# Conspiratorial beliefs observed through entropy principles

Nataša Golo and Serge Galam  Cevipof, Center for political research,
SciencesPo and CNRS, Paris, France

Abstract: We propose a novel approach framed in terms of information theory
and entropy to tackle the issue of conspiracy theories propagation. We start
with the report of an event (such as 9/11 terroristic attack) represented as a
series of individual strings of information denoted respectively by two-state
variable $E_{i}=\pm 1,i=1,…,N$. Assigning $E_{i}$ value to all strings, the
initial order parameter and entropy are determined. Conspiracy theorists
comment on the report, focusing repeatedly on several strings $E_{k}$ and
changing their meaning (from $-1$ to $+1$). The reading of the event is turned
fuzzy with an increased entropy value. Beyond some threshold value of entropy,
chosen by simplicity to its maximum value, meaning $N/2$ variables with
$E_{i}=1$, doubt prevails in the reading of the event and the chance is
created that an alternative theory might prevail. Therefore, the evolution of
the associated entropy is a way to measure the degree of penetration of a
conspiracy theory. Our general framework relies on online content made
voluntarily available by crowds of people, in response to some news or blog
articles published by official news agencies. We apply different aggregation
levels (comment, person, discussion thread) and discuss the associated
patterns of entropy change.

##  1 Introduction

While conspiracy theories have existed for centuries, the phenomenon stayed
confined to small groups of people due to the slow process of spreading based
mainly on word-of-mouth propagation. However, the recent use of internet by
conspiracists has altered the nature of the phenomenon, providing both the
increased speed of propagation and the dramatic extension of people exposed to
the rumor. Moreover, in the new context of global terrorism, the existence of
conspiracy theories disseminated in the population became an issue in the
ability of democratic societies to fight against terrorism. The propagation of
conspiratorial beliefs has been previously renowned in [1], [2] in terms of
minority opinion spreading. The subject of conspiracy formation is becoming of
strategic importance at different levels of the globalized society. Of special
interest is the shift in public opinion that the presence of conspiracy
theories can activate [3]. Therefore, we search for a robust and generic
technique to distinguish between opinions for and against such theories. There
are a number of polls and barometers which are questioning the public opinion
about conspiracy theories, others about the usage of violence in political and
social conflicts, and there are many other indicators of political attitudes.
However, it is difficult to find a way to combine these. We propose a novel
approach framed in terms of information theory and entropy to tackle the issue
of the propagation of conspiracy theories. We start with the report of an
terroristic event (such as the 9/11 terroristic attack in the USA) represented
as a series of individual strings of information denoted respectively by a
two-state variable $E_{i}=\pm 1,i=1,…,N$. An official report would be
characterized by value $E_{i}=-1$ assigned to all strings, resulting in the
order parameter $E=\overline{E_{i}}=-1$, i.e. a fully ordered state, with
minimum entropy. This would result in a clear and unambiguous reading and
understanding of the event. However, conspiracy theorists might seize this
initial presentation based on the series of information strings $E_{i}=-1$ to
question their content and changing their meaning (i.e. from $-1$ to $+1$). A
number of articles would then appear propagating different level of
conspiratorial beliefs and accompanied with the comments of the readers who
argue in favor or against the new content. The entropy of such textual
corpuses is of non-minimum value. Beyond some threshold value, which by
simplicity we choose equal to $E=0$, meaning $N/2$ variables with $E_{i}=1$,
i.e. probability $p_{i}=1/2$, a total doubt is prevailing in the reading of
the event. On this basis, knowing the evolution of the associated entropy
offers a way to measure the degree of penetration of a conspiracy theory. Our
general framework relies on online content made voluntarily available by
crowds of people, in response to some news or blog articles published by
official news agencies. We apply different aggregation levels (comment,
person, discussion thread) when analyzing the change of entropy of the entire
corpus. As opposed to studies where the interactions between agents are more
or less complex versions of contagion, the opinion dynamics uncovered by our
study displays a variety of interactions that differ by the number of
participants, their attitudes and the nature of their arguments. In particular
some of the arguments have a more information-like logical character while
others originate in, or address, subjective states and experiences. Thus the
modeling of the debate process and its outcome will have to transcend the
mechanical models of percolation time and yet include elements precise enough
to allow quantitative evaluations and predictions. In particular the
information theoretical aspects are only a part of the debate dynamics,
aspects which have to do more with the message’s language, positioning and
connotations turn out to be equally relevant. The data used for this pilot
study are introduced in Sections 2 and 3. The results have to be taken as
qualitative information used for the construction of the model, and not as its
quantitative confirmation. Once enough data is gathered, and the proposed
model is developed to the level of an application, we hope to obtain more
quantitative results. We are thus proposing the first steps for a formalism
that parallels the Shannon entropy for communication in order to encode the
amount of congeniality of the commentaries with conspiracy theories.

##  2 Background information about 9/11 conspiracy theories

After a terroristic attack, a society has to interpret it. Among other goals
of violent attacks in the context of political conflict, one is to affect a
broader audience beyond the physically targeted victims. People use various
cognitive resources to explain an unexpected and disturbing event like this.
They are searching for information and arguments that can explain the event,
while, at the same time, they seek for the explanation that will help them to
protect themselves as much as possible from further attacks. This might be
interpreted as teleological (consequentialist) ethics, which derives duty or
moral obligation from what is good or desirable as an end to be achieved. It
is opposed to deontological ethics, which holds that the basic standards for
an action’s being morally right are independent of the good or evil generated.
Such conditions open a path for the diffusion of conspiracy theories. Just
recently, after the terrorist attack on the French satirical weekly magazine
Charlie Hebdo and the Kosher supermarket in Paris in January 2015, with 17
people being killed, the results of the public opinion poll [4] have shown
that in the weeks after the attack a significant number of French citizens
held conspiratorial beliefs about it $(17\%)$. Following the instructions and
advice by Osama Bin Laden [5] in 2002/3, presently active terrorist groups
realized that the targeted attacks have a better chance to win support of
possible followers, while violence against innocents usually hurts public
support. Similarly, an inadequate official response could guide public opinion
to their advantage. Conspiracy theories on their own can have an additional
leveraging effect: in case they are successful, and the violence cannot be
clearly attached to the terrorist, the entire population of peaceful political
activists may decide to give an advantage to the terrorist supporters. The
conspiracy theories that we analyze in this paper concern the September 11
attack, when al-Qaeda terrorists killed 2,996 people and caused at least
$\$10$ billion in property and infrastructure damage to the United States. It
is rather well known that there was a vast number of conspiracy theories which
developed after this attack [6]. There is no consistent polling about the
popularity of this theory. However, we will analyze the available data from
official opinion polls, but we will also use opinion mining tools and extract
data from the internet about the latest events. An example of the dynamics of
the polled opinion about 9/11 conspiracy worldwide is given in Figure 1 [7].

Figure 1: The illustrative fitting lines (2nd and 3rd order polynomial) show
dynamics of opinion polls concerning the question “Who did the 9/11 terrorist
attacks?” in the US. In the first years after the attack, the majority was not
convinced that Al-Qaeda was responsible for the attack. This however has
changed and the confidence peaked in 2009; since then it started dropping
while at the same time more and more people declare that they do not know who
did it.

We should mention that the first advertiser of a 9/11 conspiracy theory in
France was Thierry Meyssan in his book [8]. The book was criticized by the
centre and left oriented publishers, such as Libération, Le Nouvel
Observateur, etc. However, the rumors hold similarly as in the US.
Unfortunately, in France, there are much fewer official polls about this
conspiracy [9], [10], [11], [12]. The WorldPublicOpinion.org poll [10],
conducted during the summer of 2008, revealed that about $63\%$ of the polled
French citizens think that for "Al Qaeda" was behind the 9/11 attack,$8\%$
think it was "the US government", $7\%$ said "Other" and $23\%$ said they “Do
Not Know”. However, the same poll, in which 17 nations were polled, found that
majorities believe al-Qaeda carried out the attacks in only 9 of the 17
countries. We, unfortunately, were not able to collect sufficient number of
polls to create valid statistics about the public opinion either in France or
in the rest of the world. However, even from the limited data, we can conclude
that the spreading of this conspiracy was a slow process which took a couple
of years before it became more or less stable. It requires frequent and
prolonged monitoring before it could be fully understood. Therefore we suggest
an alternative way to search for opinions about this particular conspiracy.

##  3 Analysis of the web-based comments about possible 9/11 conspiracy

The internet is a very convenient place for conspiracy theories. It provides
many so called “proofs” that conspiracy theories rely on, and which are used
to convince others of their validity. Therefore we opt to think that the
conspiracies and their followers are more likely to express themselves through
on-line tools then with conventional opinion polls.

###  3.1 Example No. 1, BBC news article

We analyze comments of the readers of the BBC site to an article that BBC
published on conspiracy theories that refers to some of the conspiracies about
September 11 [13]. There were 755 comments of the readers. The content of the
article and the content of all the comments are presented by the two semantic
clouds in Figure LABEL:fig:2a and LABEL:fig:2b respectively. The semantic
clouds were constructed using the software Iramuteq [14]. The program selects
active words (i.e. nouns, verbs, adjectives) and creates a graph based on the
words co-occurrences. Comparing the two clouds, one can see that the readers
frequently commented on two subjects: “collapse” (of the building next to the
Twin towers that were hit by the terrorists) and “person” (who might believe
in a conspiracy theory). These two clusters of words create two sides of a
conspiracy theory. The “person” cluster implies that the political interests
and our own ‘human nature’ are some of the possible incentives for the
conspiracy theory. The second cluster of words reflects on the conspiracy
theories listed in the BBC article and the discussions about the 9/11 ‘facts’
and whether they match with the reality.

(a) The main text.

(b) The comments.

Figure 2: (a) Semantic cloud showing the subjects that are discussed in the
BBC article: conspiracy, plane, collapse etc. (b) Semantic cloud showing the
subjects that are discussed in the comments on the BBC article: conspiracy,
person, collapse etc. The edges represent a selection the co-occurrences of
those words between comments (frequency > 10).

We further attempt to establish a connection between the comments that refer
one to the other. The longest thread of comments in our sample is shown in
Figure 3. We look further at the interactions in this thread of messages and
we would like to know whether these messages are supporting or opposing a
conspiracy theory. We need this in order to determine the possible interaction
patterns of a person who propagates a conspiracy theory.

Figure 3: Thread of comments that mention one to the other (establishing the
possibility that they share a common ideology) from the BBC article. Comments
in the same color were posted under the same pseudo name. The size of the
nodes reflects the rating (number of “likes”) by the readers (the larger node,
the better the rating). The numbers on the edges give the time when a comment
was posted. The shape gives their position: a circle for the ones supporting
conspiracy, diamonds for the ones opposing it and a triangle for some comments
that are vague or undecided.

The selected network of comments shown in Figure 3 starts with the comment No.
42: “In the 50s, the CIA did a lot more questionable things and had more power
than they do now, and they got ’outed’ for much of it. Nixon couldn’t even
keep a break-in to an office a secret, I’m supposed to believe a massive
coverup with thousands of conspirators and nobody’s said *anything*? Soldiers
were leaking secrets on Lady Gaga CD’s! The government isn’t made up of mind-
controlled zombies.”  The community defending conspiracy theories is giving
some reasonable and highly ranked/liked arguments. Such is, for example,
comment No. 130: “I have no agenda other than to find the truth. Not all CTs
as you are calling them are lunatics. Many people wouldn’t call themselves CTs
but are still not sure we have heard the whole truth. Building 7’s collapse is
very strange and unique and there do seem to be many unanswered questions
throughout. Stop attacking people asking questions. Being closed minded is not
a virtue.”.

Another selected network of comments (can be found in the Appendix), is the
star-shaped one, starting with the comment No. 436: “The one thing that made
me sit up and wonder, were the pictures of the Pentagon before it collapsed.
The hole (allegedly made by the plane) was just not big enough, and after all
these years I still haven’t seen a good enough explanation for that anomaly.”

These examples bring out the two possible main motifs to believe in the
conspiracy theory. The first theorist points out the political incentives and
the ‘human nature’ of the conspiracy. The second is more concerned about the
‘facts’ and whether they match with the expectations.

In order to understand in how far the commentaries section reflects the
opinion of an average reader of the BBC site, we need to know some more
information about the activity patterns of the commentators. For this reason,
we analyze the emergent social network of the persons who commented on the
given BBC article, which is shown in Figure 4. We distinguished 341 unique
pseudo names that appear in the discussions. The network structure in Figure 4
shows that a small-world network is created instantly (as all the comments
were posted during a single day). This is also confirmed by the measured
degree distribution, which very accurately follows a power law with the slope
of $-1.3$.

Figure 4: Network of persons who commented on the BBC article on 9/11
conspiracy. The size of the nodes marks the Out-degree, while the colors
correspond to In-degree. It shows that the In- and the Out-degree are fairly
symmetric. The nodes that are largest (Out-degree/the number of posted
comments) are characterized by the strong red color (highest In-degree, the
number of comments that a person received).

(a)

(b)

(c)

Figure 5: Sub-networks of the $3$ most active commentators. (a) The first on
the left has the largest degree. Most of its comments are supporting
conspiracy theories (value 1); the comments that he/she receives are opposing
them. (b) The middle panel shows the second most engaged person; the comments
he/she is posting are equally labeled $-1$ or 1, meaning that he/she has not
strong opinion about conspiracy theories. This person comments often on the
people otherwise isolated. (c) The third person is against conspiracy
theories, commenting always against it (value 1).

Figure 5 shows the details of the activity of the three persons who discussed
the most frequently. Their patterns of activity are different: the person in
Figure 5(a) was mostly engaged with two other persons, while the commentators
in Figures 5(b) and 5(c) had interaction with a large number of persons.

Participants in the discussion are also allowed to give a positive or negative
vote to each comment. This results in the final ‘rating’ which is available
for each comment on the web site. Unfortunately, we do not have access to the
exact procedure used for the generation of rating. An analytical approach of
dealing with this problem in the context of crowdsourcing has been discussed
in [15]. They analyzed a crowdsourcing system consisting of a set of users and
a set of binary choice questions under assumption that each user has an
unknown, fixed, reliability that determines the user’s error rate in answering
questions. However, we have more interest in the analysis of rating per
discussion thread then per an individual comment. For this reason, we have
analyzed the total rating (sum of ratings of all comments in a discussion
thread). This analysis shows that the comments with the highest $(+34)$ and
the lowest rating $(-32)$ did not result in long discussions. As a matter of
fact, they have remained as individual comments. The statistics also shows
that as the thread becomes longer, the probability is higher that the total
rating will be positive. By pure chance, it happened so that the longest
discussion thread (which is shown in Figure 4) has also the largest total
rating $(+101)$. As we can see in Figure 4, where the rating is visualized as
the size of the nodes, there are 3 comments (No. 42, 78, 130) with large
positive rating. These comments might be against or for conspiracy theories as
the shape of the comments in Figure 4 implies. As we know that the number of
participants in the discussion is $12$, we can conclude that the discussion
was interesting also to many other participants and is not a product of mutual
support of the discussion participants.

Finally, and most importantly, we analyze the dynamics and the alternation of
opinions in the discussion threads: how the discussion impacts the opinion of
the people regarding conspiracy theories around 9/11? It is obvious that the
discussion usually means a repeated interchange of two different opinions
between two or more persons. The most important empirical take away is
twofold:

  * •

The dynamics of the interchange is very fast. All discussions are happening
within 1 day with very seldom post-reactions after that. The longest
discussion thread is finished within 3 hours.

  * •

The size of the groups in which people take part in discussions is very small.
It is distributed by power law, as shown in Figure 6. Therefore it is
reasonable to assume that all participants in the online discussions have an
equal chance to present their opinion to the other participants in that
discussion.

Figure 6: Rank of the size of groups in which people held discussions. It
follows a Zipf law of $-0.4$, which is equivalent to a power law of $-2.25$.

As opposed to studies where the interactions between agents are more or less
complex versions of contagion, the opinion dynamics uncovered by our study
displays a variety of interactions that differ by the number of participants,
their attitudes and the nature of their arguments. In particular some of the
arguments have a more information-like logical character while others
originate in or address to subjective states and experiences. Thus the
modeling of the discussion process and its outcome will have to transcend the
mechanical models of the percolation time and yet include precise enough
elements as to allow quantitative evaluations and predictions.

###  3.2 Example No. 2, The Telegraph blog

In our second example, we examine the article written by a historian/writer
Tim Stanley and published on The Telegraph Blogs in September 2011. The author
justifies the existence and the reasoning of conspiracy theorists by claiming
that “conspiracy thinking is a natural part of political discourse”. There are
858 comments in the commentary section.

The content of the comments is represented by the semantic cloud in Figure 7.
Obviously, the main tags are similar to the ones in the first example in
Figure 3 (conspiracy, person). We need to underline that although the subject
of the Telegraph blog is similar with that of the BBC article, the tone is
different. It is more ‘philosophical’ in looking for the human side of the
conspiracy and less concerned about the facts about it. As such, it is more
general and allows a larger variety of opinions to be discussed.

(a) The main text.

(b) The comments.

Figure 7: (a) Semantic cloud showing the subjects that are discussed in the
Telegraph blog. (b) Semantic cloud showing the subjects that are discussed in
the comments on the Telegraph blog: conspiracy, fact, person etc. The edges
represent the words co-occurrences. The cloud demonstrates the symmetry of
arguments used in the comments: Muslims, Christians and Jews and other similar
words branch out from the main nodes in a very symmetrical way.

The comments on the blog site have somewhat different structure then the
comments in the previous example (news site). For example: even though the
number of comments is larger, the number of persons involved is smaller. In
this thread of comments we identified 222 different pseudo names. The bloggers
are more familiar one with another then the discussants of the BBC article,
and their readers are perhaps also more used to take part in discussions. This
is obvious from, for example, the comment 145: “I’ve been rude about plenty of
Tim’s [author of the blog] blogs so I ought to say that this one is really
quite illuminating. One thing I’d like to know though is whether Truthers
favour one side of the political spectrum, or both, or none.” Therefore, the
discussions take somewhat different flow then what we saw in the Example
$No.1$. The discussion threads are longer than the ones that appeared in
response to the BBC article, and the isolated comments (without any reaction)
are less frequent. One example of a thread of comments is given in Figure 8.

Figure 8: One thread of comments from the Telegraph blog. The colors, size and
shape of the nodes have the same meaning as in the BBC example (Figure 3).

The network of discussants, Figure 9, shows typical characteristics of a
small-world network. The IN- and OUT-degrees are quite symmetrical and the
degree of the nodes follows power law with the slope $-1$. The activity
patterns of the most active persons are visualized by their sub-networks in
Figure 10.

Figure 9: Network of discussants who commented on the Telegraph blog on 9/11
conspiracy. The size of the nodes marks the Out-degree and the colors
correspond to In-degree. It demonstrates that the In- and the Out-degree are
correlated: the red nodes, which have the largest In-degree are at the same
time the ones that have the largest Out-degree (as the nodes are the largest
in size).

(a)

(b)

(c)

Figure 10: Sub-networks of the most frequently engaged commentators in The
Telegraph blog. (a) The first on the left has the largest degree. Most of its
comments are supporting conspiracy theories; the comments that he/she receives
are opposing them. (b) The middle panel shows the second most active person;
the comments he/she posting are mostly labeled -1, meaning that he/she is
opposing conspiracy theories. (c) The third person, same as the second one,
opposes the conspiracy theories.

Finally, we examine how many people participate in a discussion thread. The
result (Figure 11) shows that even when the length of the thread is very
large, the number of participants in each discussion is very small. The slope
of the Zipf/Pareto law fitting line $(\sim 0.5)$ tells us that the probability
that a discussion will engage a more then $x$ participants scales with $x$ to
the power of approximately $-2$ $\big{(}p[x>X]\sim x^{-2}\big{)}$.

Figure 11: Rank of the size of groups in which people held discussions. It
follows a Zipf law of -0.48, which is equivalent to a power law of -2.08.

###  3.3 Observations common to both experiments and their relation with
existing literature

A social psychological study of online discussion of 9/11 conspiracy theories
[16] brought up several observations, defining some characteristics of a
person involved in those dicussions. They found out that the conspiracist
commenters are more likely to argue against opposing interpretation and less
likely to argue in favor of their own interpretation, while the opposite would
be true for conventionalist commenters. In addition, conspiracists are more
likely to express mistrust and make more positive and fewer negative
references to other conspiracy theories. They also indicate that conspiracists
are largely unwilling to apply the “conspiracy theory” label to their own
beliefs and object when others did so, lending support to the long-held
suggestion that conspiracy belief carries a social stigma. Finally,
conventionalist (defenders of the official report) tend to have a more hostile
tone. These tendencies in persuasive communication can be understood as a
reflection of an underlying conspiracist worldview in which the details of
individual conspiracy theories are less important than a generalized rejection
of official explanations.

We generally agree with the above findings. More specifically, we confirm that
the rationalist arguments tend to have a more hostile tone. We also agree that
the conspiracist do not change their view even after exposed to very clear
arguments, which confirms their generalized and unanimous rejection of
official explanations.

When it comes to the analysis of rating (the number of ‘Likes’), we refer to
the work of Muchnik et al in [17], who discovered that prior ratings create
significant bias in individual rating behavior, and positive and negative
social influences created asymmetric effects. In other words, the voting is
path dependent. We actually find out that the comments with a negative rating
have fewer responses then the comments with a positive rating. The most
popular comments (with a high probability to promote conspiracy theories) are
also the most frequently contradicted ones. Further, our research brings out
that the on-line discussions about conspiracy theories are held in small
groups of people who are exchanging their opinions. Maximum size of the
discussion group is $14$ and usually the discussion groups are much smaller,
often containing only a single opinion. The size of the groups follows a power
law approximately, with the slope of $-2$.

The emergent social network of the commentators shows small-world network
properties, characterized by a power law distribution of the commentators’
activity. The In- and Out-degrees, i.e. the number of the received and the
number of the posted comments per person are fairly balanced. The data
indicates that the dynamics of the collective opinion is determined not only
by the informational content of the debate, but aspects which have to do with
the semantics and the network connectivity of the message turn out to be
equally relevant. Once enough data will be gathered, we will disentangle those
aspects and evaluate further their effects.

##  4 Estimation of entropy of the comments on the articles reporting
conspiracy

Entropy is a measure of the information content, or rather, its uncertainty.
It can be assessed in various ways and here we propose a possible approach.
Let $E_{i}$ stand for the state variable, $E_{i}=-1$ (true) or $1$ (false),
$i=1,…,N$. Assigning $E_{i}=-1$ to all strings, the resulting order parameter
$E=-1$ marks a fully ordered state, with minimum entropy $(H=0)$.

Conspiracy theorists comment to the report, focusing on one string $E_{k}$ and
changing its meaning (i.e. from $-1$ to $+1$). Entropy here characterizes our
uncertainty about our source of information, and increases with adding more
comments of greater randomness. The source is also characterized by the
probability distribution of the samples drawn from it. The idea behind the
Shannon entropy is that the less likely an event is, the more information it
provides when it occurs. The amount of information contained in a string
$E_{i}$ is not equally weighted in the formation of the opinion about
conspiracy theory defined by the series of $E_{i}$’s. The expected value
(average) of the information changes with each comments received. The entropy
of the entire series $E_{i}$ is therefore given by:

$H=-\sum_{i=1}^{n}p_{i}log_{2}(p_{i}),$ (1)

where $p_{i}$ are coefficients that define different significance of the
information provided in the strings $E_{i}$ and belongs to a probability
distribution which satisfies the condition:

$\sum_{i=1}^{n}p_{i}=1.$ (2)

It is far from trivial to bring the above ideas to realization and provide an
automatized procedure for the measure of entropy and the estimation of the
discrete probabilities $p_{i}$.

In the subsequent subsections we will apply as the data previously introduced
in the BBC and The Telegraph blog examples in an attempt to estimate $p_{i}$.
The data might be organized is several ways, as it is illustrated in Table 1
and will be discussed in the sequel.

| Discussion | Discussion | … | Discussion |  $p_{i}$ per
---|---|---|---|---|---
| thread 1 | thread 2 |  | thread Z | person
Person | (-1, 0, 1) | (-1, 0, 1) | … | (-1, 0, 1) | $p_{i}\sim\overline{E_{i}}$
1 |  |  |  |  | $=f(INdegree)$
Person | (-1, 0, 1) | (-1, 0, 1) | … | (-1, 0, 1) | $p_{i}\sim\overline{E_{i}}$
2 |  |  |  |  | $=f(INdegree)$
… | … | … |  | … | …
Person | (-1, 0, 1) | (-1, 0, 1) | … | (-1, 0, 1) | $p_{i}\sim\overline{E_{i}}$
W |  |  |  |  | $=f(INdegree)$
$p_{i}$ per | $p_{i}\sim\overline{E_{i}}=$ | $p_{i}\sim\overline{E_{i}}=$ | … | $p_{i}\sim\overline{E_{i}}=$ |
thread |  $f(Thread$ $Size)$ |  $f(Thread$ $Size)$ |  |  $f(Thread$ $Size)$ | $H\sim-\sum p_{i}log(p_{i})$

Table 1: Observed content of comments could be organized according to persons
or discussions. Both rows and columns are expected to satisfy a power law
distribution, which is a statistical simplification that in the next stages of
this work might be used exactly the other way around, i.e. in order to
predict/confirm whether the mesages are supporting or opposing conspiracy
theory.

A common way to define entropy for a text is based on the Markov model which
indeed might also appropriate for the comments-level entropy. This approach
will be considered in sub-section 4.1 when will monitor the average value of
$E_{i}$ of all previously (and randomly) posted comments. We however are
looking for a model that would include the aspects which have to do with the
properties of networks of messages and persons. Therefore we vary the approach
to the “discretization” of the system or rather, to the aggregation of
individual comments. In sub-section 4.2 we estimate the average value of
$E_{i}$ per discussion thread. In the third case, sub-section 4.3, we monitor
the average $\overline{E_{i}}$ value per person.

Combining these distributions of the $\overline{E_{i}}$ measured as a function
of discussion thread or a person IN/OUT-degree with the previously estimated
distributions of the discussion size and the persons connectivity, it would be
possible to estimate $p_{i}$.

An interesting model that applies minimax entropy model to improve the quality
of the noisy labels obtained during crowdsourcing acquisition has been
recently reported in [18]. They have assumed a probability distribution over
workers, items and labels. The minimax entropy approach is assuming the ground
truth from the minimum entropy and generation the entire probability
distribution of the worker-item label models from the maximum entropy. To make
a remote parallel with such model, we could compare workers with discussants,
items with the comments under discussions and labels with rating. We intend to
work on this formalization in the future. In the current paper we develop the
concept and we empirically validate many aspects of the procedure. However, a
full integration of those basic steps in a coherent and feasible procedure has
to be developed yet.

###  4.1 Monitoring the state variable of comments as they are appearing (in
time)

Before we start with the analysis of the comments, the order variable value
$\overline{E_{i}}$ of the original article/blog needs to be estimated. The BBC
article is listing different 9/11 conspiracy theories in a factual style. This
article is critical about conspiracy theorists, as the following sentence
copied from this article implies: “Numerous official reports have been
published since the Twin Towers fell, but just when a piece of evidence casts
doubt on one theory, the focus then shifts to the next "unanswered question"”.
Therefore the value of its order variable $\overline{E_{i}}$ is negative. The
Telegraph article is also listing a number of conspiracy theories, but in a
less judgmental style (“…in fact conspiracy thinking is a natural part of
political discourse. It represents an effort to make sense of apparently
senseless events. People conquer their fears by drawing connections between
unconnected tragedies to create a unified theory that brings order out of
chaos.”). Therefore this article has a positive $\overline{E_{i}}$. Starting
from these two different initial conditions, we monitor the dynamics in
$\overline{E_{i}}$ of the commentaries.

Assigning to each comment a value $-1$ or $+1$ (personal judgement of the
author) and linking it to one of the original $E_{i}$ strings, to which the
comment refers to, the value of the argument posted in the original article
deteriorates. After a certain number of comments, the entropy of the system
can be recalculated. This process can be continued by taking a measure of
entropy after each X comments. To give an impression of the expected dynamics,
we apply this procedure with a time step randomly chosen to 20 $(X=20)$. This
is shown in Figure 12(a) for the BBC example and Figure 12(b) for The
Telegraph example.

(a) The BBB example.

(b) The Telegraph example.

Figure 12: (a) The figure shows the $\overline{E_{i}}$ dynamics over the
entire corpus of 775 comments on BBC article. Each black circle (full line,
primary axis on the left) represents an incremental average value recalculated
after the constant size window of 20 comments. The white circles (dashed line,
secondary axis on the right) represent the ratio of the new over the
previously involved commentators, for each set of 20 comments. (b) The figure
shows the dynamics of $\overline{E_{i}}$ over the entire set of 858 comments
on The Telegraph article. Each black circle (full line, primary axis on the
left side) represents the average incremental value recalculated after the
constant size window of 20 comments. The white circles (dashed line, secondary
axis on the right side) represent the ratio of the new and previously involved
commentators, for each set of 20 comments.

Although those two examples show different behavior, they do show some
similarities:

  * •

The first set of 20 commentaries is opposing/challenging the opinion promoted
in the original article. In the BBC example, whose $\overline{E_{i}}<0$, the
first set of comments has an $\overline{E_{i}}=0.4$. In the Telegraph example,
average(Ei)>0, the first set of comments has an$\overline{E_{i}}=-0.2$.

  * •

Increasing the number of comments, the order variable bounces back to the
opinion/side that the original article has promoted.

  * •

After a number of fluctuations the $\overline{E_{i}}$ saturates to a value of
-0.07 for the BBC article and +0.07 for the Telegraph article.

  * •

The process of saturation of the order variable $\overline{E_{i}}$ is
accompanied with the low level of interest of the readers (when the ratio
New/Known pseudo names is less than 1).

The oscillatory dynamics of $\overline{E_{i}}$ as a function of time is
significantly different in the two examples, and therefore we cannot
approximate it by a mathematical function necessary for the estimation of
$p_{i}$. We might be able to find the indicators for the ‘jumps’ in the order
parameter from the semantic similarity analysis applied to the level of the
sets of comments. The string similarity measures such as Jaccard or Tversky
distance as well as the Kullback-Leibler distance have already been tested for
this purpose and their applicability for this purpose seems to be feasible.

###  4.2 Monitoring the state variable of discussion threads

The length (size) and the balance of opinions in a discussion is determining
the impact that the discussion has on the entropy of the system. Single
comment would have much less impact then the comments given in a discussion
engaging a number of persons. From the power law distribution of the
discussion groups, the weight distribution can easily be recalculated and it
will also be a power law function. Also, the rating per discussion can be a
parameter used for this purpose. To illustrate the shape of the statistics
underlying the discussion thread structure, we use the BBC example data, as
shown in Figure 13. The data are very ‘promising‘ showing a tent shaped
distribution. Assuming that such distribution exists would indeed shorten the
entropy calculations significantly comparing with the situation presented as
case 1.

(a) The BBC example.

(b) The Telegraph example.

Figure 13: (a) The figure shows the order parameter $\overline{E_{i}}$ as a
function of the discussion thread length Li, measured for the discussions on
the BBC site. Each circle of the minimum size represents only one discussion
(such as the longest discussion of the length 42, which appears only once).
The circle of the largest size represents 198 comments for which there was no
reaction (-1, against conspiracy theories). The graph is rather symmetrical in
the upper and the lower part. Clearly, there is a tendency to have stronger
polarization of $\overline{E_{i}}$) for the shorter discussions, while the
longer ones are more ‘neutral‘, the resulting order parameter is closer to
zero. This tendency is visualized by the red line defined by Eq. 3. (b) The
figure shows the distribution of the $\overline{E_{i}}$ of the discussions on
the Telegraph example. The circles of minimum size represent only one
discussion (such as the longest discussion of the length 35). The circle of
the largest size (1, pro conspiracy theories) represents 41 comments to which
there was no reaction. The red line is following the exponential law from Eq.
3.

The red line in Figure 13 is an approximate border line and it is an
exponential function of the discussion length $L_{i}$:

$\overline{E_{i}}=\pm\exp({\frac{L_{i}}{const}}).$ (3)

The constant is chosen such that $L_{i}=1$ returns $E_{i}$ value of $\pm 1$.
This border line represent the maximal (absolute) expected value that the
$\overline{E_{i}}$ of a discussion. Knowing this relation and knowing the
distribution of the discussion thread legths, the $p_{i}$ distribution could
be recalculated. Of course it would be necessary to apply some sort of
normalization (providing $\sum_{i}p_{i}=1$).

###  4.3 Monitoring the state variable of the persons

Our intuition suggests that the comments of the persons who are only involved
in a single discussion would be weighted less then the comments of the persons
who are very active and able to constantly provide strong opinion. From the
power law distribution of the discussants activity, the weight distribution
can easily be recalculated and it will also be a power law function. In this
case rating could also be used to select the persons who are able to receive
the support of the audience.

We have tested whether it would be useful to add additional weight coefficient
to the comments of the persons that are involved in discussions more
frequently by measuring the stability of the opinion of the discussants as
presented in Figure 14. The weight factor could be estimated from the data
distribution. This graph shows that the discussants are typically very
polarized (-1 or 1). This holds for most of the persons. The change of entropy
is in such cases almost ‘neutralized’ by the opposite attitudes – as long as
the distribution that describes those persons it is symmetric. However, there
is also a number of persons whose opinion is not -1 or 1, but somewhere in
between. Finally, there is also a number of ‘internet trolls’ who are posting
off-topic messages. They are characterized with the $E_{i}=0$ and the IN-
degree=0 and should not be taken into consideration when estimating the change
of the order variable.

(a) The BBC example.

(b) The Telegraph example.

Figure 14: (a) The figure shows the ‘stability’ of discussants‘ opinion, using
the BBC example data. Each circle of minimum size represents the average
opinion only one person. The circle of the largest size represents the average
opinions of 113 persons (as a matter of fact that persons gave only one
comment valued -1 (against conspiracy theories). (b) The figure shows The
Telegraphs commentators’ opinion as a function of their IN-degree ratio.

The red border line in Figure 14 is a power law function of the persons’ IN-
degree (number of received comments):

$\overline{E_{i}}=\pm\Big{(}\frac{INdegree}{const}\Big{)}^{0.5}.$ (4)

The constant in Eq. 4 equals to the maximum IN-degree of all persons. The red
border line in Figure 14 represents the minimum absolute expected value
$\overline{E_{i}}$ of a person. As a first approximation, and in combination
with the distribution of the persons IN-degree estimated in Section 3, this
simplified relation could also be used to determine $p_{i}$ (after applying
some sort of normalization which would provide that $\sum_{i}p_{i}=1$).

An interesting side-observation from Figure 14 is that the persons with the
largest IN-degree are at the same time the “stubborn” persons who are not
changing their original view (and therefore have a highly polarized average
$E_{i}$). It looks as if they get involved in discussions while trying to
convince the others to change their opinion. This behavior in the on-line
environment is, according to the recent work of Fisher and Keil [19], not
surprising. Namely, they claim that the nature and context of argumentative
exchange depends on the social context and distinguish two different modes of
argumentation: arguing to win and arguing to learn. They experimentally proved
that arguing in private prompts a mindset of arguing to learn, while arguing
in public prompts a mindset of arguing to win. Further on, they also showed
that the choice of person with whom participants chose to argue varied as a
function of either a public or private social setting. Namely, there is a
preference to argue with the less knowledgeable person in public and with the
more knowledgeable person in private. From this perspective, all on-line
discussions, being placed in a public setting, although behind the pseudo
names, are accommodating ‘argue to win’ discussion modus.

##  5 Conclusions

In this paper, we apply the concept of entropy as a measure of the penetration
depth of a conspiracy theory (i.e. a measure for the information uncertainty
that the conspiracy theorists introduce through various on-line content). Our
basic model and the pioneering empirical studies point out to the necessity to
extend the common measures of information with some aspects of the social
networks theory and some semantic string similarity measures in order to
enable such a qualitative analysis of a large data corpus. In this particular
application, we deal with the information which carry subjective and personal
rather than purely factual content and contains aspects of mental processing
which are not governed by regular Boolean logics.

More precisely, in addition to the information content measured by Shannon-
like methods presented in Section 4 one needs to employ the knowledge of the
interaction patterns in order to classify the content of the messages and the
traces of conspiratorial beliefs. At the collective level, one should
recognize and measure the common vocabulary that labels and singles out
/separates certain semantic proximity, symmetry or idiomatic expressions (e.g.
revolution vs. violence, jihad vs. terrorism, freedom of expression vs
blasphemy, multicultural vs apostate, free thinker / nonbeliever etc). While
the use of one word or another does not affect much the information content,
it does label and affect the emotional content and dynamics of the
interaction.

Tagging the above signs could be used for machine learning training in order
to classify the order parameter of the individual comments. The large training
sets produced on the articles of similar content in different social or
geographical environments, supported by the underlying assumptions of the
predefined statistics based on the social and comments networks can provide
information theory measures, and bring entropy concepts in dealing with
subjective / social / human dynamics. This is a long overdue move which may
largely benefit both the exact sciences as well as the social / human
sciences. We hope that our first stage calculations will trigger additional
investigations towards a full integration of the various ideas here discussed
towards a coherent and feasible procedure.

## Acknowledgments

We acknowledge all participants of the Workshop “From Opinion Dynamics to
Voting, Conflict and Terrorism” held 30/31 March 2015 in Paris for useful
interactions. This work has been performed under the DGA Grant «L’impact des
supporters passifs dans l’action terroriste : une approche à partir de la
sociophysique», coordinator Serge Galam, DGA-2012 60 0013 00470 75 0.

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