military exercise, was perceived on the Internet as the
beginning of a new civil war in the United States (9).
Recent works (10–12) have shown that increasing the exposure
of users to unsubstantiated rumors increases their tendency to
be credulous.
According to ref. 13, beliefs formation and revision is influ-
enced by the way communities attempt to make sense of events or
facts. Such a phenomenon is particularly evident on the WWW
where users, embedded in homogeneous clusters (14–16), process
information through a shared system of meaning (10, 11, 17, 18)
and trigger collective framing of narratives that are often biased
toward self-confirmation.
In this work, through a thorough quantitative analysis on a
massive dataset, we study the determinants behind misinformation
diffusion. In particular, we analyze the cascade dynamics of Face-
book users when the content is related to very distinct narratives:
conspiracy theories and scientific information. On the one hand,
conspiracy theories simplify causation, reduce the complexity of
reality, and are formulated in a way that is able to tolerate a certain
level of uncertainty (19–21). On the other hand, scientific in-
formation disseminates scientific advances and exhibits the process
of scientific thinking. Notice that we do not focus on the quality of
the information but rather on the possibility of verification. Indeed,
the main difference between the two is content verifiability. The gen-
erators of scientific information and their data, methods, and out-
comes are readily identifiable and available. The origins of conspiracy
theories are often unknown and their content is strongly disengaged
from mainstream society and sharply divergent from recommended
practices (22), e.g., the belief that vaccines cause autism.
Massive digital misinformation is becoming pervasive in online
social media to the extent that it has been listed by the World
Economic Forum (WEF) as one of the main threats to our so-
ciety (23). To counteract this trend, algorithmic-driven solutions
have been proposed (24–29), e.g., Google (30) is developing a
trustworthiness score to rank the results of queries. Similarly,
Facebook has proposed a community-driven approach where
users can flag false content to correct the newsfeed algorithm.
This issue is controversial, however, because it raises fears that
the free circulation of content may be threatened and that the
proposed algorithms may not be accurate or effective (10, 11,
31). Often conspiracists will denounce attempts to debunk false
information as acts of misinformation.
Whether a claim (either substantiated or not) is accepted by
an individual is strongly influenced by social norms and by the
claim’s coherence with the individual’s belief system––i.e., con-
firmation bias (32, 33). Many mechanisms animate the flow of
false information that generates false beliefs in an individual,
which, once adopted, are rarely corrected (34–37).
In this work we provide important insights toward the un-
derstanding of cascade dynamics in online social media and in
particular about misinformation spreading.
We show that content-selective exposure is the primary driver
of content diffusion and generates the formation of homogeneous
Significance
The wide availability of user-provided content in online social
media facilitates the aggregation of people around common
interests, worldviews, and narratives. However, the World
Wide Web is a fruitful environment for the massive diffusion of
unverified rumors. In this work, using a massive quantitative
analysis of Facebook, we show that information related to
distinct narratives––conspiracy theories and scientific news––
generates homogeneous and polarized communities (i.e., echo
chambers) having similar information consumption patterns.
Then, we derive a data-driven percolation model of rumor
spreading that demonstrates that homogeneity and polariza-
tion are the main determinants for predicting cascades’size.
Author contributions: M.D.V., A.B., F.Z., A.S., G.C., H.E.S., and W.Q. designed research;
M.D.V., A.B., F.Z., H.E.S., and W.Q. performed research; M.D.V., A.B., F.Z., F.P., and W.Q.
contributed new reagents/analytic tools; M.D.V., A.B., F.Z., A.S., G.C., H.E.S., and W.Q.
analyzed data; and M.D.V., A.B., F.Z., A.S., G.C., H.E.S., and W.Q. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission. M.P. is a guest editor invited by the Editorial
Board.
Freely available online through the PNAS open access option.
1
To whom correspondence should be addressed. Email: walterquattrociocchi@gmail.com.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.
1073/pnas.1517441113/-/DCSupplemental.
www.pnas.org/cgi/doi/10.1073/pnas.1517441113 PNAS Early Edition
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clusters, i.e., “echo chambers”(10, 11, 38, 39). Indeed, our analysis
reveals that two well-formed and highly segregated communities
exist around conspiracy and scientific topics. We also find that
although consumers of scientific information and conspiracy
theories exhibit similar consumption patterns with respect to con-
tent, the cascade patterns of the two differ. Homogeneity appears
to be the preferential driver for the diffusion of content, yet each
echo chamber has its own cascade dynamics. To account for these
features we provide an accurate data-driven percolation model of
rumor spreading showing that homogeneity and polarization are
the main determinants for predicting cascade size.
The paper is structured as follows. First we provide the pre-
liminary definitions and details concerning data collection. We
then provide a comparative analysis and characterize the statistical
signatures of cascades of the different kinds of content. Finally,
we introduce a data-driven model that replicates the analyzed
cascade dynamics.
Methods
Ethics Statement. Approval and informed consent were not needed because
the data collection process has been carried out using the Facebook Graph
application program interface (API) (40), which is publicly available. For the
analysis (according to the specification settings of the API) we only used
publicly available data (thus users with privacy restrictions are not included in
the dataset). The pages from which we download data are public Facebook
entities and can be accessed by anyone. User content contributing to these
pages is also public unless the user’s privacy settings specify otherwise, and in
that case it is not available to us.
Data Collection. Debate about social issues continues to expand across the
Web, and unprecedented social phenomena such as the massive recruitment
of people around common interests, ideas, and political visions are emergi ng.
Using the approach described in ref. 10, we define the space of our in-
vestigation with the support of diverse Facebook groups that are active in
the debunking of misinformation.
The resulting dataset is composed of 67 public pages divided between 32
about conspiracy theories and 35 about science news. A second set, composed
of two troll pages, is used as a benchmark to fit our data-driven model.
The first category (conspiracy theories) includes the pages that disseminate
alternative, controversial information, often lacking supporting evidence
and frequently advancing conspiracy theories. The second category (science
news) includes the pages that disseminate scientific information. The third
category (trolls) includes those pages that intentionally disseminate sarcastic
false information on the Web with the aim of mocking the collective
credulity online.
For the three sets of pages we download all of the posts (and their
respective user interactions) across a 5-y time span (2010–2014). We
perform the data collection process by using the Facebook Graph API (40),
which is publicly available and accessible through any personal Facebook
user account. The exact breakdown of the data is presented in SI Appendix,
section 1.
Preliminaries and Definitions. A tree is an undirected simple graph that is
connected and has no simple cycles. An oriented tree is a directed acyclic
graph whose underlying undirected graph is a tree. A sharing tree, in the
context of our research, is an oriented tree made up of the successive sharing
of a news item through the Facebook system. The root of the sharing tree is
the node that performs the first share. We define the size of the sharing tree
as the number of nodes (and hence the number of news sharers) in the tree
and the height of the sharing tree as the maximum path length from the root.
We define the user polarization σ=2ϱ−1, where 0 ≤ϱ≤1 is the fraction of
“likes”a user puts on conspiracy-related content, and hence −1≤σ≤1. From
user polarization, we define the edge homogeneity, for any edge eij be-
tween nodes iand j,as
σij =σiσj,
with −1≤σij ≤1. Edge homogeneity reflects the similarity level between
the polarization of the two sharing nodes. A link in the sharing tree is
0.000
0.025
0.050
0.075
0 1020304050
Lifetime(hours)
PDF
Science
Conspirac
y
Fig. 1. PDF of lifetime computed on science news and conspiracy theories,
where the lifetime is here computed as the temporal distance (in hours) be-
tween the first and last share of a post. Both categories show a similar behavior.
0
200
400
600
0 500 1000 1500 2000 2500
Conspiracy Cascade Size
Lifetime (hours)
0
200
400
0 250 500 750
Science Cascade Size
Lifetime (hours)
Fig. 2. Lifetime as a function of the cascade size for conspiracy news (Left) and science news (Right). Science news quickly reaches a higher diffusion; a longer
lifetime does not correspond to a higher level of interest. Conspiracy rumors are assimilated more slowly and show a positive relation between lifetime
and size.
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homogeneous if its edge homogeneity is positive. We then define a
sharing path to be any path from the root to one of the leaves of the
sharing tree. A homogeneous path is a sharing path for which the edge
homogeneity of each edge is positive, i.e.,asharingpathcomposedonly
of homogeneous links.
Results and Discussion
Anatomy of Cascades. We begin our analysis by characterizing the
statistical signature of cascades as they relate to information
type. We analyze the three types—science news, conspiracy ru-
mors, and trolling—and find that size and maximum degree are
power-law distributed for all three categories. The maximum cas-
cade size values are 952 for science news, 2,422 for conspiracy
news, and 3,945 for trolling, and the estimated exponents γfor the
power-law distributions are 2.21 for science news, 2.47 for con-
spiracy, and 2.44 for trolling posts. Tree height values range from 1
to 5, with a maximum height of 5 for science news and conspiracy
theories and a maximum height of 4 for trolling. The resulting
network is very dense. Notice that such a feature weakens the role
of hubs in rumor-spreading dynamics. For further information see
SI Appendix,section2.1.
Fig. 1 shows the probability density function (PDF) of the
cascade lifetime (using hours as time units) for science and
conspiracy. We compute the lifetime as the length of time be-
tween the first user and the last user sharing a post. In both
categories we find a first peak at ∼1–2 h and a second at ∼20 h,
indicating that the temporal sharing patterns are similar irre-
spective of the difference in topic. We also find that a significant
percentage of the information diffuses rapidly (24.42% of the
science news and 20.76% of the conspiracy rumors diffuse in less
than 2 h, and 39.45% of science news and 40.78% of conspiracy
theories in less than 5 h). Only 26.82% of the diffusion of science
news and 17.79% of conspiracy lasts more than 1 d.
In Fig. 2 we show the lifetime as a function of the cascade size.
For science news we have a peak in the lifetime corresponding to
a cascade size value of ≈200, and higher cascade size values
correspond to high lifetime variability. For conspiracy-related
content the lifetime increases with cascade size.
These results suggest that news assimilation differs according
to the categories. Science news is usually assimilated, i.e., it reaches
a higher level of diffusion quickly, and a longer lifetime does not
correspond to a higher level of interest. Conversely, conspiracy
rumors are assimilated more slowly and show a positive relation
between lifetime and size. For both science and conspiracy news, we
compute the size as a function of the lifetime and confirm that
differentiation in the sharing patterns is content-driven, and that for
conspiracy there is a positive relation between size and lifetime (see
SI Appendix,section2.1for further details).
Homogeneous Clusters. We next examine the social determinants
that drive sharing patterns and we focus on the role of homo-
geneity in friendship networks.
Fig. 3 shows the PDF of the mean-edge homogeneity, com-
puted for all cascades of science news and conspiracy theories. It
shows that the majority of links between consecutively sharing
users is homogeneous. In particular, the average edge homoge-
neity value of the entire sharing cascade is always greater than or
equal to zero, indicating that either the information transmission
occurs inside homogeneous clusters in which all links are ho-
mogeneous or it occurs inside mixed neighborhoods in which the
balance between homogeneous and nonhomogeneous links is
favorable toward the former ones. However, the probability of
close to zero mean-edge homogeneity is quite small. Contents
tend to circulate only inside the echo chamber.
Hence, to further characterize the role of homogeneity in
shaping sharing cascades, we compute cascade size as a function
of mean-edge homogeneity for both science and conspiracy news
(Fig. 4). In science news, higher levels of mean-edge homogeneity in
the interval (0.5, 0.8) correspond to larger cascades, but in
conspiracy theories lower levels of mean-edge homogeneity
(∼0.25) correspond to larger cascades. Notice that, although
viral patterns related to distinct contents differ, homogeneity is
clearly the driver of information diffusion. In other words, dif-
ferent contents generate different echo chambers, characterized
by a high level of homogeneity inside them. The PDF of the edge
homogeneity, computed for science and conspiracy news as well
as the two taken together—both in the unconditional case and in
the conditional case (in the event that the user that made the
first share in the couple has a positive or negative polarization)—
confirms the roughly null probability of a negative edge homo-
geneity (SI Appendix, section 2.1).
We record the complementary cumulative distribution func-
tion (CCDF) of the number of all sharing paths* on each tree
compared with the CCDF of the number of homogeneous paths
for science and conspiracy news, and the two together. A Kol-
mogorov–Smirnov test and Q-Q plots confirm that for all three
pairs of distributions considered there is no significant statistical
difference (see SI Appendix, section 2.2 for more details). We
confirm the pervasiveness of homogeneous paths.
Indeed, cascades’lifetimes of science and conspiracy news
exhibit a probability peak in the first 2 h, and then in the fol-
lowing hours they rapidly decrease. Despite the similar con-
sumption patterns, cascade lifetime expressed as a function of
the cascade size differs greatly for the different content sets.
However, homogeneity remains the main driver of cascades’
propagation. The distributions of the number of total and ho-
mogeneous sharing paths are very similar for both content cat-
egories. Viral patterns related to contents belonging to different
narratives differ, but homogeneity is the primary driver of con-
tent diffusion.
0
1
2
3
4
0.1 1.0
Mean Edge Homogeneity
PDF
Science
Conspirac
y
Fig. 3. PDF of edge homogeneity for science (orange) and conspiracy (blue)
news. Homogeneity paths are dominant on the whole cascades for both
scientific and conspiracy news.
*Recall that a sharing path is here defined as any path from the root to one of the leaves
of the sharing tree. A homogeneous path is a sharing path for which the edge homo-
geneity of each edge is positive.
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The Model. Our findings show that users mostly tend to select and
share content according to a specific narrative and to ignore the
rest. This suggests that the determinant for the formation of echo
chambers is confirmation bias. To model this mechanism we now
introduce a percolation model of rumor spreading to account for
homogeneity and polarization. We consider nusers connected by
a small-world network (41) with rewiring probability r.Everynode
has an opinion ωi,i∈f1, nguniformly distributed between ½0,1
and is exposed to mnews items with a content ϑj, j∈f1, mg
uniformly distributed in ½0,1. At each step the news items are
diffused and initially shared by a group of first sharers. After the
first step, the news recursively passes to the neighborhoods of
previous step sharers, e.g., those of the first sharers during the
second step. If a friend of the previous step sharers has an opinion
close to the fitness of the news, then she shares the news again.
When
ωi−ϑj
≤δ,
user ishares news j;δis the sharing threshold.
Because δby itself cannot capture the homogeneous clusters
observed in the data, we model the connectivity pattern as a
signed network (4, 42) considering different fractions of homo-
geneous links and hence restricting diffusion of news only to
homogeneous links. We define ϕHL as the fraction of homoge-
neous links in the network, Mas the number of total links, and nh
as the number of homogeneous links; thus, we have
ϕHL =nh
M, 0≤nh≤M.
Notice that 0 ≤ϕHL ≤1 and that 1 −ϕHL, the fraction of nonho-
mogeneous links, is complementary to ϕHL. In particular, we can
reduce the parameters space to ϕHL ∈½0.5, 1as we would restrict
our attention to either one of the two complementary clusters.
The model can be seen as a branching process where the
sharing threshold δand neighborhood dimension zare the key
parameters. More formally, let the fitness θjof the jth news and
the opinion ωiof a the ith user be uniformly independent
identically distributed (i.i.d.) between ½0,1. Then the probability
pthat a user ishares a post jis defined by a probability
p=minð1, θ+δÞ−maxð0, θ−δÞ≈2δ, because θand ωare uni-
formly i.i.d. In general, if ωand θhave distributions fðωÞand
fðθÞ, then pwill depend on θ,
pθ=fðθÞZ
minð1, θ+δÞ
maxð0, θ−δÞ
fðωÞdω.
If we are on a tree of degree z(or on a sparse lattice of degree
z+1), the average number of sharers (the branching ratio) is
defined by
μ=zp ≈2δ z,
with a critical cascade size S=ð1−μÞ−1. If we assume that the
distribution of the number mof the first sharers is fðmÞ, then the
average cascade size is
S=X
m
fðmÞmð1−μÞ−1=
hmif
1−μ≈hmif
1−2δz,
where h...if=Pm...fðmÞis the average with respect to f. In the
simulations we fixed neighborhood dimension z=8 because
the branching ratio μdepends upon the product of zand δand,
without loss of generality, we can consider the variation of just one
of them.
If we allow a probability qthat a neighbor of a user has a
different polarization, then the branching ratio becomes
μ=zð1−qÞp. If a lattice has a degree distribution dðkÞ(k=z+1),
we can then assume a usual percolation process that provides a
critical branching ratio and that is linear in hk2id=hkid(μ≈
ð1−qÞphz2i=hzi).
Simulation Results. We explore the model parameters space using
n=5,000 nodes and m=1,000 news items with the number of first
sharers distributed as (i) inverse Gaussian, (ii)lognormal,(iii)
Poisson, (iv) uniform distribution, and as the real-data distribution
(from the science and conspiracy news sample). In Table 1 we
show a summary of relevant statistics (min value, first quantile,
median, mean, third quantile, and max value) to compare the real-
data first sharers distribution with the fitted distributions.
†
Along with the first sharers distribution, we vary the sharing
threshold δin the interval ½0.01, 0.05and the fraction of ho-
mogeneous links ϕHL in the interval ½0.5, 1. To avoid biases in-
duced by statistical fluctuations in the stochastic process, each
point of the parameter space is averaged over 100 iterations.
ϕHL ∼0.5 provides a good estimate of real-data values. In par-
ticular, consistently with the division of in two echo chambers
(science and conspiracy), the network is divided into two clusters
in which news items remain inside and are transmitted solely
within each community’s echo chamber (see SI Appendix, section
3.2 for the details of the simulation results).
In addition to the science and conspiracy content sharing
trees, we downloaded a set of 1,072 sharing trees of intentionally
false information from troll pages. Frequently troll information,
e.g., parodies of conspiracy theories such as chem-trails containing
the active principle of Viagra, is picked up and shared by habitual
conspiracy theory consumers. We computed the mean and SD of
size and height of all trolling sharing trees, and reproduced the data
using our model.
‡
We used fixed parameters from trolling messages
0
20
40
60
0.00 0.25 0.50 0.75 1.00
Mean Edge Homogeneity
Cascade Size
Science
Conspirac
y
Fig. 4. Cascade size as a function of edge homogeneity for science (orange)
and conspiracy (dashed blue) news.
†
For details on the parameters of the fitted distributions used, see SI Appendix,section3.2.
‡
Note that the real-data values for the mean (and SD) of size and height on the troll posts
are, respectively, 23.54 ð122.32Þand 1.78 ð0.73Þ.
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sample (the number of nodes in the system and the number of news
items) and varied the fraction of homogeneous links ϕHL,the
rewiring probability r, and sharing threshold δ.SeeSI Appendix,
section 3.2 for the distribution of first sharers used and for addi-
tional simulation results of the fit on trolling messages.
We simulated the model dynamics with the best combination
of parameters obtained from the simulations and the number of
first sharers distributed as an inverse Gaussian. Fig. 5 shows the
CCDF of cascades’size and the cumulative distribution function
(CDF) of their height. A summary of relevant statistics (min
value, first quantile, median, mean, third quantile, and max
value) to compare the real-data size and height distributions with
the fitted ones is reported in SI Appendix, section 3.2.
We find that the inverse Gaussian is the distribution that best
fits the data both for science and conspiracy news, and for troll
messages. For this reason, we performed one more simulation
using the inverse Gaussian as distribution of the number of first
sharers, 1,072 news items, 16,889 users, and the best parameters
combination obtained in the simulations.
§
The CCDF of size and
the CDF of height for the above parameters combination, as well
as basic statistics considered, fit real data well.
Conclusions
Digital misinformation has become so pervasive in online social
media that it has been listed by the WEF as one of the main threats
to human society. Whether a news item, either substantiated or not,
is accepted as true by a user may be strongly affected by social
norms or by how much it coheres with the user’ssystemofbeliefs
(32, 33). Many mechanisms cause false information to gain accep-
tance, which in turn generate false beliefs that, once adopted by an
individual, are highly resistant to correction (34–37). In this work,
using extensive quantitative analysis and data-driven modeling, we
provide important insights toward the understanding of the mech-
anism behind rumor spreading. Our findings show that users mostly
tend to select and share content related to a specific narrative and
to ignore the rest. In particular, we show that social homogeneity is
the primary driver of content diffusion, and one frequent result is
the formation of homogeneous, polarized clusters. Most of the
times the information is taken by a friend having the same profile
(polarization)––i.e., belonging to the same echo chamber.
We also find that although consumers of science news and
conspiracy theories show similar consumption patterns with re-
spect to content, their cascades differ.
Our analysis shows that for science and conspiracy news a
cascade’s lifetime has a probability peak in the first 2 h, followed
by a rapid decrease. Although the consumption patterns are
similar, cascade lifetime as a function of the size differs greatly.
These results suggest that news assimilation differs according
to the categories. Science news is usually assimilated, i.e., it
reaches a higher level of diffusion, quickly, and a longer lifetime
does not correspond to a higher level of interest. Conversely,
conspiracy rumors are assimilated more slowly and show a pos-
itive relation between lifetime and size.
The PDF of the mean-edge homogeneity indicates that ho-
mogeneity is present in the linking step of sharing cascades. The
distributions of the number of total sharing paths and homoge-
neous sharing paths are similar in both content categories.
Viral patterns related to distinct contents are different but
homogeneity drives content diffusion. To mimic these dynamics,
we introduce a simple data-driven percolation model of signed
networks, i.e., networks composed of signed edges accounting for
nodes preferences toward specific contents. Our model repro-
duces the observed dynamics with high accuracy.
Users tend to aggregate in communities of interest, which
causes reinforcement and fosters confirmation bias, segregation,
and polarization. This comes at the expense of the quality of the
information and leads to proliferation of biased narratives
fomented by unsubstantiated rumors, mistrust, and paranoia.
According to these settings algorithmic solutions do not seem
to be the best options in breaking such a symmetry. Next envi-
sioned steps of our research are to study efficient communication
strategies accounting for social and cognitive determinants be-
hind massive digital misinformation.
10 3
10 2.5
10 2
10 1.5
10 1
10 0.5
100
100100.5 101101.5 102102.5 103
Size
CCDF
Data
Simulated
0.00
0.25
0.50
0.75
1.00
0246
Height
CDF
Data
Simulated
Fig. 5. CCDF of size (Left) and CDF of height (Right) for the best parameters combination that fits real-data values,ðϕHL ,r,δÞ=ð0.56, 0.01, 0.015Þ, and first
sharers distributed as IGð18.73, 9.63Þ.
Table 1. Summary of relevant statistics comparing synthetic
data with the real ones
Values Data IG Lognormal Poisson
Min 1 0.36 0.10 20
First quantile 5 4.16 3.16 35
Median 10 10.45 6.99 39
Mean 39.34 39.28 13.04 39.24
Third quantile 27 31.59 14.85 43
Max 3,033 1814 486.10 66
The inverse Gaussian (IG) shows the best fit for the distribution of first
sharers with respect to all of the considered statistics.
§
The best parameters combinations is ϕHL =0.56, r=0.01, δ=0.015. In this case we have a
mean size equal to 23.42 ð33.43Þand a mean height 1.28 ð0.88Þ, and it is indeed a good
approximation; see SI Appendix, section 3.2.
Del Vicario et al. PNAS Early Edition
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ACKNOWLEDGMENTS. Special thanks go to Delia Mocanu, “Protesi di Pro-
tesi di Complotto,”“Che vuol dire reale,”“La menzogna diventa verita
e passa alla storia,”“Simply Humans,”“Semplicemente me,”Salvatore
Previti, Elio Gabalo, Sandro Forgione, Francesco Pertini, and “The rooster
on the trash”for their valuable suggestions and discussions. Funding
for this work was provided by the EU FET Project MULTIPLEX, 317532,
SIMPOL, 610704, the FET Project DOLFINS 640772, SoBigData 654024, and
CoeGSS 676547.
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