Prevalence and Propagation of Fake News111Preprint submitted for publication
review on 7 May 2021

Banafsheh Behzada,222Corresponding author
<EMAIL_ADDRESS>
Bhavana Bheemb
<EMAIL_ADDRESS>
Daniela Elizondob
<EMAIL_ADDRESS>
Deyana Marshb
<EMAIL_ADDRESS>
Susan Martonosib
<EMAIL_ADDRESS>
aDepartment of Information Systems, College of Business Administration at
California State University, Long Beach
bDepartment of Mathematics, Harvey Mudd College

Abstract

In recent years, scholars have raised concerns on the effects that unreliable
news, or “fake news,” has on our political sphere, and our democracy as a
whole. For example, the propagation of fake news on social media is widely
believed to have influenced the outcome of national elections, including the
2016 U.S. Presidential Election, and the 2020 COVID-19 pandemic. What drives
the propagation of fake news on an individual level, and which interventions
could effectively reduce the propagation rate? Our model disentangles bias
from truthfulness of an article and examines the relationship between these
two parameters and a reader’s own beliefs. Using the model, we create policy
recommendations for both social media platforms and individual social media
users to reduce the spread of untruthful or highly biased news. We recommend
that platforms sponsor unbiased truthful news, focus fact-checking efforts on
mild to moderately biased news, recommend friend suggestions across the
political spectrum, and provide users with reports about the political
alignment of their feed. We recommend that individual social media users fact
check news that strongly aligns with their political bias and read articles of
opposing political bias.

##  1 Introduction

In recent years, social media networks such as Facebook and Twitter have
become major news sources; an estimated $62\%$ of Americans claim to get news
from social media ([13]). Concurrently, social media has seen a surge of fake
news which, for the purpose of this paper, is news containing a high frequency
of falsehoods. In March and April 2021, the following examples were some of
the top fake news claims circulating through social media.

> “Per the CDC There Are Nearly Twice As Many Vaccine Related Deaths SO FAR in
> 2021 (1,755) Than All the Vaccine Deaths this Past Decade (994)”-March 2021,
> ([17]), and “Stanford Study Results: Face masks are Ineffective to Block
> Transmission of COVID-19 and Actually Can Cause Health Deterioration and
> Premature Death.”-April 2021 ([18])

Both of these claims were proven to be false, and their only purpose is to
misinform people. Considering the high ramifications of just one of these
claims propagating, mitigating the spread of untruthful news is of paramount
importance ([14, 24, 41]).

Social media has the unique characteristic of its users playing a decisive
role in which content gets propagated. Alarmingly, research shows that users
are less likely to fact check the news they receive on social media than the
news they receive through other sources ([2, 23]). Furthermore, fake news
takes advantage of the political echo chambers within social media to further
its propagation ([41]). With active users continuously making decisions on
whether to propagate news (e.g. share, re-tweet), unverified and unreliable
sources can potentially reach as large an audience on social media as national
news sources ([2]).

Current techniques for mitigating the spread of fake news include increasing
the spread of truthful news and flagging untruthful news articles or accounts
([11, 24]). The former technique increases the amount of truthful news in a
social media user’s feed, and the latter diminishes the credibility of fake
news sources. One major source of fake news propagation are social bots,
computer generated and controlled social media accounts. Since social bots
target particular social media users whose usage characteristics make them
more likely to further propagate fake news ([38, 39]), one tactic for fake
news mitigation is having more methods of detecting the difference between bot
accounts and human accounts in order to deactivate bot accounts ([7, 29, 19]).
Although current techniques are beneficial, social media users are still being
exposed to fake news during crucial times like the 2016 US presidential
election and the 2020 COVID-19 pandemic ([2, 30]).

The initial step in finding techniques to mitigate fake news is understanding
how fake news spreads on both the macroscopic and microscopic levels. The
macroscopic spread refers to the network level propagation of content. One
study uses a network graph model to analyze how people believe and spread
news; they conclude that social media cultivates echo chambers in which low
quality news is likely shared ([5]). Vicario et al. find that cascade dynamics
of fake news depend on polarized echo chambers ([41]). Since early detection
is vital to diminishing the effects of fake news, Louni and Subbalakshmi focus
on detecting the source of news within a cluster ([27]). Other studies
effectively detect fake news on a network level by using propagation paths to
classify different types of news ([26, 33]). Vosoughi et al. graph data from
Twitter and conclude through that fake news spreads faster and more broadly
than true news when analyzing a social media network ([42]). Jang et al.
utilize evolutionary tree analysis to similarly conclude that fake news
spreads deeper through social media networks and undergoes frequent
modifications ([21]). Research on the macroscopic spread of content provides
an understanding of which clusters facilitate fake news spread and also of
news propagation patterns. Both findings inform techniques for early detection
of fake news which is crucial for mitigation.

Research on content spread at the microscopic level focuses on an individual
user’s decision to further propagate information from their social media
feeds. Findings at the microscopic level on fake news propagation do not
necessarily align with findings at a macroscopic level since an individual
level decision to share news does not take into consideration the network
effects. Some microscopic level studies utilize user surveys to understand
user behavior, and other studies create models to pinpoint social media
attributes that correlate to a higher likelihood of propagating fake news.
Guess et al. use Poisson regressions to link a representative online survey to
the respondents’ sharing history on Facebook. They find that fake news is more
likely to propagate through conservatives and older users ([16]). Allcott and
Gentzkow similarly conduct surveys on a portion of the voting population and
find that people with higher education, who spend more time on social media,
and have no partisan attachment are the most accurate in distinguishing fake
from real news ([2]). Furthermore, Papanastasiou utilizes an agent-based
sequential model to conclude that social media users tend to correlate
popularity with credibility ([32]). Lee et al. successfully create a feature-
based prediction model and a time estimation model to characterize the
likelihood an account retweets a particular message based on certain traits of
the Twitter account, such as its sociability and follower base ([25]).
Characterizing the probability that a social media user will further spread
untruthful or polarizing news based on individual factors, like political
ideology, is vital in understanding how to stop the spread.

This paper further expands the research within the microscopic level. We
develop a model for the probability an individual social media user with a
certain political belief chooses to share a news article having a certain
political bias and truthfulness. Different from previous work, our model
defines both truthfulness and bias on a continuous numerical spectrum and also
does not conflate them for a given article. Our model estimates the
probability that a particular population will share a certain article.
Numerical probabilities provide a more accurate understanding of how different
social media populations share content relative to each other which helps
inform recommendations for platforms and individual users to reduce the spread
of untruthful or highly biased news.

From the analysis of the data, we find that if social media users engage only
with users of similar political beliefs, then the probability of sharing news,
fake or true, increases. If users engage with users of opposing political
beliefs, then the probability of sharing decreases, and the decrease is more
pronounced for fake news. This paper recommends social media platforms:
sponsor unbiased truthful news; focus existing fact-checking and flagging
algorithms on moderately liberal- and conservative-biased news, rather than
flagging extremely biased articles; suggest content and connections from
across the political spectrum; and supply social media users with a score
reflecting the political alignment of their feed. For individual users, this
paper recommends fact-checking news that aligns strongly with their belief, as
people are more susceptible to propagating this type of news, and to engage
with users with opposing political beliefs.

The methodology of this paper follows a three step approach: developing a
probabilistic model for the propagation of political news; analyzing the model
to determine which attributes of content and user population contribute to the
propagation of untruthful or biased content; finally, using the findings to
recommend interventions against fake news. In the next section, we explain our
model and outline the assumptions that are built into it. Section 3 analyzes
the model from an optimization perspective to determine characteristics of
content that a malicious agent seeking to propagate untruthful or highly
biased news might target. Section 4 details the data set used to validate our
model. Section 5 presents an empirical analysis of the propagation rate of
political news within populations of readers that have varying political
beliefs. The analyses of Sections 3 and 5 inform the recommended techniques
for mitigating the spread of fake news discussed in Section 6. Lastly, we
provide our ideas for future work and conclusions in Sections 7 and 8,
respectively.

##  2 Probability Model of Propagation

This section describes how we model the probability $p$ that a reader
propagates an encountered article (e.g. “share,” “retweet,” “like,” or
“upvote,” depending on the social media platform of interest) as a function of
the article’s bias and truthfulness, and the reader’s belief.

An article $i$ is characterized by two attributes: bias $b_{i}$ and
truthfulness $t_{i}$. Bias is a continuous value on $[-1,1]$ which reflects
the political ideology represented by the topics, facts, and claims included
in the article. A bias equal to $-1$ corresponds to a very liberal (“left-
biased”) article, a value of $+1$ corresponds to a very conservative (“right-
biased”) article, and 0 corresponds to a politically unbiased article. The
truthfulness $t_{i}$ representing both the frequency and severity of
falsehoods in the article, ranges from $0$ (completely false news) to $1$
(perfectly true). The article’s reader $j$ is characterized by their political
belief $B_{j}$ which also ranges from $-1$ (very liberal, or “left-political”)
to $+1$ (very conservative, or “right-political”).

The shape of the function is based on two assumptions, which are informed by
previous works, as noted below:

  1. 1.

A person will be more likely to propagate an article when its bias is close to
their beliefs, even when the article exhibits low truthfulness ([3, 14, 20,
28]). That is, $p(b_{i},t_{i},B_{j})$ should increase as $\lvert
b_{i}-B_{j}\rvert$ decreases. Users often share content as a means of
political persuasion, and we assume that a user would not wish to disseminate
content that could persuade others to adopt a dramatically different political
belief.

  2. 2.

The probability of propagation increases with article truthfulness, $t_{i}$.
Articles that are too untruthful are unlikely to be propagated because they
are too outlandish ([14]). In other words, we assume that given two articles
exhibiting the same bias, a reader will be more likely to share the more
truthful article111Readers may not always be able to distinguish truth from
untruth with high precision. However, we are modeling the likelihood of
sharing an article. External factors such as the reputability of the source
can serve as signals to readers as to the truthfulness of content.. However,
we also assume diminishing marginal increases in sharing rate as $t_{i}$
increases. Because probabilities are bounded by 1, it is natural to assume
this rate of increase will decline.

Additionally, we require a valid probability function, bounded between $0$ and
$1$ for $(b_{i},t_{i},B_{j})\in[-1,1]\times[0,1]\times[-1,1]$. A probability
function that satisfies these assumptions and is reasonably tractable is the
logistic curve:

$p(b_{t},t_{i},B_{j})=\begin{cases}\frac{f_{l}}{1+e^{-k_{l}(t_{i}-(b_{i}-B_{j})^{2})}}&\quad\mbox{
if }B_{j}<0\\\ \frac{f_{r}}{1+e^{-k_{r}(t_{i}-(b_{i}-B_{j})^{2})}}&\quad\mbox{
if }B_{j}\geq 0\end{cases}$ (1)

where $f_{l},f_{r},k_{l},$ and $k_{r}$ are scaling parameters. $f_{l}$ and
$f_{r}$ scale the overall rate of sharing for left-political and right-
political readers, respectively. $k_{l}$ and $k_{r}$ scale the rate of decay
in sharing probability associated with the squared deviation between article
bias and reader belief and the truthfulness of the article. Several plots of
level curves of the probability function are shown in Figure 1.

(a) Probability of sharing as a function of truthfulness; $B_{j}$ = 0.45,
$f_{r}$ = 1, $k_{r}$ = 10

(b) Probability of sharing as a function of bias; $B_{j}$ = 0.45, $f_{r}$ = 1,
$k_{r}$ = 10

(c) Probability of sharing as a function of reader belief; $t_{i}$ = 0.45,
$f_{r}$ = 1, $k_{r}$ = 10

Figure 1: Level curves of the probability model of equation (1)

Because of its tractability and suitability in modeling a probability function
satisfying our assumptions, we will use the model given by equation (1)
throughout the remainder of the paper.

##  3 Characterizing the Optimal Behavior of a Malicious Agent

We first examine the model from a mathematical perspective to characterize the
optimal behavior of a malicious agent wishing to spread “fake” news. We assume
that a malicious agent seeks to propagate untruthful (low $t_{i}$) or highly
biased (high $|b_{i}|$) news at the highest possible rate, assuming a single
reader having belief $B_{j}$ (in Section 3.1), or a population of readers
having a belief distribution (in Section 3.2). Specifically, we answer three
questions:

  1. 1.

What general characteristics of articles (as reflected by $(b_{i},t_{i})$
pair) achieve the highest propagation rate?

  2. 2.

For untruthful articles, what degree of bias achieves the highest propagation
rate?

  3. 3.

For highly biased articles, what degree of truthfulness achieves the highest
propagation rate?

By answering these questions, we can understand how a malicious agent might
select or create political content to maximize propagation by readers of a
particular belief. Content that bears these optimal characteristics could then
be prioritized for fact-checking or other interventions.

Although not an assumption integrated into the model, for the purposes of
optimization, this section assumes that more biased news is inherently less
truthful ([43]). Thus, we constrain the feasible region of valid
$(b_{i},t_{i})$ pairs to the region $\lvert b_{i}\rvert+t_{i}\leq 1$. This
assumption is not only reasonable but also necessary to make the optimization
nontrivial; otherwise, the optimal strategy for maximizing propagation would
always be to produce perfectly truthful articles, which contradicts what we
observe on social media.

###  3.1 Single Reader

We first address each of the three questions above in the context of a
malicious agent seeking to propagate content to a single reader having belief
$B_{j}$.

  * •

What general characteristics of articles achieve the highest propagation rate?
(Single reader case)

The probability function of equation (1) has no local maximum, indicating that
the global maximum will lie along the boundaries of the feasible region
defined by $\lvert b_{i}\rvert+t_{i}\leq 1$, $-1\leq b_{i}\leq 1$, and $0\leq
t_{i}\leq 1$. To find the bias and truth combination that maximizes
propagation for a single reader having a given belief, we optimize the
function on all three boundaries separately.

For the constraint $t_{i}=1-\lvert b_{i}\rvert$, the function’s maximum is a
piecewise function split at $\lvert B_{j}\rvert=0.5$. For extreme conservative
beliefs ($B_{j}\geq 0.5)$, the function’s maximum is at
$b_{i}=-\frac{1}{2}+B_{j}$ and $t_{i}=\frac{3}{2}-B_{j}$. As the function is
symmetric, for extreme liberal beliefs ($B_{j}\leq-0.5$), the function’s
maximum is at $b_{i}=\frac{1}{2}+B_{j}$ and $t_{i}=\frac{3}{2}+B_{j}$. When
reader belief is more moderate, i.e., $\lvert B_{j}\rvert<0.5$, the maximum
value of the function occurs at maximum truthfulness: $b_{i}=0$ and $t_{i}=1$.
Although the probability of sharing increases as $\lvert b_{i}-B_{j}\rvert$
decreases, the constraint $\lvert b_{i}\rvert+t_{i}\leq 1$ causes the optimal
article bias to be substantially lower in magnitude than the reader’s belief.

On the constraint boundary $b_{i}=0$, the maximum propagation rate is attained
when $t_{i}=1$. Unbiased news achieves maximum propagation when it is
perfectly truthful. On the other hand, along the constraint $t_{i}=0$, the
maximum is at $b_{i}=B_{j}$. Untruthful news achieves maximum propagation when
its bias matches the belief of the reader.

From this analysis, we provide the first of our recommendations for social
media platforms (P) and users (U)222A summary of all recommendations made in
this paper can be found in Table 4 in Section 6..

Recommendation P1: Populate users’ feeds with unbiased truthful news as
sponsored articles. Such articles achieve the highest propagation rate among
users with moderately biased political views.

  * •

For untruthful articles that have low $t_{i}$, what degree of bias achieves
the highest propagation rate? (Single reader case)

We saw above that when $t_{i}=0$, maximum propagation is achieved when
$b_{i}=B_{j}$. This holds more generally for fixed $t_{i}$, provided that
$t_{i}+|B_{j}|<=1$. When $t_{i}+|B_{j}|>1$, there is no local maximum along
the cross sectional probability curve. Thus, maximum propagation is attained
at the endpoint $|b_{i}|=1-t_{i}$ (with the sign of $b_{i}$ chosen to match
the sign of $B_{j}$). To propagate a low-truth article to a single user, the
malicious agent will try to match the article’s bias to the user’s belief,
unless the user’s belief is so extreme that doing so would violate the truth-
bias constraint. In this latter case, the optimal bias of the article will be
chosen along the constraint boundary (i.e., as extreme as possible). From
this, we infer our next recommendation.

Recommendation U1: Fact check news that aligns strongly with a user’s beliefs
by using a third party fact-checking website for individual facts within the
article. Because a malicious agent acting optimally would align false content
with a user’s political beliefs, users are more susceptible to sharing
untruthful news that aligns with their beliefs.

  * •

For polarizing articles that have high $|b_{i}|$, what degree of truthfulness
achieves the highest propagation rate? (Single reader case)

For any fixed reader belief and article bias, the probability of sharing an
article increases with article truthfulness, by assumption. Thus in our
constrained optimization framework, the malicious agent wishing to propagate a
highly biased article would select article truthfulness along the constraint
boundary: $t_{i}=1-|b_{i}|$.

###  3.2 Distribution of Readers

Expanding on the case of a single reader, we examine the maximal propagation
rate over a distribution of readers. To start, given a discrete distribution
of reader belief $f(B_{j})$, the probability of a reader sharing an article
having bias $b_{i}$ and truthfulness $t_{i}$ is given by:

$p(b_{i},t_{i})=\sum_{B_{j}}{p(b_{i},t_{i},B_{j})*f(B_{j})}.$ (2)

With the probability function described in equation (2), we solve numerically
for the values of $b_{i}$ and $t_{i}$ that attain maximum propagation over
systematically varied distributions of user belief having a variety of
expectations and variances. For each distribution, we find the combination of
article bias and truthfulness that maximizes the probability of propagation to
answer each of the three questions posed earlier. We use the BARON solver
([37, 40]) on the NEOS server ([8, 9, 15]) to obtain the results.

  * •

What general characteristics of articles achieve the highest propagation rate?
(Distribution of readers case)

Plots of numerical solutions, shown in Figure 2, display the effects of the
reader belief distribution’s expected value and variance on both optimal bias
and the optimal probability of sharing. As the expected value of $B_{j}$
increases, the optimal bias increases and the probability of sharing
decreases. Additionally, in all cases examined, the optimal truthfulness of
the article was found to lie on the constraint boundary $|b_{i}|+t_{i}=1$.
Interestingly, the relationship between the expected value and optimal bias is
approximately the same as the closed form solution for the single reader case,
but variance also affects optimal bias. As the variance of the distribution of
$B_{j}$ increases, the optimal bias increases modestly for expected values
greater than $0.5$, and the probability of sharing decreases for all expected
values.

Thus, the propagation-maximizing article parameter values for the population
of readers resemble those of the case of a single reader. However, increasing
the variance of reader beliefs decreases the overall sharing probability and
modestly increases the bias required to attain maximum propagation. This
allows us to make our second recommendation for social media platforms:

Recommendation P2: Recommend friends and connections across the political
spectrum. Social media platforms generally recommend connections, content and
advertising based on similarities. The opposite tendency, however, may help
reduce the sharing rate of biased, untruthful news by increasing the variation
in political belief of a user population.

(a) Optimal article bias $b_{i}$ as a function of the expectation ($x$-axis)
and variance (gradient) of the distribution of reader beliefs

(b) Probability of sharing an article as a function of expectation ($x$-axis)
and variance (gradient) of the distribution of reader beliefs

Figure 2: Optimal article bias and probability of sharing over all
$(b_{i},t_{i})$ pairs, for numerous discrete probability distributions of
reader belief having given expected value and variance.

  * •

For untruthful articles that have low $t_{i}$, what degree of bias achieves
the highest propagation rate? (Distribution of readers case)

In the single-user case, we saw that for content having a low $t_{i}$ the
choice of $b_{i}$ that maximizes propagation is equal to the user’s belief
$B_{j}$ unless constrained by the boundary $|B_{j}|+t_{i}\leq 1$. For the case
of a distribution of readers, Figure 3(a) displays the optimal article bias,
when $t_{i}=0.1$ as a function of the expected reader belief, shaded by the
distribution variance; Figure 3(b) does the same for optimal probability of
sharing. We see that when truth is fixed, the optimal bias equals $E[B_{j}]$
until the constraint boundary is reached, at which point the optimal bias
equals $1-t_{i}$. The optimal bias is unaffected by variance. As in Figure
2(b), we see that the optimal sharing rate decreases as reader belief variance
increases. Once again, we conclude that recommending contacts across the
political spectrum could reduce the rate at which untruthful content is shared
(Recommendation P2).

(a) Optimal article bias $b_{i}$ with low $t_{i}$ as a function of the
expectation ($x$-axis) and variance (gradient) of the distribution of reader
beliefs

(b) Probability of sharing an article as a function of expectation ($x$-axis)
and variance (gradient) of the distribution of reader beliefs

Figure 3: Optimal article bias and probability of sharing over all $b_{i}$
values when $t_{i}=0.1$, for numerous discrete probability distributions of
reader belief having given expected value and variance

  * •

For polarizing articles that have high $|b_{i}|$, what degree of truthfulness
achieves the highest propagation rate? (Distribution of readers case)

As in the single reader case, sharing probability, for any value of $|b_{i}|$,
increases with $t_{i}$, independent of the beliefs of the readers. Thus, for
highly biased content, the degree of truthfulness achieving the highest
propagation rate lies along the constraint boundary, $t_{i}=1-|b_{i}|$.

Having explored the mathematical properties of our model in the context of a
malicious agent seeking to propagate untruthful or polarizing content, we now
introduce the data set we use to conduct an empirical analysis of the model.

##  4 Model Validation

We validate our model using social media data from [14] and Media Bias Monitor
(MBM) ([34]) to estimate sharing probabilities and the three model parameters:
bias, truthfulness and reader belief. We then compare the proposed probability
function to the propagation rates of articles found on social media and verify
that the model assumptions and general shape of the functional curve are
satisfied.

###  4.1 Data

To obtain a set of readers along with their political beliefs, we utilize the
data from Grinberg et al. which contains files with information about panel
members, a set of Twitter users whose Twitter profiles can be uniquely linked
to public voting records. There are 572 panel members, who are categorized by
political affiliation: extreme left, left, lean left, center, lean right,
right, or extreme right. In the original data set, the categories lean left
and left were combined to form the category left, and the same was done for
the right side; the groups are not merged together in this paper. We assign a
belief value $B_{j}$ to each panelist equal to the average political
affiliation of reader $j$’s assigned belief group, which can be found in the
last column of Table 3.

In the raw data set, the panel members are heterogeneous in their social media
behaviors, with some panelists categorized as supersharers (sharing more than
922 political articles), superconsumers (having more than 45,128 exposures to
political articles), bots (nonhuman automated users responsible for mass
distribution of content), and apolitical (having fewer than 100 exposures to
political articles). As in Grinberg et al., to mitigate against outliers, we
filter out these users, leaving a more homogeneous population of panel members
for our analysis. As shown in Appendix A, the overall sharing rate of the
excluded panelists (governed by parameter $f$ in Equation (1)) is
statistically different from that of the remaining panelists. However, the
model parameter $k$ governing the sharing probability curve’s overall shape is
not statistically different between the excluded and remaining panelists.
Therefore, to simplify our analysis we restrict our data to panel members of
an average activity rate.

The data from Grinberg et al. also holds a collection of tweets containing
URLs and the political news sources of the URLs. The data has 3,141,106 tweets
that the panel members were potentially exposed to, meaning the tweet was
shared by an account the user follows. The URLs from these tweets are given
unique identifiers making the actual link anonymous. Thus, it is impossible to
attribute $b_{i}$ and $t_{i}$ values to individual URLs shared by the panel
member. Instead, we use the recorded domain names of the news website from
which the URL originates ([14]). We make the assumption that every article on
a particular news website has the same bias and truthfulness value as the
entire domain. We can make this assumption without losing much detail, as many
news sources have an editorial bias and targeted audiences which influence
their published articles ([6, 14]). As with user political beliefs, we
categorize the article bias as extreme left (i.e., very liberal, having values
of $b_{i}$ close to $-1$), left, lean left, center, lean right, right, or
extreme right (i.e., very conservative, having values of $b_{i}$ close to
$+1$).

To our knowledge there is currently no established research that calculates a
numeric value for the content-based semantic bias for a wide range of domains.
Therefore, we have to use other ways of estimating bias of an article. The
Grinberg et al. data includes alignment scores, the weighted average of
panelists exposed to a website that are registered with either the Democratic
or Republican party, for only 245 distinct websites. Instead, we estimate
article bias using MBM, a database that has over 20,000 news websites ([34]).
Ribeiro et al. develop MBM as a novel scalable way of calculating domain bias
that uses audience demographics ([34]). Ribeiro et al. find correlation
coefficients between their bias estimates and those of Budak et al. who use
machine learning and crowdsourcing techniques to determine the bias of
articles from fifteen domains ([6]). The strong correlation makes MBM’s bias
estimates a sufficient proxy for article bias, $b_{i}$.

In order to estimate a numerical truthfulness score for each URL, we expand on
color codes provided by Grinberg et al. that assess the overall truthfulness
of the domain ([14]). Grinberg et al. provide justifications for each domain’s
assigned coding which we use to develop a numerical conversion for each color.
Domains that are coded as black are given a truthfulness value of zero. All
other domains are assigned a value of $t_{i}$ from a range of values. These
conversions are shown in Table 1. Note that a truthfulness rating above green
refers to domains with rigorous editorial processes, such as academic
journals. Since the data set did not have such domains, we assign no domain a
$t_{i}$ in the interval $[0.8,1]$. By assigning truthfulness ratings to each
justification from Grinberg et al., we are able to average the domain’s
justification ratings to assign a $t_{i}$ value to each URL. Lastly, for the
purpose of the panel plot given in Figure 4, we redistribute the $t_{i}$
values into the truthfulness categories of very low, low, mixed, high, and
very high.

Truthfulness Color |  Grinberg et al. Description ([14]) |  Numeric Range |  Figure 4 Category
---|---|---|---
Black |  “Sites that published almost exclusively fabricated stories” |  $0.0-0.1$ |  Very Low
Red |  Sites that “spread falsehoods that clearly reflected a flawed editorial process” |  $0.1-0.2$ |  Low
Orange |  Sites “where annotators were less certain that the falsehoods stemmed from a systematically flawed process” |  $0.2-0.3$ |  Low
Yellow |  “Mild or rare inaccuracies” |  $0.3-0.6$ |  Mixed ($0.3-0.5$) High ($0.5-0.6$)
Green |  “Factual and sourced reporting” |  $0.6-0.8$ |  High ($0.6-0.7$) Very High ($0.7-0.8$)
Table 1: Conversion of colors provided in Grinberg et al. ([14]) into a
numerical range compatible with the truthfulness scale of the model. To evenly
distribute the data, the numerical scale was further categorized into
truthfulness bins for the purposes of Figure 4

Lastly, we estimate the probability of propagation by dividing the number of
people in each political affiliation group who shared URLs from a given
website by the number of people in that group who were exposed to URLs from
that website. The data from Grinberg et al. categorizes tweets as being a
retweet, quote, or original posting. On Twitter, retweets and quotes function
as shares. When a URL is shared as an original post on Twitter, there is no
record of the level of exposure the user had to the news source. So, in
calculating the probability, we consider only retweets and quotes. The panel
members shared 1,566 URL links from thirty-seven distinct websites.

###  4.2 Validating Model Assumptions

The data set, as described above, is used to validate the assumptions built
into the model. Figure 4 plots the propagation rate as a function of reader
belief, across combinations of article truthfulness and bias. We use these
plots as well as statistical comparisons to assert the validity of our
assumptions outlined in Section 2.

The first assumption is that the probability of sharing increases as the bias
of an article becomes more aligned with reader belief. This can be seen in
Figure 4 as panels in the top two rows (corresponding to liberal article bias)
show a decreasing trend in sharing rate as reader belief increases (becomes
more conservative), and panels in the bottom three rows (corresponding to
conservative article bias) show an increasing trend as reader belief becomes
more conservative. A statistical analysis of the slope coefficient describing
sharing rate as a function of $|b_{i}-B_{j}|$ reveals that this slope is
statistically negative for both liberal ($p$-value: 0.00002) and conservative
readers ($p$-value: 0.00001) further supporting this assumption. Moreover,
readers are less likely to share an article having an opposing political bias.
The sharing rate of liberals sharing liberal news is statistically higher than
the sharing rate of liberals sharing conservative news ($p$-value: 0.00054),
and likewise for the respective comparison applied to conservatives
($p$-value: 0.00856).

The model’s second assumption has two components, the first being that higher
truthfulness increases the probability of sharing, and the second being that
the rate of increase with increasing truthfulness itself decreases
(diminishing marginal increases in sharing rate). Figure 4 confirms that
articles in the “High” and “Very High” truthfulness columns have higher
probabilities of sharing than in the “Low” and “Very Low” truthfulness
columns. This is also supported by a statistical test of the slope coefficient
describing sharing rate as a function of $t_{i}$. This coefficient is
statistically positive for both liberal and conservative readers ($p$-values
0.00000 and 0.02535, respectively). However, the assumption of diminishing
marginal increases in sharing rate on propagation rate as truthfulness
increases cannot be validated here.

Figure 4: Probability of sharing an article depending on the political belief
of a reader ($x$-axis), truthfulness of an article (columns), and bias of an
article (rows) ([14, 34])

Having validated the model on real social media data, we can now begin an
empirical analysis of the model.

##  5 Empirical Analysis of Sharing Rate Over Populations of Users

In this section, we examine qualitative trends in model-estimated sharing
probabilities over a population of users as a function of the bias and
truthfulness of the article being propagated. The purpose in this analysis is
not to use the model for statistical prediction of propagation rates but to
understand general characteristics of the population and the content being
disseminated that result in larger or smaller propagation rates, as estimated
by the model. We interrogate the types of populations over which the
propagation of untruthful news will be highest, and for a given population,
the types of news that the model suggests will be widely shared. Understanding
which content is more likely to be shared, and the characteristics of
populations that are more susceptible to the sharing of untruthful or highly
biased news, provides insights into how users and platforms could prioritize
their interventions to mitigate the propagation of untruthful or highly biased
news. Using these qualitative results, we are able to make several
recommendations for social media users and platforms.

###  5.1 Estimating Model Parameters

To examine trends in model-estimated propagation rates of political news over
a population of users, we must first estimate reasonable values for the
scaling parameters $f_{l}$, $k_{l}$, $f_{r}$ and $k_{r}$. The values used in
the remainder of this section are given in the column labeled “Base Scenario”
in Table 2 and are obtained from a nonlinear least squares regression model
fitted to the observed values of $(b_{i},\ t_{i},\ B_{j})$ in our data set.
(The fit summary for this regression is provided in Appendix B.) To account
for uncertainty in these parameter estimates, the column labeled “Sensitivity
Analysis” provides low and high values of each parameter used to test the
sensitivity of our recommendations to the parameters. We find the
recommendations resulting from our model analysis to be robust to changes in
these parameter values; Appendix C contains the results of this sensitivity
analysis.

| Base | Sensitivity
---|---|---
Parameter | Scenario | Analysis
$f_{l}$ | 0.010 | [0.005, 0.014]
$k_{l}$ | 4.465 | [2.232, 6.697]
$f_{r}$ | 0.007 | [0.004, 0.011]
$k_{r}$ | 5.581 | [2.791, 8.372]

Table 2: Estimated parameter values used for model analysis. The Base Case
values are used in the analysis of Section 5.3. The sensitivity analysis is
found in Appendix C.

.

###  5.2 Empirical and Synthetic Population Distributions

Using these parameter estimates, we compute the model-estimated probability of
sharing an article for all possible combinations of $b_{i}$ and $t_{i}$, and
over a distribution of readers, using equation (2). The probability
distribution function $f(B_{j})$ is the fraction of the population in the
belief group having average political belief $B_{j}$. This process of
computing probabilities is done separately for articles having left- and
right-bias using their respective estimated parameters.

We use the distribution found in the data from Grinberg et al., henceforth
referred to as the empirical distribution, and five more constructed
distributions: two bimodal distributions (representing partisan and
hyperpartisan populations) and three unimodal distributions (representing
left-unimodal, centrist, and right-unimodal populations). Table 3 gives the
proportions of each distribution type having each political belief.

The five constructed distributions of readers represent different types of
social media populations. The two bimodal distributions reflect populations
with a higher concentration of partisan and hyperpartisan beliefs. The three
unimodal distributions represent populations concentrated on the left side of
the political spectrum, in the center of the spectrum, or on the right side of
the spectrum. Comparing the trends in the model-estimated propagation rate
over bimodal and unimodal distributions of readers’ political beliefs will
help us understand the effects of partisanship on the overall propagation rate
of different types of political content.

|  Empirical Distribution |  Partisan Distribution (Bimodal on the Left and Right) |  Hyperparti-san Distribution (Bimodal on the Extreme Left and Extreme Right) |  Left-Unimodal Distribution |  Centrist-Unimodal Distribution |  Right-Unimodal Distribution |  Belief Center Point
---|---|---|---|---|---|---|---
Extreme Left |  0.092 |  0.080 |  0.400 |  0.200 |  0.020 |  0.020 |  -0.857
Left |  0.230 |  0.400 |  0.080 |  0.400 |  0.080 |  0.040 |  -0.571
Lean Left |  0.225 |  0.020 |  0.020 |  0.200 |  0.200 |  0.060 |  -0.286
Center |  0.184 |  0 |  0 |  0.080 |  0.400 |  0.080 |  0
Lean Right |  0.131 |  0.020 |  0.020 |  0.060 |  0.200 |  0.200 |  0.286
Right |  0.091 |  0.400 |  0.080 |  0.040 |  0.080 |  0.400 |  0.571
Extreme Right |  0.046 |  0.080 |  0.400 |  0.020 |  0.020 |  0.200 |  0.857
Expectation |  -0.146 |  0 |  0 |  -0.400 |  0 |  0.400 |
Variance |  0.214 |  0.382 |  0.643 |  0.167 |  0.114 |  0.167 |

Table 3: The distribution of readers’ political belief for the empirical and
synthetic distributions

###  5.3 Results

In the sections that follow, we present population-wide model-estimated
sharing rates of political content as a function of the content’s bias and
truthfulness, for each of the six population distributions described above. We
focus our analysis on the propagation of right-biased articles. The analysis
for left-biased articles is symmetric and therefore not presented in this
paper. Figures 5 and 6 present the probability of article propagation as a
function of bias and truth, respectively, for right-biased articles over the
six distributions.

We examine the same three questions as were posed previously, now focused on
both content and population characteristics: 1) What general characteristics
of articles and populations achieve the highest propagation rate? 2) For
untruthful articles, what characteristics yield the highest propagation rate?
3) For highly biased articles, what characteristics yield the highest
propagation rate?

(a) The empirical distribution of readers’ political belief

(b) A partisan bimodal distribution of readers’ political belief

(c) A hyperpartisan bimodal distribution of readers’ political belief

(d) A right-unimodal distribution of readers’ political belief

(e) A centrist-unimodal distribution of readers’ political belief

(f) A left-unimodal distribution of readers’ political belief

Figure 5: The probability of sharing an article as a function of right
political bias ($x$-axis) and truthfulness (gradient)

(a) The empirical distribution of readers’ political belief

(b) A partisan bimodal distribution of readers’ political belief

(c) A hyperpartisan bimodal distribution of readers’ political belief

(d) A right-unimodal distribution of readers’ political belief  (e) A
centrist-unimodal distribution of readers’ political belief

(f) A left-unimodal distribution of readers’ political belief

Figure 6: The probability of sharing an article as a function of truthfulness
($x$-axis) and right political bias (gradient)

####  5.3.1 What general characteristics of articles and populations achieve
the highest propagation rate?

To determine the conditions under which the highest overall propagation rate
is achieved, we find the location of the maximum of each graph. As seen most
clearly in Figure 5, in all distributions except the right-unimodal
distribution (Fig. 5(d)), overall propagation rate is maximized for unbiased,
high-truth content. The centrist-unimodal population (Fig. 5(e)) exhibits the
highest sharing rate of this type of content. In the right-unimodal
population, the sharing rate is highest for mildly biased, moderately truthful
content. Thus we conclude that unbiased, truthful content can achieve high
propagation rates in most population types. This is consistent with
Recommendation P1. However, bias in the population of readers promotes the
propagation of biased and less truthful news.

We can compare the centrist population to the partisan and hyperpartisan
populations to understand how the variability of beliefs in a population
impacts propagation rate even as the average belief is held constant at $0$.
Contrasting Figures 5(b), (c), and (e), we see that partisan and hyperpartisan
populations share biased news at the highest rate of the distributions
studied, while centrist populations share unbiased news at the highest rate.
Importantly, as seen in Figures 6(b), (c), and (e), a centrist population
shares highly truthful news at a higher rate than partisan and hyperpartisan
populations. This leads to the conclusion that partisanism incentivizes the
propagation of biased, untruthful news, even when the mean belief of the
population is centered at zero.

Based on this analysis we can conclude that if users engage only with users of
similar political beliefs, then the probability of sharing news, fake or true,
increases. If users engage with users of opposing political beliefs, then the
probability of sharing decreases, and the decrease is more pronounced for fake
news 333This result can be supported mathematically; a proof is given in
Appendix D.. This suggests a trade-off between mitigating the spread of
untruthful news and increasing the flow of information. On one hand,
dissemination and flow of information is considered to be positive ([35]), and
based on the aforementioned result, it increases when users engage only with
users of similar political beliefs. On the other hand, spread of fake news is
harmful ([12]) and this as well increases when users engage only with users of
similar political beliefs. Thus in the extreme, in an echo chamber the level
of information flow might be high but it might be predominated by
misinformation. This result lends additional support for Recommendation P2
(Recommend friends and connections across the political spectrum.)

####  5.3.2 For untruthful articles, what characteristics yield the highest
propagation rate?

To determine the conditions under which the propagation of untruthful (low
$t_{i}$) content is highest, we can examine the dark regions of Figure 5
forming the bottom curve of each plot, and identify where the maximum occurs
along this curve.

We see that for the empirical (Fig. 5(a)), centrist-unimodal (Fig. 5(e)), and
left-unimodal (Fig. 5(f)) distributions, the maximum of the bottom curve
occurs at the left side of the graph: propagation of untruthful right-leaning
content is highest in these populations when the content is unbiased. Thus, a
unimodal population with centrist beliefs will prioritize truthful unbiased
news, but such a population is also susceptible to propagating untruthful
unbiased news. This suggests that fact-checking efforts should not focus only
on extremely biased content; the greatest propagation of untruthful content in
these populations will occur with unbiased content.

However, for the partisan bimodal (Fig. 5(b)) and right-unimodal (Fig. 5(d))
distributions, propagation of untruthful content is highest for moderately
biased content ($|b_{i}|\approx 0.5$). And for the hyperpartisan bimodal
population (Fig. 5(c)), propagation of untruthful content is highest for
strongly biased content ($|b_{i}|\approx 0.75$). The probability of sharing
any partisan news, true or fake, increases when the expected belief is
similarly partisan. Thus, a malicious agent trying to maximize the probability
of sharing an untruthful article in a partisan population would choose to
propagate biased content, with the optimal bias increasing with the
partisanism of the population.

Recommendation P3: Prioritize the fact-checking of moderately biased news,
based on the user population. In unimodal populations, untruthful content is
most likely to be shared if it exhibits little to no bias. However, in
partisan bimodal populations, untruthful content is most likely to be shared
if it exhibits moderate bias. Strongly biased untruthful content is highly
likely to be shared only in hyperpartisan populations.

####  5.3.3 For highly biased articles, what characteristics yield the highest
propagation rate?

The probability of sharing strongly-right-biased content in populations having
the empirical, centrist-unimodal and left-unimodal distributions is low
overall. Moreover, there is a general decreasing trend in propagation rate of
strongly-right-biased content as we look across the tails of the graphs from
right-unimodal (Fig. 5(d)), to centrist-unimodal (Fig. 5(e)), to left-unimodal
populations (Fig. 5(f)). This behavior is expected as it demonstrates an
assumption built into the model: people are less likely to share articles that
oppose their political beliefs. The hyperpartisan population (Fig. 5(c)) has a
higher propagation rate of extremely biased, low truthfulness news, and a
lower overall propagation rate of unbiased news of all truthfulness levels.
This further supports Recommendation P2 above (Recommend friends and
connections across the political spectrum) to reduce extreme bimodality in the
population. This also supports three additional recommendations:

Recommendation P4: Suggest related articles that present an issue from a
diversity of viewpoints. In addition to permitting a user to assess the verity
of the content when presented multiple ways, untruthful content will be less
likely to be shared.

Recommendation P5: Give users a report about the alignment of their feed. This
will help users understand if they are consuming a sufficiently varied diet of
political content.

Recommendation U2: Read articles with opposing beliefs to mitigate the effects
of confirmation bias. Once presented with content from across the political
spectrum and given a metric on the political alignment of their own media use,
users should consciously engage with content that runs counter to their
beliefs.

From this analysis it is clear that the choice of content propagated by a
malicious agent will be influenced by the population it is targeting. Even
when the average population belief is centered at zero, bimodal partisanism
incentivizes the propagation of biased, untruthful news, whereas a centrist
population is more likely to share truthful news. This suggests that
interventions to reduce bimodality in the social media user population could
reduce the propagation of biased and untruthful content. Moreover, fact-
checking mildly or moderately biased news could have more impact than fact-
checking strongly biased news, as users are more susceptible to sharing
untruthful news when it exhibits mild-to-moderate bias.

##  6 Discussion

In this section, we further develop our recommendations to mitigate fake news.
The spread of fake news is intricate and occurs at both the individual and
network level. Our work is novel in that it characterizes the effect of a news
article’s bias and truthfulness on its likelihood of being propagated by
individual users. While we do not consider how political content travels
through a network of social media users, we have characterized how a
population of individuals interacts with news once received. Our policy
recommendations come in two forms: recommendations for social media platforms
and individual users. These recommendations, with references to the sections
of this paper that support the recommendations, are summarized in Table 4.

Platform |  |  Paper
---|---|---
Recommendation |  Description |  Section(s)
P1 |  Populate users’ feeds with unbiased truthful news as sponsored articles. |  3.1, 3.2, 5.3.1
P2 |  Recommend friends and connections across the political spectrum. |  3.2, 5.3.1, 5.3.3
P3 |  Prioritize the fact-checking of moderately biased news, based on the user population. |  5.3.2
P4 |  Suggest related articles that present an issue from a diversity of viewpoints. |  5.3.3
P5 |  Give users a report about the alignment of their feed |  5.3.3
User |  |  Paper
Recommendation |  Description |  Section(s)
U1 |  Fact check news that aligns strongly with a user’s beliefs. |  3.1, 3.2
U2 |  Read articles with opposing beliefs to mitigate the effects of confirmation bias. |  5.3.3
Table 4: Platform and user recommendations derived from the analysis of this
paper, with references to sections of this paper’s analysis that generate
each.

Recommendations for platforms: We can further divide the recommendations for
platforms into two groups. The first group seeks to change/decrease the supply
of biased news. P1, P2, and P4 are in this group. With these interventions in
place, the platform minimizes the access of the users to highly biased news.
The second group operates through information. P3 and P5 are in this group.
With these platform informational interventions, the users are being provided
information intended to modify their behavior. In many domains informational
interventions have been shown to result in substantial changes in behavior of
individuals ([4]).

  * •

[P1]: Populate users’ feeds with unbiased truthful news as sponsored articles.
An unbiased truthful article maximizes the probability of sharing on social
media for all distributions except the non-centrist-unimodal and hyperpartisan
bimodal distributions. However, even in these distributions, the optimal
article is only moderately biased with a relatively high truthfulness. An
unbiased and highly truthful article will attain a high propagation rate among
many populations of readers. By sponsoring unbiased, truthful content, users
will have more opportunities to engage with, and potentially share, legitimate
content.

  * •

[P2]: Recommend friends and connections across the political spectrum.
From the analysis, bimodal and centrist-unimodal distributions decrease the
overall probability of sharing content as compared to the non-centrist-
unimodal distributions, particularly for untruthful news. This result suggests
that a population that has readers with opposing beliefs or moderate beliefs
will exhibit a lower probability of sharing untruthful news. When suggesting
new contacts to a user (e.g. a friend of a friend), platforms should suggest
contacts that are across the political spectrum, as this will give users a
chance to expose themselves to different political ideologies. This
recommendation is counter to the current practice of many social media
platforms to rank recommended connections based on similarity with the user.

  * •

[P3]: Prioritize the fact-checking of moderately biased news, based on the
user population.
In populations having a centrist unimodal or partisan bimodal belief
distribution, a malicious agent wishing to maximize the probability of sharing
untruthful news will choose moderately biased, rather than extremely biased,
content. Algorithms should fact-check articles of mild or moderate bias, as
these are the articles, when untruthful, readers are more susceptible to. Only
in hyperpartisan bimodal populations do untruthful articles exhibiting strong
bias achieve maximum propagation.

It is worth noting that this suggestion aligns with research on persuasion.
According to [22], in environments with one-sided flow of information,
significant persuasive effects would be expected. However, it is expected that
the level of persuasion decreases in environments with users across the
political spectrum and more specifically among users with opposing beliefs. In
such environments, an extremely biased article is unlikely to influence the
user’s views, because its contents are improbable under the user’s prior,
whereas mildly biased news is significantly more likely to persuade the user.
In other words, mildly biased news might update user’s belief, frictionlessly
([4]). With this recommendation, our model contributes to the studies which
analyze the evolution of beliefs/opinions in social media (see [31] among
all).

  * •

[P4]: Suggest related articles that present an issue from a diversity of
viewpoints.
When a user accesses an article on their social media feed, the platform can
recommend related articles from multiple news sources, ideally from a variety
of political alignments. This allows the user to assess whether multiple news
sources report similar statements as the initial article they have read, and
directly assess the bias and truthfulness of the statements. It is noteworthy
to mention that the current algorithm of the majority of the social media
platforms including Facebook, Twitter, and Instagram is based on the objective
of showing users the content that is relevant to them ([1]). Our suggestion
runs counter to this practice, and represents a tradeoff a social media
platform might have to make between profitability and social good.

  * •

[P5]: Give users a report about the alignment of their feed.
To reduce the polarity of a population, platforms can provide each user with a
score measuring the alignment of their feed. This will help users to be more
cognizant of whether they expose themselves to different types of news and
allow them to broaden their networks and content beyond those that only work
to confirm their beliefs.

Recommendations for users:

  * •

[U1]: Fact check news that aligns strongly with a user’s beliefs by using a
third party fact-checking website for individual facts within the article.
Users are more susceptible to untruthful news that aligns with their beliefs.
In response, users should be more proactive with the news that they read, and
fact-check articles of this nature. If the platform does not suggest related
content, as suggested in Recommendation P4 above, users can also seek multiple
news sources to corroborate the story.

  * •

[U2]: Read articles with opposing beliefs to mitigate the effects of
confirmation bias.
When users read articles that oppose their beliefs the probability of sharing
an untruthful news article is relatively low, and they prioritize sharing
truthful news.

Our work supports the conclusion that both unimodal skew or bimodal
partisanism in the reader population contributes to the spread of biased,
untruthful news. Some platforms already have ways to mitigate the spread of
fake news. Facebook has fact-checkers that review content and rate whether it
is false or true ([10]). Twitter has recently implemented a feature that flags
tweets that could be misconstruing information or be misleading ([36]). These
are important steps to mitigate spread of fake news, and our model provides a
mechanism for prioritizing this fact-checking effort. Additionally, while
fact-checking content that has already been shared is a retroactive endeavor,
several of our recommendations above serve to more proactively prevent the
sharing of fake news.

##  7 Limitations and Future Directions

The model developed in this paper is a novel way of characterizing the spread
of political content on social media. Unlike previous models, it disentangles
the effects of article truthfulness and bias on the sharing rate. Whereas
previous work treats bias and truthfulness as categorical or binary values,
our model allows for more granularity as the parameters are continuous and are
permitted to vary individually. This aspect of our model enables us to examine
each parameter’s respective relationship to the spread of fake news in more
detail.

The model and similar work are limited by the lack of available data sources
for measuring bias and truthfulness of news articles. Currently, work in the
literature on calculating content-based semantic bias is not scalable, which
limits the size of the data set used in this paper. Although alignment has a
strong correlation with bias, future papers could continue the process of
finding credible and scalable ways to calculate content-based semantic bias.

Future work with the model provided in this paper could increase its
dimensionality. Currently, the model describes the probability an individual
user will propagate a certain article based on three parameters: bias of the
article, truthfulness of the article and political belief of the reader. The
model is based on accurate assumptions, as validated with data by Grinberg et
al. and MBM ([34]); however, political news propagation is based on more than
just these three variables. For example, the popularity of an article has a
direct correlation to the probability that the article will be shared ([32]).
Additionally, the model makes an assumption that a reader’s belief stays
static even as they are consuming news. The model also does not differentiate
between potentially different motivation behind sharing (i.e. retweeting,
liking, or commenting). For example, a user can share a tweet in order to
express their opinion, support a person/institution, or persuade others. The
act of sharing, regardless of its motive, is treated the same in the model.

Lastly, the current model does not address how fake news spreads in a network,
nor how sharing behavior might change in response to changes in content. In
reality, fake news is largely caused and spread by social bots among a network
of users. Future research could use our model to characterize the propagation
of fake news over a social network that includes social bots. The network
effects also govern the types of content an individual is likely to be exposed
to, whereas our analysis has examined the conditional sharing rate assuming
that an exposure to certain content has occurred. Moreover, the
recommendations we make consider short-term impacts of interventions. If the
bias distribution of articles to which a reader is exposed changes as a result
of interventions, it is possible that sharing behavior will change over time
in response. Such evolution is not captured by our model; future research
could explore this further.

##  8 Conclusion

This paper provides a probabilistic model that describes the likelihood of an
individual social media user sharing an online news article based on the
article’s bias and truthfulness, as well as their own political belief. The
model is validated using data from [14] and [34]. We are able to examine the
immediate potential impact of interventions to reduce the likelihood of
untruthful news being shared, by modeling the decision of an individual user
to share (or not) political content. Future work would incorporate this model
into a network framework. Doing so would permit us to examine the time
evolution of intervention effects.

We characterize a malicious agent as someone who wants to spread untruthful
and/or highly biased news, and we use the model to determine the agent’s
choice of bias and truthfulness that maximizes the probability of propagation.
Understanding the content and population characteristics that a malicious
agent would target permits us to prioritize among intervention strategies,
such as which content to fact-check. Additionally, we examine model-estimated
propagation rates in six different populations to characterize the conditions
under which “fake” news is most likely to propagate.

Across all population types studied, our model suggests that readers generally
prefer to share highly truthful news. Thus social media platforms should
promote unbiased, truthful news. Since users are more susceptible to sharing
untruthful content when it is mildly or moderately biased, platforms should
prioritize allocating fact-checking resources to this type of content, rather
than extremely biased content. Social media platforms should suggest content
and contacts that reflect differing political opinions, so that users can
avoid politically homogeneous clusters and extricate themselves from
populations that only confirm their beliefs. Additionally, platforms should
provide users with information about the political alignment of their feed to
encourage users to seek out articles with opposing beliefs from their own.
Individuals should also attempt to fact check news that aligns strongly with
their beliefs.

Although these recommendations are relatively straightforward to implement,
two in particular run counter to the business model of most social media
platforms: P2 (Recommend friends and connections across the political
spectrum) and P4 (Suggest related articles that present an issue from a
diversity of viewpoints). Platforms such as Facebook, Twitter, and Instagram
make content and contact recommendations based on similarity with the user; in
doing so, they provide content that is perceived as being relevant to the
user, which increases the user’s enjoyment and use of the platform. However,
the recommendation that social media platforms should suggest content and
contacts that oppose a user’s belief presents a tradeoff that a platform must
make in order to mitigate fake news spread.

In summary, our model characterizes an individual user’s decision of whether
or not to share political news on social media. Using this model, we present
policy recommendations for social media platforms and users. Mitigating the
spread of fake news has become increasingly urgent worldwide, as we’ve seen in
recent elections and the global Covid-19 pandemic. Our model provides a useful
tool for identifying realistic actions that platforms and users can take to
mitigate the spread of fake news.

##  Appendix A Comparison of Model Parameters between Typical and Atypical
Users

To determine the influence of heterogeneous sharing rates, we conduct a
statistical test on the fitting parameters of panel members with average
sharing rates and extreme sharing rates (e.g. supersharers and superconsumers,
as described in Section 4.1). We use indicator variables to differentiate
between panel members with average and extreme rates of sharing, as defined by
[14]. Then we fit a modified version of the model using non-linear least
squares to understand if atypical users exhibit statistically different
sharing rates. We fit equation (3) in our analysis, where $f$ and $k$ are our
fitting parameters for panel members with an average sharing rate and $f_{e}$
and $k_{e}$ are the effects on those fitting parameters of our panel members
with extreme sharing rates. We use the indicator variable $I_{e}$ to flag when
a panelist has an extreme sharing rate.

$p(b_{i},t_{i},B_{j})=\frac{f+f_{e}*I_{e}}{1+e^{(-(k+k_{e}*I_{e})*(t_{i}-(b_{i}-B_{j})^{2}))}}$
(3)

From our non-linear least squares regression, which can be found in Table 5,
we see that $f_{e}$ for both left-political and right-political readers is
statistically different from zero, meaning that the extreme users exhibit a
different sharing rate from typical users, as expected. Parameter $k_{e}$ is
not statistically different from zero. Since $k$ governs the shape of the
probability curve we infer the general shape of the curve is unaffected by the
type of user. Therefore, we restrict our analysis to panel members having an
average activity rate.

| Left Reader Belief | Right Reader Belief
---|---|---
| $f_{e}$ | $k_{e}$ | $f_{e}$ | $k_{e}$
Estimate | 0.152 | -1.129 | 0.253 | -3.395
$p$ value | 1.32e-15 *** | 0.958 | 2e-16 *** | 0.945
Table 5: The fitted parameters found using non-linear least squares regression
for panel members with extreme sharing rates; Significance value: 0 ‘***’

##  Appendix B Nonlinear Least Squares Fit

To find estimates of the fitting parameters for our model we use non-linear
least squares regression. Table 6 includes these estimates, their $p$-values,
standard errors, and the overall residual standard errors. These values are
used in our analysis of population distributions as well as within our
analysis of typical and atypical social media users.

| $f_{l}$ | $k_{l}$ | $f_{r}$ | $k_{r}$
---|---|---|---|---
Estimate | 0.010 | 4.465 | 0.007 | 5.581
p value | 1.12e-12 *** | 0.036 * | 1.25e-9 *** | 0.11
Standard Error | 0.001 | 2.108 | 0.001 | 3.475
Residual Standard Error | 0.007 | 0.007
Table 6: The fitted parameters found using a non-linear least squares
regression; Significance value: 0 ‘***’, 0.001 ‘**’

##  Appendix C Sensitivity of Population Results to Estimated Model Parameters

We use our model to understand general trends of sharing political news on
social media. The fitting parameters found using non-linear least squares may
have some unknown bias. Therefore we conduct sensitivity analysis on the
fitting parameters to understand how the trends identified in Section 5.3
change. The levels we use for $f_{l}$, $k_{l}$, $f_{r}$, and $k_{r}$ can be
found in Table 2. For each synthetic distribution described in Table 3 we find
the probability of sharing using sixteen different combinations of $f_{l}$,
$k_{l}$, $f_{r}$, and $k_{r}$. As seen in the graphs below, even though the
actual probability of sharing does change, the conclusions we draw rarely
change. The only combinations that result in a (modest) change in the
conclusions are when we use the two combinations in which $f_{l}$ is at its
high level, $k_{l}$ and $f_{r}$ are at their low levels, and $k_{r}$ is at
either level (Figures 15 and 16). In these cases, the interpretation changes
as follows: To propagate untruthful right-biased news, an adversary should
propagate low-bias news to all populations except the right-centric
population, to which it should propagate low-to-moderately biased news. This
change is because for these parameter settings, the overall sharing rate of
left-biased readers has increased while that of right-biased readers has
decreased. Thus, the sharing rate in the population is more heavily affected
by the behavior of left-political readers when encountering right-biased news.

(a) The empirical distribution of readers’ political belief (b) A partisan
bimodal distribution of readers’ political belief  (c) A hyperpartisan bimodal
distribution of readers’ political belief  (d) A right-unimodal distribution
of readers’ political belief

(e) A centrist-unimodal distribution of readers’ political belief  (f) A left-
unimodal distribution of readers’ political belief

Figure 7: The probability of sharing an article as a function of right
political bias ($x$-axis) and truthfulness (gradient). Fitting parameters
combination: $f_{l}$ \- low, $k_{l}$ \- low, $f_{r}$ \- low, $k_{r}$ \- low

(a) The empirical distribution of readers’ political belief (b) A partisan
bimodal distribution of readers’ political belief  (c) A hyperpartisan bimodal
distribution of readers’ political belief  (d) A right-unimodal distribution
of readers’ political belief

(e) A centrist-unimodal distribution of readers’ political belief  (f) A left-
unimodal distribution of readers’ political belief

Figure 8: The probability of sharing an article as a function of right
political bias ($x$-axis) and truthfulness (gradient). Fitting parameters
combination: $f_{l}$ \- low, $k_{l}$ \- low, $f_{r}$ \- low, $k_{r}$ \- high

(a) The empirical distribution of readers’ political belief (b) A partisan
bimodal distribution of readers’ political belief  (c) A hyperpartisan bimodal
distribution of readers’ political belief  (d) A right-unimodal distribution
of readers’ political belief

(e) A centrist-unimodal distribution of readers’ political belief  (f) A left-
unimodal distribution of readers’ political belief

Figure 9: The probability of sharing an article as a function of right
political bias ($x$-axis) and truthfulness (gradient). Fitting parameters
combination: $f_{l}$ \- low, $k_{l}$ \- low, $f_{r}$ \- high, $k_{r}$ \- low

(a) The empirical distribution of readers’ political belief (b) A partisan
bimodal distribution of readers’ political belief  (c) A hyperpartisan bimodal
distribution of readers’ political belief  (d) A right-unimodal distribution
of readers’ political belief

(e) A centrist-unimodal distribution of readers’ political belief  (f) A left-
unimodal distribution of readers’ political belief

Figure 10: The probability of sharing an article as a function of right
political bias ($x$-axis) and truthfulness (gradient). Fitting parameters
combination: $f_{l}$ \- low, $k_{l}$ \- low, $f_{r}$ \- high, $k_{r}$ \- high

(a) The empirical distribution of readers’ political belief (b) A partisan
bimodal distribution of readers’ political belief  (c) A hyperpartisan bimodal
distribution of readers’ political belief  (d) A right-unimodal distribution
of readers’ political belief

(e) A centrist-unimodal distribution of readers’ political belief  (f) A left-
unimodal distribution of readers’ political belief

Figure 11: The probability of sharing an article as a function of right
political bias ($x$-axis) and truthfulness (gradient). Fitting parameters
combination: $f_{l}$ \- low, $k_{l}$ \- high, $f_{r}$ \- low, $k_{r}$ \- low

(a) The empirical distribution of readers’ political belief (b) A partisan
bimodal distribution of readers’ political belief  (c) A hyperpartisan bimodal
distribution of readers’ political belief  (d) A right-unimodal distribution
of readers’ political belief

(e) A centrist-unimodal distribution of readers’ political belief  (f) A left-
unimodal distribution of readers’ political belief

Figure 12: The probability of sharing an article as a function of right
political bias ($x$-axis) and truthfulness (gradient). Fitting parameters
combination: $f_{l}$ \- low, $k_{l}$ \- high, $f_{r}$ \- low, $k_{r}$ \- high

(a) The empirical distribution of readers’ political belief (b) A partisan
bimodal distribution of readers’ political belief  (c) A hyperpartisan bimodal
distribution of readers’ political belief  (d) A right-unimodal distribution
of readers’ political belief

(e) A centrist-unimodal distribution of readers’ political belief  (f) A left-
unimodal distribution of readers’ political belief

Figure 13: The probability of sharing an article as a function of right
political bias ($x$-axis) and truthfulness (gradient). Fitting parameters
combination: $f_{l}$ \- low, $k_{l}$ \- high, $f_{r}$ \- high, $k_{r}$ \- low

(a) The empirical distribution of readers’ political belief (b) A partisan
bimodal distribution of readers’ political belief  (c) A hyperpartisan bimodal
distribution of readers’ political belief  (d) A right-unimodal distribution
of readers’ political belief

(e) A centrist-unimodal distribution of readers’ political belief  (f) A left-
unimodal distribution of readers’ political belief

Figure 14: The probability of sharing an article as a function of right
political bias ($x$-axis) and truthfulness (gradient). Fitting parameters
combination: $f_{l}$ \- low, $k_{l}$ \- high, $f_{r}$ \- high, $k_{r}$ \- high

(a) The empirical distribution of readers’ political belief (b) A partisan
bimodal distribution of readers’ political belief  (c) A hyperpartisan bimodal
distribution of readers’ political belief  (d) A right-unimodal distribution
of readers’ political belief

(e) A centrist-unimodal distribution of readers’ political belief  (f) A left-
unimodal distribution of readers’ political belief

Figure 15: The probability of sharing an article as a function of right
political bias ($x$-axis) and truthfulness (gradient). Fitting parameters
combination: $f_{l}$ \- high, $k_{l}$ \- low, $f_{r}$ \- low, $k_{r}$ \- low

(a) The empirical distribution of readers’ political belief (b) A partisan
bimodal distribution of readers’ political belief  (c) A hyperpartisan bimodal
distribution of readers’ political belief  (d) A right-unimodal distribution
of readers’ political belief

(e) A centrist-unimodal distribution of readers’ political belief  (f) A left-
unimodal distribution of readers’ political belief

Figure 16: The probability of sharing an article as a function of right
political bias ($x$-axis) and truthfulness (gradient). Fitting parameters
combination: $f_{l}$ \- high, $k_{l}$ \- low, $f_{r}$ \- low, $k_{r}$ \- high

(a) The empirical distribution of readers’ political belief (b) A partisan
bimodal distribution of readers’ political belief  (c) A hyperpartisan bimodal
distribution of readers’ political belief  (d) A right-unimodal distribution
of readers’ political belief

(e) A centrist-unimodal distribution of readers’ political belief  (f) A left-
unimodal distribution of readers’ political belief

Figure 17: The probability of sharing an article as a function of right
political bias ($x$-axis) and truthfulness (gradient). Fitting parameters
combination: $f_{l}$ \- high, $k_{l}$ \- low, $f_{r}$ \- high, $k_{r}$ \- low

(a) The empirical distribution of readers’ political belief (b) A partisan
bimodal distribution of readers’ political belief  (c) A hyperpartisan bimodal
distribution of readers’ political belief  (d) A right-unimodal distribution
of readers’ political belief

(e) A centrist-unimodal distribution of readers’ political belief  (f) A left-
unimodal distribution of readers’ political belief

Figure 18: The probability of sharing an article as a function of right
political bias ($x$-axis) and truthfulness (gradient). Fitting parameters
combination: $f_{l}$ \- high, $k_{l}$ \- low, $f_{r}$ \- high, $k_{r}$ \- high

(a) The empirical distribution of readers’ political belief (b) A partisan
bimodal distribution of readers’ political belief  (c) A hyperpartisan bimodal
distribution of readers’ political belief  (d) A right-unimodal distribution
of readers’ political belief

(e) A centrist-unimodal distribution of readers’ political belief  (f) A left-
unimodal distribution of readers’ political belief

Figure 19: The probability of sharing an article as a function of right
political bias ($x$-axis) and truthfulness (gradient). Fitting parameters
combination: $f_{l}$ \- high, $k_{l}$ \- high, $f_{r}$ \- low, $k_{r}$ \- low

(a) The empirical distribution of readers’ political belief (b) A partisan
bimodal distribution of readers’ political belief  (c) A hyperpartisan bimodal
distribution of readers’ political belief  (d) A right-unimodal distribution
of readers’ political belief

(e) A centrist-unimodal distribution of readers’ political belief  (f) A left-
unimodal distribution of readers’ political belief

Figure 20: The probability of sharing an article as a function of right
political bias ($x$-axis) and truthfulness (gradient). Fitting parameters
combination: $f_{l}$ \- high, $k_{l}$ \- high, $f_{r}$ \- low, $k_{r}$ \- high

(a) The empirical distribution of readers’ political belief (b) A partisan
bimodal distribution of readers’ political belief  (c) A hyperpartisan bimodal
distribution of readers’ political belief  (d) A right-unimodal distribution
of readers’ political belief

(e) A centrist-unimodal distribution of readers’ political belief  (f) A left-
unimodal distribution of readers’ political belief

Figure 21: The probability of sharing an article as a function of right
political bias ($x$-axis) and truthfulness (gradient). Fitting parameters
combination: $f_{l}$ \- high, $k_{l}$ \- high, $f_{r}$ \- high, $k_{r}$ \- low

(a) The empirical distribution of readers’ political belief (b) A partisan
bimodal distribution of readers’ political belief  (c) A hyperpartisan bimodal
distribution of readers’ political belief  (d) A right-unimodal distribution
of readers’ political belief

(e) A centrist-unimodal distribution of readers’ political belief  (f) A left-
unimodal distribution of readers’ political belief

Figure 22: The probability of sharing an article as a function of right
political bias ($x$-axis) and truthfulness (gradient). Fitting parameters
combination: $f_{l}$ \- high, $k_{l}$ \- high, $f_{r}$ \- high, $k_{r}$ \-
high

##  Appendix D Proof that Population Diversity Reduces Sharing Rates

In Section 6 we conclude from our empirical analysis that if “users engage
only with users of similar political beliefs, then the probability of sharing
news, fake or true, increases. If users engage with users of opposing
political beliefs, then the probability of sharing decreases, and the decrease
is more pronounced for fake news.” We can support this assertion
mathematically with a toy example involving two, simple, hypothetical
populations.

Consider population $U$ (for unimodal) wherein $100\%$ of the population has
political belief $B_{j}$, and population $P$ (for partisan) wherein proportion
$q$ of the population has political belief $B_{j}$ and the other proportion
$1-q$ has political belief $-B_{j}$. Consider an article having truthfulness
$t_{i}$ and political bias $b_{i}$ such that $sign(b_{i})=sign(B_{j})$. Assume
that the probability function parameters are equal on both sides of the
political spectrum: $f_{l}=f_{r}$ and $k_{l}=k_{r}$.

We use equation (2) to compare the probability of sharing this article in
population $U$ to that in population $P$.

$p^{U}(b_{i},t_{i})=p(b_{i},t_{i},B_{j})=qp(b_{i},t_{i},B_{j})+(1-q)p(b_{i},t_{i},B_{j}).$
$p^{P}(b_{i},t_{i})=qp(b_{i},t_{i},B_{j})+(1-q)p(b_{i},t_{i},-B_{j}).$

Thus to compare $p^{U}(b_{i},t_{i})$ and $p^{P}(b_{i},t_{i})$, we need only
compare $p(b_{i},t_{i},B_{j})$ and $p(b_{i},t_{i},-B_{j})$.

By the first assumption of our probability function, the probability of
sharing an article decreases as the difference between the article’s bias and
the reader’s belief increases.

$|b_{i}-B_{j}|\leq|b_{i}|+|-B_{j}|=|b_{i}-(-B_{j})|,\text{ when $b_{i}$ and
$B_{j}$ have the same sign.}$

Therefore $p(b_{i},t_{i},B_{j})\geq p(b_{i},t_{i},-B_{j})$, demonstrating that
$p^{U}(b_{i},t_{i})\geq p^{P}(b_{i},t_{i})$. Propagation is higher, for any
truthfulness value, in the unimodal population than in the partisan
population. Moreover, because the deviation between $b_{i}$ and $B_{j}$ is
attenuated multiplicatively by the term $e^{-kt_{i}}$ in equation (1), the
decrease in sharing probability between the partisan and unimodal populations
is greatest for low-truth content.

Acknowledgement. This material is based upon work supported by the National
Science Foundation under Grant No. DMS-1757952. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the
authors and do not necessarily reflect the views of the National Science
Foundation. The authors also acknowledge financial support from Harvey Mudd
College and the California State University at Long Beach. The authors would
like to thank Michael Gao and Steven Witkin for their contributions to earlier
versions of this work. Additionally, the authors thank David Lazer and Nir
Grinberg for providing access to the data set from ([14]). Finally, the
authors thank three anonymous reviewers for detailed feedback that greatly
improved this manuscript.

## References

  * [1] AJ Agarwal  “What Do Social Media Algorithms Mean For You?” Accessed on 04 May 2021.  In _Forbes_ , 2016  URL: https://www.forbes.com/sites/insights-capitalone/2021/03/10/growth-now-4-opportunities-for-cxos-to-build-their-companies-in-the-coming-year/?sh=18d5f8d05558
  * [2] Hunt Allcott and Matthew Gentzkow  “Social Media and Fake News in the 2016 Election”  In _Journal of Economic Perspectives_ 31.2, 2017, pp. 211–236  DOI: 10.3386/w23089
  * [3] Eytan Bakshy, Solomon Messing and Lada A. Adamic  “Exposure to ideologically diverse news and opinion on Facebook”  In _Science_ 348.6239 American Association for the Advancement of Science, 2015, pp. 1130–1132  DOI: 10.1126/science.aaa1160
  * [4] Eric Bettinger, Nina Cunha, Guilherme Lichand and Ricardo Madeira  “Are the effects of informational interventions driven by salience?”, 2020  URL: https://ideas.repec.org/p/zur/econwp/350.html
  * [5] Heather Z. Brooks and Mason A. Porter  “A Model for the Influence of Media on the Ideology of Content in Online Social Networks”  In _ArXiv_ abs/1904.09238, 2019
  * [6] Ceren Budak, Sharad Goel and Justin M. Rao  “Fair and Balanced? Quantifying Media Bias Through Crowdsourced Content Analysis”  In _Public Opinion Quarterly_ , 2016, pp. 250–271  DOI: 10.1093/poq/nfw007
  * [7] Z. Chu, S. Gianvecchio, H. Wang and S. Jajodia  “Detecting Automation of Twitter Accounts: Are You a Human, Bot, or Cyborg?”  In _IEEE Transactions on Dependable and Secure Computing_ 9.6, 2012, pp. 811–824  DOI: 10.1109/TDSC.2012.75
  * [8] Joseph Czyzyk, Michael P. Mesnier and Jorge J. Moré  “The NEOS Server”  In _IEEE Journal on Computational Science and Engineering_ 5.3, 1998, pp. 68 —–75
  * [9] Elizabeth D. Dolan  “The NEOS Server 4.0 Administrative Guide”, 2001
  * [10] Facebook  “What Publishers Should Know About Third Party Fact Checking on Facebook” Accessed on 29 July 2020.  In _Facebook Business Help Center_ , 2020  URL: https://www.facebook.com/business/help/182222309230722
  * [11] Mehrdad Farajtabar et al.  “Fake News Mitigation via Point Process Based Intervention”  In _Proceedings of the 34th International Conference on Machine Learning_ 70, Proceedings of Machine Learning Research  International Convention Centre, Sydney, Australia: PMLR, 2017, pp. 1097–1106  URL: http://proceedings.mlr.press/v70/farajtabar17a.html
  * [12] Anastasia Giachanou and Paolo Rosso  “The Battle Against Online Harmful Information: The Cases of Fake News and Hate Speech”  New York, NY, USA: Association for Computing Machinery, 2020  DOI: 10.1145/3340531.3412169
  * [13] Jeffrey Gottfried and Elisa Shearer  “News Use Across Social Media Platforms 2016”  In _Pew Research Center’s Journalism Project_ Pew Research Center, 2017  URL: https://www.journalism.org/2016/05/26/news-use-across-social-media-platforms-2016/
  * [14] Nir Grinberg et al.  “Fake news on Twitter during the 2016 U.S. presidential election”  In _Science_ 363.6425, 2019, pp. 374  DOI: 10.1126/science.aau2706
  * [15] William Gropp and Jorge J. Moré  “Optimization Environments and the NEOS Server”  In _Approximation Theory and Optimization_ Cambridge University Press, 1997, pp. 167 –182
  * [16] Andrew Guess, Jonathan Nagler and Joshua Tucker  “Less than you think: Prevalence and predictors of fake news dissemination on Facebook”  In _Science Advances_ 5.1 American Association for the Advancement of Science, 2019  DOI: 10.1126/sciadv.aau4586
  * [17] Joe Hoft  “Per the CDC There Are Nearly Twice As Many Vaccine Related Deaths SO FAR in 2021 (1,755) Than All the Vaccine Deaths this Past Decade (994)” Accessed on 04 May 2021.  In _Gateway Pundit_ , 2021  URL: https://archive.is/mqXNN#selection-595.11-595.150
  * [18] Joe Hoft  “Stanford Study Results: Facemasks are Ineffective to Block Transmission of COVID-19 and Actually Can Cause Health Deterioration and Premature Death” Accessed on 04 May 2021.  In _Gateway Pundit_ , 2021  URL: https://archive.is/LmMTo#selection-603.0-603.147
  * [19] Isa Inuwa-Dutse, Mark Liptrott and Ioannis Korkontzelos  “Detection of spam-posting accounts on Twitter”  In _Neurocomputing_ 315, 2018, pp. 496 –511  DOI: https://doi.org/10.1016/j.neucom.2018.07.044
  * [20] Shanto Iyengar and Kyu S Hahn  “Red Media, Blue Media: Evidence of Ideological Selectivity in Media Use”  In _Journal of Communication_ 59.1, 2009, pp. 19–39  DOI: 10.1111/j.1460-2466.2008.01402.x
  * [21] S. Mo Jang et al.  “A computational approach for examining the roots and spreading patterns of fake news: Evolution tree analysis”  In _Computers in Human Behavior_ 84, 2018, pp. 103–113  DOI: 10.1016/j.chb.2018.02.032
  * [22] Morgen S. Johansen and Mark R. Joslyn  “Political Persuasion during Times of Crisis: The Effects of Education and News Media on Citizens’ Factual Information about Iraq”  In _Journalism & Mass Communication Quarterly_ 85.3, 2008, pp. 591–608  DOI: 10.1177/107769900808500307
  * [23] Youjung Jun, Rachel Meng and Gita Venkataramani Johar  “Perceived social presence reduces fact-checking”  In _Proceedings of the National Academy of Sciences_ 114.23 National Academy of Sciences, 2017, pp. 5976–5981  DOI: 10.1073/pnas.1700175114
  * [24] David M Lazer, Matthew A Baum and Yochai Benkler  “The Science of Fake News”  In _Science_ 359.6380, 2018, pp. 1094–1096  DOI: 10.1126/science.aao2998
  * [25] Kyumin Lee et al.  “Who Will Retweet This?: Automatically Identifying and Engaging Strangers on Twitter to Spread Information”  In _Proceedings of the 19th International Conference on Intelligent User Interfaces_ , IUI ’14  Haifa, Israel: ACM, 2014, pp. 247–256  DOI: 10.1145/2557500.2557502
  * [26] Yang Liu and Yi-Fang Wu  “Early Detection of Fake News on Social Media Through Propagation Path Classification with Recurrent and Convolutional Networks”  In _AAAI Conference on Artificial Intelligence_ , 2018  URL: https://www.aaai.org/ocs/index.php/AAAI/AAAI18/paper/view/16826/15707
  * [27] Alireza Louni and K. P. Subbalakshmi  “A two-stage algorithm to estimate the source of information diffusion in social media networks”  In _2014 IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS)_ , 2014  DOI: 10.1109/infcomw.2014.6849253
  * [28] Indu Manickam, Andrew S Lan, Gautam Dasarathy and Richard G Baraniuk  “IdeoTrace: A Framework for Ideology Tracing with a Case Study on the 2016 U.S. Presidential Election”, 2019  URL: arXiv:1905.08831v2
  * [29] Nicolas Guenon Mesnards and Tauhid Zaman  “Detecting Influence Campaigns in Social Networks Using the Ising Model”  In _CoRR_ abs/1805.10244, 2018  arXiv: http://arxiv.org/abs/1805.10244
  * [30] David Mikkelson  “Are Non-Respirator Masks Ineffective at Protecting Against COVID-19?” Accessed on 04 May 2021.  In _Snopes.com_ , 2020  URL: https://www.snopes.com/fact-check/cloth-masks-ineffective-covid19/
  * [31] Alessandro Nordio, Alberto Tarable, Carla-Fabiana Chiasserini and Emilio Leonardi  “Belief Dynamics in Social Networks: A Fluid-Based Analysis”  In _IEEE Transactions on Network Science and Engineering_ PP, 2017  DOI: 10.1109/TNSE.2017.2760016
  * [32] Yiangos Papanastasiou  “Fake News Propagation and Detection: A Sequential Model”  In _SSRN Electronic Journal_ , 2017  DOI: 10.2139/ssrn.3028354
  * [33] Francesco Pierri, Carlo Piccardi and Stefano Ceri  “Topology comparison of Twitter diffusion networks reliably reveals disinformation news”  In _arXiv preprint arXiv:1905.03043_ , 2019
  * [34] Filipe Ribeiro et al.  “Media Bias Monitor: Quantifying Biases of Social Media News Outlets at Large-Scale”  International AAAI Conference on WebSocial Media, 2018  URL: https://aaai.org/ocs/index.php/ICWSM/ICWSM18/paper/view/17878
  * [35] Hans Rosling  “Factfulness: ten reasons we’re wrong about the world - and why things are better than you think”  New York: Flatiron Books, 2018
  * [36] Yoel Roth and Nick Pickles  “Updating our Approach to Misleading Information” Accessed on 04 May 2021.  In _Twitter_ Twitter, 2020  URL: https://blog.twitter.com/en_us/topics/product/2020/updating-our-approach-to-misleading-information.html
  * [37] N. V. Sahinidis  “BARON 17.8.9: Global Optimization of Mixed-Integer Nonlinear Programs, User’s Manual”, 2017  URL: {http://www.minlp.com/downloads/docs/baron%20manual.pdf}
  * [38] Karishma Sharma et al.  “Combating Fake News: A Survey on Identification and Mitigation Techniques”  In _ACM Transactions on Intelligent Systems and Technology_ 10.3, 2019  URL: https://doi.org/10.1145/3305260
  * [39] Kai Shu, H. Russell Bernard and Huan Liu  “Studying Fake News via Network Analysis: Detection and Mitigation”  In _Lecture Notes in Social Networks Emerging Research Challenges and Opportunities in Computational Social Network Analysis and Mining_ , 2018, pp. 43–65  DOI: 10.1007/978-3-319-94105-9˙3
  * [40] M. Tawarmalani and N. V. Sahinidis  “A polyhedral branch-and-cut approach to global optimization”  In _Mathematical Programming_ 103, 2005, pp. 225–249
  * [41] Michela Del Vicario et al.  “The spreading of misinformation online”  In _Proceedings of the National Academy of Sciences_ 113.3, 2016, pp. 554–559  DOI: 10.1073/pnas.1517441113
  * [42] Soroush Vosoughi, Deb Roy and Sinan Aral  “The spread of true and false news online”  In _Science_ 359.6380, 2018, pp. 1146–1151  DOI: 10.1126/science.aap9559
  * [43] Yi Xiang and Miklos Sarvary  “News Consumption and Media Bias”  In _Marketing Science_ 26.5, 2007, pp. 611–628  DOI: 10.1287/mksc.1070.0279