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American Academic Scientific Research Journal for Engineering, Technology, and Sciences   

ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

 

http://asrjetsjournal.org/  

 

Fake News and Post-truth: Numerical Simulations of 

Information Diffusion in Social Networks 

Gabriela Fernandes Teixeira
a
, Ana Carolina Carius

b*
 

a,b
Universidade Católica de Petrópolis, Rua Barão do Amazonas, 124, Petrópolis 25685-100, Brasil 

a
Email: gabriela.11710043@ucp.br 

b
Email: ana.carius@ucp.br 

 

 

Abstract 

The year 2016 was crucial in terms of the use of social networks as tools for disseminating information for 

political purposes. The election of candidate Donald Trump for the presidency of the United States of America 

lit a warning signal about the influence that the information disseminated through social networks exerted in the 

choice of candidates by the American people. Since then, researchers from different areas have focused on the 

topic, which involves different aspects: computing, social sciences, mathematics, among others. Therefore, the 

object of study of this work is the phenomenon of behavioral changes brought about by the new social 

relationships established by digital social networks, under the scope of the spread of fake news through them. To 

guarantee the intended study, the general objective was to adapt mathematical models consisting of ordinary 

differential equations for the dissemination of information on social networks for the spread of fake news. As 

specific objectives, the contribution of the mathematical models proposed in the mitigation, through algorithms, 

of the spread of false or distorted information was discussed, as well as the discussion on the concepts of fake 

news and post-truth from a social point of view, in a way that individuals can also distinguish true information 

from disinformation through individual interpretation tools. As a research methodology, bibliographic research 

was chosen and a systematic literature review was carried out, to consider published works on the proposed 

research object. For the numerical simulations, a numerical code was developed in MATLAB, which was able 

to carry out the desirable experiments. It is concluded that innovation diffusion models can adapt to fake news 

dissemination models. However, such models are not able to robustly simulate the mitigation of fake news. 

Keywords: Logistic Models of Ordinary Differential Equations; Numerical Simulations; Fake News; Post-Truth; 

Dissemination of Information. 

------------------------------------------------------------------------ 

* Corresponding author.  

 

http://asrjetsjournal.org/


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1. Introduction 

The understanding of the current social scenario permeates, without questioning, the influences that digital 

information and communication technologies exert on people's daily lives. In fact, it is difficult to think about 

today's society without the services available through technological-digital artifacts with internet access. The 

COVID-19 pandemic was an immersive experience in this intimately digital landscape [1,2]. 

Several authors carry out robust studies on the social transformations triggered by the internet [3–5]. In Manuel 

Castells' view, the technological advance represented, above all, by Big Techs has been leading to paradigmatic 

changes from an economic and social point of view.  

In an optimistic view, the authors Pierre Lévy and André Lemos [6] considered that the expansion of access to 

communication between individuals and the creation of an interplanetary space such as cyberspace would be 

enough to create a wide public square for debates, including and, in this way, expand the concept of democracy 

also in the virtual space. For the authors, the possibility of a greater number of people having a common space 

for exchanging ideas would foster a more plural space, in which divergent voices would have the chance to 

dialogue and, consequently, more “people” would be exercising their political rights. People are understood to 

be the set of citizens of a given territory who participate in the political and legal processes of the State [7]. 

However, what was seen after the election of President Donald Trump in the United States of America and 

Brexit in the United Kingdom, both in 2016, was the incorporation of digital social networks as tools for 

political manipulation of a certain population. Such facts sparked an alert to social researchers: how can the 

algorithms of social networking platforms, in fact, manipulate information and communication, worldwide? [8] 

It is in this context of insecurity and uncertainty that two similar concepts are coined: fake news and post-truth. 

According to Rodrigo Santos [9, p.124] ‘A fake news is a lie told to obtain some gain on the part of its 

perpetrator’, that is, when talking about fake news in the political scenario, it refers to lies that are deliberately 

disclosed with the objective of obtaining some gain, mainly about electoral issues. 

The concept of post-truth is a little more complex. For Santaella [8], there are two possible interpretations for 

the concept. One, related to the meaning of the expression itself, that is, it would be a concept after the fact that 

the truth was already known. The other, more recent, means that the truth is no longer relevant. For this work, 

the second perspective is adopted: the non-relevant truth in the context of generalized disinformation generated 

by digital social networks. 

Considering the current scenario of influence of quick information on society, including political decisions 

based on inaccurate information, this work is situated. Therefore, this research intends to answer the following 

research question: ‘Is it possible to simulate the propagation of disinformation from ordinary differential 

equations, in order to predict the impact of fake news in the communicational scope?’ In order to answer the 

previous research question, the general objective is the study of models generated from ordinary differential 

equations which, in fact, reflect simulations of information propagation. As specific objectives, we intend to 

analyze such models from the perspective of fake news, observing the social impact of disinformation on a 



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certain population of individuals. Finally, it is intended to propose strategies from a computational and 

educational point of view that can, in fact, mitigate the effects of linking fake news today. 

2. Modeling the spread of information 

Digital social networks such as Facebook, Twitter or YouTube, managed by algorithms based on artificial 

intelligence, have gained extreme relevance in the social life of individuals, especially since the 2010s. Such 

importance is largely due to the increase in internet coverage worldwide, driven by internet access by individual 

devices such as smartphones and tablets [10]. From these advances, several researchers, in different areas, 

became interested in modeling the phenomenon, in an attempt to observe the virtual behavior of individuals 

from their actions through digital social networks. 

According to Wang and his colleagues [11], digital social networks emerge as ‘model organisms’ of Big Data 

and, for the most part, are dynamic models based on information diffusion through Ordinary Differential 

Equations (ODE). However, in more recent models, the inclusion of Partial Differential Equations (PDE) is 

perceived as integral parts of the mathematical models which describe the problem in question. 

Studies on the dissemination of information through digital social networks have gained extreme relevance in 

the academic and research environment. The interdisciplinary character of such studies is observed, since the 

phenomenon in focus can be analyzed from a mathematical point of view, from computer science, from social 

communication, from sociology, marketing, among others [11]. 

It is observed that the first models for the dissemination of information were thought of by the academic Rogers 

[12] and his theory of the diffusion of innovation gained strength in areas such as marketing and administration, 

culminating in the dialogicity between digital social networks and information dissemination. Therefore, as a 

pioneering theory on the issue of information diffusion, the mathematical models derived from this theory were 

considered in this work. 

Diffusion of innovation is then defined as the theory that seeks to explain how new ideas and technologies 

spread across different cultures. [11]. For Rogers [12] there are four elements that influence the diffusion of a 

new idea: innovation itself, communication channels, time, and the social system. For the same author, 

innovation is conceptualized as a new idea, practice, or object if it is perceived as new by individuals or other 

adoption units (companies, institutions, among others). 

From the definitions by Rogers [12] it is intended, under the ordinary differential equations, to identify factors 

that influence the spread of information from a mathematical point of view and how such models adapt to the 

process of diffusion of false information. 

For this work, the classification in levels of adoption of innovations defined by Rogers [12] is considered: 

innovators, representing 2.5% of the total; early adopters, representing 13.5%; initial majority, representing 

34%; late majority, representing 34% and, finally, latecomers, representing 16%. In addition to the definitions 

by Rogers [12], mathematical curves were observed which would adequately model the phenomenon of 



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information scattering. Among the possible models, those that adopted the logistic curve were chosen, which is 

characterized by an inflection point where the curve changes its concavity downwards to concavity upwards. 

After the change of inflection, few individuals would adopt the innovation. 

It is important to note that the influence of characters on social networks is not identical for each profile. There 

are individuals who exert greater influence on the population, who are usually more experienced and have a 

higher social status. To better understand this process, in the context of mass communications, the two-step flow 

theory of Katz and Lazarsfeld [13] is considered. First, opinion leaders should be willing to accept the 

information. Next, opinion leaders pass on their own interpretations or opinions to others, as illustrated in Figure 

1. 

 

Figure 1: Mass Communication Flow. 

From the theory of the flow of mass communication in two steps, it is possible to observe the fundamental role 

of opinion leaders in the dissemination of information. Considering the context of social networks, we have the 

so-called digital influencers: people who gather a significant number of followers on their social networks and, 

therefore, gain relevance in their posts, driven, above all, by the algorithms of social networks. Therefore, the 

post on a certain subject by a digital influencer is much more relevant than posts by the public, that is, it is the 

digital version of the opinion leader. However, the dynamics of social networks is quite different from the 

dynamics of mass media and, in this sense, the study of robust mathematical models that translate this 

phenomenon is of paramount importance. 

3. Research Methodology 

From the two-step mass communication flow theory [13] and the definitions for the different profiles of 

innovation adoption [12] we tried to adopt mathematical models that reflected the logistic behavior, which is the 

most suitable for the phenomenon. Therefore, the present work is outlined as exploratory research, with a 

quantitative character. The research is characterized as exploratory because the mathematical models studied 

have not been used, so far, to model the dissemination of fake news in social networks [14]. 



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The research problem in question arises from the need, on the part of researchers, to understand the social 

phenomenon of fake news. The post-positivist perspective was chosen, which defends a deterministic 

philosophy in which causes determine effects [15]. By hypothesis, it is believed that the dissemination of fake 

news occurs through a process like the dissemination of information, that is, it is intended to validate the models 

of information diffusion, under the cut of ordinary differential equations, as descriptors, also, of the 

phenomenon of spreading fake news. 

 First, a literature review on the subject was carried out, observing the existence of publications by Brazilian 

researchers involving mathematical models for the dissemination of information in social networks. As these 

works were not found, we searched for works published in English that addressed the research object. The 

works of Wang [16], [17], Xu [18] and Wang [11] presented several approaches involving differential 

equations, both ordinary and partial, to which they modeled the diffusion of information from known 

phenomena, epidemiological problems. 

4. Results and discussions 

The mathematical model of innovation diffusion chosen was represented by the following equation [11] 

𝑑𝑁

𝑑𝑡
= 𝑔(𝑁, 𝑡)[𝑁 − 𝑁(𝑡)], 𝑁(0) = 𝑁0, (1) 

where the function N is the cumulative number of adopters at time t in a network. 𝑁 denotes the total number of 

potential adopters. 𝑔(𝑁, 𝑡) represents the diffusion coefficient, that is, the way in which the innovation spreads. 

Therefore, the information dissemination rate as a function of time is proportional to the difference between the 

total number of potential adopters and the total number of adopters at any instant of time t. 

The diffusion coefficient 𝑔(𝑁, 𝑡)  depends on several aspects, namely: the nature of the information; the 

available communication channels and attributes of the social system. Considering the cut in the dissemination 

of fake news, the influence of the nature of information occurs in several ways. Information can be purposely 

biased, with the aim of causing controversy, or misinformation with the aim of ensuring relevance on social 

networks (the more clicks, the greater relevance). About communication channels, it is observed that digital 

social networks are an efficient channel due to the speed with which information is disseminated. Facebook, 

Twitter, Whatsapp are tools used ostensibly by Brazilians and, in this sense, they are important communication 

channels. Regarding the attributes of the social system, there is information on who are, in fact, the social actors 

that are acting in cyberspace, that is, who are the people who have access to fake news, who work for its 

dissemination and who are the individuals influenced for them. At this point, the so-called bots stand out, which 

are fake profiles created on digital social networks with the aim of generating relevance to certain information 

through clicks and views. Twitter is one of the platforms that houses this type of profile. Contrary to this 

discussion, there are real individuals who are not part of the universe of digital social networks, either by choice 

or by lack of digital ability to communicate through the platform. Therefore, the attributes of the social system 

under study in the proposed research problem are not simple. 



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Despite the difficulties described, we considered, like Wang [18], three cases of innovation diffusion. 

Case I: 𝒈(𝑵, 𝒕) = 𝜶 

Considering the diffusion of information as a constant, a diffusion coefficient independent of the number of 

adopters is being represented, that is, a model of strictly external influence. The function that represents the 

number of innovation adopters as a function of time t is given by 

𝑁(𝑡) = 𝑁 − (𝑁 − 𝑁0)𝑒−𝛼𝑡 (2) 

Considering the analytical solution (2) and a numerical approximation to equation (1) from 𝑔(𝑁, 𝑡) = 𝛼 given 

by Euler's method, the following comparisons between the analytical and numerical graphs are presented. 

Numerical simulations and the analytical graph were generated from MATLAB. The h for Euler's method was 

0.05 for both experiments. 

In the first experiment, 𝑁0 = 1, 𝛼 = 0.05 and 𝑁 = 1000000 were considered as initial parameters, that is, if it 

starts with just one individual who adopts the innovation and if it intends to reach 1 million adepts. In the second 

experiment, the same initial parameters were considered, with a different value for α, that is, 𝛼 = 0.0001. 

Figure 2 presents the exact and approximate solutions for the parameters described above. 

 

( a )                                                                                         ( b ) 

Figure 2: Spreading fake news by strictly external influence. 

( a ) Diffusion coefficient α=0.05; ( b ) Diffusion coefficient α=0.0001. 

In Figure 2 (a) it is observed that, in 20 minutes, a little more than 600000 people are reached by fake news 

while in Figure 2 (b) 2000 people are reached by fake news in 20 minutes. It should be noted that, with this 

diffusion coefficient, an individual affected by fake news does not become a multiplier, that is, he does not 

influence other people. This is the main criticism of this model: it is not realistic, as the network influences the 

propagation of fake news. 



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Case II: 𝒈(𝑵, 𝒕) = 𝜷𝑵(𝒕) 

For this model, it is assumed that the diffusion coefficient is linearly proportional to the current number of 

adopters. The crucial difference of this model in relation to Case I is the influence of innovation adopters in 

obtaining new adopters. In the context of the dissemination of fake news, everyone affected by the fake news 

can become a replicator of it, depending, of course, on the value of the β coefficient. The function that 

represents the solution to equation (1) considering the function 𝑔(𝑁, 𝑡) described above is 

𝑁(𝑡) =
𝑁

�̅�−𝑁0
𝑁0

𝑒−𝛽�̅�𝑡+1
     (3) 

Again, experiments were carried out using two different values for β, considering again the parameters 𝑁0 =

1,N 𝑁 = 1000000 and β=0.0001 using the analytical solution (3) and the numerical approximation via Euler's 

method with h =0, 05. Figure 3 presents the simulations for the parameters described above through MATLAB. 

 

Figure 3: Spreading fake news by internal linear spreading. 

It is possible to observe that, in this case, the total number of adopters of fake news is reached in approximately 

2.35 minutes, that is, much faster than the previous case. It can be deduced that if a part of the adopters of fake 

news becomes a multiplier of the same, the fake news is quickly disseminated. 

Case III: 𝒈(𝑵, 𝒕) = 𝜶 + 𝜷𝑵(𝒕) 

This model admits a mixed diffusion coefficient, that is, composed of a part referring to external influence and 

another part referring to internal influence. In this way, the diffusion of innovation takes place through external 

factors in the same way that adopters become multipliers of it. 

The analytical solution for this model is given by 

𝑁(𝑡) =
𝑁−

𝛼(�̅�−𝑁0)

𝛼+𝛽𝑁0
𝑒−(𝛼+𝛽�̅�)𝑡

1+
𝛽(�̅�−𝑁0)

𝛼+𝛽𝑁0
𝑒−(𝛼+𝛽�̅�)𝑡

,  (4) 



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In order to carry out the experiments with the analytical solution (4) and the numerical solution by Euler's 

method for h=0.05 for the previous function 𝑔(𝑁, 𝑡), we used as input data 𝛼 = 0.001, β =0.0001, 𝑁 = 100000 

and 𝑁0 = 1. Figure 4 is the exact and approximate solution for the above data. 

 

Figure 4: Spreading fake news by mixed spread. 

It is possible to observe that in 2.05 minutes the dissemination of fake news reaches the total number of people 

intended, that is, when there is a mixed mechanism for the dissemination of fake news. It is concluded, 

therefore, that the most efficient mechanism of dissemination is the one that uses internal and external 

influences. 

5. Final Considerations 

The present work aimed to analyze the mathematical models of innovation diffusion as a modeling proposal for 

the dissemination of fake news. 

Based on a model of innovation diffusion, in which individuals are divided between innovators and laggards, 

totaling 100% of the population, the relevance of innovation diffusion was observed in three scenarios: only by 

external influence, only by the influence of those that adopted the innovation and in a mixed way, that is, by 

external or internal influence. 

Translating the model of diffusion of innovation to the dissemination of fake news, it is possible to observe that, 

not always, the models presented in this work translate reality. The model that only contemplates external 

influence was the one that presented the lowest speed of propagation of innovation, that is, it would have the 

least impact on the dissemination of fake news. However, this model is not realistic since it does not consider 

the influence of the information network itself on disclosure. This model would simulate, for example, a post on 

Twitter that doesn't receive any likes or retweets, and users see it and don't comment to anyone else about it. In 

fact, this is an unrealistic situation. 

Among the situations of internal influence and mixed influence, the simulation of mixed influence is the most 



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realistic: individuals can be informed from comments, likes or retweets (due to the influence of someone's 

profile or someone's comments), by a posting an unknown profile or an ad. This case presents the highest speed 

of dissemination of fake news, reaching the total number of individuals that is estimated in the shortest period. 

From the simulations presented, more refined mathematical models can be proposed in future works, to try to 

absorb the different profiles of people who are affected by fake news according to their profile of adoption of 

innovations, a fact that was not discussed in the model introduced. It is concluded, therefore, that the innovation 

diffusion model can be adapted for the dissemination of fake news. However, the model does not have 

characteristics that can mitigate the spread of fake news. It is still believed that an education for social networks 

is the most effective instrument in the fight against disinformation. 

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