**Advancements in the Practical Applications of Agents, Multi-Agent Systems and Simulating Complex Systems**

Editors

**Philippe Mathieu Juan M. Corchado Alfonso Gonz´alez-Briones Fernando De la Prieta Pintado**

*Editors* Philippe Mathieu Department of Computer Science Lille University Villeneuve d'Ascq France

Juan M. Corchado Department of Computer Science and Automation University of Salamanca Salamanca Spain

Alfonso Gonzalez-Briones ´ Department of Computer Science and Automation University of Salamanca Salamanca Spain

Fernando De la Prieta Pintado Department of Computer Science and Automation University of Salamanca Salamanca Spain

*Editorial Office* MDPI St. Alban-Anlage 66 4052 Basel, Switzerland

This is a reprint of articles from the Special Issue published online in the open access journal *Systems* (ISSN 2079-8954) (available at: www.mdpi.com/journal/systems/special issues/paams22).

For citation purposes, cite each article independently as indicated on the article page online and as indicated below:

Lastname, A.A.; Lastname, B.B. Article Title. *Journal Name* **Year**, *Volume Number*, Page Range.

**ISBN 978-3-0365-9309-8 (Hbk) ISBN 978-3-0365-9308-1 (PDF) doi.org/10.3390/books978-3-0365-9308-1**

© 2023 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs (CC BY-NC-ND) license.

## **Contents**

## **About the Editors**

#### **Philippe Mathieu**

Philippe Mathieu received his Ph.D. degree in computer science from the University of Lille, France, in 1991. He is now full professor at the University of Lille. He is the team leader of the SMAC team in CRIStAL Lab, UMR 9189 CNRS since 22 years. His research focuses on Artificial Intelligence, Multi-agent Systems, Computational game theory and Computational economics. He is the author of numerous publications and communications on these research fields.

#### **Juan M. Corchado**

Juan Manuel Corchado (15 May 1971, Salamanca, Spain) is Professor at the University of Salamanca, Director of the BISITE Research Group (Bioinformatics, Intelligent Systems and Educational Technology) and President of the AIR Institute. He is also the Director of the IOT Digital Innovation Hub, President of the AIR Institute, and Principal Investigator of the Project funded by DIGIS3 (Digital Innovation Hub of Castilla y Leon). He is also Visiting Professor at the Osaka Institute ´ of Technology and Visiting Professor at the Universiti Malaysia Kelantan.

Juan M. Corchado has been Vice-Rector for Research from 2013 to 2017 and Director of the Science Park of the University of Salamanca. He was elected twice as Dean of the Faculty of Science, has been president of the IEEE Systems, Man and Cybernetics association, and academic coordinator of the University Institute for Research in Art and Animation Technology of the University of Salamanca, as well as researcher at the Universities of Paisley (UK), Vigo (Spain), and the Plymouth Marine Laboratory (UK). He has also been a member of the Advisory Group on Online Terrorist Propaganda of the European Counter Terrorism Centre (EUROPOL) and has been a visiting professor at the University of Technology Malaysia.

#### **Alfonso Gonz´alez-Briones**

Alfonso Gonzalez-Briones earned a Ph.D. in Computer Engineering in 2018 at the University ´ of Salamanca; his thesis obtained the second place in the 1st SENSORS+CIRTI Award for the best national thesis in Smart Cities (CAEPIA 2018). At the same University, he had also obtained a Bachelor of Technical Engineering in Computer Engineering (2012), a Bachelor's Degree in Computer Engineering (2013), and a Master's Degree in Intelligent Systems (2014). Since 2014, Alfonso Gonzalez ´ Briones has worked in different public research centres, such as the BISITE Research Group, and at Complutense University of Madrid as a "Juan De La Cierva" Postdoc. Moreover, he has worked at private research centres, such as Virtual Power Solutions (VPS) in Coimbra, (Portugal), where he was involved in research on decision tree algorithms with a high percentage of efficiency in the recognition of devices that consumed energy, as well as parameters that helped to improve the accuracy of recognition and detection; in the ISLab Research Group (Synthetic Intelligence Lab) of Universidade do Minho, Braga (Portugal), he was involved in a research on the methodologies and algorithms that achieve energy efficiency through the use of Artificial Intelligence techniques. He has also worked as Project Manager in Industry 4.0 and IoT projects in the AIR Institute, and Lecturer at the International University of La Rioja (UNIR).

Currently, Alfonso Gonzalez Briones is Associate Professor at the University of Salamanca in ´ the Department of Computer Science and Automation.

#### **Fernando De la Prieta Pintado**

Fernando de la Prieta Pintado is Associate Professor at the University of Salamanca Department of Computer Science and Automation, where he currently is Deputy Secretary-General of the University of Salamanca. Dr. De la Prieta is equally well experienced in research and teaching, as evident in his curriculum.

Over the past years, he has followed a clearly defined line of research, focusing on the integration of multi-agent organisations, machine learning and advanced architectures in different fields. He applied the resulting knowledge toward both his doctoral thesis (for which he obtained an international PhD mention and an extraordinary PhD award) and in the projects he has been involved in. He has more than 50 publications in international journals, many of which have a JCR impact factor on the Web of Science database. His H index in Google Scholar is 27. Furthermore, he has published more than 100 articles in books and in the proceedings of prestigious international conferences, around thirty of these publications have been published in conferences indexed according to the CORE ranking. He has worked on more than 90 research projects (16 of them were international and in several he has been the principal investigator). In addition, he has participated in more than 30 research contracts (Art. 83), in some of them as the principal investigator. As a result of his work, around 40 intellectual properties have been registered. He has performed several stays abroad (pre- and post-doctoral) in Portugal, Japan and South Korea. He has also taken an active part in the organisation of international conferences, some of them included in the CORE ranking: IEEE-GLOBECOM (core B), ICCBR (Core B), CEDI, PAAMS (core C), ACM-SAC (core B), IEEE-FUSION (core C), and others.

## **Preface**

The relationship between individuality and aggregation is an important topic in complex systems science, as both aspects are facets of emergence. This problem has generally been addressed by adopting a classical individual- versus population-level approach in which boundaries emerge in segregated communities. More specifically, boundaries delimiting and interconnecting aggregates are at play. It is, therefore, crucial to define the properties of complex systems correctly, such as generic agent-based models, with which to simulate communities situated in grid- and scale-free network environments. To do this, complexities may be resolved through simulation, modeling and analysis techniques, which help provide confidence regarding the behavior of such systems, especially of those operating in dynamic environments or under unexpected constraints. Moreover, modeling and simulation help reduce the risks and costs involved in the design and development of validation tests.

Understanding the emergent behaviors of complex systems, and ensuring their correct performance in different environments, will allow for their evolution, as well as that of the methodologies integrated in them.

Although various methodologies are being used for the development of Complex Systems Simulation, one of the most widely adopted approaches is based on the agent paradigm, which may be used to create simulations for dynamic continuous time systems and discrete event systems. Agent-based systems may be applied in all sorts of areas, including home automation, industry, smart cities and automotive sectors.

This Special Issue invited researchers to submit original, quality studies regarding the domain of Complex Systems Simulation and urged them to address its main sub-disciplines.

### **Philippe Mathieu, Juan M. Corchado, Alfonso Gonz´alez-Briones, and Fernando De la Prieta Pintado**

*Editors*

## *Editorial* **Advancements in the Practical Applications of Agents, Multi-Agent Systems and Simulating Complex Systems**

**Philippe Mathieu 1, Juan Manuel Corchado 2,3,4, Alfonso González-Briones 2 and Fernando De la Prieta 2,\***

#### **Introduction**

This Editorial summarizes the content of the Special Issue entitled *Advancements in The Practical Applications of* Agents, Multi-Agent Systems *and Simulating Complex Systems*, published in the "Complex Systems" section of *Systems* (ISSN 2079-8954).

Complex systems have played a fundamental role in the simulation, modeling, and analysis of information in dynamic environments and under unexpected constraints [1–3]. These agent-based systems have evolved significantly throughout history, providing increasingly sophisticated solutions to address the complex challenges encountered across multiple fields. The history of complex systems dates back to early research in systems theory and cybernetics in the 1940s [4]. These disciplines laid the foundation for understanding and addressing problems involving complex and emergent interactions between multiple components. As computer technology advanced, the first agent-based modeling and simulation approaches emerged, allowing complex systems to be represented through the interaction of multiple autonomous entities [5,6].

At the heart of complex systems are agents, which can be individuals, organizations, robots, or any entity with the ability to make decisions and respond to its environment. These agents interact with each other and with their environment, generating emergent patterns and collective behaviors that cannot be attributed solely to the individual characteristics of the agents. Agent-based systems technology has been advancing rapidly, enabling greater sophistication in the representation and simulation of complex systems [7]. The importance of complex systems lies in their ability to address real-world problems in a wide range of disciplines, including economics, biology, ecology, logistics, and supply chain management, among others. These systems can model and simulate complex phenomena such as crowd behavior, traffic flow, the spread of disease, climate change, and the evolution of ecosystems [8–10].

The simulation and modeling of complex systems offer several significant advantages. First, they enable the evaluation of different scenarios and strategies without incurring the costs and risks associated with real-world implementation. This is especially valuable when dealing with unpredictable or highly complex environments wherein it is difficult to obtain empirical data or conduct controlled experiments [11]. Complex systems provide a deeper understanding of the underlying mechanisms and interactions that shape the system being studied. This helps to identify emerging patterns, hotspots, and non-intuitive behaviors, which in turn can guide decision-making and strategic planning. By better understanding complex systems, it is possible to reduce the risks and costs associated with the design and development of real-world validation tests [12–14].

**Citation:** Mathieu, P.; Corchado, J.M.; González-Briones, A.; De la Prieta, F. Advancements in the Practical Applications of Agents, Multi-Agent Systems and Simulating Complex Systems. *Systems* **2023**, *11*, 525. https://doi.org/10.3390/ systems11100525

Received: 6 September 2023 Accepted: 19 October 2023 Published: 21 October 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

1

#### **An Overview of Published Articles**

This Special Issue consists of fifteen practical papers covering key topics in the field of multi-agent systems and complex systems. These articles, presented during the 20th International Conference on Practical Applications of Agents and Multi-Agent Systems (PAAMS'22) (https://www.paams.net, 6 September 2023), are noteworthy for their highly innovative results and trends [15]. The conference was held in L'Aquila, Italy, and it was organized by the University of L'Aquila (Italy), Umeå University (Sweden), the University of Lille (France), and the University of Salamanca (Spain)

The first three articles address the relevance of using multi-agent systems for the analysis of information. Such an analysis is carried out through the application of natural language processing techniques. These articles aimed to understand public opinion, detect false information, or identify accounts that provide misleading information. Guzmán Rincon et al. (Contribution 1) present a mathematical model to simulate scenarios of disinformation propagation in social networks caused by bots, trolls, and others. The authors carried out simulations related to the increase in the rate of the activation and deactivation of disinformation agents and the disinformation caused by this mechanism. Ye et al. (Contribution 2) explored the specific attributes of individuals and opinion network nodes by incorporating parameters such as individual conformity and the strength of individual online relationships for the purpose of identifying an online opinion polarization of a group. Through simulations, the authors found that individual conformity and the difference in environmental attitude greatly influence the trajectory of opinion polarization events. Similarly, the analysis of shared beliefs, opinions, and views in groups is a topic of great interest that has been debated in sociology, political science, communication, and organizational science. Koponen (Contribution 3) performed an analysis of consensus group formation through an agent-based model. Agents' views were described as complex, and they have extensive structures, similar to semantic networks, i.e., belief networks. In the agent-based model presented by the author, the agents' interactions and their participation in the sharing of their views depend on the similarity of the agents' belief webs; the higher the similarity, the more likely the interaction and the sharing of webs of belief elements.

In the areas of economics, finance, and e-commerce, complex systems have also had a major impact. Zhao et al. (Contribution 4) present an agent-based model created using empirical data from a number of cities as sample data to simulate the evolutionary trajectory of eco-protection and high-quality development under different policy scenarios, such as green innovation, ecological constraints on the environment, ecological compensation, etc. The model shows how, depending on the existing development model, the economic development of cities will be subject to different degrees of ecological and resource constraints and that different policy scenarios significantly affect the evolutionary trends of economic development. Other authors, such as Bae et al. (Contribution 5), introduce a formalism or multi-resolution translational discrete event system specification (MRT-DEVS) intended to facilitate the implementation of simulations and reduce simulation execution costs. MRT-DEVS embeds state and event translation functions into the model's specifications so that it enables multi-resolution modeling with less complex mechanisms in terms of operations. Wang et al. (Contribution 6) studied the product encroachment behavior of composite e-commerce platforms with double-differentiated multi-product competition and constructed a game model of product innovation by an independent seller and product encroachment by the platform owner. Using multi-agent simulation, the authors simulated the bounded rational decision-making and interaction process of multiple agents in multiple periods and analyzed the influence of the main parameters. Moreover, Castañón-Puga et al. (Contribution 7) illustrate how earned value management (EVM) is an efficient method for measuring a project's performance by comparing actual progress against planned activities, thus facilitating the formulation of more accurate predicted estimations using an agent-based simulation model.

Researchers have also focused on applying complex system algorithms to facilitate problem solving in the field of transport. Karalakou et al. (Contribution 8) propose the

design of autonomous vehicles using deep reinforcement learning and the combination between various reward components that are able to gradually learn effective policies in environments with different levels of difficulty, especially when all the proposed reward components are appropriately combined. Spanoudakis et al. (Contribution 9) have designed an open system for the V2G/G2V power transfer problem domain using an agent-based architecture involving flexible microservices that are interconnected via an IoT platform. Gómez Vilchez et al. (Contribution 10) describe a simulation model that facilitates the analysis of potential emission penalties in the broader context of the financial position of original equipment manufacturers. Through their simulation, the authors aim to understand the channels through which money flows (e.g., to promote R&D in cleaner vehicles and to finance zero-emission powertrain sales) between market players.

On the other hand, agent systems have demonstrated successful performance in the application of Cartesian genetic programming to solve a series of use cases, such as complete enumeration in local agent decisions. In this context, Bremer et al. (Contribution 11) present the adaptation of a distributed optimization heuristic protocol for Cartesian genetic programming and an extension using CMA-ES (Covariant Matrix Adaption Evolution Strategy) to improve local agent decisions. By decomposing the evolution on an algorithmic level, it becomes possible to distribute the nodes and regard the evolution process as a parallel, asynchronous execution of an individual coordinate's descent.

Atrazhev et al. (Contribution 12) address the issue of choosing an appropriate reward function in multi-agent reinforcement learning. Among the traditional approaches to employing joint rewards for team performance, this one is questioned because of its lack of theoretical support. Thus, the authors explore the impact of changing the reward function from joint to individual on learning centralized–decentralized execution algorithms in a level-based foraging environment. The results show that different algorithms are affected differently, with value factorization and proximal policy optimization (PPO)-based methods taking advantage of the increased variance to achieve better performance. This study sheds light on the importance of considering the choice of a reward function and its impact on multi-agent reinforcement learning systems.

Within the area of optimization, Pincheira et al. (Contribution 13) present a framework for evaluating the infrastructure costs and benefits of blockchain applications. The framework includes a taxonomy that classifies relevant transactions, a model to evaluate the infrastructure costs and application benefits using public or private blockchains, and guidance on how to use the model. Another research work focusing on optimization comes in the form of the paper by Esmaelii et al. (Contribution 14). The authors of this paper introduce an agent-based collaborative technique for finding near-optimal values for any arbitrary set of hyperparameters (or decision variables) in a machine learning model (or a blackbox function optimization problem). The developed method forms a hierarchical agent-based architecture for the distribution of the searching operations at different dimensions and employs a cooperative searching procedure based on an adaptive width-based random sampling technique to locate the optima.

Finally, within this Special Issue, Roussel et al. (Contribution 15) address the issue of conflicting bundle allocation and weighted directed acyclic graphs. The authors propose several models for novel resource allocation problems where agents express their preferences over conflicting bundles of items as edge-weighted on a directed acyclic graph (directed path allocation problem, or DPAP), particularizing conflicts on vertices (V-DPAP) and conflicts on resources (R-DPAP). The multi-agent system proposed by the authors allows for the search of path allocation. Conflicting bundle allocation and weighted directed acyclic graphs are also commonly simulated using complex systems.

#### **Conclusions**

This Special Issue showcases a variety of research papers on practical approaches to the use of complex systems and complementary agent-based AI models, facilitating the parallel use of data treatment and knowledge processing algorithms.

#### **List of Contributions**

**Author Contributions:** P.M., J.M.C., A.G.-B. and F.D.l.P. worked together throughout the entire editorial process of this Special Issue entitled "Advancements in The Practical Applications of Agents, Multi-Agent Systems and Simulating Complex Systems", published by the *Systems* Journal. A.G.-B. and F.D.l.P. drafted this editorial summary. P.M., F.D.l.P. and J.M.C. reviewed, edited, and finalized the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the project "Coordinated intelligent Services for Adaptive Smart areaS (COSASS)", Reference: PID2021-123673OB-C33, financed by MCIN/AEI/10.13039/<PHONE_NUMBER>33/FEDER, UE.

**Acknowledgments:** First and foremost, we would like to thank all the researchers who submitted articles to this Special Issue for their excellent contributions. We are also grateful to all the reviewers who helped in the evaluation of the manuscripts and made very valuable suggestions to improve the quality of the contributions. We would like to acknowledge the editorial board of *Systems*, who invited us to guest edit this Special Issue. We are also grateful to the *Systems* Editorial Office staff, who worked thoroughly to maintain the rigorous peer-review schedule and ensure the timely publication of this Special Issue.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

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## *Article* **Disinformation in Social Networks and Bots: Simulated Scenarios of Its Spread from System Dynamics**

**Alfredo Guzmán Rincón 1,\*, Ruby Lorena Carrillo Barbosa 2, Nuria Segovia-García 1 and David Ricardo Africano Franco 2**

**Abstract:** Social networks have become the scenario with the greatest potential for the circulation of disinformation, hence there is a growing interest in understanding how this type of information is spread, especially in relation to the mechanisms used by disinformation agents such as bots and trolls, among others. In this scenario, the potential of bots to facilitate the spread of disinformation is recognised, however, the analysis of how they do this is still in its initial stages. Taking into consideration what was previously stated, this paper aimed to model and simulate scenarios of disinformation propagation in social networks caused by bots based on the dynamics of this mechanism documented in the literature. For achieving the purpose, System dynamics was used as the main modelling technique. The results present a mathematical model, as far as disinformation by this mechanism is concerned, and the simulations carried out against the increase in the rate of activation and deactivation of bots. Thus, the preponderant role of social networks in controlling disinformation through this mechanism, and the potential of bots to affect citizens, is recognised.

**Keywords:** disinformation; social networks; bots; model

#### **1. Introduction**

The academic community has shown widespread interest in understanding how disinformation spreads in virtual media, including social networks, e.g., [1–7], due to the potential of disinformation to trigger various problems for governments, citizens, and other social actors [2]. Thus, the state approach has attributed multiple consequences to disinformation on social networks, such as: the polarisation of citizens' opinions [4], the destruction of the credibility of traditional media [8], the mobility of citizens to prevent the development of public policies [9], among others.

Recently, the spread of misinformation has been growing exponentially [1] as a result of the massive use of social networks. An example of this was the case of COVID-19, when the Russian media RT and Sputnik accused NATO and the United States of America of creating the virus in order to destabilise the Chinese economy, and this information was widely spread on social networks such as Facebook, Twitter and Tik Tok [3,10], or, in the case of the vaccines developed for COVID-19, where the anti-vaccine movement sought to attribute effects such as autism and possible genetic malformations to their use, triggering mistrust on the part of the population and preventing the control of the virus and the mitigation of its transmission [11]. In view of these examples, one of the main problems for social actors, in particular states, as well as the academic community, is the lack of awareness of the existence of this type of information and the lack of understanding of the strategies used by the disinformation agent to ensure the propagation of misinformation on social networks [12–14].

**Citation:** Guzmán Rincón, A.; Carrillo Barbosa, R.L.; Segovia-García, N.; Africano Franco, D.R. Disinformation in Social Networks and Bots: Simulated Scenarios of Its Spread from System Dynamics. *Systems* **2022**, *10*, 34. https://doi.org/10.3390/ systems10020034

Academic Editors: Philippe Mathieu, Fernando De la Prieta, Alfonso González-Briones and Juan M. Corchado

Received: 16 February 2022 Accepted: 4 March 2022 Published: 10 March 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

This article focuses on the second problem area. Thus, it is evident from the literature that disinformation can be propagated by the direct intervention of individuals either consciously or unconsciously, as well as by automated accounts known as bots [15]. These bots are present in all social networks, however, in some of them their presence is more noticeable. An example of this is Twitter, where, given the importance of this network in issues related to politics, it has been estimated that between 9% and 15% of active accounts are bots [15–18].

Due to the proliferation of bots used in social networks to misinform and the interest in this mechanism, the literature has focused on establishing the role of bots in the spread of misinformation, with a focus on describing case studies, e.g., [15,18–24], however, the development of models to understand the patterns of propagation of disinformation caused by this mechanism is still in its beginnings, with most of the advances being generalist and not specialised in bots such as the works of Lazer et al. [24], Shao et al. [25]; Vosoughi et al. [26] and Shao et al. [27]. The limited amount of work is due to the difficulty of finding the initial sources of disinformation [4], as well as the absence of robust and easily accessible tools for identifying bots, and thus correctly identifying their activities [15].

In this context, the aim of this article was to model and simulate scenarios of disinformation propagation in social networks caused by bots based on the dynamics of this mechanism documented in the literature. This will strengthen the understanding of the phenomenon of disinformation through bots by making it possible to establish patterns of behaviour in the system and to evaluate the effects of the various decisions made by the actors involved in disinformation, especially in relation to social network policies.

It is important to highlight that, although there is no unified meaning of disinformation due to the large number of definitions and intermediate terms found in the literature, particularly in the literature from Anglo-Saxon countries, (e.g., fake news, misinformation, etc.), this work assumes disinformation to be any deviant information that is intended to distort and mislead a target audience in a predetermined way [3,28]. In this way, disinformation refers to a wide range of content, including fake news, misinformation, misleading content, hate speech and deliberate misreporting, among others [29–31]. Additionally, disinformation is not only about the message itself, but as a practice it has the potential to discredit the messenger and true information due to its close relationship with multiple social sectors, especially politics.

This article is structured in five main sections. The first one broadens the conceptualisation of bots and their features; the second one establishes the behaviour of the disinformation caused by this mechanism in social networks and introduces the diagram of causal loops with their respective dynamic hypotheses; the third section shows the methodology used while the fourth section describes the results with emphasis on the mathematical model developed and the simulations defined in the methodology; finally, the fifth section presents the conclusions of the study.

#### **2. Theoretical Framework and Dynamic Model**

#### *2.1. What Are the Bots?*

The use of the word bots corresponds to an abbreviation of the Anglicism "software robot" [15], which has permeated the disinformation literature, including that of Latin American origin. Thus, the bots as machines are usually automated to some degree, either independently or with human intervention. The bots as machines are put at the service of external agents, in this case diplomacy. In this sense, this type of software can be categorised as either beneficial or malicious [32]. In the case of those that are beneficial, they are programmed by companies to improve the attention and service provided to their customers [33], or as in the case of the social network Twitter, some are used to inform the time, as documented by Yang et al. [32], or to support the raising of charitable resources [34].

For those called malicious, in some cases they oversee malware distribution or spam generation, but in social networks they have another purpose, which is to automatically produce content and interact with humans on social networks, trying to emulate them and possibly alter their behaviour [35–37]. According to the Academic Society for Managment and Communication [38] these bots have three strategies for influencing the behaviour of online users on social networks:

Although the strategies described above are the basis of the strategies used by the malicious bots, it is necessary to recognise that, along with the emergence of new technologies, these tend to evolve and new strategies emerge, which is why a general view of the behaviour of the system generated by this type of software is required, rather than the study of each of the strategies themselves.

#### *2.2. Disinformation, Bots and Causal Loop Diagrams*

Disinformation as a social phenomenon that occurs in social networks is not the result of chance, but of a strategic analysis developed by the disinforming agent, as related by Guzmán and Rodríguez-Canovas [2]. Therefore, the disinforming agent seeks to attract the largest possible target population in order to convert it into a population susceptible to being disinformed [39], which is where the various mechanisms for disinformation, such as bots, emerge [5]. With the linking of the population susceptible to disinformation, we proceed to the propagation of the message [40] to consolidate the disinformed population, which for the purposes of this study will be understood as the individual or subject who was exposed to the message of the disinforming agent.

Considering the above as a basis, the role of bots as a mechanism for spreading disinformation begins with their activation, where the disinformation agent establishes how many he or she wishes to have, however, maintaining a fixed number over time is difficult, given the detection mechanisms that social networks have to eliminate or block this type of account [16,41]. However, the interaction of these fake accounts is usually done with publications based on powerful hashtags, comments and by sharing content, being standardised in terms of the number of characters when providing a machine, as well as in the frequency and time of publications [42]. In view of the number of the target population, which will become a susceptible and post-informed population, bots are characterised by making exclusive use of the organic reach of the account, so that as the number of followers grows, this reach tends to decrease, requiring a greater number of bots to impact a greater number of the population [2]. Figure 1 shows the causal loop diagram that represents the behaviour of the studied phenomenon.

Finally, it should be noted that the spread of disinformation is like the way a disease is spread, where there is a population of infected people (bots), who seek to spread the disinformation message in a susceptible population, hence the previous models developed are based on the SIR model (Susceptible—Infected—Recovered), e.g., [43,44]. Thus, previous studies have sought to complement the SIR model by analysing specific disinformation mechanisms, as well as models that integrate several of these mechanisms, (e.g., bots, trolls, paid outreach, etc.). These models include the SIRaRu model, which allowed us to understand the behaviour of disinformation in homogeneous and heterogeneous communities [45], the SEIR model (Susceptible—Exposed—Infectious—Recovered), the SIR model for dynamic and complex social networks [46], among others. Hence, the model presented here both in Figure 1 and in the subsequent sections is based on the SIR model.

**Figure 1.** Diagram of causal loops. Note: B represents the balance loops of the system and R represents the reinforcement loops.

#### **3. Methodology**

The aim of this article was to model and simulate scenarios of disinformation propagation in social networks caused by bots based on the dynamics of this mechanism documented in the literature, so the main technique used for the development of the model was System Dynamics, considering Bala et al. [47] and Bianchi [48] as theoretical references. In this sense, the choice of modelling technique is based on the complexity of the disinformation caused by bots, in which various elements are involved and whose behaviour is characterised by a non-linear, multicausal and time lagged behaviour [47]. The model is based on the existing literature on the problem under study, for which the steps established by Bala et al. [47] were followed and summarised below:

With the proposed model, we proceeded to develop the simulations presented in Table 1, for which modifications were made to the parameters established in the initial model (Table 2); it should be noted that in the execution of the simulations, only the parameter indicated in Table 1 was modified, and the other parameters retained their initial values. The description of the modified parameters is presented in Table 2.

**Table 1.** Computer simulations.

**Table 2.** Variables required for model development and initial parameters.

The analyses of the simulations were carried out descriptively, and in order to determine the existence of statistically significant differences in the behaviour of the system, the medians of the PobD level were compared (see Table 2). The Kolmogorov-Smirnov statistic was applied to check whether the data fit a normal distribution (*p*-value > 0.05), and it was found that the data did not follow a normal distribution. Thus, to establish the difference in averages between the behaviour of the system with the initial parameters and the modified parameters, the Wilcoxon test was used, considering this difference with a *p*-value < 0.05.

Finally, the computational work on the model and the simulations were implemented in Stella Architect software version 1.9.5. The following model settings were considered: *ti* = 0, *tf* = 360, Δ*t* = 4, where *t* represents time in days; additionally, Euler was used as the integration method. SPSS software version 25 was used for the statistical analyses.

#### **4. Results**

The results are presented in two sections. The first one describes the model proposed with the capacity to replicate the behaviour of the system based on the evidence from the literature review; and the second one describes the simulations obtained and the corresponding statistical analyses.

#### *4.1. Model*

Figure 2 represents the flow and level diagram [1], which is based on the SIR model. Thus, it consisted of five level variables, five flow variables and 10 auxiliary variables. The green section represents the process of disinformation of the target population, and the blue section how the activation and deactivation of bots behaves.

**Figure 2.** Diagram of flows and levels of disinformation in social networks by means of bots.

The proposed model explains the behaviour of the phenomenon studied here, as long as the following assumptions are met:

Under the technical conditions of non-negativity of the level variables, (i.e., their domain is restricted to 0 or positive numbers) and that *t* = 0, 1, 2 ... , 360, the system was represented by the following differential equations.

Target population:

$$PobO\_t = \left[PolyO\_{t-1} + \left(PobO\_{t-1} \times TCpob\mathbb{C}\right) - \left(PobO\_{t-1} \times AO \times TEC \times Bots\right)\right]dt \tag{1}$$

Susceptible population:

$$PobS\_t = \left[PolyS\_{t-1} + \left(ProbO\_{t-1} \times AO \times TEC\right) - \left(ProbS\_{t-1} \times f(X\_{t\prime} X\_{t-\tau\prime} dt; t \ge t\_0)\right)\right]dt \tag{2}$$

where:

$$X\_t = \left[PolyS\_{t-1} \times Bots \times AO\right]dt\tag{3}$$

Misinformed population:

$$PobD\_{l} = \left[PolyD\_{l-1} + \left(PolyS\_{l-1} \times f(\mathbf{X}\_{l}, \mathbf{X}\_{l-\tau\_{l}}, dt; t \ge t\_{0})\right)\right]dt\tag{4}$$

Bots:

$$Bots\_l = \left[ Bots\_{l-1} + \left( Bots\_{l-1} \times TAB \times \frac{METB - Bots}{METB} \right) - \left( Bots\_{l-1} \times f(Y, Y\_{l-1}, dt; t \ge t\_0) \right) \right] dt \tag{5}$$

where:

$$Y\_t = \left[Bots\_{t-1} \times TDB \times RTDB\right]dt\tag{6}$$

Deactivated bots:

$$Bots\_t = \left[ Bots\_{t-1} + \left( Bots\_{t-1} \times f(\mathbf{Y}, \mathbf{Y}\_{t-\tau}, dt; t \ge t\_0) \right) \right] dt \tag{7}$$

Table 2 presents the description of the variables and the initial parameters for the operationalisation of the model. Because of the generalist nature of the proposed model, which is applicable to any social network, the initial parameters are susceptible to modification. In this particular case, those of Twitter were used, so parameters such as organic reach, paid reach, effective contact rate, among others, must be modified for its use in other social networks.

#### *4.2. Simulations*

Compared to the results obtained in SIM-1, it was found that under the initial parameters for *t* = 360 *PobO* increased by 230,000 people, with 159,000 being effectively uninformed, which represented 15.9% of the initial *PobO*. On the other hand, for *t* = 90, period in which the disinformation of *PobS* begins, this was close to 28,200 people, increasing until *t* = 111, after this day the *PobS* begins to decrease until it reaches the value of 0. In the case of the bots for *t* = 133 the highest number of activated automata was 54.8 ≈ 55 and for *t* = 360 the number of deactivated bots was 158. Figure 3 shows the behaviour of the system for SIM-1.

**Figure 3.** Results SIM-1. Note: (**a**) behaviours for the levels of the disinformation process and (**b**) behaviour of the activation and deactivation of bots.

The results for SIM-2 show that with a higher activation rate *bot*, the *PobO* for *t* = 360 would be equal to 1,150,000 people, with *PobD* being 228,000, which represented an increase of 43.39% compared to SIM-1. For the bots in this simulation for *t* = 132 the highest number of activated automata would be reached with a total of 81.2 ≈ 82. Similarly, due to the increase in the activation rate for *t* = 360, a total of 238 *bots* would have been deactivated, showing an increase of 50.63% in relation to SIM-1. On the other hand, a comparison between the *PobD* between SIM-1 and SIM-2 showed statistically significant differences with *z* = −16.42, *p*-value = 0.00 . Figure 4 shows the behaviour of the system for SIM-2.

For SIM-3, which sought to simulate a higher rate of bots' deactivation by social networks, a change in system behaviour was evident (see Figure 5). Thus, the disinformation target population *PobO* for *t* = 360 would be equal to 1,260,000 people with a total of 109,000 people being disinformed, decreasing the *PobD* by 31.44 % in relation to SIM-1. Similarly, in the case of the susceptible population, there are two peaks at *t* = 107 and *t* = 360 (see Figure 5a), correlated with the number of active bots (see Figure 5b). Regarding

the *PobD* level between SIM-3 and SIM-1, it was determined that there are statistically significant differences with *z* = −14.00 *p*-value = 0.00 .

**Figure 4.** Results SIM-2. Note: (**a**) behaviours for the levels of the disinformation process and (**b**) behaviour of the activation and deactivation of bots.

**Figure 5.** Results SIM-3. Note: (**a**) behaviours for the levels of the disinformation process and (**b**) behaviour of the activation and deactivation of bots.

Now, in the case that disinformation starts to circulate at *t* = 30, it was observed that the target population of disinformation *PobO* for *t* = 360 would be equal to 1,240,000, being similar to the behaviour derived from SIM-1. For the population that managed to be uninformed, it was determined that for *t* = 360 the *PobD* was 145,000. Similarly, in the case of bots, and due to the ability of social networks to deactivate them, after *t* = 170 the number of active automatons tends to stabilise. Thus, for *t* = 360 a total of 23.3 ≈ 24 active bots and 147 deactivated bots were evident. On the other hand, the comparison between the *PobD* between SIM-1 and SIM-2 established statistically significant differences with *z* = −14.66, *p*-value = 0.00 . Figure 6 shows the behaviour of the system for SIM-4.

**Figure 6.** Results SIM-4. Note: (**a**) behaviours for the levels of the disinformation process and (**b**) behaviour of the activation and deactivation of bots.

#### **5. Conclusions**

The objective of this study, which was to model and simulate scenarios of the propagation of disinformation in social networks caused by bots based on the dynamics of this mechanism documented in the literature, was achieved. The model presented has the capacity to replicate the behaviour of the system, being consistent with the dynamic

hypotheses set out in Figure 1, complementing previous studies such as those developed by de Lazer et al. [24], Shao et al. [25]; Vosoughi et al. [26] and Shao et al. [27], in relation to the use of bots as a mechanism to propagate false information.

This model allows actors involved in disinformation to analyse in a more objective way the behavioural patterns of disinformation caused by bots for decision making, based on three assumptions. The first one relates to the delay in the start of disinformation; the second one to the limited number of bots that the disinformation agent can put in place; and the third one to the limitations of social network systems to detect and deactivate automated accounts.

With that said, the simulations developed clarify that the system of disinformation using bots is susceptible to policies that are conducive to better detection, blocking and elimination of these types of accounts. In this sense, the uninformed population is smaller, hence the responsibility of social networks to design better detection mechanisms and of citizens to report these types of accounts, to increase the effective rate of deactivation of bots. However, if the disinformation agent starts its activity early, the impact over time will not be on the amount of the disinformed population, but on the number of bots required to achieve its purpose, since the number of bots tends to stabilise over time.

From the proposed model and the simulations developed, it is necessary to recognise the role of bots in aggravating existing social problems as a result of the propagation of false information, hence the need to delve deeper into various analyses such as the evolution of this type of mechanism, the new technologies they incorporate to circumvent the security systems of social networks, the use of artificial intelligence in these, among other aspects. On the other hand, it is necessary to urge the academic community to make use of the model, to complement it and, above all, to eliminate the current barriers to the study of disinformation, such as the difficulties of access to declassified information that would allow the model to be operationalised under conditions different from those expressed in this article and which are based on secondary data from other studies.

**Author Contributions:** Conceptualization, A.G.R., R.L.C.B., D.R.A.F. and N.S.-G.; methodology, A.G.R.; software, A.G.R. and N.S.-G.; validation, R.L.C.B.; formal analysis, A.G.R.; investigation, A.G.R. and D.R.A.F.; resources, A.G.R.; data curation, A.G.R. and R.L.C.B.; writing—original draft preparation, A.G.R.; writing—review and editing, N.S.-G.; supervision, R.L.C.B.; project administration, A.G.R. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding, and the APC was funded by Corporación Universitaria de Asturias.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Thea data and the model are available at: https://exchange.iseesystems. com/models/player/alfredoguzmanrincon/disinformation-and-bots (accessed on 4 March 2022).

**Acknowledgments:** To Cecilia Carabajal who, with her unconditional support, made the style correction and translation of this article.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

## *Article* **A Novel Public Opinion Polarization Model Based on BA Network**

**Yuanjian Ye 1, Renjie Zhang 2, Yiqing Zhao 3, Yuanyuan Yu 4, Wenxin Du 3 and Tinggui Chen 1,\***

	-<EMAIL_ADDRESS>(Y.Z<EMAIL_ADDRESS>(W.D.)

**Abstract:** At present, the polarization of online public opinion is becoming more frequent, and individuals actively participate in attitude interactions more and more frequently. Thus, online views have become the dominant force in current public opinion. However, the rapid fermentation of polarized public opinion makes it very easy for actual topic views to go to extremes. Significantly, negative information seriously affects the healthy development of the social opinion ecology. Therefore, it is beneficial to maintain national credibility, social peace, and stability by exploring the communication structure of online public opinions, analyzing the logical model of extreme public attitudes, and guiding the communication of public opinions in a timely and reasonable manner. Starting from the J–A model and BA network, this paper explores the specific attributes of individuals and opinion network nodes. By incorporating parameters such as individual conformity and the strength of individual online relationships, we established a model of online group attitude polarization, then conducted simulation experiments on the phenomenon of online opinion polarization. Through simulations, we found that individual conformity and the difference in environmental attitude greatly influence the direction of opinion polarization events. In addition, crowd mentality makes individuals spontaneously choose the side of a particular, extreme view, which makes it easier for polarization to form and reach its peak.

**Keywords:** online public opinion; group polarization; influencing factors; power relations

#### **1. Introduction**

In recent years, with the rapid development of China's self-media platforms, the polarization of online public opinion has become more frequent, for example, the Tesla car brake failure incident, the self-explosion "0 sugar" incident in Genki Forest, and the China Express blind box pet incident, all of which have aroused widespread social concern. It can be noted that in the process of spreading online opinions, due to the reduction of transmission cost, the amount of information received by individuals increases. At the same time, information homogenization and fragmentation are serious, which makes it difficult for individuals to maintain a neutral and objective attitude toward their actual output. As a result, the information views of the surrounding environment tend to be consistent and then become a driver of polarizing events in online public opinion. In fact, in the process of information sharing and decision making, individuals' actual behaviors are generated by their objective cognition together with psychological activities. Consequently, it is very easy to collide with the surrounding environment and group views. Furthermore, the original decision is biased, leading to different degrees of polarization. Moreover, individuals have differences in age, occupation, family, education level and other various aspects. These differences make them have various sensitivities to the polarization of public opinion and

**Citation:** Ye, Y.; Zhang, R.; Zhao, Y.; Yu, Y.; Du, W.; Chen, T. A Novel Public Opinion Polarization Model Based on BA Network. *Systems* **2022**, *10*, 46. https://doi.org/10.3390/ systems10020046

Academic Editors: Philippe Mathieu, Fernando De la Prieta, Alfonso González-Briones and Juan M. Corchado

Received: 13 March 2022 Accepted: 7 April 2022 Published: 9 April 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

fluctuations of their own opinions and attitudes, resulting in a complex trend of public opinion polarization events.

Existing studies on opinion polarization usually use a relatively small and simple network structure to analyze changes in the attitudinal values of social groups and focus on the influence of individual heterogeneity on the connections between individuals. In fact, the polarization process of individual attitudes relies on a complex network structure in which complex mechanisms of opinion polarization arise.

Distinguishing from existing research tools, this paper considers the heterogeneity properties of nodes and defines them specifically based on the J–A model, as well as the threshold changes between individuals, adding parameters such as individual consistency and strength of network relationships, forming a complex network structure of the group, and using its basic characteristics of growth and preferential attachment, the BA scale-free network is selected for the study, which can be used to investigate the sensitivity problem of the model correction.

We denote the number of edges formed by individuals and their neighbors as 'degree' (P) and use the BA scale-free network to present the power law of the distribution of complex networks. After matching the degree, relationship strength, and attitude values with the relevant data, it is confirmed that the group communication behavior between individuals will make the opposing parties continuously reinforce their own views, and the trend of bifurcation of online opinions is obvious.

Further, we try to put the connections between nodes into the set network model, fully discuss the relationship between the polarization process of individual attitudes and the complex network structure, and comprehensively consider the network public opinion propagation mechanism and polarization prediction laws. Based on the opinion polarization model of the BA network, we simulated the collision process of individual opinions on the network and predicted the values of public attitudes after more than 400 collisions. In the evolutionary process, it was found that most individuals would actively choose extreme views to battle in the evolution of time. Moreover, when they are in a network group with the same interests as their own, they will keep looking for similar views to their own in the mutual communication with members to reinforce their original ideas.

In terms of social organization, the negative impact of public opinion polarization reversal will intensify contradictions in a disguised way. This makes the output effect of superimposed views one-sided and extreme, with less output space for the positive and effective viewpoint information. Based on the BA network's opinion polarization model, this paper predicts the trend of public attitudes toward online events. In general, it is conducive to control and govern online opinion polarization events by clearly understanding the dynamics and process of online opinion polarization, simulating public opinion and attitude, and controlling the heat of online public opinion in a timely fashion.

#### **2. Literature Review**

#### *2.1. Literature Review Based on J–A Model*

The J–A model refers to a new social attitude judgment model proposed by Jager and Amblard [1], which shows that the subject's attitude structure determines the occurrence of assimilation and alienation effects, which in turn lead to the phenomenon of consensus and polarization. Since then, there have been many studies related to the J–A model. Barash et al. [2] found that the complex infection model could produce highly nonlinear infection diffusion dynamics, and its critical mass had potential practical significance for the prediction of the early stage of transmission activities. Li and Tang [3] proposed the threshold model of group behavior and considered group spatial factors and the strength of social influence relationships among individuals. Based on the group polarization effect, Gabbay et al. [4] added a new explanation, that was, the interaction between individuals with the same interests will trigger the change of attitude to extremes in disguise. Chen et al. [5] used the J–A model to study the rumor diffusion process with

the consideration of individual heterogeneity. Subsequently, they took the imported food safety issue as an example during the COVID-19 pandemic and testified to the efficiency of the proposed model.

From the above analysis, it can be seen that the existing literature uses a relatively small and simple network structure to analyze the changes in social group attitude value, which is different from the structure in complex social networks. In addition, many articles do not give node heterogeneity attributes and do not consider the change of threshold between individuals, which is also far from reality. Based on this, this paper integrates the parameters such as individual conformity and network individual relationship strength into the classical J–A model, which makes the model well adapted to complex, real-world events.

#### *2.2. Literature Review of BA Models*

The BA model, or scale-free model, was proposed by Barabasi and Albert in 1999 [6]. They pointed out that the network produced by the BA model had the characteristic of no scale, and the distribution of its network degree values followed a power-law distribution, which was closer to most actual networks. Liu et al. [7] conducted further research and found that the BA model could only generate a network model in which the distribution of degrees follows a power index of 3, while the value in the actual network was usually between 1 and 3. Chen et al. [8] explored a multi-dimensional public opinion process based on a complex network dynamics model in the context of derived topics, and they found that information intensity was the most important influence factor. Zhou et al. [9] found that the network generated by the BA model did not have obvious small-world characteristics, while the actual network usually had both unscale and small-world characteristics. In addition, a large number of scholars have found that the BA model is prone to isolated nodes in the application process and has the characteristics of only "first rich" and not "later rich", which are not in line with the evolutionary characteristics of the actual network.

Combined with existing research, we find that most scholars focus on the interaction between nodes, emphasizing that the heterogeneous characteristics of individuals themselves will affect the connections between individuals. However, we notice that the connections between nodes are not only related to the properties of the nodes but also the network structure. Taking the BA model as the background, the model degree distribution is generally similar to the power-law distribution, and the connection between nodes has the characteristics of merit. This study will try to give a specific definition of the node's own attributes according to the characteristics of the actual network and emphasize the node characteristics in the network background in order to improve the adaptability of the model in the existing research.

#### *2.3. The Prediction Law of Online Public Opinion Dissemination and Polarization*

More and more people have been connected to the world through digital technology in recent years. As a result, public opinion can spread quickly. It is difficult for the public to identify and judge what they want from a large amount of data. At present, more scholars have already conducted in-depth studies on social network structures and the phenomenon of opinion polarization. For example, Wang [10] dissected the dynamic relationship between the factors influencing group attitudes. Chen et al. [11] analyzed the panic emotion propagation process and further identified the emergence process of group panic buying behavior under the COVID-19 pandemic. Wang et al. [12] considered the components of group polarization formation of online public opinion, quantitatively analyzed the mechanism of public opinion polarization dynamics and regulation strategies, and strongly argued the relevance of the main factors of public opinion development through an example simulation. Zhang et al. [13] proposed the intertextual characteristics of the process of generation, diffusion, and polarization in self-media online public opinion. Hatton [14] proposed that preference and significance are related to different individuallevel characteristics through the analysis of the European Social Survey and European barometer data. Heizler and Israeli [15] proposed that the tragedy of a specific individual

is more likely to cause the polarization of public opinion than the tragedy of a group. Blake et al. [16] believe that the neutrality and polarization of people's views vary according to sociodemographic characteristics, including age, gender, and education.

The above-mentioned literature has summarized the general law of the polarization phenomenon of online public opinion groups. However, the influencing factors and network structure in the polarization process are seldom analyzed. Based on this, this paper relies on a specific network structure to study the complexity of the polarization mechanism and the process of individual attitude polarization. By fully discussing the relationship between the two, we can understand the communication mechanism of network public opinion and the law of polarization prediction.

#### **3. A Novel Public Opinion Polarization Model Based on BA Network**

#### *3.1. Basic J–A Model*

Much of the existing research is discussed based on the D–W or J–A models. Both originate from social judgment theory. Social judgment theory analyzes the phenomenon of how an individual's position changes when confronted with different points of view. It is founded on the idea that a person's attitude changes depending on the information that causes the change. If the positive information is close to the individual's initial position, then the information is within the individual's range of acceptance. The view is that the individual is likely to move to the advocated position. That is, individuals are more likely to assimilate similar information. We brought this perspective to the J–A model as an example and obtained the following conclusions.

Individual *i* and individual *j* interact with information. The attitude values are based on the distance between them. The rule of attitude value change is related to the difference between the two attitude values. Individuals tend to prefer information close to themselves and reject information farther away, although the quality of attitudes affects the degree of individual interaction. The specific rules are as follows [1].

$$|If|\mathbf{x}\_i - \mathbf{x}\_j|  t\_i d\mathbf{x}\_i = \mu \cdot (\mathbf{x}\_i - \mathbf{x}\_j) \tag{2}$$

where *ui* is the threshold when individual *i* decides to accept the message, *ti* is the threshold when individual *i* rejects the message, and *μ* is the intensity of the control influence.

#### *3.2. Improved Ideas*

The J–A model provides a theoretical basis for information exchange simulation. However, the model does not consider factors such as environmental climate, individual affinity, and individual subordination. This deviates from the actual situation. For example, when the individual's herding is strong, the individual will move towards the stronger party. If the individual's herding is weak, they will adjust and move in a specific direction according to their own and the environmental attitude value. Obviously, the J–A model does not consider the population characteristics and individual attributes, and it does not have practical application value.

At the same time, we assigned the corresponding initial network structure, which aims to meet the environmental conditions in the process of individual interaction. Society is intricate and complex, with varying views on opinion events. In existing studies, smallworld networks and BA scale-free networks (from now on referred to as BA networks) are often invoked to simulate realistic social networks to restore real individual attitudinal interaction processes. Small-world networks are derived from the regular network model, in which *N* nodes relate to probability *p* on broken edges. Its "degree" distribution is in line with normal distribution. The BA network has a power-law distribution of degrees characterized by a growth mechanism and meritocratic connectivity. The BA network grows while the nodes move to the nodes with a higher degree. In general, both network

structures are closer to reality, and both preserve the diversity in real networks. They both guarantee faster convergence of the algorithm and meet the requirements of the model.

Based on the above considerations, the network group attitude polarization model is improved based on the J–A model. For the attributes of individuals and networks, parameters such as individual followership and strength of personal network relationships are added to the J–A model. The model can be adapted for actual complex events. Moreover, in real society, the network distribution law is mostly reflected in the power-law distribution, and the BA network is used as the agent adjacency model. In addition, we set the effect interval parameters *d*1 and *d*2 to illustrate the positive or negative effects of relationship strength distribution and followership parameters on group attitude polarization.

#### *3.3. Methodology*

#### 3.3.1. J–A Model

The J–A model refers to the new model of social attitude judgment proposed by Jager and Amblard. The main conclusions of the J–A model are as follows: first, the attitude structure of the subject determines the inevitability of its assimilation effect and alienation effect; second, the assimilation effect and the alienation effect have a counter-effect, which will lead to the subject reaching consensus, polarization, and other phenomena. The core idea of the J–A model is based on the theory of social judgment, whereby a person's attitude changes depending on the location of the persuasive information he receives. For example, commentators will be more inclined to make statements with similar views. The idea of this study is to explore the polarization of network public opinion, and the idea is to create a model adapted to different group characteristics and individual attributes, specifically by integrating parameters such as individual conformity and network individual relationship strength into the classical J–A model, so that the model is more suitable for complex, realworld events. The method of model simulation can more intuitively see the assimilation and alienation effects that occur in individual attitudes and the final polarization results.

#### 3.3.2. BA Network

The BA network refers to the scaleless network proposed by Barabasi and Albert that follows power-law distribution. The BA network is based on the growth mechanism and the preferential connection; that is, the size of the BA network shows an increasing trend, and the network nodes will be connected to the nodes with higher proximity. In this study, under the rules of individual attitude interaction, the corresponding initial network structure is assigned to meet the simulation environment. Compared with the intricate interactive networks in reality, the BA network not only retains the diversity of the actual network but also standardizes and simplifies the individual interaction process.

#### 3.3.3. Multi-Agent System

A multi-agent system is a collection of multiple agents that coordinate and serve each other to complete a task together. Its goal is to build large, complex systems into small, easily managed systems that communicate and coordinate with each other and has wide uses in many fields such as platform management [17], the effect of policy implementation [18,19], and so on. A multi-agent system has the following characteristics: first, each agent is independent, autonomous, and can solve a given sub-problem and affect the environment in a specific way; second, agents communicate and coordinate with each other.

The reason why a multi-agent system is selected for this study is precisely because it is suitable for complex and open distributed systems and meets the setting conditions of this paper.

#### *3.4. The Novel Public Opinion Polarization Model*

#### 3.4.1. Model Construction

The individuals and connections between the individuals form a population-complex network structure. We define the parameters and features in the network. The model parameters are shown in Table 1 as follows.

**Table 1.** Model parameters.

(1) Degree (*Pi*)

The number of edges formed by individuals and their neighbors is called the degree. The size of the degree reflects the number of individuals in the nearby area. The higher the number of nearby individuals, the higher the importance of individuals. In social relationships, the higher the importance of the individual, the higher the level of information, with considerable power of speech and definition.

#### (2) Strength of relationship (*kij*)

The strength of the relationship describes the closeness of the relationship between individual *i* and individual *j*. The model assigns a value to *k* by a random function. The *k*-value reflects the extent to which individuals influence each other. The range of the *k*-value is between integers 1 and 4. The strength of the relationship increases sequentially as the value increases.

#### (3) Individual attitude value (*Xi*(*t*))

The individual attitude value is a quantitative indicator of the individual's attitude at the moment *t*. *Si*(*t*) is the average of all individual attitude values near individual *i* at the moment, also known as the integrated environmental attitude value. The expression for *Si*(*t*) is as follows.

$$S\_{\bar{l}}(t) = \sum\_{j=1}^{n} \frac{2k\_{\bar{l}\bar{j}} - 1}{4(n-1)} X\_{\bar{j}}(t) \tag{3}$$

where *Si* +(*t*) is the summation of positive attitude values, and *Si*(*t*) is the summation of negative attitude values. The distribution of *Xi*(*t*) conforms to the Gaussian distribution.

(4) The clustering coefficient of individuals (*Ci*)

The clustering coefficient of individuals is the ratio of the actual number of edges formed by individual *i* and neighboring individuals to the maximum number of possible edges. The maximum number of possible sides is (*n*2 − *n*)/2. *Ci* reflects the aggregation of individuals. In general, individuals tend to build groups with a high degree of collection. The expression is as follows.

$$\mathcal{C}\_{i} = \frac{2n}{n(n-1)}\tag{4}$$

#### (5) The clustering coefficient of the network (*C*)

The clustering coefficient of the network *C* is the average of the clustering coefficients of all individuals in the network, which quantifies the degree of individual aggregation. The expression is as follows.

$$C = \frac{1}{n-2} \sum\_{i=1}^{n} C\_i \tag{5}$$

#### (6) Impact threshold (*Mi*)

The impact threshold *Mi* determines whether an individual's attitude has changed, directly responding to the level of information in the neighborhood. If *Mi* ≥ 1, the individual attitude value has changed. Otherwise, the individual does not change. *α* is the adjustment parameter. The expressions are as follows.

$$\text{If } \mathcal{S}\_i(t) \ge 0 \, M\_i = \alpha \mathcal{S}\_i^+(t) + \mathcal{C}\_i \tag{6}$$

$$\text{If } S\_i(t) < 0 \text{ } M\_i = \alpha \mathbb{S}\_i^-(t) + \mathbb{C}\_i \tag{7}$$

The interpretation of *Mi* is as follows. According to the rule, whether an individual's attitude value changes depends on its subordination and the degree of environmental influence. There are three main scenarios. In the first case, the individual is highly submissive, entirely influenced by the environment. The individual will always follow the environment and adjust their attitude. In the second case, the environment around the individual is unbalanced, and there will be a view recognized and dominated by more individuals. In this case, the individual will also favor the strong side. In the third case, the individual's subordination combined with the environment drives the individual to move towards a particular side of the camp.

#### (7) Effect interval parameters (*d*1/*d*2)

Effect interval parameters specify the range of individual attitude value changes. If the distance between *Xi*(*t*) and *Si*(*t*) is less than *d*1, the individual does not follow the rule of exclusion. Otherwise, individuals do not follow the rules of assimilation.

#### (8) Assimilation/exclusion degree coefficient (*β*/*γ*)

The assimilation/exclusion degree coefficient is the degree of control over the value of individual attitude change. *β* is the degree coefficient of the assimilation rule, and *γ* is the degree coefficient of the exclusion rule: both range between 0 and 1.

#### (9) Average distance length (*L*)

The average distance length is the average number of distances between individuals in the network [20]. The distance between individuals is the sum of the edges connecting both. The maximum distance is the diameter of the network. The *L*-value reflects the ability and efficiency of information transfer between individuals. Let the path length between individual *i* and individual *j* be *lij*. The expression of *lij* is as follows.

$$L = \frac{2}{n(n-2)} \sum\_{i=1}^{n-1} \sum\_{j=i+1}^{n-1} l\_{ij} \tag{8}$$

#### 3.4.2. Simulation Process

To reveal the mechanism of individual attitude polarization, we established a social networking platform. Research has shown that most complex, real-world networks exhibit power-law distribution, which indicates that most individuals have a small degree, and only a few offer a large degree. Barabasi and Albert proposed BA scale-free networks to study this class of networks that exhibit power-law distributions. The basis of the network is Growth and Preferential attachment. Growth means that the complex network structure will continue to expand. Preferential attachment means that the additional individuals are more inclined to connect with individuals of a higher degree. The specific construction method is as follows.

Step1. Growth: We randomly construct the initial network containing *m*0 individuals. Next, we constantly increase the number of individuals, and individuals are randomly connected to the original model.

Step2. Preferential attachment: The probability (*πi*) that an individual is connected to the network is positively correlated with the degree (*pi*) of nearby individuals. The expression is as follows [6].

$$
\pi\_i = \sum\_{j=1}^n \frac{p\_i}{p\_j} \tag{9}
$$

BA scale-free networks conform to the characteristics of self-organization, synchronization, and emergence mechanisms in actual society. Therefore, we choose the BA scale-free network to study and make corrections for issues such as model sensitivity.

#### 3.4.3. Interaction Rules

In individual interaction, the impact threshold *Mi* is calculated by first considering the environmental attitude value, relationship strength, and clustering coefficient. Next, a judgment is made: if *Mi* ≥ 1, the interaction takes place; otherwise, the individual attitude value does not change in any way.

We set the effect interval *d*1/*d*2 as the discriminate condition. A discussion of the interaction process follows.

#### (1) Assimilation rules

If the distance between *Xi*(*t*) and *Si*(*t*) is less than *d*1, it is considered that assimilation of individual and environmental attitudes occurs. The rules of attitude value evolution follow the following rules.

$$X\_{i}(t+1) = (1 - \beta)X\_{i}(t) + \beta S\_{i}(t) \tag{10}$$

#### (2) Exclusionary rule

If the distance between *Xi*(*t*) and *Si*(*t*) is greater than *d*2, the individual and the environmental attitude values are considered in exclusion. The rules of attitude value evolution follow the following rules.

$$X\_i(t+1) = (1 - \gamma)X\_i(t) + \gamma S\_i(t) \tag{11}$$

(3) Neutrality rules

If none of the above conditions are met, the individual is considered not to make any changes.

The following flow chart (shown in Figure 1) outlines the discriminatory process of the polarization model.

**Figure 1.** Network population attitude polarization model discrimination process.

#### **4. Experiment Simulation**

Because the BA network can present the social network well, this paper defines the BA network as the basis of evolution. By setting different parameter values, this paper makes an intensive study of the evolution process. First, this paper sets the scale of network nodes as 100 and takes *d*1 = 0.3, *d*2 = 0.7, *β* = 0.1, *γ* = 0.2. Through practical operations, this paper finds that after 400 interactions, the individual's attitude will tend to polarize with the surrounding environment, and their attitude value will gradually shift to the two extreme directions of −1 and 1. However, some individuals will still maintain their original attitude. Furthermore, some individuals will constantly adjust their attitude value in the range of −1 to 1 to achieve a balanced state by adapting to the external environment. Specifically, in the process of attitude evolution, the quantitative distribution of different attitude values under different interaction times is shown in Figure 2 below:

**Figure 2.** Quantitative distribution of attitude values under different interaction times.

In the initial state, time = 0: the individual attitude value distribution diagram is shown in Figure 2. The abscissa in the diagram represents the individual attitude value, and the ordinate represents the number of individuals corresponding to the attitude value. The simulation results show that in the initial state, the individual attitude value is relatively scattered and evenly distributed. In the initial state, individuals in the group hold their views on events, and there is no clear view of which is right or wrong, or there is a relatively unified opinion. Everyone makes judgments and forms attitude values purely through their views on events. Therefore, in the early stage of event development, there will be no

obvious extreme phenomenon in the attitude value of the group towards an event. With continuous interaction between individuals, when the time is 50, 100, and 400, the attitude value of individuals begins to show a differentiation trend. The specific simulation results are shown in Figure 2.

The number of individuals with a neutral view decreases, while the number of individuals close to 1 and −1 attitude values increases. These changes can obviously show a polarization phenomenon. In the process of increasing the number of interactions, it can be found from the four simulation results that the attitude distribution diagram presented in Figure 2 has been relatively stable. Even a few individuals did not change their attitude values. This paper lists two reasons:

In a real event, after each event is polarized, some people will always define the event according to their judgment to maintain their original point of view. Similarly, some individuals will hold a wait-and-see attitude because they cannot understand the truth of the event. However, as the simulation results show, driven by herd mentality, most individuals actively choose an extreme point of view to stand in line, which shows that most individuals show a phenomenon of joining the powerful party to seek security in the face of group events to avoid isolation.

#### **5. An Empirical Case**

In this paper, the public opinion polarization model based on the BA network is used to predict the trends in public attitudes towards network events. Based on the 4.1 simulation study, this paper selects the network event of "Hua Chenyu and Zhang Bichen having children unmarried" as a research sample to predict the attitude of online groups. The original data of the case sample is shown in Table 2.

**Table 2.** Case Sample Public Raw Attitude Values.

Data source: Zhang Bichen's long article posted on Weibo at 17:51 on 22 January 2021.

On 21 January 2021, an unknown netizen broke the news on the Internet: a top male star in the entertainment industry married and had children, the woman was also an insider, and the child was registered when he was one year old. Another netizen revealed that the male star was Hua. On the same day, Hua's cousin posted a denial. At 17:45 on 22 January 2021, Hua admitted to having a child with Zhang. At 17:51, Zhang also confirmed this by posting a long article on Weibo under his real name.

The incident of "Hua and Zhang having a child out of wedlock" caused an uproar on the Internet. With the continuous revelation of news related to the incident, netizens had a heated discussion, and the public view gradually became distinct and polarized. In this paper, the BA network simulates the state of a real social network, and we use the polarization model of public opinion to simulate and predict the evolution of this event. Through web crawlers, this article obtained the original data set of public attitudes under Zhang's long post on Weibo at 17:51 on 22 January 2021. In this paper, Python

NLP natural language processing and machine learning are used to obtain 16,579 valid data, thereby determining the size of the instance network nodes. According to the actual situation of the case, this paper determines that the assimilation degree coefficient is 0.005, the repulsion degree coefficient is 0.01, the assimilation effect band distance is 0.3, and the repulsion effect band distance is 0.7. Based on existing stop word rules and machine learning recognition methods, this article assigns a positive or negative attitude value to the initial valid comment. With the soaring heat of the incident, the matter has aroused heated discussion among the public. In the environment of constantly revising the direction of public opinion, the views of network individuals collide, resulting in different degrees of change in their attitudes. This article regards this transformation as a process of individual interaction. Based on the polarization model of public opinion based on the BA network, this paper simulates the process of the collision of individual views on a network and predicts the value of public attitude after the occurrence of 10, 50, 100, and 400 such situations. The forecast statistics are shown in Table 3.

**Table 3.** Polarization predictions of public attitude values after interaction.

We captured 16,579 valid comments from Zhang's statement at 17:51 on 22 January 2021. Figure 3 is valid comments from eight hours after the long article was published. This paper uses certain rules to assign different attitude values to different comments, and the distribution of individual attitude values is shown in Figure 3. The original public attitude value from the case sample was analyzed as follows: 2062 people held a neutral attitude towards the incident, and 4055 people held an absolute positive or negative attitude. At this time, the distribution of public attitude values was relatively even. Network individuals expressed their opinions on the event, and there was no obvious polarization tendency in network public opinion and no clear and unified view. For the incident, many network users still held a wait-and-see attitude and looked forward to the follow-up development of the event; at the same time, there were also a considerable number of netizens who held a "blessing" support attitude or a "not optimistic" opposition attitude.

Based on the model above, the prediction results of the attitude value distribution of the sample dataset after different interactions are shown in Figure 4. After 5 h, 1 day, 2 days, and 8 days of simulated interaction, the polarization trend of public attitudes gradually became obvious, and the number of neutral network individuals began to decrease. From the forecast results, it can be seen that from the beginning of Zhang's statement at 17:51 on 22 January 2021 to 8 days after the statement was released, the proportion of network users with an absolute positive or absolute negative attitude rose from 24.46% to 44.27%, while the proportion of neutral internet users decreased from only 12.44% to 11.26%. Obviously, the proportion of network individuals who show an absolute attitude has increased significantly, and the polarization trend of network public opinion has become more and more obvious, while the numbers of netizens who indicate a neutral attitude has remained at a low level, and the range of changes is small. With the clarity of the incident, the views of netizens have become more distinct. However, there are still some who hold a neutral attitude, such as "eating melons", and do not express personal views with a clear attitude.

**Figure 3.** Distribution of public raw attitude values.

**Figure 4.** Prediction of attitude value distribution under different interactions.

From the prediction results of attitude value distribution, it was found that with the continuous occurrence of interaction, the growth rate of the proportion of network

individuals with an absolute attitude is from fast to slow, while the proportion of someone with a neutral attitude does not change much. From the simulated interaction results of the four time nodes of 5 h, 1 day, 2 days, and 8 days, it can be seen that from the beginning of Zhang's long article at 17:51 on 22 January 2021 to one day after the incident, the growth rate of the proportion of users who showed an absolute attitude increased from 24.45% to 39.64%, and after the incident, the growth rate slowed down from 39.64% to 0.71%. In summary, after one day of simulated interaction, the trend of the attitude distribution map stabilized.

Individuals participating in the evolution of network public opinion have the characteristics of subjective judgment and labeling processing, as well as passive acceptance and loss of subjectivity. There are countless exchanges of opinions between individuals, which eventually form group behavior, and individual behavior is affected by group behavior. The phenomenon of polarization of online groups mostly occurs in the field of opinions, and the result is mostly that the views are further differentiated and opposed, and the opposing parties continue to strengthen their views in the group discussion, and it is obvious that they cannot merge. At the same time, when a person is in a network group with similar interests or views as a link, he will exchange common ideas and understandings with other members of the group or constantly look for views like his, trying to obtain psychological comfort and strengthen his original concepts.

#### **6. Conclusions**

#### *6.1. Summary*

In order to explore the propagation structure of online public opinion and analyze the logic of extreme public opinion in the social context of the big data era and the developed self-media network, we constructed a BA network model, simulated and analyzed the trend of public attitudes toward online events and the polarization mechanism of individual attitudes, verified the propagation mechanism and polarization prediction law of online public opinion through experiments, confirmed the validity of the BA model, and obtained the following conclusions through simulation experiments.

network individuals with absolute attitudes increases significantly, and the trend of polarization of double-linked opinions becomes more and more obvious. On the other hand, the proportion of Internet individuals who expressed neutral attitudes remained low and slightly changed. With the development of the event, the Internet users' ideas about the event become more and more distinct. Based on the results of this simulation, we give policy recommendations and discuss the problems in the experimental process in the following sections.

#### *6.2. Policy Recommendations*

With the prevalence of the trend of network intelligence and the expansion of network coverage, the predicament of information blocking has changed, and human networked society has risen rapidly. At the same time, a large amount of true and false information causes confusion, and false information spreads to the public, misleading moral values, laying down hidden danger for the maintenance of a harmonious environment for online public opinion. The government should take timely measures for the real-time, changing, networked environment to create a just and harmonious network environment for citizens and eradicate some unsettling hidden social dangers. To better cope with the polarization of network public opinion, this paper puts forward the following suggestions:

(1) Improve the public opinion monitoring mechanism and build a harmonious network order

Internet public opinion is easy to use to guide the views of the masses, and if it is not properly supervised, it is easy to mislead the masses. Even the evolution of online public opinion may cause the masses to fall into a vicious circle of emotional or even group polarization. At the same time, freedom of speech on the Internet promotes the interaction of people's information and the collision of thinking and produces a situation in which false information and misinformation affect the emotions and thoughts of viewers to achieve the publisher's personal, bad goals. Therefore, the normality of online public opinion requires the cooperation of a strong monitoring mechanism to ensure the safety and order of cyberspace to some extent.

Although there are some online information reporting platforms, the government's use of them is inefficient, and even the processing and feedback of reporting information is not timely. The government should further improve the monitoring mechanism of network public opinion, not only relying on computer keyword recognition and big data processing technology but also mobilizing social forces to help network monitoring. Reporting information is more accurate than keyword recognition technology. Only by screening false information and reasonably guiding the direction of public opinion can some netizens who have difficulty judging information be protected and not misled or suffer some losses due to being deceived. The government actively participates in the governance of cyberspace. It will contribute to the harmonious co-construction and sharing of the network.

(2) Rebuild the accountability mechanism for public opinion and crackdown on online anomie

Every occurrence of online public opinion polarization is a test of the government's credibility. Whether the government's accountability mechanism is sound and whether other aftermath measures are appropriate and timely will affect the government's image. As a public servant of the people, the government should actively investigate disharmonious factors or improper regulation by the government itself after the negative impact of online public opinion and safeguard the legitimate interests of the people. Only in this way can the credibility of the government be maintained.

The reconstruction of the network public opinion accountability mechanism is a necessary part of the government's governance of the network environment. Relevant government departments can start by conducting satisfaction surveys on network individuals related to the governance of the network environment. Through this post-mortem investigation, the government can clarify its image positioning in the eyes of the public and

understand its own problems. Only when the crisis is handled properly and the masses are satisfied will the negative influence of online public opinion be weakened, and the rebound will be avoided. At the same time, relevant government departments should also be held accountable for illegal acts that maliciously affect public order and damage the interests of others. Only by thoroughly cracking down on online anomie can we give a warning to criminals and, at the same time, put an end to attempts to conduct anomie because of luck.

(3) Guide netizens' values and transmit positive energy of public opinion

Internet public opinion has both positive and negative effects, and in the contemporary era, when the Internet closely links everyone, positive and negative emotions are more likely to spread and affect the public. Therefore, it is particularly important to guide netizens' values in a timely and positive manner and transmit positive energy of public opinion.

For different network groups, different measures should be taken to guide and pass on the "right medicine". Neutral internet users with many fans and high membership levels not only have a stable stance but also have a relatively large fan base, a high degree of activity, and a strong potential to control public opinion. Therefore, such network users can be used as a key group for public opinion dissemination guidance and polarization intervention, and the background of social platforms should increase efforts to maintain key groups, promote content that is conducive to guiding the development of netizens' values in a positive direction, and transmit positive public opinion. For some network users who pay less attention to hot events because their sources of information are relatively closed and single and passive, they can push comprehensive information to this group in a targeted manner, which helps the group form an objective and comprehensive understanding of hot events. At the same time, increasing the frequency of pushing positive content to enhance the positive experience of network users helps to transmit positive public opinion. In addition to paying attention to the above two parts of network users, highimpact and highly active groups can be found through background big data, and advanced technology can be used to seize the opportunity of positive information exposure and play and enhance leadership.

(4) Enhance the image of the government and maximize the interests of society

Internet public opinion is usually inextricably linked to civil rights, people's livelihood, and real society and the problems it exposes or the focus of discussion are related to this. Therefore, the government plays an important role in the management of network public opinion, which is conducive to enhancing the image and playing a more decisive role in maximizing the interests of society.

The government can use advanced technology and big data platforms to strengthen the management of the two major sources of information dissemination, official media and self-media. First, to standardize the operation and management mechanism of official media, we should put social benefits in the first place, standardize and restrain professional journalists, and correct the one-sided pursuit of traffic realization by some bad official media; at the same time, increase support for official media in terms of policies, funds and talent introduction. Second, through the public or industry associations to regulate the development of self-media in the right direction, guide them to carry out activities to produce and disseminate positive energy information, and create a positive atmosphere of public opinion and emotion among the public.

The government focuses on governing online public opinion to further consolidate its position and enhance its image. At the same time, in the process of standardization and guidance, the interests of all parties maximize social interests after the game.

This paper discusses the extreme model of public opinion based on the BA network, enriches the theory and method of polarization of online group attitudes, and predicts the network public opinion of hot events through empirical analysis, providing practical guidance for the intervention and guidance of network public opinion, which is of great significance for promoting the modernization of national governance capabilities.

#### *6.3. Limitations and Future Research Directions*

The present study has limitations in some respects. First, the model is a poor fit for the phenomenon of group attitude reversal. Internet public opinion changes rapidly, and as events develop, the final direction may not always be consistent with the initial state. People's attitudes will undergo drastic shifts in the process, which is often difficult to simulate by polarization models. Later, we will enrich and extend the model to address the conditions and trends of public opinion reversal. Second, the example simulation process includes only one public opinion event with a small sample size. This study can conduct practical simulation experiments by collecting different events and a larger sample size. By simulating multiple occasions, we can effectively improve the model's generalizability. Third, there is some bias in analyzing attitude values during the experiment. This study uses a machine-learning algorithm to assign attitude values to event comments. It is crucial to extract group attitudes from the text effectively. The algorithm's limitations primarily influence the encoding operation of the training set. If the algorithm can accurately extract attitude values from buzzwords, expressions, and punctuation, the error of the model will be significantly reduced. There is still room for improvement in the example algorithm piece.

**Author Contributions:** Conceptualization, T.C.; data curation, R.Z.; formal analysis, Y.Z.; software, W.D.; visualization, Y.Y. (Yuanyuan Yu); writing—original draft, Y.Y. (Yuanjian Ye). All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LY22G010003), the 2020 National Innovation and Entrepreneurship Training Program (No.202010353008), the 2020 Science and Technology Innovation Activity Program for College Students and New Seedling Talent Program of Zhejiang Province (No.2021R408050), and the 2019 University-level College Student Innovation and Entrepreneurship Project of Zhejiang Gongshang University (No.CX202002021).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data used to support the findings of this study are available from the corresponding author upon request.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

## *Article* **Agent-Based Modeling of Consensus Group Formation with Complex Webs of Beliefs**

**Ismo T. Koponen**

Department of Physics, University of Helsinki, 00014 Helsinki, Finland<EMAIL_ADDRESS>
**Abstract:** Formation of consensus groups with shared opinions or views is a common feature of human social life and also a well-known phenomenon in cases when views are complex, as in the case of the formation of scholarly disciplines. In such cases, shared views are not simple sets of opinions but rather complex webs of beliefs (WoBs). Here, we approach such consensus group formation through the agent-based model (ABM). Agents' views are described as complex, extensive web-like structures resembling semantic networks, i.e., webs of beliefs. In the ABM introduced here, the agents' interactions and participation in sharing their views are dependent on the similarity of the agents' webs of beliefs; the greater the similarity, the more likely the interaction and sharing of elements of WoBs. In interactions, the WoBs are altered when agents seek consensus and consensus groups are formed. The consensus group formation depends on the agents' sensitivity to the similarity of their WoBs. If their sensitivity is low, only one large and diffuse group is formed, while with high sensitivity, many separated and segregated consensus groups emerge. To conclude, we discuss how such results resemble the formation of disciplinary, scholarly consensus groups.

**Keywords:** consensus groups; agent-based model; web of beliefs

#### **1. Introduction**

The formation of groups with shared beliefs, opinions, and views has been and continues to be a topic of great interest, discussed in sociology, political science, communication, and organizational science, as well as studies focusing on structure of science (see, e.g., [1–5] for the diversity of topics and areas of studies). In all these cases, one key issue is to understand the dynamics which drive the segregation and consolidation of groups, even in conditions where communication and sharing of beliefs is common and frequent (for reviews, see [3–5]).

The computational modeling of opinion group formation [5–7], the formation of collaborative groups and collective decision making [2,8,9], as well as disciplinary fragmentation and progress in science [10], has shed light on the social dynamics behind group formation, segregation, and consolidation, often revealing the unexpectedly simple but self-reinforcing interactions behind such complex phenomena. Consequently, consensus group formation and opinion adoption, and their change and evolution, have been modeled using a variety of idealized models, many of them founded on one or another theoretical view about social influence and interaction or social learning, some of them being computational renderings of empirical findings. The models of opinion dynamics are as diverse as are the their theoretical underpinnings and intended scopes of applications. However, many of the models of opinion dynamics, where formation of consensus groups is of interest, can be classified in three groups [3]: (1) models of assimilative social influence; (2) models with similarity biased influence, and; (3) models with repulsive influence. In addition, it is possible to recognize models that are hybrids of these three classes [3]. Different well-known models in each class, their theoretical background and motivation, and most well-known computational implementations and empirical applications and justification (when it exists) are reviewed in detail elsewhere [3–5], and therefore we provide here only a brief summary

**Citation:** Koponen, I.T. Agent-Based Modeling of Consensus Group Formation with Complex Webs of Beliefs. *Systems* **2022**, *10*, 212. https://doi.org/10.3390/ systems10060212

Academic Editors: Philippe Mathieu, Juan M. Corchado, Alfonso González-Briones and Fernando De la Prieta

Received: 11 October 2022 Accepted: 8 November 2022 Published: 9 November 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

of the most important aspects of models with similarity bias to the extent needed to put in perspective the model introduced here.

Models with similarity bias do not assume a structurally fixed connection between agents, but instead, agents can interact if they have sufficiently similar opinions (or beliefs) or if they are sufficiently similar regarding some other pertinent feature. If the similarity (opinions or other preferred feature) is too distant, interactions are no longer possible. Such a threshold of interaction can be interpreted as a confidence to interact and modify one's opinion and can be assumed to arise from a variety of psychological or sociological reasons [3–5]. Therefore, many similarity bias models are referred to as bounded confidence models [11,12] owing to the existence of a kind of a confidence threshold to interact. These models typically give rise to persistent opinion clusters, where agents are similar and dissimilar to agents in other clusters. In these models, the cluster formation is an outcome of similarity bias homophily; the more stringent the threshold for required similarity, the more numerous are the segregated non-interacting clusters. In most bounded confidence models, the threshold is sharp and deterministic, and stochastic variation is included by adding a stochastic noise [11,12]. Interestingly, the addition of noise may fundamentally affect the consensus cluster formation [3–5]. The role of noise and its non-trivial effects in deterministically thresholded bounded confidence models suggests that to model the bounded regions of interactions due to similarity bias, genuinely stochastic and probabilistic rules to decide whether or not the interactions happen are sometimes preferable [13,14]. Finally, important and interesting recent generalizations of bounded confidence models are models where opinions or beliefs are multidimensional [15–17], or in one way or another, more complex structures of beliefs and related opinion elements [2,7,9,18]. Such models allow more realistic modeling of complex opinions, give rise to richer dynamics than onedimensional bounded confidence models, and, moreover, the emerging clusters of opinions are in these cases more diverse than those found in one-dimensional models.

Paralleling the above briefly summarized recent generalizations of similarity-biased models of consensus formation, the agent-based model (ABM) proposed here is meant to be another step towards generalizations applicable to model complex sets of beliefs, where interactions between agents and dynamics of the change in beliefs are modeled as a genuine stochastic process. The model, however, adopts the simple view where agents modify their views by acquiring and accommodating their sets of views present in to a collection of all agents; no new elements emerge.

The ABM proposed here, given its restriction to describe the creation of new elements of opinion beliefs, and instead describing the evolution of consensus groups with complex sets of beliefs, has two possible areas of applications. One area of interest is the formation of disciplinary scholarly groups and schools, along with their characteristic ways of using scientific terms and forming various research programs [19–22]. Research fields always contain disciplinary groups where key scientific terms differ, and the same terms may be used differently in discussing and framing the key problems within the field [21,22]. In this case, within the established paradigms of research, new disciplinary groups are often formed within the existing fields. Such strong disciplinary fragmentation seems to be particularly apparent and typical in the human and behavioral sciences [23–25]. Another situation of interest, where the creation and generation of new knowledge is not necessarily of primary interest, but where differing disciplinary views about thematic topics can be recognized, is related to the disciplinary views of science education scholars [26,27] as well as science students, where student groups may have consensus views that differ from those of other student groups views, even when they have used the same study materials [28,29]. To address situations in which the knowledge or meaning structures of interest are complex systems of terms, concepts, or conceptions characteristic of the disciplinary group, it seems appropriate to use the expression "webs of beliefs" (compare e.g., ref. [30]), to be referred to briefly as WoBs in what follows. In this study, we approach such a disciplinary group formation through convergence and consolidation of WoBs, where the dynamics are driven by similarity and consensus seeking, without posing explicit bounds of confidence to

constrain interactions (as in so-called bounded confidence models, [1–3]). In the ABM presented here, agents possess generic WoBs in the form of a complex network. Agents compare their WoBs and exchange bits and pieces of them, guided by similarity-seeking dynamics (i.e., homophily). The comparison-triggered adjustment of WoBs leads to their convergence but also divergence, and thus to the formation of segregated disciplinary consensus groups. The model is highly idealized, generic, and simple, but as discussed in the final section, it has features that resemble the real situations of disciplinary group formation. Thus, the model presented here is a first attempt towards agent-based modeling of consensus group formation where views and opinions are complex webs of beliefs.

#### **2. Materials and Methods**

The computational model presented here is an agent-based multi-optional model for the formation, consolidation, and segregation of consensus groups based on agents' webs of beliefs (WoBs). The dynamics of the model are driven by agents' repeated comparison of their WoBs, guided by the utility of adjusting the WoB for better mutual similarity. We describe first the constructions of agents' WoBs, second the multi-state probabilistic model of selection of partners for comparison, and third, update dynamics for the change in WoBs. Simulations are carried out using an event-based roulette-wheel method [31,32]. Symbols and their meaning in the study are summarized in Table 1.

**Table 1.** Summary of symbols and abbreviations used recurrently in the text and figures. In the sub-indexes, *ξ* and *γ* refer to a pair of agents, Ens. avrg. denotes ensemble averages, and Dim[{ · }] refers to a dimension of set.

#### *2.1. Model of Webs of Beliefs (WoBs)*

The WoBs of interest, which mimic real scholarly conceptual or semantic structures, can be characterized as complex in the sense that they consist of several elements (vertexes or nodes) connected (by edges or links) in complex ways to their other elements. Such WoBs have a distribution of vertexes with broad difference in their connectivity; some vertexes have a high connectivity, while many are loosely connected (see e.g., [28,29,33,34]). Consequently, appropriate WoBs can be characterized as sparse complex networks with about 100 vertexes, with low average degrees of about 3 or 4 but with very few nodes having larger degrees up to 10.

Generic initial WoBs of agents are obtained by sampling a larger template network with desired properties, generated by a generative model previously introduced to produce networks with broad distributions of connectivities of vertexes [35]. In constructing the generic networks, they are pruned by removing auxiliary vertexes possessing only one edge, so that to only two cores (nodes with at least two edges) are taken into account. The template network has approximately 200 vertexes (i.e., nodes) and 400 edges (i.e., links), with the degrees (i.e., number of attached links) of the vertexes distributed according to inverse power law, with an inverse power close to 3.0 (see Appendix A for details of rationalization of this choice). With these parameters, the average degree of vertexes is about 3 to 4, with very few vertexes having a degree of about 8 to 12. From the template networks, a set of initial edges *E*0 = 120 ± 20 are drawn at random, and connected networks (discarding unconnected parts) are formed to serve as individual WoBs for each agent. With these choices, WoBs contain about 40–100 vertexes (in some rare cases only about 20) and are always sub-networks of the template. The WoBs thus obtained from the template are stochastically generated and have in each case a slightly different detailed structure. The details of the generative model, briefly summarized in Appendix A, are reported elsewhere [35] and are of no further interest here.

#### *2.2. Interaction Dynamics of Agents*

The interaction between agents in the agent-based model (ABM) introduced here comprises *N* agents, where one agent *ξ* selects an agent *γ* to interact among *N* − 1 other agents. The selection is based on the similarity *Sξγ* of lexicons between the agents. The probability *Pξγ* that agent *ξ* selects agent *γ* is assumed to follow the (Gibbs-like) probability distribution (compare with refs. [36–38])

$$P\_{\mathfrak{F}\gamma} = \frac{S\_{\mathfrak{F}\gamma} \exp\left[\beta \, \mathcal{S}\_{\mathfrak{F}\gamma}\right]}{\sum\_{i \neq \mathfrak{k}} \mathcal{S}\_{\mathfrak{F}i} \, \exp\left[\beta \, \mathcal{S}\_{\mathfrak{F}i}\right]}, \quad \mathfrak{F} \neq \gamma \tag{1}$$

where *β* is a parameter related to the *sensitivity* to similarity, with *β* < 1 (here, in practice *β* ≈ 1) indicating low sensitivity (i.e., high noise or randomness) and *β* 1 high sensitivity (i.e., low noise or randomness). The prefactors in Equation (1) ensure that at the limit *β* → 0, the decision probabilities attain values corresponding to the ordinary rational, non-probabilistic choice (for details, see [37,38]).

The similarity of the agents is defined simply as the ratio of the number of shared elements (vertexes and edges) to all elements. Denoting the set of elements as {X} = {v} for vertexes and {X} = {e}, the similarity of agents *ξ* and *γ* based on either edges or vertexes is given simply as a ratio of non-shared to shared elements,

$$S\_{\tilde{\xi}\gamma}\left[\mathbb{X}\right] = \frac{\text{Dim}\left[\{\mathbb{X}\}\_{\tilde{\xi}} \cap \{\mathbb{X}\}\_{\gamma}\right]}{\text{Dim}\left[\{\mathbb{X}\}\_{\tilde{\xi}} \cup \{\mathbb{X}\}\_{\gamma}\right]},\tag{2}$$

where X denotes either edges (e) or vertexes (v) and Dim[ ·] means dimension (number of elements) in a given set. In defining the similarity, we chose to keep it symmetric and as simple as possible, although more elaborate definitions are possible, for example, taking into account the role of asymmetry and different number of non-shared elements. However, the definition of similarity is always a matter of choice, and no unambiguous definition seems to be possible because choices need to be made about what features are taken into account in similarity, as discussed in detail in, e.g., ref. [39].

We assume that both vertex and edge similarities need to have high values for a high similarity, and thus, in the effective similarity used in simulations, the effective similarity is taken as a geometric mean of vertex and edge similarities.

$$\mathcal{S}\_{\mathfrak{F}\gamma} = \sqrt{\mathcal{S}\_{\mathfrak{F}\gamma}\begin{bmatrix} \mathbf{e} \end{bmatrix} \ \mathcal{S}\_{\mathfrak{F}\gamma}\begin{bmatrix} \mathbf{v} \end{bmatrix}} \tag{3}$$

This choice of defining the similarity is a trade-off between simplicity and taking enough detail into account to characterize the WoBs. Here, given the generic nature of the WoBs, a simple symmetric similarity *Sξγ* = *Sγξ* based on the counting of elements and constrained in the range of values from zero to one is satisfactory for the present purposes. In fact, we tried out more elaborated similarity definitions, but in the present ABM, they had little effect on the final results.

The selection criterion in Equation (1) with similarity defined as in Equation (2) prefers similar agents (i.e., homophilic preference) and, in that sense, it closely resembles the selection criteria in bounded confidence models and their variants [1,3], where homophilic cut-off criterion constrains the possibility of interactions. The present model, however, belongs to a class of probabilistic similarity-biased models [3,13], where no interaction possibilities are ruled out a priori, but partners with high similarity are prioritized.

#### *2.3. Updating the WoBs*

In the interaction events, agents update their WoBs, seeking to improve their consensus. When agent *ξ* has decided on the partner *γ* it will interact with, the update of WoBs for optimized communication takes place in two steps. First, agents *ξ* and *γ* pick out a common vertex, which is a center element in their interaction (i.e., shared term or concept for their belief). Second, they check the neighborhood of that selected vertex and check the number of available new edges to neighborhood vertexes their partner possesses but the agent itself does not yet possess (i.e., possible new elements and connections the agent can acquire to adopt a new edge and possibly a new vertex connected to it). Both agents have then two choices of how to increase mutual similarity: either to add a new edge, or to delete an edge not possessed by the interaction partner. The decision to add or delete an edge is made on the basis of the utilities, based on the changes in the similarity, which are taken to be changes in similarity by addition or deletion of an edge or a vertex (contained in the neighborhood of the common vertex). The advantage of such a simplified procedure is that utilities of addition can be estimated through changes in similarities, when number of elements for one agent changes while remaining intact for the other agent. Now, four possibilities are available, either agent *γ* changes its WoB or agent *ξ* does, but only one event happens at a time. In both cases, however, the changes are computable as a difference Δ±[*X*] = *S*[X ± 1] − *S*[X] for addition (+) and deletion (−), respectively, where indexes referring to agents are dropped because of symmetry in regard to the agents. For faster computation, the differences are approximated by the first linear terms, which are obtained through direct calculation, in the form

$$
\Delta\_+[\chi] \quad = \quad (1-\mathcal{S}[X]) \text{ Dim}[\;\{\mathcal{X}\}\_{\mathcal{\xi}} \cup \{\mathcal{X}\}\_{\mathcal{\gamma}}\;]^{-1} \tag{4}
$$

$$
\Delta\_{-}[\lambda]\_{-}=S[X]\,\mathrm{Max}[1,\mathrm{Dim}[\{\lambda\}\_{\xi}^{\mathsf{C}}]+\mathrm{Dim}[\{\lambda\}\_{\gamma}^{\mathsf{C}}]]^{-1}\tag{5}
$$

where X denotes either vertexes or edges and notation {·}*C* complements of sets. Note that in all expressions containing *S* and Δ± or related to them, indexes referring to agents are dropped in what follows. In Equation (5), the minimum value allowed in the denominator is 1, in order to prevent division by zero (this occurs rarely and has no consequences on results). In case only an edge is added/deleted but no new vertex becomes added/deleted, utilities are simply for edges (X = e). In the case that the added/deleted edge contains a new vertex, the effective utility Δ± corresponding to addition (+) or deletion (−) of a vertex and its neighborhood is taken to be the geometric average of utilities for X = e and X = v.

$$
\Delta\_{\pm} = \sqrt{\Delta\_{\pm}[\mathbf{e}] \cdot \Delta\_{\pm}[\mathbf{v}]} \tag{6}
$$

Probabilities of addition and deletion are then given as

$$P\_{\pm} = Z^{-1} \,\, \Delta\_{\pm} \exp[\beta^\* \,\, \Delta\_{\pm}] \tag{7}$$

where parameter *β*∗ has the role of confidence of the decision to add or delete on the basis of utility. The factor *Z* = Δ− exp[ *β*∗ Δ−] + Δ+ exp[ *β*∗ Δ+] is the normalization factor. The rationalization to include prefactors is similar to the case of probability in Equation (1) selecting the partner for interactions. The choice is always between addition and deletion; staying intact (which is not counted as an event) is not included. This is related to the choice that simulations are event-driven; only events that make a difference (i.e., change the state of the agent) are taken into account. Note that in each interaction event, only one outcome of the four possibilities is allowed.

#### *2.4. Simulations*

Simulations of the ABM are based on the probabilities of selection *Pξγ* in Equation (1) an *P*± in Equation (7), with each event of an agents' interaction consisting of both updates. Simulations are event-based, consisting of sequences of events *τ* = 1, 2, ... *τ*MAX events. At

each instant, when the value of *τ* increases by 1, it is decided: (1) which agents are going to interact and (2) whether an edge (and possibly, a vertex) is added or deleted. Each of the event selections is carried out by the roulette-wheel method [31,32]. In the roulettewheel method, a discrete set of *N* possible events *k* with probabilities *pk* is arranged with cumulative probability

$$\Phi\_k = \sum\_{i=1}^k p\_i / \sum\_{i=1}^N p\_i \,. \tag{8}$$

The event *k* is selected if a random number 0 < *r* < 1 falls in the slot Φ*k*−1 < *r* < Φ*k*. In case (1), the probabilities *pk* are given by *Pξγ* in Equation (1), while in case (2), one has only two probabilities *pk* = *P*± defined by Equation (7). The roulette-wheel method is thus entirely event-based, where the occurrence of events is predicted on the basis of cumulative distribution. Consequently, there are no constant time-like intervals between events. All simulations are carried out for *N* = 25 agents, with *τ*MAX = 50 · 103 events, corresponding to about 30 updates for each edge in a set of WoBs of 25 agents. However, only a fraction, usually about 2–3%, eventually leads to addition or deletion of and edge, since after the onset of formation of consensus groups, many neighborhoods are already identical. In practice, this means from 40 to 60 changes per agent in the course of simulations, and as is seen, a stabilized situation is reached well before *τ*MAX is reached. In the simulations, only one affinity distribution is used, corresponding to an inverse power of 2.9, close to the marginal value of 3 (see Appendix A for details). The simulations are repeated for the ensemble of 36 = 6 × 6 different initial states, sampled for six different initial templates and each six times for different initial WoBs. Simulations and numerical computations are realized using Mathematica [40].

#### *2.5. Representation of Data*

The simulations track the evolution of agents' WoBs, and on that basis, agents' similarities based on shared edges and vertexes are the outcome of the simulations. The agents are classified in clusters according to the effective similarity *S* = √*S*E *S*V and geometric average of size √E V, where E = Dim[{e}] and V = Dim[{v}]. In finding the clusters, we use Mathematica's [40] DBSCAN algorithm, which partitions the datasets into clusters using density-based classification with noise [41]. DBSCAN proved to be a reliable and fast method for finding clusters not constrained by a prefixed number of clusters for the data generated by simulations. When clusters are detected, the number of clusters *N* and agents in a given cluster are counted to obtain the relative occupancy *R* as the average value of the fraction of agents that belongs to a given cluster.

In addition to cluster statistics condensed in average values *N* and *r* and their standard deviations, the Shannon entropy *H* of cluster distribution is calculated, given by

$$H = -\sum\_{i,j} p\_{ij} \log[p\_{ij}] \tag{9}$$

where *pij* = 0 is the probability density of agents with discretized values of similarity and size, (*S*, √E V). The entropy *H* is useful to monitor the consolidation of the cluster distribution.

In what follows, cluster formation, consolidation, and segregation are monitored by using the quantites *N*, *R*, *S*, and *H*. Of these, *N*, *R*, and *H* are of the greatest interest, because owing to the choice of simulation parameters to maintain average similarity close to a constant, similarity *S* is always nearly the same in the stabilized states, and thus, contains little information about the differences of cluster distributions in the stabilized state.

#### **3. Results**

The simulations are carried out for 25 agents, which possess different but partially overlapping WoBs about *E*0 ± Δ*E*0 (here *E*0 = 120 and Δ*E*0 = 20) edges, which are randomly

selected parts of the same template network consisting of 200 vertexes (i.e., nodes) and 400 edges (i.e., links). However, only 2-core (i.e., each vertex has at least two edges attached) and fully connected WoBs are included as initial states, and thus the initial number of vertexes *V*0 in the WoBs, when pruned to connected networks, varies roughly from 40 to 100 (with a few exceptional cases of 20 to fewer than 40 vertexes). The parameter affecting mostly the evolution of WoBs during the simulations is the sensitivity *β* to similarity of partners (i.e., sensitivity to homophily). Note that simulations are carried out for this parameter in the range from 10 to 300, but results are reported for scaled values *β* → *β*/10 to allow easy presentation in logarithmic scale. The parameters *E*0 and Δ*E*0 are the next most important in affecting the outcome of simulations. The effect of these parameters is tested by keeping ratio *E*0/Δ*E*0 fixed and changing the absolute values by ±%20 by scaling them with a factor of *η* = 0.8 and 1.2.

In simulations with unfolding interaction events *τ*, the agent selects another agent to compare and adjust its WoB to better match the other agent's WoB. The selection probability of the agent to interact depends on the similarity between the agents and the agent's sensitivity *β* to similarity in that selection. In comparison events of WoBs, interacting agents either adopt or delete an edge to increase their mutual similarity.

In Figure 1, examples of initial and final WoBs are shown. In all cases, the number of vertexes *V* (and edges) increase because the utility function favors the addition of edges and vertexes, thus favoring the growth of WoBs. However, initial and resulting final WoBs may have quite different sizes. Note that the initial WoBs are always 2-core networks, but during their evolution, singly connected vertexes may appear. In the final state ensemble, averaged values characterizing WoBs are not changing anymore, although slight changes in details of the structure may take place.

**Figure 1.** Three examples of agents' webs of beliefs (WoBs) (**a**–**c**) are shown for initial WoBs (at left), which in interaction events evolve to the final WoBs (at right). In all cases, WoBs are projected on a common aggregated template (with unconnected vertexes shown).

The dynamic changes in WoBs, driven by the similarity bias of interactions, eventually leads to a formation of consensus clusters. In Figure 2, an example of a similarity cluster in the initial stage is shown, and then in the stabilized final stage for agents making confident choices of interaction partners with *β* = 30 (recall that this is a scaled value *β*/10). The

stabilization of the cluster distributions with *β* = 30, as shown in Figure 2b, is obtained with about 30,000 interaction events in a group of 25 agents, each of them possessing about 100 edges on average (originating from a sampled set 120 edges on average). Roughly, this means that agents need about 10 interaction events for each edge if they are to reach consensus states. Note that in what follows, we report interaction events *τ* scaled by a factor of 1000.