# Conspiratorial Beliefs Observed through Entropy Principles

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## Abstract

**:**

_{i}= ±1, i = 1, …, N. If the values of the E

_{i}are set to −1 for all strings, a state of minimum entropy is achieved. Comments on the report, focusing repeatedly on several strings E

_{k}, might alternate their meaning (from −1 to +1). The representation 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, the chance is created that a conspiracy theory might be initiated/propagated. 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

_{i}= ±1 with i = 1, …, N. An official report would be characterized by value E

_{i}= −1 assigned to all strings, resulting in the average order parameter E = <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, focusing on a series of information strings E

_{i}= −1 to question their content and change their classification (i.e., from −1 to +1).

_{i}> per set of 20 consecutive comments, set of comments of one person, set of comments that belong to one discussion thread) and discuss the associated patterns of entropy change in Section 4.1, Section 4.2 and Section 4.3. In Section 5, we discuss the results and outline future work. We also recognize that while the proposed model looks very promising, much more empirical evidence is needed to show the feasibility of the proposed model. Once more data has been gathered, and the proposed model has been developed to the level of an application, we hope to obtain more quantitative results.

## 2. Opinion Polls about 9/11 Conspiracy Theories

**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.

## 3. Analysis of the Web-Based Comments about Possible 9/11 Conspiracy

#### 3.1. Example No. 1, BBC News Article

**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 commentary on the BBC article: conspiracy, person, collapse, etc. The edges are constructed upon the minimum number of ten co-occurrences.

**Figure 3.**Thread of comments that mention one to the other (establishing the possibility that they share/develop 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 (Figure 6); the larger node, the better its 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. (Note: the sub-networks of all of the comments from the BBC site are in the Appendix).

**Figure 4.**Network of people who commented on the BBC article for the 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).

**Figure 5.**Social sub-networks of three 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 no strong opinion about conspiracy theories. This person comments often on the people otherwise isolated (small nodes). (

**c**) The third person is against conspiracy theories and comments always against them (value −1).

- The dynamics of the interchange is very fast. All discussions happen on the same day with very few post-reactions after that. The longest discussion thread takes only three hours.
- The size of the groups that take part in each discussion thread 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 had 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.

#### 3.2. Example No. 2, the Telegraph Blog

**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. Edges represent word 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.

**Figure 8.**One thread of comments from the Telegraph blog. 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 its 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. (Note: the sub-networks of all of the comments from the Telegraph site are in the Appendix).

**Figure 9.**Network of discussants who commented on the Telegraph blog about the 9/11 conspiracy. The size of the nodes marks the Out-degree (the number of posted comments) and the colors correspond to In-degree (the number of comments that a person received). It demonstrates that the In- and the Out-degree are correlated: 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 largest in size).

**Figure 10.**Social sub-networks of three most frequently engaged commentators on 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 posted 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.

^{−2}).

**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

## 4. Estimation of Entropy of Commentaries on the Articles Reporting Conspiracy

_{i}stand for the state variable, E

_{i}= −1 (no conspiracy) or 1 (conspiracy), 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).

_{k}and changing its meaning (i.e., from −1 to +1). Entropy here characterizes uncertainty about the source of information, and increases by adding more comments of greater randomness. (The source is also characterized by the probability distribution of the strings drawn from it, but this is not part of the current model.) 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 comment received. The entropy of the entire series E

_{i}is therefore given by:

_{i}are coefficients that define different significance of the information provided in the strings E

_{i}and belong to a probability distribution which satisfies the condition:

_{i}. Working with p

_{i}on the level of individual comments (p

_{i}= E

_{i}) would be the common way to define entropy for a text based on the Markov model. However, this procedure would be not only cumbersome, but it could also result in an impractical and inconclusive result. A procedure with an intermediate aggregation would however lead to an accelerated process and to a result that could be of use in building and moderating discussion platforms. Our model indeed includes the relevant aspects that have to do with the properties of networks of comments and persons who are posting them.

_{i}distributions. The data might be organized in several ways, per person (rows in Table 1), per discussion (columns in Table 1), or according to the time they appeared on the website, as will be discussed in subsequent sub-sections.

**Table 1.**Observed content of comments can be organized according to persons or discussions. Both rows and columns are expected to satisfy a preliminary distribution, which is a necessary simplification in order to counterbalance the large uncertainty of opinion mining. Symbol L

_{i}is used for the length of discussions. Estimates of preliminary distributions are proposed in Sub-section 4.2 and Sub-section 4.3 <…> = average.

Discussion 1 | Discussion 2 | Discussion 3 | … | Discussion Z | p_{i} per person | |
---|---|---|---|---|---|---|

Discussant 1 | {−1, 0, 1} | {−1, 0, 1} | {−1, 0, 1} | {−1, 0, 1} | <E_{i}> = f(IN-degree_{1}) | |

Discussant 2 | {−1, 0, 1} | {−1, 0, 1} | {−1, 0, 1} | {−1, 0, 1} | <E_{i}> = f(IN-degree_{2}) | |

Discussant 3 | {−1, 0, 1} | {−1, 0, 1} | {−1, 0, 1} | {−1, 0, 1} | <E_{i}> = f(IN-degree_{3}) | |

… | ||||||

Discussant W | {−1, 0, 1} | {−1, 0, 1} | {−1, 0, 1} | {−1, 0, 1} | <E_{i}> = f(IN-degree_{W}) | |

p_{i} per discussion | <E_{i}> = f(L_{1}) | <E_{i}> = f(L_{2}) | <E_{i}> = f(L_{3}) | <E_{i}> = f(L_{Z}) | H = − sum(p_{i}log_{2}p_{i}) |

_{i}for each set of 20 consecutively (and randomly) posted 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 E

_{i}value per person.

_{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 would make possible to estimate p

_{i}distributions and Shannon entropy associated with it.

_{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 <Ei> 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 <E

_{i}>. Starting from these two different initial conditions, we monitor the dynamics in <E

_{i}> of the commentaries.

**Figure 12.**(

**a**) The figure shows the <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 <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.

#### 4.1. Monitoring the State Variable of Sets of Comments as They Are Appearing in Time

_{i}strings, to which the comment refers to, the value of the argument 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 comment. To give an impression of the expected dynamics, we apply this procedure with a time step chosen to 20 (X = 20) in order to (approximately) match the size of the original blog/article. This is shown in Figure 12a for the BBC example and Figure 12b for The Telegraph example.Although those two examples show different behavior, they do show some similarities:

- -
- The first set of 20 comments is opposing/challenging the opinion promoted in the original article. In the BBC example, whose <E
_{i}> < 0, the first set of comments has an <E_{i}> = 0.4. In the Telegraph example, <E_{i}> > 0, the first set of comments has an <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 <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 <E
_{i}> is accompanied with the low level of interest of the readers (when the ratio New/Known pseudo names < 1).

_{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 seems to be feasible.

#### 4.2. Monitoring the State Variable of Discussion Threads

_{i}> as a function of the discussion length L

_{i}. The data are very ‘promising,’ showing a tent-shaped distribution. Assuming that such a distribution exists would indeed shorten the entropy calculations significantly compared with the situation presented as case 1.

_{i}. It is expected that the value of <E

_{i}> will fall between the border lines, described by Equation (3). In the given example, the upper and the lower part of the line are symmetrical, i.e., for any discussion length L

_{i}, the chance is equal that <E

_{i}> will have a positive or a negative sign.

_{i}= 1 returns E

_{i}value of +/−1. This border line represents the maximal (absolute) expected value for the <E

_{i}> of a discussion. Knowing this relation and knowing the distribution of the discussion thread lengths, the p

_{i}distribution could be recalculated. However, it would be necessary to apply some sort of normalization (providing ∑(p

_{i}) = 1).

**Figure 13.**(

**a**) The figure shows the order parameter <E

_{i}> as a function of the discussion length L

_{i}, 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 <E

_{i}>, for the shorter discussions, while the longer ones are more “neutral”, the resulting order parameter being closer to zero. This tendency is visualized by the red line defined by Equation (3). (

**b**) The figure shows the distribution of the <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 Equation (3).

#### 4.3. Monitoring the Persons’ State Variable

_{i}= 0 and the IN-degree = 0 and should not be taken into consideration when estimating the change of the order variable.

**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 of only one person. The circle of the largest size represents the average opinions of 113 persons (those giving only one comment valued −1, i.e., against conspiracy theories). (

**b**) The figure shows The Telegraph commentators’ opinion as a function of their IN-degree ratio.

_{i}> of a person. As a first approximation, and in combination with the distribution of the persons IN-degree estimated of Section 3, this simplified relation could also be used to determine p

_{i}(after applying some sort of normalization which would provide that ∑(p

_{i}) = 1).

_{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 [18], 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, they also show that the choice of the 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 pseudo names, are accommodating “argue to win” discussion modus.

## 5. Conclusions

## Supplementary Files

Supplementary File 1## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Golo, N.; Galam, S.
Conspiratorial Beliefs Observed through Entropy Principles. *Entropy* **2015**, *17*, 5611-5634.
https://doi.org/10.3390/e17085611

**AMA Style**

Golo N, Galam S.
Conspiratorial Beliefs Observed through Entropy Principles. *Entropy*. 2015; 17(8):5611-5634.
https://doi.org/10.3390/e17085611

**Chicago/Turabian Style**

Golo, Nataša, and Serge Galam.
2015. "Conspiratorial Beliefs Observed through Entropy Principles" *Entropy* 17, no. 8: 5611-5634.
https://doi.org/10.3390/e17085611