Mathematical Modeling of Individual Behavior Based on Viewpoint Dynamics: Analyzing Group Cohesion Effects Induced by Individual Potential Power Differences

Abstract

The group cohesion effect refers to individuals’ identification with the group’s viewpoint, resulting from behavioral and cognitive changes during interactions, and is crucial for group development. However, individual differences in intrinsic characteristics lead to varied group behaviors and cohesion. This paper uses a mathematical model based on viewpoint dynamics to explore how these differences shape group cohesion. The primary consideration is the potential power inherent in individual characteristics, which can be understood as symmetry-breaking concepts. In the model, individuals are classified into two types, each supporting one of two viewpoints. The potential power reflects the individuals’ degree of firmness regarding their viewpoint and their perceptual range. Differences in the potential power, both within and between types, drive shifts in viewpoints and behaviors, generating diverse cohesion effects. Additionally, the model also incorporates the influence of group size and external factors, such as individuals with no viewpoints and those holding public opinion viewpoints. The results indicate that group size has no significant effect on group cohesion, while individuals with no viewpoints contribute to stabilizing it, whereas individuals with public opinions weaken it. These findings highlight the complex relationship between individual differences in potential power and group cohesion, suggesting that symmetry-breaking dynamics can effectively explain group cohesion effects.

1. Introduction

Groups are fundamental units within the social environment, offering numerous advantages over individuals working alone for accomplishing tasks [1,2]. The group effect is the collective behavioral phenomenon that emerges when individuals within a group are influenced and changed in terms of their behavior, attitudes, opinions, and perspectives. Group effects involve the process by which individuals interact with each other to form collective behaviors, including primarily the consistency effect, the herd effect, and the cohesion effect. Whether it is swarming insects [3] and schools of fish in biology [4,5], active substance groups in physics [6], or swarm robots in engineering [7], group effects and collective behaviors are the outcomes of coordination among individuals [8].

In group systems, individual characteristics play a critical role in shaping overall group dynamics [9]. Understanding these characteristics and their impact on group dynamics and group effects is key to understanding the ecology of these systems [10] and designing control rules in complex systems [11]. Individual characteristics encompass a broad range of factors, including, but not limited to, physical characteristics (e.g., speed, direction, mass), cognitive characteristics (e.g., knowledge, potential powers), physiological characteristics (e.g., hunger, health status), and psychological characteristics (e.g., attitudes). The interaction of these characteristics influences individual behavior and directly impacts various group behaviors and effects. For example, in the butterfly effect phenomenon [12], a single individual’s behavioral perturbation can propagate through an interconnected network, potentially leading to a collective effect. Recent studies on animal groups have shown that when the velocity or acceleration of individuals undergoes specific changes, the group enters various critical behavioral states [13,14,15,16]. External perturbations (e.g., the presence of a predator or the discovery of a critical resource) can induce sudden changes in the velocity of individual characteristics, prompting the entire group to adopt survival strategies such as resistance, aggression, or avoidance [13,15,16]. Furthermore, the information and judgment abilities of individuals can lead to resource preferences and competition among individuals, resulting in variations in group consistency and disorder [17]. However, such information can also regulate group adaptation and maintain group order [18,19]. For instance, individuals in a colony exhibiting more pronounced hunger characteristics may display more aggressive food-searching behaviors, which can improve the overall foraging efficiency of the group [19].

Within groups, networks represent complex individual interactions among individuals [20,21,22,23]. Individuals with the characteristic of having varying interests exhibit cooperative and non-cooperative behavior within social groups, and their opinions fluctuate between cooperative and non-cooperative. Concurrently, the dissemination of opinions across the network ultimately influences the behavior of neighboring individuals and impacts both group cohesion and network density [21]. In multi-domain groups (i.e., groups with distinct network layers), even if individuals within each domain exhibit non-cooperative or resentful tendencies, the presence of cooperative individuals across domains can foster consistency and enhance cohesion throughout the entire group [22]. When an individual with an asymmetric potential power (which may arise from group structure, personal information, or interaction rules [24]) is present, they may assume a leadership role, significantly influencing the group’s trajectory and behavior [25]. Similarly, analyzing the characteristics of the individuals capable of altering group states, such as cohesion, polarization, or vortex formations, allows for precise adjustments of individual behaviors to achieve controllable transitions between different group states and effects [26]. This paper focuses on the potential power inherent in individual characteristics, which influences individual behavior and roles within the group, thereby affecting group collective behavior and the cohesion effect.

Individual differences are prevalent across various group types, including animal and social groups. Examining this differentiation is essential for group cooperation and development. A growing body of research has emphasized the role of individual character differences, as they directly lead to varying group effects [16,17,21,27,28,29,30,31]. For example, the Axelrod model [30] illustrates how local interactions lead to the emergence of cultural domains. In this model, an individual’s cultural characteristics are represented as a finite vector of pluralistic opinion states. After interacting with their neighbors, individuals copy some of their neighbors’ cultural traits, with the rate of change being proportional to the degree of similarity between the two individuals. Consequently, individuals tend to associate with others who share similar traits, leading to homogeneity. Building on this, Reia [31] improved the model by introducing the concept of comfort level, which reflects an individual’s tendency to remain in a neighborhood based on the similarity of its neighboring individuals’ cultural characteristics. These models explain the varying group effects arising from individual characteristic differences through simple interaction rules.

Interestingly, most studies on group effects have focused on individual viewpoint shifts and decision making [32,33]. The dynamics of viewpoints is a critical area in group dynamics and group effects. In particular, external factors (e.g., public opinion dynamics, etc.) impact group effects and may lead to changes in the entire group’s behavior [32,33,34,35]. This paper explores how potential power differences in individual characteristics drive shifts in individual viewpoints and their subsequent impact on group cohesion. Here, we are mainly concerned with the group cohesion effect, which arises from the degree of firmness and identification with supported viewpoints following individual interactions. Group cohesion is crucial for maintaining and developing the group and directly influences the achievement of group goals [36,37,38].

This paper builds on previous group dynamics models [30,31,39,40,41,42,43] and presents an enhanced mathematical model to describe individual behavior in groups. In the model, we consider two types of individuals who support each of the two viewpoints. From the perspective of symmetry-breaking, we examine how differences in the potential power stemming from individual characteristics affect viewpoint shifts, decision making, and group cohesion for each viewpoint. An individual’s potential power is reflected in the firmness of their viewpoints and their perceptual range. Furthermore, we introduce an external perturbation variable to the model, representing individuals with no viewpoints and those with public opinion viewpoints. The main contribution of this paper lies in the mathematical model, designed to reflect the interaction of viewpoints and behavioral decisions among individuals, and the resulting different group behaviors and effects.

2. Preliminaries

2.1. Complex Network

Complex interactions among individuals in groups resemble a complex relationship network (as shown in Figure 1a), where edges represent connections between individuals. These connections are often bidirectional, facilitating the exchange of information, exploration, suggestions, and more [44]. Thus, complex networks are an indispensable theoretical and methodological tool for analyzing and understanding the complex structures and relationships within groups [45]. Complex networks are composed of multiple undirected graphs (as shown in Figure 1b), with “nodes” representing individuals and “edges” representing the links between them. The mathematical definition of the complex relational network is as follows:

Definition 1.

For an undirected graph, G(t) = (V, E(t), A(t)), containing m individuals, where V = {1, 2, …, m} represents the set of individuals, and E(t) denotes the set of edges between individuals, that is, the set of interactions among individuals. $A(t)=[a_{ij}(t)]\in R^{m\times m}$ represents the adjacency matrix, and if there is a link between individuals, then a_ij(t) = 1; otherwise, a_ij(t) = 0. The Laplacian matrix L = (l_ij(t)) is defined as L(t) = D(t) − A(t), where D(t) = diag(d_i) is the degree matrix, and d_i is the degree of node i. l_ij (t) = −a_ij(t), $\forall i\ne j$.

2.2. Individual Modeling

In groups, each individual needs to have a perceptual range in order to be able to interact with others around them. However, for most individuals in groups, there are limitations to the range of communication for each individual, and they are not always in contact with others. Consequently, each individual’s self-perception range is limited. Additionally, to avoid collisions in physical space during interactions, individuals need to maintain a safe distance from each other. Considering the possible actual field of view of an individual, we introduced this concept into the individual model. In summary, the individual model is illustrated in Figure 1c. The arrow indicates the direction of the individual movement.

In Figure 1c, d_s represents the safe distance required to avoid crowding and collisions. r represents the perception range of an individual. The perceptual range, r, is finite and always greater than d_s. In this model, the ratio of r to d_s is fixed at 1.4 [46]. θ represents the blind field of view range of an individual.

2.3. Inter-Individual Interaction

When individuals need to communicate and interact with each other, they need to approach each other to establish contact within their perceptual range, while simultaneously maintaining a safe distance to avoid collisions. Interestingly, the attraction and repulsion potentials in the potential function of molecular dynamics can accurately represent these processes. When two individuals are within the perceptual range, the attraction potential allows them to attract and move closer to each other. When the distance is smaller than is safe, the repulsion potential causes them to repel each other. In many studies on group dynamics, adopting the potential function to depict inter-individual interactions is both representative and effective [46,47,48,49,50]. Therefore, in combination with the individual model presented in this paper, the potential function needs to satisfy the following conditions:

• When the distance between individuals is less than the safe distance, d_s, the potential function exhibits a repulsive potential to avoid collisions, even within an individual’s blind field of view θ.
• When the distance between individuals is greater than the safe distance, d_s, but smaller than the perception range, r, and within the individual’s field of view (2π − θ), the potential function exhibits an attractive potential to draw the two individuals closer to facilitate interaction and information exchange. If individuals are within the blind field of view range, θ, they do not attract each other.
• When the distance between individuals exceeds their respective perceptual ranges, r, they are likewise independent of each other and do not influence each other.

Based on the above description, the potential function constructed in this paper is shown in Equation (1). The image of the potential function is presented in Figure 2. Here, we set d_s = 5 and r = 7, where o_ij denotes the distance between individual i and individual j. w denotes the potential function’s well depth. The schematic diagram of inter-individual interaction is shown in Figure 3.

$$P(o_{ij})=\left\{\begin{array}{cc}{k}_1({\displaystyle \frac{o_{ij}}{d_s}}-\log (o_{ij})+\log (d_s)-1)& 0<o_{ij}\le d_s\\ k_2(\log (o_{ij}-d_s+r)-\log (r))& d_s<o_{ij}\le r\\ 0& r<o_{ij}\end{array}\right.$$

(1)

In Figure 3, f_a is the attractive force, and f_r is the repulsive force. dir denotes the current moving direction of individual i. β denotes the angle between the other individual j and the current moving direction of individual i. The specific expression of β is as follows:

$$\beta =\arccos \left|{\displaystyle \frac{v_{ix}{‖o_{ij}‖}_x+v_{iy}{‖o_{ij}‖}_y}{‖o_{ij}‖}}\right|$$

(2)

where ‖ ‖ denotes the Euclidean paradigm. When β < θ/2, it indicates that individual j is within the blind field of view of individual i, meaning they do not approach each other for interaction. Conversely, when β ≥ θ/2, individual j is within the field of view of individual i and can interact and exchange information.

2.4. γ-Lattices and Quasi-γ-Lattices

One of our objectives is to design group mathematical models that enable a set of dynamic individuals to maintain a specific distance within a network G(o). G(o) is denoted as a spatially induced graph determined by the individual’s position, o, i.e., an undirected graph. Without the loss of generality, we assume that all individuals in the network will maintain the same distance, which leads us to consider the following set of algebraic constraints on inter-individual interactions:

$$‖o_j-o_i‖=d_s,\forall j\in N_i(t)$$

(3)

$$N_i(t)=\{j\in V:‖o_j-o_i‖<r\}$$

(4)

where $N_i(t)$ denotes the set of neighbors of individual i. The position, o, of an individual is a solution to the constraint set in Equation (3). Therefore, it is convenient to define these solutions as lattice-type objects.

Definition 2.

γ-lattice: a γ-lattice is a configuration, o, that satisfies the set of constraints in Equation (3). We denote d_s and α = r/d_s as the scales and ratios of the γ-lattice.

It is important to note that the γ-lattice-induced undirected network, G(o), does not necessarily need to be connected. Common examples of two-dimensional γ-lattices (where the interaction range r = αd_s, so α = 1 + ϑ and ϑ > 0) include the square lattice, the hexagonal lattice, the set of vertices of the orthogonal polygon, and the Platonic cubic. An example of γ-lattices is illustrated in Figure 4.

Additionally, we consider slightly deformed γ-lattices, which are defined as shown below:

Definition 3.

Quasi-γ-lattice: a quasi-γ-lattice is a configuration, o, that satisfies the following set of inequality constraints.

$$-\tau \le ‖o_j-o_i‖-d_s\le \tau ,\forall (i,j)\in E(o)$$

(5)

where $\tau \ll d_s$ is the edge length uncertainty, d_s is the scale, and α = r/d_s is the ratio of the quasi-γ-lattice. E(o) is the set of undirected graph edges of graph G(o).

To measure the degree of difference between the configurations o and γ-lattices, we use the following deviation energies [42].

$$E_g(o)={\displaystyle \frac{1}{\left|E(o)\right|+1}}{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in N_i}\varphi (‖o_j-o_i‖-d_s)}}$$

(6)

$\varphi (x)=x^2$ is the pair potential function (other scalar potentials can also be used). The deviation energy can be viewed as a non-smooth function of the m-particle system. However, the γ-lattice represents the global minimum of the potential function and reaches 0. For a quasi-γ-lattice, o, with an uncertainty in the edge length of τ, the deviation energy is shown below:

$$E_g(o)\le {\displaystyle \frac{\left|E(o)\right|}{\left|E(o)\right|+1}}{\tau }^2\le {\tau }^2={\vartheta }^2{d}_s^2,\vartheta \ll 1$$

(7)

This implies that the quasi-γ-lattice is a low-energy conformation of m nodes. The order of magnitude of the quasi-γ-lattice deflection energy in Figure 4b is ${10}^{-3}$. Therefore, α = 1.4 is justified. Next, we explain the reason for choosing α = 1.4. For γ-lattices, the induced undirected graph G(o) in D dimensional space satisfies the following theorem.

Theorem 1

[42]. Let o be a γ-lattice with m different positional nodes of scale d_s > 1 and a ratio of α > 1. Then

(1) 

The graph G(o), induced by o, is a planar graph with D = 2, 3 dimensions.

(2) 

The graph G(o) has, at most, 3m-6 connected edges in D = 2 dimensions.

(3) 

The graph G(o), induced by a γ-lattice with the number of nodes m > D + 1 in D of dimensions 1, 2, and 3, will not constitute a complete graph.

According to part (3) of Theorem 1, the γ-lattice has the following characterization: If there is a moderate or large number of nodes in the lattice (m ≥ 4 nodes for D = 2), this leads to the fact that its spatially induced graphs G(o) are never complete graphs. This feature prevents us from using all-to-all networks to construct groups after the formation of individual interactions. Also, according to part (2) of Theorem 1, the total number of connected edges of the γ-lattice in D = 2 dimensions is linearly related to m.

Meanwhile, Theorem 1 also restricts the maximum ratio of the planarity of the graph induced by the γ-lattice to the perception range, r, and the safe distance, d_s. For example, a cubic lattice, o, in dimension D is an effective γ-lattice with a ratio of α = 1 + ϑ if $0<\vartheta <\sqrt{D}$. Otherwise, the two nodes on either side of the hypercube’s diagonal will be neighbors, which will make o become an ineffective γ-lattice. Therefore, each node is unable to maintain an equal distance between itself and all its neighbors (i.e., safety distance constraints). We show this result in the form of the D = 2 dimensions in Figure 5.

In summary, without the loss of generality, in order to allow individuals to interact fully in space and to satisfy the above conditions ($\vartheta \ll 1$ in Equation (7), 1 + ϑ < $\sqrt{2}$), we set the ratio of the perception range, r, to the safe distance, d_s, to be 1.4.

2.5. Individual Potential Power Differences

The potential power of an individual refers to the influence and capacity that the person can exert, stemming from factors such as skills, knowledge, and experience. Within a group, differences in individuals’ potential power can lead to differences in knowledge and spheres of influence, which in turn shape their roles and abilities. Individuals with extensive knowledge and a broad sphere of influence often act as leaders, guiding others in decision-making processes. In the context of this paper, potential power differences are reflected in the degree of firmness of individuals’ viewpoints and their perceptual range. Those with a higher potential power generally have deeper insights and a more comprehensive understanding of their perspectives, demonstrating greater firmness to their viewpoint. Conversely, individuals with a low potential power have less knowledge and exhibit weaker firmness with their viewpoint. These differences in potential power contribute to variations in individual behavior, thereby shaping group dynamics. When individuals with different potential powers interact, they communicate about their viewpoints and knowledge reserves. Those with higher potential power tend to persuade individuals with lower potential power, leading them to align with and “follow” the more powerful individual in decision making. In other words, differences in individuals’ potential power correspond to differences in leadership ability.

We represent this state of “following” as an attractive force between individuals. Specifically, the more powerful individual attracts the less powerful individual to make their decision. In the mathematical model, the well depth in the potential function represents the magnitude of potential power. In [49], it was demonstrated that the well depth is positively correlated with the strength of attraction. Additionally, an individual’s leadership ability is determined by both their potential power and their perceptual range [46]. The greater an individual’s potential power and perceptual range, the stronger their leadership. Therefore, considering a group containing m individuals, we define the well depth parameter for each individual as $X_i=\{i\in m|X_1,X_2,\cdots ,X_m\}$ and X_i ∈ [0, 1]. X_iw represents the potential power of individual i. The larger the X_i is, the higher the individual’s firmness in their viewpoint and potential power. The perceptual range and safe distance of individual i are r_i = ξ(X_i + k) and d_si = r_i/α, respectively, where $\xi$ and $k$ are positive numbers. Since the ratio of d_s to r is fixed, the potential function’s well depth, w, is fixed for each individual according to Equation (1), as shown in Remark 1.

Remark 1.

The attraction potential used for interactions between individuals depends on the potential function’s well depth. For the potential function P(o_ij) (as in Equation (1)), the well depth is the vertical distance from the minimum point of the potential energy (global minimum) to the steady state point (limit) of the potential energy within the range of individual interactions [49], denoted as

$$\begin{array}{ll}w& =\underset{o_{ij}\to r}{\lim }P(o_{ij})-P({\nabla }_{o_{ij}\in (0,r]}P(o_{ij})=0)\\ & =k_2(\log (r-d_s+r)-\log (r))\\ & =k_2(\log ((1-\vartheta )r)-\log (r))\\ & =k_2(\log (1-\vartheta ))\end{array}$$

where k₂ is a constant. Thus, the value of the well depth corresponding to each individual is fixed when the perception range, r, is proportional to the safe distance, d_s.

3. Mathematical Models Describing Individual Behavior

In this section, we present a detailed mathematical model and corresponding formulas describing individual motions. In our model, all the individuals in the group are classified into two types, each supporting one of two viewpoints, denoted as Op₁ and Op₂. The individuals supporting Op₁ are labeled as type C₁ individuals, and the individuals supporting Op₂ are labeled as type C₂ individuals. Within each type, there are potential power differences, as well as differences in the potential power between the individuals of different types. The two types of individuals’ initial positions are randomly distributed within a finite space and obey a Gaussian distribution. The interactions between individuals influence their final decisions. Through the interaction results, we can observe the impact of the potential power differences on the structure and cohesion effect of each viewpoint group. It is important to note that viewpoints Op₁ and Op₂ are not antagonistic or mutually exclusive; instead, they represent different approaches to solving the same problem.

In terms of external perturbations, we introduce two additional types of individuals: individuals with no viewpoints (type B individuals) and individuals with public opinion viewpoints (type P individuals). Individuals with no viewpoints lack specific viewpoints and, thus, have no potential power in the model, yet they retain the perceptual range to interact with other individuals. Individuals with public opinion viewpoints possess public opinion power, which can influence the potential power of other individuals, thereby affecting their decision-making processes. Consequently, both types of external perturbations affect type C₁ and C₂ individuals, leading to varied behaviors and ultimately affecting the group cohesion effect with each viewpoint. The analysis schematic of this study is depicted in Figure 6. The detailed interaction rules and mathematical models describing the interactions between these different types of individuals are presented in the following sections.

3.1. Only Two Types of Individuals—C₁, C₂

When there are only two types of individuals and no external perturbations, the total number of individuals, m, is the sum of type C₁ and C₂ individuals (i.e., m = C₁ + C₂). Both individuals of the same type and different types can interact with each other. The interaction diagram and the interaction rules between individuals are illustrated in Figure 7 and defined as follows:

Interaction rules of type C₁ and C₂ individuals:

• When two individuals, i and j, in the same type interact ($i,j\in C_1$ or $i,j\in C_2$), the individual with the lower potential power will always obey and “follow” the individual with the higher potential power, thereby supporting the same viewpoint (Figure 7b,c)
• When two individuals, i and j, belong to different types, if the potential power of individual i (of type C₁, initially supporting Op₁) is stronger than that of individual j (of type C₂, supporting Op₂), then after the information exchange and communication during the interaction, individual j will “follow” individual i and ultimately support Op₁ (Figure 7d). In this process, if a higher potential power individual, k, who supports Op₂, interacts with i and j (i.e., $i\in C_1,j,k\in C_2$), then individuals i and j will also “follow” that higher potential power individual, k, and this will ultimately lead to individuals i, j, and k supporting Op₂ (Figure 7e).

According to the above rules, we build upon the Olfati model [42,51,52,53] and integrate it with the individual model described in Section 2 to establish this paper’s mathematical framework. This framework [30,31,41,46,51,52] encompasses diverse forms of interactions, including, but not limited to, aligned movement, collision avoidance between individuals, and guidance mechanisms for individuals to navigate toward the corresponding viewpoint. To describe the motion of each individual in two-dimensional space, we employ a second-order continuous motion equation, as described by the following formula:

$$\left\{\begin{array}{c}{\dot{o}}_i=v_i\\ {\dot{v}}_i=g_i\end{array}\right.,i=1,2,\cdots ,m$$

(8)

where $o_i$ represents the position of individual i and $v_i$ represents the velocity of individual i. $g_i$ represents the control input for the movement of individual i, which is expressed as follows:

$$g_i=g_i^a+g_i^{\beta }+g_i^{\gamma }$$

(9)

$g_i^a$ is the interaction term used to represent the interaction process between individuals. For the case of only two types of individuals (type C₁ and C₂ individuals), the specific expression is

$$\underset{(i,j\in U_1)}{g_i^{\alpha }}=\left\{\begin{array}{cc}{\displaystyle \sum _{j\in N_i(t)}[X_j\nabla P_{att}(‖o_{ij}‖)+\nabla P_{re}(‖o_{ij}‖)]e_{ij}}& \beta \ge {\displaystyle \frac{\theta }{2}}\\ {\displaystyle \sum _{j\in N_i(t)}\nabla P_{re}(‖o_{ij}‖)e_{ij}}& \beta <{\displaystyle \frac{\theta }{2}}\end{array}\right.$$

(10)

$$U_1=\left\{i,j|i,j\in C_1\right\}\cup \left\{i,j\in C_2\right\}\cup \left\{i\in C_1,j\in C_2\right\}\cup \left\{i\in C_2,j\in C_1\right\}$$

(11)

They can only attract each other for interaction within the individual’s field of view. $N_i(t)$ denotes the set of other individuals within the perceptual range of individual i at time t. Since $\nabla P_{att}\propto w$, $X_j\nabla P_{att}$ can represent the potential power of individual j within the perceptual range of individual i. $\nabla P_{att}(x)$ and $\nabla P_{re}(x)$ denote the attraction and repulsion potentials in Equation (1), respectively.

$$\begin{array}{l}\nabla P_{re}(‖o_{ij}‖)=k_1({\displaystyle \frac{o_{ij}}{d_s}}-\log (o_{ij})+\log (d_s)-1)\\ \nabla P_{att}(‖o_{ij}‖)=k_2(\log (o_{ij}-d_s+r)-\log (r))\end{array}$$

(12)

$e_{ij}=(o_j-o_i)/\sqrt{1+l{‖o_j-o_i‖}^2}$ is the connection vector between individual i and individual j, $l\in (0,1)$.

$g_i^{\beta }$ is the speed-matching term between individuals. When individuals interact with each other, individuals with a lower potential power will “follow” individuals with a higher potential power and support the final viewpoint. In order to sustain this “following” state and connection, it is necessary to ensure that individuals keep a fixed safe distance from one another, which requires matching speeds. Specifically, the speed of the individual with lower potential power will actively adjust to match the speed of the individual with higher potential power. The expression for $g_i^{\beta }$ is shown below:

$$\underset{(i,j\in U_1)}{g_i^{\beta }}=\left\{\begin{array}{cc}{\displaystyle \sum _{j\in N_i(t)}\max (X_i,X_j)a_{ij}(t)(v_j-v_i)}& \beta \ge {\displaystyle \frac{\theta }{2}}\\ 0& \beta <{\displaystyle \frac{\theta }{2}}\end{array}\right.$$

(13)

Similarly, speed matching is only possible within the individual’s field of view and is not possible within each other’s blind field of view. $a_{ij}(t)$ is the adjacency matrix between individuals i and j (as shown in Definition 1).

$g_i^{\gamma }$ denotes the initial viewpoint support term for each individual. In this model, two viewpoints, Op₁ and Op₂, are introduced, and while type C₁ and C₂ individuals initially support these viewpoints, their initial viewpoint support terms are the same—only the types differ. In the mathematical model, we represent this support for the viewpoint as the individual’s position converges to the viewpoint position. Then, the initial viewpoint support term for each type of individual is shown in Equation (14).

$$\begin{array}{l}{g}_i^{\gamma }(i\in C_1)={\displaystyle \frac{o_{op1}-o_i}{‖o_{op1}-o_i‖}},i\in 1,2,\cdots C_1\\ g_i^{\gamma }(i\in C_2)={\displaystyle \frac{o_{op2}-o_i}{‖o_{op2}-o_i‖}},i\in 1,2,\cdots C_2\end{array}$$

(14)

where $o_{op1}$ and $o_{op2}$ are the positions in two dimensions of the Op₁ and Op₂, respectively.

3.2. Three Types of Individuals—C₁, C₂, and B

In this section, we introduce an external perturbation, i.e., the individuals with no viewpoints, B. The total number of individuals becomes m = C₁ + C₂ + B. Since individuals with no viewpoints do not have any viewpoints, they are influenced only by type C₁ and C₂ individuals, and do not impact them in return. So, type B individuals have no potential power, i.e., no attraction potential or well depth in the potential function. However, type B individuals still retain their perceptual range and safe distance, allowing them to interact with others (being guided) while avoiding collisions. Therefore, the interaction diagram and the interaction rules for these three types of individuals are shown in Figure 8 and are defined as follows:

Interaction rules of type C₁, C₂, and B individuals:

• Type C₁ and type C₂ individuals interact with each other in the same manner as described in Section 3.1 (Figure 7).
• When type C₁ individuals interact solely with type B individuals, or type C₂ individuals interact solely with type B individuals ($i\in C_1/C_2,j\in B$), type B individuals will “follow” the type C₁ or C₂ individuals and eventually support the corresponding viewpoint (Figure 8b,c).
• When type B individuals interact with both type C₁ and C₂ individuals ($i\in C_1,j\in C_2,k\in B$), type B individuals will “follow” the high potential power individual of the interacting type C₁ or C₂ individuals, supporting the corresponding viewpoint. Thus, individual i ultimately leads to individuals j and k supporting Op₁ (Figure 8d).
• When type B individuals interact with each other ($i,j\in B$), they do not actively approach one another to exchange information (Figure 8e).

According to the above rule, when there are three types of individuals, their interaction term $g_i^a=g_i^{a1}+g_i^{a2}$ is shown below:

$$\underset{(i,j\in U_2)}{g_i^{\alpha 1}}=\left\{\begin{array}{cc}{\displaystyle \sum _{j\in N_i(t)}[X_j\nabla P_{att}(‖o_{ij}‖)+\nabla P_{re}(‖o_{ij}‖)]e_{ij}}& \beta \ge {\displaystyle \frac{\theta }{2}}\\ {\displaystyle \sum _{j\in N_i(t)}\nabla P_{re}(‖o_{ij}‖)e_{ij}}& \beta <{\displaystyle \frac{\theta }{2}}\end{array}\right.$$

(15)

$$g_i^{\alpha 2}(i,j\in U_3)={\displaystyle \sum _{j\in N_i(t)}\nabla P_{re}(‖o_{ij}‖)e_{ij}}$$

(16)

$$U_2=U_1\cup \left\{i,j|i\in C_1,C_2,j\in B\right\}$$

(17)

$$U_3=\left\{i,j|i,j\in B\right\}$$

(18)

However, the speeds of individuals belonging to the same type B can be matched, thus creating the “no viewpoints subgroups”. The speed match terms $g_i^{\beta }=g_i^{\beta 1}+g_i^{\beta 2}$ are shown below:

$$\underset{(i,j\in U_2)}{g_i^{\beta 1}}=\left\{\begin{array}{cc}{\displaystyle \sum _{j\in N_i(t)}\max (X_i,X_j)a_{ij}(t)(v_j-v_i)}& \beta \ge {\displaystyle \frac{\theta }{2}}\\ 0& \beta <{\displaystyle \frac{\theta }{2}}\end{array}\right.$$

(19)

$$g_i^{\beta 2}(i,j\in U_3)={\displaystyle \sum _{j\in N_i(t)}{a}_{ij}(t)(v_j-v_i)}$$

(20)

Since type B individuals have no initial viewpoint, there is no individual initial viewpoint support term. Only type C₁ and C₂ individuals have viewpoint support terms. Therefore, in the case of three types of individuals, $g_i^{\gamma }$ is the same as in Equation (14).

3.3. Four Types of Individuals—C₁, C₂, B, and P

In this section, we introduce another external perturbation: the individuals with public opinion viewpoints, P. Thus, the total number of individuals becomes m = C₁+ C₂ + B + P. The individuals with public opinion viewpoints have their own potential power, i.e., the public opinion power, which is determined by the magnitude of their well depth parameter. It is crucial to note that type P individuals can only influence the potential power of type C₁, C₂, and B individuals but do not “follow” them to support their respective viewpoints. Furthermore, the perceptual range of type P individuals increases with the degree of their public opinion power. The interaction diagram of these four types of individuals and the interaction rules are shown in Figure 9 and are defined as follows:

Interaction rules of type C₁, C₂, B, and P individuals:

• Type C₁ and type C₂ individuals interact with each other in the same manner as described in Section 3.1 (Figure 7).
• Type B individuals interact with type C₁ and C₂ individuals and with the same individuals with no viewpoints, type B, in the same manner as described in Section 3.2 (Figure 8).
• When type P individuals interact with type C₁ and C₂ individuals, type C₁ and C₂ individuals compare their own potential power magnitude with type P individuals. (1) If the potential power of type C₁ and C₂ individuals is less than that of type P individuals ($X_{C_1,C_2}<X_P$), it indicates that public opinion has influenced them, weakening the firmness of their respective viewpoints, thereby reducing their potential power ($i\in C_1,j\in C_2,k\in P$, Figure 9b). (2) If the potential power of type C₁ and C₂ individuals is greater than that of type P individuals ($X_{C_1,C_2}>X_P$), type C₁ and C₂ individuals will remain unaffected by public opinion (Figure 9c), i.e.,

  $$\left\{\begin{array}{cc}{X}_{C_1,C_2}=X_{C_1,C_2}-\left|X_P-X_{C_1,C_2}\right|& X_{C_1,C_2}<X_P\\ X_{C_1,C_2}=X_{C_1,C_2}& X_{C_1,C_2}>X_P\end{array}\right.$$

  where $X_{C_1,C_2}>0$. If $X_{C_1,C_2}-\left|X_P-X_{C_1,C_2}\right|<0$, then $X_{C_1,C_2}=0.01$.
• When type P individuals interact with type B individuals, the lack of viewpoints in type B individuals results in them acquiring a degree of stubbornness (potential power) from type P individuals, making them less susceptible to influence by type C₁ and C₂ individuals. We label the type B individuals who are not influenced by the type P individuals as B₀ and those who are influenced by the type P individuals and possess a degree of stubbornness as B₁ (Figure 9d). Then, $X_{B_1}=X_P$, $X_{B_0}=0$.
• Type B individuals with this acquired stubbornness exhibit the following behaviors when interacting with type C₁ and C₂ individuals: (1) If the degree of stubbornness of type B₁ individuals exceeds the potential power of type C₁ and C₂ individuals, they remain uninfluenced and do not support any viewpoint (Figure 9e). (2) If the degree of stubbornness of type B₁ individuals is less than the potential power of type C₁ and C₂ individuals, they are guided by these individuals and “follow” them in supporting the corresponding viewpoints (Figure 9f). Of course, the final support still depends on the potential power of the interacting type C₁ and C₂ individuals.

According to the above rule, when there are four types of individuals, their interaction terms $g_i^a=g_i^{a3}+g_i^{a4}$ are shown as follows:

$$\underset{(i,j\in U_4)}{g_i^{\alpha 3}}=\left\{\begin{array}{cc}{\displaystyle \sum _{j\in N_i(t)}[X_j\nabla P_{att}(‖o_{ij}‖)+\nabla P_{re}(‖o_{ij}‖)]e_{ij}}& \beta \ge {\displaystyle \frac{\theta }{2}}\\ {\displaystyle \sum _{j\in N_i(t)}\nabla P_{re}(‖o_{ij}‖)e_{ij}}& \beta <{\displaystyle \frac{\theta }{2}}\end{array}\right.$$

(21)

$$g_i^{\alpha 4}(i,j\in U_5)={\displaystyle \sum _{j\in N_i(t)}\nabla P_{re}(‖o_{ij}‖)e_{ij}}$$

(22)

$$U_4=U_1\cup \left\{i,j|i\in C_1,C_2,j\in B_1\right\}$$

(23)

$$U_5=\left\{i,j|i,j\in B_0\right\}\cup \left\{i,j|i,j\in P\right\}\cup \left\{i,j|i\in P,j\in C_1,C_2,B_0\right\}\cup \left\{i,j|i\in P,j\in C_1,C_2,B_1\right\}$$

(24)

Similarly, speeds can be matched between the individuals of the same type, P, thus creating “public opinion subgroups”. The speed match terms $g_i^{\beta }=g_i^{\beta 3}+g_i^{\beta 4}$ are shown below:

$$\underset{(i,j\in U_4)}{g_i^{\beta 3}}=\left\{\begin{array}{cc}{\displaystyle \sum _{j\in N_i(t)}\max (X_i,X_j)a_{ij}(t)(v_j-v_i)}& \beta \ge {\displaystyle \frac{\theta }{2}}\\ 0& \beta <{\displaystyle \frac{\theta }{2}}\end{array}\right.$$

(25)

$$g_i^{\beta 4}(i,j\in U_5)={\displaystyle \sum _{j\in N_i(t)}{a}_{ij}(t)(v_j-v_i)}$$

(26)

Since type P individuals only influence the potential power of type C₁, C₂, and B individuals and do not follow them in supporting the corresponding viewpoints, there is no individual initial viewpoint support term. In the case of four types of individuals, $g_i^{\gamma }$ is the same as in Equation (14).

3.4. Group Cohesion Effect

The group cohesion effect refers to the strength of individuals’ connections within a group, which is driven by their identification with a particular viewpoint. In the context of this paper, the group cohesion effect is conceptualized as group cohesion in support of a viewpoint (Op₁ or Op₂), which manifests as the intensity of the connections between individuals. This intensity is evidenced by the attraction between individuals. Consequently, the formula for calculating the average group cohesion $C_o(O{p}_1/O{p}_2)$ that supports each viewpoint after individual interactions is provided in Equation (27). The average individual potential power of all the individuals that support each viewpoint is provided in Equation (28).

$$C_o(O{p}_1/O{p}_2)={\displaystyle \frac{1}{m_{O{p}_1/O{p}_2}}}{\displaystyle \sum _{i=1}^{m_{O{p}_1/O{p}_2}}{\displaystyle \sum _{j\in N_i(t)}{X}_j\nabla P_{att}(‖o_j-o_i‖)}}$$

(27)

$$P_p(O{p}_1/O{p}_2)={\displaystyle \frac{1}{m_{O{p}_1/O{p}_2}}}{\displaystyle \sum _{i=1}^{m_{O{p}_1/O{p}_2}}{X}_{i}w}$$

(28)

where $m_{O{p}_1}$ and $m_{O{p}_2}$ are denoted as the number of individuals who ultimately support Op₁ and Op₂, respectively. The average potential power of individuals describes their decision-making ability and viewpoint leadership ability.

4. Experimental Results

This section presents the experimental results obtained under various conditions and external perturbations, allowing us to observe how individual behaviors and group cohesion effects vary due to differences in individuals’ potential power. Initially, we examine group cohesion when there are differences in the potential power among individuals within the same type (C₁ or C₂) while maintaining the same average potential power for both types. Subsequently, we introduce a difference in the average potential power between type C₁ and C₂ individuals and analyze the resulting changes in group cohesion. Additionally, we investigate how differences in the number of type C₁ and C₂ individuals, as well as the public opinion power magnitude of type P individuals, affect the group cohesion and the individual potential power. It is worth noting that the purpose of this paper is to explore differences in potential power. Therefore, it ignores the effects of individuals’ size and scale, i.e., all individuals are assumed to have the same size and scale.

4.1. Experimental Results for Only Two Types of Individuals—C₁ and C₂

4.1.1. Type C₁ and C₂ Individuals Have Potential Power Differences Within the Same Type and the Same Average Potential Power Within the Two Types

Before the start of the experiment, we provided detailed experimental parameter settings, as shown in

Table 1. Parameter Settings.

Parameter Name	Parameter Value
Number of all individuals	$m=100$
Number of C1 individuals	$m_{C_1}=50$
Number of C2 individuals	$m_{C_2}=50$
Values of two types of individuals obeying a Gaussian distribution $N(\mu ,{\sigma }^2)$	$\mu =0$, ${\sigma }^2=50$
Parameters of perception range and well depth $\xi$,$k$	$\xi =7$, $k=0.8$
Coordinates of viewpoints Op1 and Op2	[−50, −50]; [50, 50]
Individual blind visual field angle θ	$\theta =\pi /3$
Potential function parameters	$k_1=6,k_2=10$
Type C1 individuals’ average individual potential power	${\displaystyle \frac{1}{m_{C_1}}}{\displaystyle \sum _{i\in C_1}^{m_{C_1}}{X}_i}=0.5$
Type C2 individuals’ average individual potential power	${\displaystyle \frac{1}{m_{C_2}}}{\displaystyle \sum _{i\in C_2}^{m_{C_2}}{X}_i}=0.5$

. The experimental results are shown in Figure 10, where “Step” represents the number of steps taken by all individuals, with “Step = 0” indicating the initial moment, when the individuals were stationary.

In Figure 10a, the red individuals represent type C₁ individuals supporting Op₁, while the blue individuals represent type C₂ individuals supporting Op₂. The shades of color for the individuals in types C₁ and C₂ denote the magnitude of their potential power; darker shades indicate a greater potential power, whereas lighter shades indicate a lesser potential power. As individuals interact, their initial viewpoints may change. In Figure 10b, when a type C₁ individual interacts with a type C₂ individual, the one with a lower potential power tends to “follow” the one with a higher potential power, adopting their viewpoint. Figure 10d depicts the groups ultimately supporting each viewpoint after interactions, with both type C₁ and C₂ individuals present in each viewpoint group.

Figure 11 presents the initial and final average individual potential power and group cohesion for Op₁ and Op₂. Figure 11a visualizes the group cohesion supporting each viewpoint at the final step (Step = 1000 in Figure 10d), with the color gradient from blue to yellow representing the cohesion from low to high. In Figure 11b, IAPOp₁, and IAPOp₂ represent the average individual potential power of all the individuals initially supporting Op₁ and Op₂, respectively. FAPOp₁ and FAPOp₂ denote the average individual potential power of all the individuals that ultimately support Op₁ and Op₂. Figure 11c shows the initial average group cohesion for viewpoints Op₁ and Op₂, i.e., IACOp₁ and IACOp₂, while FACOp₁ and FACOp₂ represent the final average group cohesion after interactions.

Remark 2.

IACOp₁ and IACOp₂ represent the initial average group cohesion for each viewpoint. Specifically, IACOp₁ denotes the cohesion of the group supporting Op₁, consisting solely of type C₁ individuals. Similarly, IACOp₂ consists solely of type C₂ individuals. To obtain this data set, we built on the rules in Section 3.1 so that types C₁ and C₂ individuals could not interact with each other. The experimental results are shown in Figure 12. After the final formation of the groups, although the average potential power of the two types of individuals was the same, IACOp₁ and IACOp₂ exhibited some differences, rather than absolute consistency, due to the random initial distribution of the individuals with varying potential powers and the different interactions each individual experiences.

4.1.2. Type C₁ and C₂ Individuals Have Potential Power Differences Within the Same Type and Average Potential Power Differences Within the Two Types

In this section, the average potential power among individuals of types C₁ and C₂ is varied. Specifically, the average potential power within type C₁ individuals is greater than that of type C₂ individuals, i.e., $1/m_{C_1}{\displaystyle \sum _{i\in C_1}^{m_{C_1}}{X}_i}=0.75$, $1/m_{C_2}{\displaystyle \sum _{i\in C_2}^{m_{C_2}}{X}_i}=0.25$. All the other conditions remain the same, and the initial position matrices of all the individuals are identical to those in the previous section. The goal of this section is to observe the impact on group cohesion when there is a difference in the average potential power between the two types of individuals. The experimental results are shown in Figure 13 and Figure 14. Note: in this section of the experiment, the initial position matrix of the individuals is the same as in the previous section, so the image at Step = 0 is shown in Figure 10a.

4.1.3. Type C₁ and C₂ Individuals Differ in Number

In this section, we focus on observing the impact of the varying numbers of individuals in each viewpoint group on cohesion. We keep the number of the type C₁ individuals constant $m_{C_1}=50$ and increase the number of type C₂ individuals to 100, i.e., $m_{C_2}=100$. The experiment was conducted under two conditions: one where the average potential power of the two types was the same, and one where it was different. The results of the experiment are shown in Figure 15 and Figure 16.

Figure 15 shows that individuals with a high potential power in type C₁ can still guide the decision making of type C₂ individuals with a lower potential power, independently of the number of individuals. Combined with Figure 11, Figure 14, and Figure 16, this result is more clearly demonstrated. It is important to note that, since the initial position of all the individuals is the same in this experiment, the results corresponding to Step = 0 remain consistent whether the average potential power is equal or different.

4.1.4. Analysis of When Two Types of Individuals (C₁ and C₂) Coexist

When type C₁ and C₂ individuals have potential power differences within the same type and the same average potential power within the two types, the FACOp₁ is approximately equal to FACOp₂ and is similarly approximately equal to the IACOp₁ and IACOp₂ at the initial time for each viewpoint (Figure 11c). This is because, after the interaction between type C₁ and C₂ individuals, the lower potential power individuals in type C₁ (or C₂) may “follow” the higher potential power individuals in type C₂ (or C₁) and support Op₂ (or Op₁). Thus, the interaction process can be viewed as a “swapping” of individual potential power, resulting in an approximately equal average individual potential power and group cohesion in each viewpoint at both the initial and final moments.

When the average individual potential power differs between type C₁ and C₂ individuals, after the interaction, the average individual potential power and the final group cohesion in support of Op₁ is greater than the group supporting Op₂ (i.e., FAPOp₁ > FAPOp₂; FACOp₁ > FACOp₂). This is attributed to the higher average personal potential of type C₁ individuals. Additionally, in Figure 14c, we can see that FACOp₁ is smaller than IACOp₁, while FACOp₂ is larger than IACOp₂. This is because type C₁ contains more individuals with a high potential power, and type C₂ has more individuals with a low potential power. After the interaction, more type C₂ individuals will “follow” type C₁ individuals and ultimately support Op₁. As the number of type C₂ individuals with a low potential power in Op₁ increases, the FAPOp₁ and FACOp₁ will decrease. Simultaneously, as some individuals with a low potential power in type C₂ move to support Op₁, the FAPOp₂ and FACOp₂ will increase. Nonetheless, the overall values remain smaller than those supporting Op₁.

In the case with the different number of individuals (Figure 16), the corresponding FAPOp₁, FAPOp₂, FACOp₁, and FACOp₂ do not change significantly with the results when the number is the same (Figure 11 and Figure 14), regardless of whether the average individual potential power differs between the two types. The trend remains consistent. It indicates that group cohesion does not change with increasing numbers; hence, group size does not affect it. Only the individual potential power has a significant impact on group cohesion. Measurements of IAPOp₁, IAPOp₂, IACOp₁, and IACOp₂ in Figure 16 are consistent with Remark 2.

4.2. Experimental Results for Three Types of Individuals—C₁, C₂, and B

In this section, we introduce the individual with no viewpoints, B. According to the rules in Section 3.2, the individual with no viewpoints, B, can only be influenced by type C₁ and C₂ individuals and does not influence them in return. Thus, the potential power of type B individuals is $X_B=0$, and their number is set to $m_B=50$. The total number of individuals is $m=m_{C_1}+m_{C_2}+m_B=150$. We also consider potential power differences within the same type of C₁ and C₂ individuals, exploring two scenarios: one where the average potential power is the same for both types, and another where it differs. The goal is to explore the impact of introducing the individual with no viewpoints, B, on the interaction process and the group cohesion.

4.2.1. Type C₁ and C₂ Individuals Have Potential Power Differences Within the Same Type and the Same Average Potential Power Within the Two Types

The initial positions of type C₁ and C₂ individuals in the results of this experiment are kept the same as in Section 4.1, and 50 type B individuals are introduced. The experiment results are shown in Figure 17 and Figure 18.

4.2.2. Type C₁ and C₂ Individuals Have Potential Power Differences Within the Same Type and Average Potential Power Differences Within the Two Types

The number and initial position of all individuals in this section is the same as in Section 4.2.1, with only the average potential power of the two types of individuals (C₁ and C₂) changed (0.75, 0.25). The experiment results are shown in Figure 19 and Figure 20.

4.2.3. Analysis of When Three Types of Individuals (C₁, C₂, and B) Coexist

If the type B individuals are not influenced by any type C₁ or C₂ individuals during the interaction, they remain viewpoint-free and move freely (see Figure 17). In Figure 17d, after the interaction, the number of type B individuals supporting each viewpoint is approximately equal due to the equal average potential power of type C₁ and C₂ individuals. The results in Figure 18 similarly show that the FAPOp₁ and FAPOp₂ are approximately equal, and the FACOp₁ and FACOp₂ are also approximately equal. However, since type B individuals have zero potential power, all these values (FAPOp₁, FAPOp₂, FACOp₁, and FACOp₂) are smaller than their initial values.

When type C₁ and C₂ individuals have different average potential powers, after the interaction, the differences in both the average individual potential power and the group cohesion among the individuals supporting the two viewpoints become smaller compared to their initial differentials. Specifically, the final difference in the average potential power is less than the initial difference (FAPOp₁ − FAPOp₂ < IAPOp₁ − IAPOp₂), and the final difference in the group cohesion is also smaller than the initial difference (FACOp₁ − FACOp₂ < IACOp₁ − IACOp₂), as shown in Figure 20. Because the average individual potential power of type C₁ individuals is higher, they will “guide” more type B individuals, resulting in a large number of type B individuals ultimately supporting Op₁ (Figure 19b). Therefore, the average individual potential power and group cohesion of Op₁ will be reduced more significantly. Conversely, type C₂ individuals, with the lower potential power, guide fewer type B individuals to support Op₂, leading to a relatively smaller reduction in the potential power and cohesion for Op₂. In other words, the number of the individuals with no viewpoints, B, can balance the group cohesion supporting each viewpoint. It is worth noting that the groups with a higher average individual potential power will increase the group size that ultimately supports that viewpoint.

4.3. Experimental Results for Four Types of Individuals—C₁, C₂, B, and P

In this section, we introduce the individual with public opinion viewpoints, P, based on the coexistence of three types of individuals. According to the rules in Section 3.3, type P individuals influence the potential power of other individuals but are not influenced by them (type C₁, C₂, and B individuals). We again consider the potential power differences within the same type of C₁ and C₂ individuals and explore scenarios where the average potential power is either the same or different between the two types. The goal is to investigate the impact of external factors, such as the individual with no viewpoints, B, and the individual with a public opinion viewpoint, P, on the group cohesion that ultimately supports each viewpoint. Additionally, we consider the influence of the magnitude of the public opinion power of type P individuals on group cohesion.

4.3.1. Type C₁ and C₂ Individuals Have Potential Power Differences Within the Same Type and the Same Average Potential Power Within the Two Types

We set the number of type P individuals to be 30 and all their public opinion powers to be X_p = 0.5. The initial positions of the type C₁, C₂, and B individuals in this section of the experiment are the same as in Section 4.2. The experimental results are shown in Figure 21 and Figure 22.

In this section, type B individuals will be influenced by type P individuals, altering their degree of stubbornness. Therefore, in Figure 21, the shades of the colors of type B individuals represent the degree of stubbornness during the interaction. A lighter color indicates a lower degree of stubbornness, while a darker color indicates a higher degree of stubbornness. As shown in Figure 21b, the individual with no viewpoints, B, becomes a type B₁ individual after being influenced by type P individuals and remains a type B₀ individual if not influenced by them (note that the final average individual potential power and the group cohesion are calculated by subtracting the potential power of the type B₁ individuals).

4.3.2. Type C₁ and C₂ Individuals Have Potential Power Differences Within the Same Type and Average Potential Power Differences Within the Two Types

We similarly change the average potential power of the two types of individuals (0.75, 0.25). The experimental results are shown in Figure 23 and Figure 24.

4.3.3. Type P Individuals with Differences in Public Opinion Power

This section primarily investigates the impact of the magnitude of the public opinion power held by type P individuals on the average individual potential power and the group cohesion for each viewpoint. Therefore, only type C₁, C₂, and P individuals are considered in this section. The public opinion power is adjusted to be X_p = 0.2 and X_p = 0.7, respectively. The experimental results are shown in Figure 25.

4.3.4. Analysis of When Four Types of Individuals (C₁, C₂, B, and P) Coexist

When type C₁ and C₂ individuals with the same average potential power and type P individuals who possess the same public opinion power exist, the result in Figure 22b,c remains that FAPOp₁ and FAPOp₂ are approximately equal, as are FACOp₁ and FACOp₂. All the values are smaller than the initial moment. This occurs because when the individual potential power of type C₁ and C₂ individuals is lower than the public opinion power of type P individuals, their potential power is further reduced. At the same time, the number of type B₀ and B₁ individuals guided by type C₁ and C₂ individuals is also approximately the same.

When type C₁ and C₂ individuals have different average potential powers, the results indicate that the average individual potential power and the group cohesion supporting Op₁ after the interaction remain larger than those supporting Op₂ (FAPOp₁ > FAPOp₂; FACOp₁ > FACOp₂) (Figure 24b,c). This is because type C₁ individuals initially possess a higher average potential power and there are more individuals with a higher internal potential power, making them less likely to be influenced by type P individuals. Additionally, type B individuals increase their degree of stubbornness after being influenced by type P individuals. For type C₁ individuals, even if type B individuals become more stubborn (B₁), they can still “follow” the high potential power of type C₁ individuals and support Op₁. Conversely, type C₂ individuals have a lower initial average potential power, making them more susceptible to the influence of type P individuals. This makes it harder for type B₁ individuals to “follow” type C₂ individuals and support Op₂. The introduction of type B₀ and B₁ individuals further reduces the group cohesion, with a more significant reduction in type C₁ individuals. Thus, ultimately, the group cohesion of Op₁ (FACOp₁−IACOp₁) decays more than that of Op₂ (FACOp₂−IACOp₂) (Figure 24c).

When type P individuals have differences in their public opinion power, the public opinion power of type P individuals negatively correlates with the average individual potential power and the group cohesion. The stronger the public opinion power, the smaller the group cohesion and the individual decision-making ability. Conversely, the weaker the public opinion power, the larger the group cohesion and the individual decision-making ability. (Figure 25).

5. Discussion

In this section, we better align the individual model with real-world scenarios and relate the analyzed results to practical applications, particularly in the context of social networks and organizational behavior.

5.1. Connection Between the Individual Modeling and the Real-World

In this paper, we define individuals with characteristics such as a perception range, safe distance, field of view, and potential power to facilitate scaling to real-world applications. In a practical context, such as individuals within a social network, their perception range can be regarded as the influence area in which an individual can both receive and exert information. Specifically, it is the range of social connections that an individual uses to reach and receive information from his friends, followers, and other group members in a social network. The safe distance can be understood as the psychological distance for individuals to avoid excessive proximity and maintain personal space in social interactions. In social networks, this psychological “comfort zone” (safe distance) manifests as an individual choosing to keep a certain distance when interacting with others to mitigate psychological pressure from overly close interactions. For example, individuals may avoid interacting with strangers or overly aggressive individuals to maintain their psychological boundaries. Additionally, maintaining a fixed ratio between the perception range and the safe distance suggests that as an individual’s perception range expands, their psychological safety distance similarly grows. This dynamic explains why, in certain social networks, relationships may tend to become more cautious or decentralized as the interaction frequency increases. For instance, when an individual has more social media friends, they may tend to share information more cautiously and maintain a certain “social distance” to avoid being influenced or psychologically pressured by too many individuals.

The field of view range represents the breadth of information an individual can receive, with some information being captured by the individual while others are ignored. The potential power signifies an individual’s capacity or influence. In a social network, an individual with a high potential power, such as a public figure or expert, tends to possess greater credibility and a higher level of influence.

The individual model is also applicable in various practical engineering applications. For example, in systems such as clustered robots, swarm UAVs, or intelligent transportation systems, the perception range of each robot or vehicle can be defined as the maximum distance at which it can detect other individuals or receive data from sensors (e.g., radar, laser radar, or cameras). Within this range, individuals can engage in coordinated movement, flight, or information sharing. The safety distance is intended to avoid collisions or affect the stability of the swarm during cluster control. For example, if the distance is too close, it may result in physical collisions or data interference, impairing the group’s overall performance. Maintaining a fixed ratio between the perception range and the safety distance implies that in certain applications, as the flight speed of an individual or swarm increases, the perception range may need to be extended, which in turn requires an increase in the safety distance to avoid collisions at higher speeds. The field of view represents the area covered by its sensors. At the same time, the potential power reflects the leadership and decision-making ability of the individual, i.e., by guiding other robots to perform a specific behavior, such as rounding up a target, tracking an object, or forming a formation.

5.2. Connection Between the Mathematical Model and Real-World Applications

In the model, we examined how the average potential power of two types of individuals affects group cohesion and individual decision making. We then extended this analysis by introducing individuals with no viewpoints and individuals with public opinion viewpoints. Next, we will draw connections to real-world applications. The specific analysis is as follows.

5.2.1. The Case of Two Types of Individuals

For the case of only two types of individuals (type C₁ and C₂ individuals), the model demonstrates that in social groups, when individuals supporting different viewpoints make decisions after interactive discussion and negotiation, the individuals with the higher potential power tend to have a stronger influence on group decision making. For example, in organizational teams or social networks, more authoritative or expert individuals (such as KOLs, i.e., Key Opinion Leaders) often guide the direction of discussions, leading to a greater consensus around their viewpoints.

In the context of elections, certain voter groups may exhibit strong, unwavering support (i.e., potential power) for a particular candidate, and their votes remain steadfast in the face of a campaign opponent. If these voter groups have a higher average potential power, their collective decisions can significantly impact the election’s outcome. For example, in a U.S. election, where voters in certain regions may have long supported a partisan candidate, these voter groups display high voter loyalty and may form a cohesive group that ultimately sways the election results in that region.

In marketing and consumer behavior, consumer decisions are not only influenced by product quality but also by emotional attachment and brand loyalty (i.e., potential power). Groups of consumers with strong brand loyalty wield a greater potential power, thus influencing the purchasing decisions of other consumers. A notable example is Apple’s loyal customer base, whose strong brand identification and unwavering support confer significant influence in the market. To enhance market competitiveness, a brand marketing team could strategically appeal to these loyal consumer groups to influence more potential consumers through social communication and word-of-mouth effects.

This analysis highlights the importance of rationally distributing individual potential power within a group, as it directly impacts the group’s cohesion and influences its future trajectory. These insights are particularly relevant for organizational behavior, where balancing power dynamics can impact team performance and decision outcomes.

5.2.2. The Case of Individuals with No Viewpoints Exists

With the introduction of a third type of individual (type B, representing those without viewpoints or new participants), the model reflects the dynamics in social networks when new individuals join an existing group. These “no-viewpoint” individuals significantly affect the overall group cohesion, often balancing the influence between groups with differing viewpoints. Furthermore, groups with a high potential power are more attractive to these new or undecided individuals than those with a weak potential power, increasing their size. This dynamic suggests that in real-world social networks, the size and cohesion of a group can be enhanced by increasing the potential power of individuals, thus attracting more members and stabilizing the group environment.

For example, in social networks, the individuals with no viewpoints may be those users who lack a clear stance on an issue, i.e., users who browse information but refrain from participating in discussions. Through their “neutral” or “ambiguous” stance, these individuals with no viewpoints can balance the group cohesion of different viewpoints in a certain way. For example, in a discussion on a social topic, a large number of individuals with no viewpoints may form a “bridge” between the two factions, avoiding the over-polarization of either side and pushing the discussion to maintain a certain balance.

In the context of election campaigns, individuals with no viewpoints can be analogized to voters who are undecided about voting or who do not have a clear preference for the outcome of the election. The inclusion of these individuals can balance the power of support more evenly between the various candidates and avoid an overconcentration of supporters on one side. If supporters of a particular candidate possess a firmer level of support (i.e., potential power) for a candidate, then they may attract more undecided voters to support that side, thus increasing the number of voters and affecting the final election outcome.

In marketing and consumer behavior, there is often a segment of “undecided” consumers who may not have a clear preference for any brand until they decide to make a purchase. These “undecided consumers” may remain somewhat neutral until their position is shaped through advertising, recommendations from friends, or interactions on social media. As these “undecided” individuals form their opinions, consumers with strong brand loyalty (i.e., high potential power) can sway and influence them to support that brand. This dynamic increases the brand’s market share and enhances its competitiveness.

5.2.3. The Case of Individuals with Public Opinion Viewpoints Exists

Public opinion events, which influence individual decisions and spread through various forms, directly affect group cohesion. Groups with individuals possessing a higher potential power are more resilient to such external influences, maintaining their cohesion and stability despite external public opinions. On the other hand, the stronger the public opinion power is, the more easily it can affect group cohesion, especially if external public opinion viewpoints influence the majority of individuals with a low potential power in the group. This insight suggests that strengthening the potential power of group members can help mitigate the disruptive effects of external public opinion, ensuring more stable group development.

For example, in social networks, when public opinion on a topic emerges, individuals with specialized skills or expertise often remain unaffected due to their ability to assess the situation critically. Similarly, in the election process, public opinion news regarding a candidate may influence low-support voters and undecided voters, while voters with higher levels of support for the candidate are less likely to be swayed. In marketing, some consumers with strong brand preferences or loyalty (i.e., high potential power) are less susceptible to the negative public opinion given by rival brands and continue to support their preferred brand.

6. Conclusions

In this paper, we investigated how the potential power differences in individuals’ characteristics induce changes in group cohesion effects, similar to symmetry-breaking phenomena in physical systems. We designed a mathematical model to describe inter-individual interactions and reflect group behavior. The differences in individuals’ potential powers (represented by the well depth of the potential function used for interactions) were represented by the degree of firmness to their viewpoint and their respective perceptual ranges, which can be understood as a form of symmetry-breaking. In the model, we categorized all the individuals into two types supporting two viewpoints, and there were potential power differences between the same and two types. Through interactions, the individuals with different potential powers were shown to potentially alter their original viewpoints and decisions, thereby causing varying group cohesion for each viewpoint. Additionally, we considered the impact of external perturbations, such as the introduction of individuals with no viewpoints and individuals with public opinion viewpoints, on the group cohesion of each viewpoint. The experimental results showed that groups with greater individual potential powers exhibited stronger decision-making capabilities and cohesion effects, contributing to group stability. In the case of involving only two types of individuals, the size of the group did not significantly affect the cohesion effect. However, the presence of individuals with no viewpoints and public opinion viewpoints further affected and reduced the group cohesion. Notably, individuals with no viewpoints balanced the cohesion effect between the different viewpoint groups, while the public opinion power of the individuals with public opinion viewpoints was negatively correlated with group cohesion.

In summary, this research provides a framework for understanding group behaviors, decision-making dynamics, and group cohesion effects through the concept of symmetry-breaking, offering insights for further studies in group research and decision-making processes.