Utilizing Potential Field Mechanisms and Distributed Learning to Discover Collective Behavior on Complex Social Systems

Abstract

This paper proposes the complex dynamics of collective behavior through an interdisciplinary approach that integrates individual cognition with potential fields. Firstly, the interaction between individual cognition and external potential fields in complex social systems is explored, integrating perspectives from physics, cognitive psychology, and social science. Subsequently, a new modeling method for the multidimensional potential field mechanism is proposed, aiming to reduce individual behavioral errors and cognitive dissonance, thereby improving system efficiency and accuracy. The approach uses cooperative control and distributed learning algorithms to simulate collective behavior, allowing individuals to iteratively adapt based on local information and collective intelligence. Simulations highlight the impact of factors such as individual density, noise intensity, communication radius, and negative potential fields on collective dynamics. For instance, in a high-density environment with 180 individuals, increased social friction and competition for resources significantly decrease collective search efficiency. Validation is achieved by comparing simulation results with existing research, showing consistency and improvements over traditional models. In noisy environments, simulations maintain higher accuracy and group cohesion compared to standard methods. Additionally, without communication, the Mean Squared Error (MSE) initially drops rapidly as individuals adapt but stabilizes over time, emphasizing the importance of communication in maintaining collective efficiency. The study concludes that collective behavior emerges from complex nonlinear interactions between individual cognition and potential fields, rather than being merely the sum of individual actions. These insights enhance the understanding of complex system dynamics, providing a foundation for future applications in adaptive urban environments and the design of autonomous robots and AI systems.

1. Introduction

The study of collective behavior is an interdisciplinary field that incorporates perspectives from computer science, sociology, physics, psychology, and biology [1,2]. This interdisciplinarity arises from simple local interactions between collective members, which may be organisms like fish shoals, bird flocks and cellular structures, artificial agents like swarms of robots, or social individuals like crowds. Collective behavior can be seen in various sizes and forms, from the synchronized movement of flocks of birds and fish, and the movement of crowds, to the volatile shifts of financial markets [3,4]. These complex systems exhibit properties that are not inherent in the individual elements but emerge from their interactions [5,6]. The complexity of collective behavior requires a fluid and responsive modeling approach. This approach should be able to adapt to the dynamic interactions between individual elements in a changing environment [7].

The exploration of collective behavior has become a critical focus across various domains, from robotics to urban planning and disaster response [8,9]. Traditional models are mainly based on rules or statistics, often ignoring the dynamic nature of individual motivation and group effects [10,11]. These models tend to either amplify individual behaviors or excessively focus on group effects, failing to adequately consider the nature of emergent interactions in complex social systems [12]. This research gap not only hinders theoretical understanding but also limits practical applications to ecological conservation, urban development, and crowd safety management. Therefore, it is necessary to develop an interdisciplinary approach with a dynamic framework such as a natural potential field [13]. Unlike artificial potential fields, natural potential fields are inherently emergent and can affect individual cognitive changes in complex social systems. In contrast, artificial potential fields commonly used in robotics and control theory are different from natural potential fields [14]. Artificial potential fields are designed to achieve specific goals (such as path planning or obstacle avoidance) by controlling individual motion [15,16]. Therefore, it is necessary to make a clear distinction between natural and artificial potential fields used in robotics and control theory. The concept of a natural potential field represents an emergent behavior between individuals, formed by natural interactions and not dependent on precise algorithmic control [17]. This concept has been extensively expanded to not only include physical influences but also cultural, economic, geographical, and psychological factors [18], offering an interdisciplinary perspective to observe individual decision-making processes and collective behaviors [19,20]. The potential field is not static but fluctuates with the changes in internal dynamics and external conditions of the social system, which requires the potential field to be both adaptive and dynamic. Social systems provide incentives, constraints, and support for individuals, thereby influencing the dynamics of collective behavior [21,22]. Thus, potential fields represent gradients in social, environmental, and cognitive psychology [23,24].

In this study, potential fields represent the multidimensional factors of complex social systems, such as cities, communities, and institutions [25]. External factors act as attractive and repulsive behaviors between individuals [26]. Internal factors imply cultural norms, economic interests, and social hierarchy [27]. For example, urban space affects the formation and dispersion of people, while economic interests lead to a stampede of market participants. From a cognitive psychology perspective, individuals navigate real environments by processing large amounts of sensory information and make decisions based on a mixture of cognitive assessments and emotional responses. This cognitive process is influenced by perceived potential fields such as risk, reward, and social identity [28]. These factors contribute to a drive towards convergence in behavior, as seen when individuals gather in areas of interest or disperse from zones of discomfort [29]. This behavior reflects a natural tendency to seek optimal states within the social system, balancing individual needs with the collective dynamics of their surroundings [30].

Collective behavior is widely used in various fields [31]. Robotics and autonomous systems seek to replicate the decentralized decision making and adaptability of natural collectives [32]. Urban planners and disaster management experts want to understand crowd dynamics to create safer, more efficient public spaces [33]. Integrating the concept of potential fields into social systems requires an understanding of environmental factors, including geographic layout, resource distribution, and building design, and how they influence both individual and collective decision making. This exploration reveals an invisible latent field that guides individuals to pursue group benefits, influencing phenomena such as migration trends, transportation patterns, and community formation [34]. In summary, the concept of potential field in social systems is different from the concept of potential field in robots. While the latter is engineered with specific objectives, social potential fields emerge from individual interactions and environmental factors. This distinction is crucial because it highlights the emergent nature of social systems. In addition, the potential field can also be considered as a mathematical function of the attractive and repulsive forces that drive the physical motion of an individual [35]. The formation of this drive is influenced by various factors, such as the desirability of a destination or the avoidance of danger Specifically, the strength of attraction or repulsion increases as the individual approaches the source of the potential field, depending on the case, and the weaker vice versa. This concept extends the physical understanding of forces to encompass influences that affect individual behavior in different contexts. This influence plays a vital role in regulating collective behavior, especially in terms of group formation, movement, and evolution. Therefore, complex social systems can use potential field mechanisms to model collective behavior in real environments, including animal migration or crowd dynamics.

The study simulates collective behavior in complex social systems by applying innovative distributed learning and swarming control approaches. In an environment with limited resources and communication range, the approach deviates from traditional control methods. Instead, each individual in a complex social system participates in data collection and iterates new information from their interactions. This collective information sharing process enables each individual to continuously update their search for potential fields. The task of each individual is to identify potential domains in the environment, which reflects the resource-seeking behavior seen in wildlife formations like fish schools and bird flocks. In uncertain environments, the system is inspired by the efficient resource-seeking behavior of animal groups, emphasizing the importance of localized communication in navigation and resource location. Simulations validated the effectiveness of this approach, showing that individuals can quickly and collaboratively identify potential areas in a social system. In the absence of cooperation and communication, individual task performance is significantly less efficient, highlighting the crucial role of information sharing and local interactions in decision making. From an interdisciplinary perspective, the research reveals the intricate interactions between social dynamics and individual behavior

In summary, a multidisciplinary framework is provided by the potential field mechanism, helping explain the complexity of collective behavior in social systems. This perspective highlights the interactions between individual behavior, cognitive psychology, and environmental factors. This aids in providing a deeper understanding of the balance between autonomy and conformity in collective behavior. The rest of this paper is structured as follows: The algorithm is detailed in Section 2 and Section 3. Section 4 and Section 5 present the convergence analysis and simulation results, respectively. The conclusion is provided in Section 6.

2. Mobile Individual Network

In this section, a complex social network model that combines concepts from social complexity and cognitive science with mathematical descriptions is used, as shown in Figure 1. In complex social systems, each individual has cognitive and collective intelligence, and individuals are constantly collecting, processing, and sharing information about their surroundings.

Let $\mathbb{Z},{\mathbb{Z}}_{\ge 0},{\mathbb{Z}}_{>0},\mathbb{R},{\mathbb{R}}_{\ge 0},{\mathbb{R}}_{>0}$ represent the collection of integer, non-negative integer, positive integer numbers, real, non-negative real, and positive real numbers, respectively. The positive definiteness and semi-definiteness of a matrix A are defined by $A\succ 0$ and $A\succcurlyeq 0$, respectively. $0_n\in {\mathbb{R}}^{n\times n}$ and $I_n\in {\mathbb{R}}^{n\times n}$ represent the zero and identity matrices. $0_{m\times n}\in {\mathbb{R}}^{m\times n}$ is defined as the m × n zero matrix [36]. According to the vector domain q, the gradient of a differentiable real function $\phi \left(q\right) : {\mathbb{R}}^{2n}\to \mathbb{R}$ is defined by:

$$\nabla \varphi (q)={\left[\begin{array}{cccc}{\displaystyle \frac{\partial \varphi {(q)}^{\mathrm{T}}}{\partial q_1}}& {\displaystyle \frac{\partial \varphi {(q)}^{\mathrm{T}}}{\partial q_2}}& \cdots & {\displaystyle \frac{\partial \varphi {(q)}^{\mathrm{T}}}{\partial q_n}}\end{array}\right]}^{\mathrm{T}}\in {\mathbb{R}}^{2n}$$

where $q=col(q_1, \dots , q_n )\in {\mathbb{R}}^{2n}$ and $q_i\in {\mathbb{R}}^2$. col(q₁, …, q_n) means creating a single column vector by stacking $q_1, \dots , q_n$ on top of each other.

2.1. Models for Mobile Perception Individual

The dynamics of individual actions and complex interactions are represented by the convex compact set. In the observation area $\mathcal{M}\subset {\mathbb{R}}^2$, n indicates the total number of social individuals that are randomly distributed. ${\displaystyle \mathcal{l}}=\{1, 2, \dots , N_s \}$ denotes the attributes of each social individual. The location of the i-th social individual at time t is defined by $q_i\left(t\right)\in \mathcal{M}$, while $q=col( q_1,q_2, \dots , q_{N_s }) \in {\mathbb{R}}^{2Ns}$ represents the swarm system’s configuration. The following equation describes the discrete-time high-order dynamic model:

$$\left\{\begin{array}{l}{q}_i\left(t+{\mathsf{\Delta }}_t\right)=q_i(t)+{\mathsf{\Delta }}_t{p}_i(t)\\ p_i\left(t+{\mathsf{\Delta }}_t\right)=p_i(t)+{\mathsf{\Delta }}_t{u}_i(t)\end{array}\right.$$

(1)

where $q_i\left(t\right)$ represents the position of the i-th social individual at time t.

The velocity of the i-th social individual at time t is denoted by $p_i\left(t\right)$. $u_i\left(t\right)$ represents the control input (acceleration or deceleration) of the i-th social individual at time t. $\Delta t$ denotes the sampling time or iteration step size. The first equation shows that the individual’s new position is determined by its current position and the velocity over the time interval $\Delta t$. The second equation updates the velocity of the individual based on its current velocity and the control input (acceleration) over the time interval $\Delta t$. It supposes that the measured value $F\left(q_i\right(t\left)\right)$ of the ith sensing individual includes the value of the potential field $F\left(q_i\right(t\left)\right)$ and sensing individual noise $w\left(t\right)$, at its location $q_i\left(t\right)$ and a sampling time t.

$$y\left(q_i(t)\right)=F\left(q_i(t)\right)+w(t)$$

(2)

where $F: \mathcal{M}\to [0, {\mu }_{max}]$ represents the maximum value of the potential field. The measured value $y\left(q_i\right(t\left)\right)$ is at the position of the i-th sensing individual, including sensing noise. $F\left(q_i\right(t\left)\right)$ is the value of the potential field at the location $q_i\left(t\right)$. w(t) represents the sensing individual noise at time t. Equations (1) and (2) form the backbone of a discrete-time dynamic model for individual behavior within a monitoring area. By iteratively updating positions and velocities based on control inputs and measuring the potential field with noise considerations, the model provides a robust foundation for simulating and examining the dynamics of sensing individuals in a swarm system.

2.2. Potential Field Model

The collective behavior of n individuals within sensing and searching potential fields in an m-dimensional space is studied. Each individual gathers social information within a limited interaction range, including identifying potential rewards, understanding social cues, and detecting environmental changes. This information gathering relies on cognitive abilities such as attention allocation, intention reasoning, and perceptual processing. Individuals use the results of cognitive processing and benefit orientation to estimate the position of the potential field, which reflects the role of cognitive mapping in environmental navigation and decision making. Therefore, the potential field can be defined as a dynamic model that adjusts and updates based on the cognitive processing abilities of individuals.

$$F=\mathsf{\Theta }{\mathsf{\Phi }}^T={\displaystyle \sum _{j=1}^K{\theta }_j{\varphi }_j}$$

(3)

where $\mathsf{\Theta }=[{\theta }_1,{\theta }_2, \dots , {\theta }_K]$, $\mathsf{\Phi }=[\phi 1, \phi 2, \dots , \phi K]$, and the index is represented as j. The total number of function distributions is denoted as K. The variables ${\phi }^T$ and $\theta$ are given as follows:

$$\begin{array}{c}{\varphi }^T(p)=\left[\begin{array}{cccc}{\varphi }_1(p)& {\varphi }_2(p)& \cdots & {\varphi }_k(p)\end{array}\right],\\ \mathsf{\Theta }={\left[\begin{array}{cccc}{\theta }_1& {\theta }_2& \cdots & {\theta }_k\end{array}\right]}^{\mathrm{T}}\in \mathsf{\Theta }\end{array}$$

(4)

where $\theta \subset R^m$ is a compact set. The Gaussian radial basis functions are denoted as $\left\{\left.{\phi }_j\left(p\right)\right\}\right.$ and are defined as follows:

$${\varphi }_i(p)={\displaystyle \frac{1}{{\beta }_j}}\exp \left({\displaystyle \frac{-{‖p-{\zeta }_j‖}^2}{{\sigma }_j^2}}\right),\forall j\in M,$$

(5)

where $M=\{1, \dots , m\}$, ${\mathsf{\beta }}_j$ and ${\sigma }_j$ denote the normalized constant and Gaussian basis width, respectively. The Gaussian radial basis function $\left\{\left.{\phi }_j\left(p\right)\right\}\right.$ has a center point ${\xi }_j$, which is included in the set $\{{\xi }_j | j \in M\}$. This set is uniformly distributed throughout the observation area Q. At the specific point $x^{*}$, the partial derivative of $\phi \left(x\right) \in {\mathbb{R}}_{m\times 1}$ regarding $x\in {\mathbb{R}}_{2\times 1}$ represented by ${\phi }^{\prime }\left(x^{*}\right)$ and is given by the following expression:

$${\varphi }^{\prime }(v^{*})={\left.{\displaystyle \frac{\partial \varphi (x)}{\partial x}}\right|}_{v=v^{*}}\in {\mathbb{R}}^{m\times 2}$$

(6)

In social systems, Gaussian basis functions represent various social stimuli or attractors in space, such as social public opinion hotspots and economic activity regions. The parameters ${\sigma }_j$ and ${\beta }_j$ can be seen as factors that determine the strength and spread of these social influences. The gradient of the potential field is given by the following expression at $q_i$:

$$\nabla F(q_i)={\left.{\displaystyle \frac{\partial {\varphi }^{\mathrm{T}}(x)}{\partial x}}\right|}_{x=q_i}\in {\mathbb{R}}^{2\times 1}$$

(7)

The estimate of $\nabla F(q_i)$ based on $\widehat{\theta }$ is represented as $\nabla \widehat{F}(q_i)$. In a two-dimensional space, the motion of n individuals is now considered [37]. The potential field within a two-dimensional space defined by (x, y) is represented by:

$$F(x,y)=\mathsf{\Theta }{\mathsf{\Phi }}^T(x,y)={\displaystyle \sum _{j=1}^K{\theta }_j{\varphi }_j(x,y)}$$

(8)

where Θ = [θ₁, θ₂, …, θ_K], and $\mathsf{\Phi }(x, y)=[{\phi }_1,{\phi }_2,...,{\phi }_K]$, where φ_j (x, y) is defined as a function of the density distribution, while θ_j indicates its weight. The index and total number of density distribution functions are represented by j and K, respectively.

2.3. Collective Behavior

In complex social systems, it is necessary for each individual to avoid obstacles to collect information and take actions. Therefore, controlled artificial potential functions and proposed smooth collective potential functions need to be introduced [36,38]. To guarantee balance and efficient data collection within the group, each individual needs to adhere to a collection of algebraic constraints as defined by $‖q_i-q_j‖=d$ for all $j\in \mathcal{N}(i,q)$

$$\begin{array}{c}{U}_1(q)={\displaystyle \sum _i}{\displaystyle \sum _{j\in \mathcal{N}(i,q),j\ne i}}{U}_{ij}\left(q_i-{q_j}^2\right)\\ ={\displaystyle \sum _i}{\displaystyle \sum _{j\in \mathcal{N}(i,q),j\ne i}}{U}_{ij}\left(r_{ij}\right)\end{array}$$

(9)

where $r_{ij}={‖q_i-q_j‖}^2$. The attractive/repulsive potential function $U_{ij}\left(r_{ij}\right)$ is represented as n [38]:

$$U_{ij}\left(r_{ij}\right)={\displaystyle \frac{1}{2}}\left(\log \left(\alpha +r_{ij}\right)+{\displaystyle \frac{\alpha +d^2}{\alpha +r_{ij}}}\right), if r_{ij}<d_0^2,$$

(10)

otherwise ($i.e., rij\ge d_0^2$), it is determined by the gradient of the potential function where $\alpha , d\in {\mathbb{R}}_{>0}$ and $d<d_0$. This represents the spatial boundary of the individual, and the individuals maintain a certain distance from others for safety or to avoid conflict. The gradient of the potential functions for individual i with respect to q_i can be represented by:

$$\begin{array}{ll}\nabla U_1\left(q_i\right)& ={\displaystyle \frac{\partial U_1(q)}{\partial q_i}}={\left.{\displaystyle \sum _{j\ne i}}{\displaystyle \frac{\partial U_{ij}(r)}{\partial r}}\right|}_{r=r_{ij}}2\left(q_i-q_j\right)\\ & =\left\{\begin{array}{ll}{\displaystyle \sum _{j\ne i}\frac{\left(r_{ij}-d^2\right)\left(q_i-q_j\right)}{{\left(\alpha +r_{ij}\right)}^2}}\hfill & if r_{ij}<d_0^2\\ {\displaystyle \sum _{j\ne i}\rho \left(\frac{\sqrt{r_{ij}}-d_0}{\left|d_1-d_0\right|}\right)\frac{d_0^2-d^2}{{\left(\alpha +d_0^2\right)}^2}\left(q_i-q_j\right)}\hfill & \mathrm{i}f r_{ij}\ge d_0^2\end{array}\right.\end{array}$$

(11)

In Equations (9)–(11), a non-zero gain factor α is implemented to prevent uncontrollable reaction forces when individuals are extremely close. When separated, the reaction force is attractive, and when too close, it is repulsive. This interaction creates an equilibrium state at distance d, where the force is neutral, reflecting the balance within the social system. Additionally, a potential U₂ is introduced to simulate the constraints and impacts of the social system on behavior, ensuring individuals remain within a connected monitoring area M. This illustrates how physical and social environments influence human behavior, such as navigating urban spaces or avoiding social conflicts. The total of the potential functions can be described as:

$$U(q)=k_1{U}_1(q)+k_2{U}_2(q)$$

(12)

where $k_1,k_2\in {\mathbb{R}}_{\geqq 0}$ are represented as weighting factors. This approach simulates individuals gathering and sharing information while adhering to social constraints. By utilizing collaborative strategies, individuals move towards areas of maximum benefit. It highlights the importance of local interactions and collective intelligence in group decision making.

2.4. Graph-Theoretic Representation

Let $G\left(q\right)=({\displaystyle \mathcal{l}},\mathcal{E}(q\left)\right)$ be an edge $(i, j)\in \mathcal{E}\left(q\right)$ that signifies that individual i can communicate and interact with another individual j, where $j\ne i$. The communication between individuals is constrained by a defined interaction range, denoted by a radius r. This means that any two individuals i and j are linked in the graph if and only if the distance between them $‖q_i\left(t\right)-q_j\left(t\right)‖\le r$ is less than or equal to r [36]. The concept of an adjacency matrix A can also be defined and adjusted based on a smooth form of q. The scalar graph Laplacian $L=\left[l_{ij}\right]\in {\mathbb{R}}^{Ns\times Ns}$ can be represented by is defined as $L=D^A-A$, where $D^A$ is the diagonal matrix formed by the sums of the rows of A, i.e., $D^A=diag\left({\sum }_{j=1}^{N_s}{a}_{ij}\right)$. Furthermore, $\widehat{L}=L\otimes I_2$ can define the two-dimensional graph Laplacian, where $\otimes$ denotes the Kronecker product. Under the topology of an undirected graph G, the state of individual i for $i\in {\displaystyle \mathcal{l}}$ is defined as $p_i\in R^2$. This state can represent various factors, such as an individual perception, viewpoint, or other attributes in social context. When p_i = p_j, two individuals i and j are in the same state. In order to quantify the degree of inconsistency within the group, a quadratic disagreement function ${\Psi }_G :{\mathbb{R}}^{{2N}_s}\to R_{\ge 0}$ is introduced. This function aims to measure the degree of inconsistency between all paired individuals, and is defined as follows:

$${\mathsf{\Psi }}_G(p)={\displaystyle \frac{1}{4}}\sum _{(i,j)\in \mathcal{E}(q)}{a}_{ij}{p}_j-{p_i}^2$$

(13)

where $p=col (p_1, p_1, \dots ,p_{N_s})\in {\mathbb{R}}^{{2N}_s}$. By the Laplacian $\widehat{L}$, a disagreement function is given by the following [38]:

$${\mathsf{\Psi }}_G(p)={\displaystyle \frac{1}{2}}{p}^{\mathrm{T}}\widehat{L}p$$

(14)

3. Distributed Learning and Cooperative Control

In a complex social system, each individual usually exchanges information with its nearby individuals. The individual uses the information of local interactions and the results of cognitive processing to continuously improve the strategy of searching the potential field. During the search, individuals use distributed mechanisms to learn from their environment and collectively move towards the most beneficial regions.

3.1. Distributed Learning

It is presented that each individual employs a distributed learning algorithm to estimate the maximum value of the potential field $F: M \to [0 , F: max]$. Let $F\left(p\right)$ be the potential field, described as:

$$F(p)=\sum _{j=1}^k{\varphi }_j(p){\theta }_j={\varphi }^{\mathrm{T}}(p)\mathsf{\Theta }$$

(15)

where ${\phi }^T\left(p\right)$ and $\theta$ are presented, respectively, as

$$\begin{array}{c}{\varphi }^{\mathrm{T}}(p)=\left[\begin{array}{cccc}{\varphi }_1(p)& {\varphi }_2(p)& \cdots & {\varphi }_k(p)\end{array}\right],\\ \mathsf{\Theta }={\left[\begin{array}{cccc}{\theta }_1& {\theta }_2& \cdots & {\theta }_k\end{array}\right]}^{\mathrm{T}}\in \mathsf{\Theta }\end{array}$$

where $\theta \subset R^m$ is a compact set. $\left\{\left.{\phi }_j\left(p\right)\right\}\right.$ is the Gaussian radial basis functions and is given by

$${\varphi }_j(p)={\displaystyle \frac{1}{{\beta }_j}}\exp \left({\displaystyle \frac{-{‖s-v_j^c‖}^2}{{\sigma }_j^2}}\right)$$

(16)

where ${\sigma }_j$ represents the width of the Gaussian basis and ${\beta }_j$ denotes a normalizing constant. The centers of the basis functions $p_j^c$ for $j\in \{1,\dots ,k\}$ are uniformly distributed throughout the observation area R. $\mathsf{\Theta }\in \mathsf{\Theta }\subset {\mathbb{R}}^m$ can be defined as the true parameter in Equation (9). The zero mean Gaussian process $\varsigma (s, t)$ is given by $\varsigma (s, t)\sim GP(0, \mathcal{K}(s, t; s^{\prime }, t^{\prime }\left)\right)$ with a covariance matrix $K(s_i, t_i;s_j, t_j)=\kappa (s_i, t_i){\delta }_{s_j, t_j}$, where $\delta (...)$ is defined as the Kronecker delta:

$$K(s_i,s_j)={\displaystyle \frac{1}{{\beta }_k}}\exp \left({\displaystyle \frac{-{‖s-v_j^c‖}^2}{{\sigma }_k^2}}\right)$$

(17)

The dynamic component of the spatiotemporal gaussian process, along with the noisy observation, is described as [39]:

$$\begin{array}{l}\theta \left(t+1\right)=F\left(t\right)\theta (t)+G(t)u\left(t\right)\in {\mathbb{R}}^n,\\ y\left(s,t\right)={\varphi }^T\left(s\right)\theta (t)+\tau (s,t)+w\left(t\right)\in \mathbb{R},\end{array}$$

(18)

A prior for the parameter $\theta by: \theta \sim \mathcal{N}({\theta }_0, {\sum }_{\theta }\left(0\right))$ is defined. Assuming that at the iteration time $t\in \mathcal{T}=\{0, 1, 2, \dots \}$, individual i is able to gather observations ${Y\left(t\right)=\left[y\right(s_1\left(t\right), t), \dots , y(s_n\left(t\right), t\left)\right]}^T$ from n sites $\left\{s_1\right(t), \dots , s_n(t\left)\right\}$, including data from its own location and the $n-1$ sites of its neighbors, the following relation is established:

$$Y\left(t\right)=\mathsf{\Phi }\left(t\right)\theta +v\left(t\right)\in {\mathbb{R}}^n$$

(19)

where $\Phi (t)={[\varnothing (s_1\left(t\right)), \dots , \varnothing (s_n\left(t\right)\left)\right]}^T$, $v\left(t\right)\sim N\left(0, {\sum }_v\left(t\right)\right)$, and rank $\mathsf{\Phi }\left(t\right)=n^1$. Subsequently, in Equation (2), the perception individuals are observed at the position $p_k$,

$$y(p_k)={\varphi }^t(p_k)+w(k)$$

(20)

where k serves as the parameter for measurement sampling. the aim is to identify the $\widehat{\mathsf{\Theta }}$ that minimizes the least-squares error using the collected data and regression coefficient ${\left\{\right(y\left(p_k\right), \varphi \left(p_k\right)\left)\right\}}_{k=1}^n$ as follow:

$${\displaystyle \sum _{k=1}^n{\left|y\left(p_k\right)-{\varphi }^T\left(p_k\right)\hat{\mathsf{\Theta }}\right|}^2}$$

(21)

The difference between the observed data and the predictions is minimized in this learning process, effectively allowing the parameter estimate $\widehat{\mathsf{\Theta }}$ that reduces the least-squares error to be sought.

Remark 1. 

The environmental model in Equation (21) is regarded as a radial basis function network, which can be used to simulate nonlinear spatial phenomena [40]. The dynamic version denotes the time-varying trend in the spatiotemporal Kalman filter model [41]. Many studies of collective behavior often use these methods because they are suitable for simulating real-world complexity [42].

Noiseless measurements

The model in Equation (2) is considered without the noise $w\left(k\right)$. The minimum variance of the estimation error can be achieved by using the spatial estimation algorithm based on noise level. The optimal least-squares estimation solution minimizes the error function in Equation (21) for a set ${\left\{\right(y\left(p_k\right),\varphi \left(p_k\right)\left)\right\}}_{k=1}^n$ [43].

$$\hat{\mathsf{\Theta }}(n)=P(n,1){\mathsf{\Phi }}^{\mathrm{T}}(n,1)Y(n,1)$$

(22)

As shown in Equation (1), a large number of measurement sample data is collected by each individual, both from their own observations and the shared information of their group, as time progresses from t to $t+△t$. We consider the data previously collected up to the last iteration as ${\left\{\right(y\left(p_k\right),\varphi \left(p_k\right)\left)\right\}}_{k=1}^{n-s}$, where n − s represents a set of prior data. The new collective measurements are then added to update the current understanding of the potential field $\widehat{F}(·)$. Let it be assumed that ${\mathsf{\Phi }}^T(t)\mathsf{\Phi }(t)$ is nonsingular for all t. The recursion algorithm is developed for the recently gathered observational sample data S and the associated regressors can be described by:

$$\begin{array}{l}K(n)=P(n-s){\mathsf{\Phi }}_{\ast }^{\mathrm{T}}{\left[I_s+{\mathsf{\Phi }}_{\ast }P(n-s){\mathsf{\Phi }}_{\ast }^{\mathrm{T}}\right]}^{-1}\\ P(n)=\left[I_m-K(n){\mathsf{\Phi }}_{\ast }\right]P(n-s)\\ \hat{\mathsf{\Theta }}(n)=\hat{\mathsf{\Theta }}(n-s)+K(n)\left[Y_{\ast }-{\mathsf{\Phi }}_{\ast }\hat{\mathsf{\Theta }}(n-s)\right]\\ \widehat{F}(p)={\varphi }^{\mathrm{T}}(p)\hat{\mathsf{\Theta }}(n)\end{array}$$

(23)

To make the notation more concise and easier to understand, we will use simplified symbols and expressions to clearly convey the mathematical concepts, let $Y_{*}=Y\left(n,n-s+1\right)\in {\mathbb{R}}^s$,${\mathsf{\Phi }}_{*}=\mathsf{\Phi }\left(n,n-s+1\right)\in {\mathbb{R}}^{s\times m}$, ${\mathsf{\Phi }}^T={\mathsf{\Phi }}^T\left(n,1\right)\in R^{n\times m}$, $Y\left(n\right)=Y(n,1)\in {\mathbb{R}}^n$, and $P\left(n\right)=P(n,1)\in {\mathbb{R}}^{m\times m}$. The least square estimation that minimizes the error function in Equation (21) can express the recursive estimation in Equation (23).

Remark 2. 

For n < m, ${\Phi }^T(n)\Phi (n)$ is constantly singular. W remains nonsingular for n ≥m, except when evaluated at just one set of measurement zeros. In the initial state of $\widehat{\Theta }(0)$ and P$\succ$ (0), the initial parameter distribution is used by each individual in a recursive LSE algorithm as described in Equation (23) [36]. This process demonstrates how individuals depend on past data or assumptions while engaging with their surroundings. As new information and data become available, the perception of the potential field by the individuals is continuously updated and enhanced.

$$P^{-1}(n)=P^{-1}(0)+{\mathsf{\Phi }}^{\mathrm{T}}(n)\mathsf{\Phi }(n)\succ 0$$

(24)

In the next subsection, the influence of noise observation on the estimation potential field and its gradient is explained in detail.

Noisy measurements

A measurement model (2) is considered, where each individual is generally disturbed by social noise, denoted by $w\left(k\right)$. This noise represents unpredictable variables in complex social systems. Thus, social noise is defined as a white noise sequence with an unknown variance W

$$\mathbb{E}(w(k))=0,\hspace{1em}\mathbb{E}(w(k)w({\displaystyle \mathcal{l}}))=\left\{\begin{array}{ll}W>0& if k={\displaystyle \mathcal{l}}\\ 0& if k\ne {\displaystyle \mathcal{l}}\end{array}\right.$$

(25)

where $\mathbb{E}$ can be represented as is the expectation operator. $\mathbb{E}$ represents the individual measurement noise at time k. Furthermore, the existence of $L<\infty$ is supposed such that the probability of $\left|w\right(k\left)\right|<L$ is

$$one (w.p.1)\forall k$$

(26)

The measurement data set is given as

$$\left\{y\left(F\right)\left|F\in S\right.\right\}$$

where $S=\left\{p_k\right| 1\le k\le n\}$. For the social noise $\left\{w\right(k) | k\in \{1, \dots , n\}\}$ defined in (25) and (26), the individual will use the recursive LSE algorithm in (24) to estimate $\widehat{\mathsf{\Theta }}\left(\mathrm{n}\right)$. Let $\widehat{\mathsf{\Theta }}\left(\mathrm{n}\right)=\widehat{\mathsf{\Theta }}\left(\mathrm{n}\right)-\mathsf{\Theta }$ represent the estimation error vector. The error values of the estimated and perceived potential field at the position $p\in \mathcal{M}$ can be obtained as follows:

$$\widehat{F}(S,p)=\widehat{F}(S,p)-F(p)={\varphi }^{\mathrm{T}}(p)\tilde{\mathsf{\Theta }}(|S|)$$

(27)

where the cardinality of the set $S$ is represented by $\left|S\right|$. The error values of the estimated and perceived potential field is described as follows:

$$\widehat{F}(S,p)=\mathbb{E}(\widehat{F}(S,p)+\in (S,p)$$

(28)

where $\left|S\right|$ denotes the total sum of collective measurement data. In the case of the continuous motivation coordination strategy, where $({\mathsf{\Phi }}_{*}^T{\mathsf{\Phi }}_{*}\succ 0)$, the value of the estimator is asymptotically unbiased.

$$\underset{\left|S\right|\to \infty }{\lim }\mathbb{E}(\widehat{F}(S,p))=0,\forall v\in \mathcal{M}$$

(29)

The variance associated with the estimation error is described as

$$\begin{array}{c}\mathbb{E}\left(ϵ(S,p){ϵ}^{\mathrm{T}}(S,p)\right)={\varphi }^{\mathrm{T}}(p)WP(|S|)\varphi (p)\\ ={\varphi }^{\mathrm{T}}(p){\displaystyle \frac{W}{\left|S\right|}}{R}^{-1}(S)\varphi (p)\end{array}$$

(30)

Remark 3. 

From Equation (30), the estimated position function v is defined by the estimation error variance, which is proportional to the variance W and diminishes at the rates determined by $1/\left|S\right|$ and R⁻¹(S). At the measurement point in S, the time average of the external product of the basis function is asymptotically influenced by R(S). This implies that locations with more frequent observations lead to a reduced estimation error variance. Therefore, the more observations or interactions an individual has at a point, the more refined their perception and understanding of that position becomes. The gradient of the potential field can be represented by

$$\nabla F(p)={\left.{\displaystyle \frac{\partial F(x)}{\partial x}}\right|}_{x=q_i=p}={\varphi }^{\prime T}(v)\theta \in {\mathbb{R}}^{2\times 1}$$

(31)

From (15),

$$\nabla F(p)={\left.{\displaystyle \frac{\partial {\varphi }^{\mathrm{T}}(x)}{\partial x}}\right|}_{x=q_i=p}\mathsf{\Theta }={\varphi }^{\prime \mathrm{T}}(p)\mathsf{\Theta }\in {\mathbb{R}}^{2\times 1}$$

(32)

is obtained, where ${\varphi }^{\prime T}\left(p\right)\in R^{2\times m}$ is given. Consequently, the gradient of the estimated potential field with respect to the observation parameters ${S=\left\{p_k\right\}}_{k=1}^n$ and $\{{y\left(F\right)\}}_{\mu \in S}$ is expressed as follows:

$$\nabla \widehat{F}(S,p)={\varphi }^{\prime \mathrm{T}}(p)\widehat{\mathsf{\Theta }}(|S|)\in R^{2\times 1}$$

(33)

At position $p$, the error of the estimated gradient is described as:

$$\begin{array}{c}\nabla \widehat{F}(S,p)={\varphi }^{\prime \mathrm{T}}(p)\widehat{\mathsf{\Theta }}(|S|)-\nabla \mu (p)={\varphi }^{\prime \mathrm{T}}(p)\tilde{\mathsf{\Theta }}(|S|)\\ =\mathbb{E}(\nabla \widehat{F}(S,p))+\nabla ϵ(S,p)\end{array}$$

(34)

Similar to (29) and (30), for ${\mathsf{\Phi }}_{*}^T{\mathsf{\Phi }}_{*}\succ 0$, it is observed that the gradient of the estimated value exhibits asymptotic unbiasedness.

$$\underset{\left|S\right|\to \infty }{\lim }\mathbb{E}(\nabla \widehat{F}(S,p))=0,{\forall }_v\in \mathcal{M}$$

(35)

and the covariance matrix $E(\nabla \in \left(S ,p\right)\nabla {\in }^T(S,p\left)\right)$ can be represented by:

$${\varphi }^{\prime \mathrm{T}}(p){\displaystyle \frac{W}{\left|S\right|}}{R}^{-1}(S){\varphi }^{\prime }(p)$$

(36)

where R(S) can be defined by:

$$R(S)=\left[{\displaystyle \frac{P^{-1}(0)}{\left|S\right|}}+{\displaystyle \frac{1}{\left|S\right|}}\sum _{v_k\in S}\varphi \left(p_k\right){\varphi }^{\mathrm{T}}\left(p_k\right)\right]$$

(37)

$R(n, S)$ functions asymptotically as the temporal mean of the outer product of the collection of basis functions at the sampling point S. Subsequently, the cooperative control strategies can be introduced.

3.2. Cooperative Control

The individual receives measurement data from its neighbors. The gradient of its estimated potential field is updated by using $\widehat{\theta }$ of a recursive algorithm in (23). Then, the algorithm updates its gradient state to determine the control of coordination. The recursive LSE algorithm introduces a new coordination time symbol, replacing $t\in \mathbb{Z}$ with $n-s\in {\mathbb{Z}}_{\ge 0}$ and $t+1\in {\mathbb{Z}}_{\ge 0}$ with $n\in {\mathbb{Z}}_{\ge 0}$ in (23). At position $q_i\left(t\right)$, a new time index for individual i is introduced by the recursive algorithm as follow:

$$\begin{array}{c}{K}_i(t+1)=P_i(t){\mathsf{\Phi }}_{\ast i}^{\mathrm{T}}{\left(I_s+{\mathsf{\Phi }}_{\ast i}{P}_i(t){\mathsf{\Phi }}_{\ast i}^{\mathrm{T}}\right)}^{-1}\\ P_i(t+1)=\left(I_m-K_i(t+1){\mathsf{\Phi }}_{\ast i}\right)P_i(t)\\ {\hat{\mathsf{\Theta }}}_i(t+1)={\hat{\mathsf{\Theta }}}_i(t)+K_i(t+1)\left[Y_{\ast i}-{\mathsf{\Phi }}_{\ast i}{\hat{\mathsf{\Theta }}}_i(t)\right]\\ \nabla {\widehat{F}}_i\left(t,q_i(t)\right)={\varphi }^{\prime \mathrm{T}}\left(q_i(t)\right){\hat{\mathsf{\Theta }}}_i(t+1)\end{array}$$

(38)

where $\nabla {\widehat{F}}_i\left(t,p\right):{\mathbb{Z}}_{\ge 0}\times \mathcal{M}⟶{\mathbb{R}}^2$ represents the gradient of the estimated potential field at location ν, as determined by the measurement sample data gathered prior to time t + 1. For an individual i, $Y_{*i}$ and ${\varphi }_{*i}$ are defined in a manner analogous to $Y_{*}$ and ${\varphi }_{*}$ in Equation (14). $Y_{*i}$ denotes the summary data obtained from the interactive measurement. For all $j\in \mathcal{N}(i,q(t\left)\right)\cup \left\{i\right\}$, it can be obtained from (2) as follows:

$$Y_{\ast i}={\mathsf{\Phi }}_{\ast i}\mathsf{\Theta }+\left[\begin{array}{c}⋮\\ w_j(k)\\ ⋮\end{array}\right]={\mathsf{\Phi }}_{\ast i}\mathsf{\Theta }+w_{\ast i}(t)$$

(39)

where during the interval between time t to t + 1, each individual collects data, with t serving as a point of reference at any time of data collection. Each individual receives a noise of uncertainty, represented by $w_j \left(k\right)$. In order to adapt to the dynamic noise changes, a new variable w_∗i(t) is introduced for following studies. Each individual updates its behavior based on the latest state, represented by the latest update of the gradient of the estimated potential field $\nabla {\widehat{F}}_i(t,q_i(t\left)\right)$. Based on this, a distributed control p_i (t + 1) is designed for each individual i as follows [44]:

$$p_i(t+1)={\displaystyle \frac{\gamma (t+1)}{{\mathsf{\Delta }}_t}}\left[{\displaystyle \frac{{\mathsf{\Delta }}_t}{\gamma (t)}}{p}_i(t)+\gamma (t)u_i(t)\right]$$

(40)

with

$$\begin{array}{l}{u}_i(t)=-\nabla U\left(q_i(t)\right)+k_4\nabla {\widehat{F}}_i\left(t,q_i(t)\right)-k_{di}{\displaystyle \frac{{\mathsf{\Delta }}_t}{\gamma (t)}}{p}_i(t)\\ +{\displaystyle \sum _{j\in \mathcal{N}(i,q(t))}}{a}_{ij}(q(t))\left({\displaystyle \frac{{\mathsf{\Delta }}_t\left(p_j(t)-p_i(t)\right)}{\gamma (t)}}\right)\end{array}$$

(41)

where $k_4\in {\mathbb{R}}_{>0}$ represents the gain factor for both the estimated and perceived gradient of the potential field. A gain of the speed feedback is denoted by $k_{di}\in {\mathbb{R}}_{\ge 0}$, and it facilitates dynamic adjustments to motion. In Equation (41), $-\nabla U\left(q_i\right(t\left)\right)$ is defined as the gradient of the artificial potential, which functions to create both attractive and repulsive forces among individuals. This mechanism not only promotes cooperative interaction but also helps to avoid collisions, thereby maintaining the integrity of collective movement. The second term denotes the damping force in Equation (41), as it moderates the individual’s momentum to facilitate steady control of the process and prevent abrupt movements. The speed of an individual is represented by the third term, which matches the speeds of its neighbors. This consistency is a form of behavior synchronization in social groups. The last term in Equation (41) is to estimate the gradient ascent of the potential field, which is used to drive individuals toward the maximum peak of the potential field according to the dynamic social environment.

In addition, to effectively search for the potential fields’ maximum peak, the intensity of collaborative control over individuals needs to be gradually reduced. Equation (40) introduces a control protocol with a standard adaptive gain sequence γ(t). These adaptive gain sequences allow individuals to maintain continuous feedback on changing potential fields, offering a dynamic approach to swarm motion and collective intelligence. The following conditions must be satisfied by the control protocol:

$$\begin{array}{l}\gamma (t)>0, {\displaystyle \sum _{t=1}^{\infty }\gamma (t)=}\infty , {\displaystyle \sum _{t=1}^{\infty }{\gamma }^2(t)=}<\infty \\ \underset{t\to \infty }{\lim } \sup [1/\gamma (t)-1/\gamma (t-1)]<\infty \end{array}$$

(42)

The gain sequence plays a crucial role and is often employed in random approximation algorithms [45,46]. This sequence is usually applied to the ordinary differential equation (ODE) approach, which is a technique for analyzing convergence [44,46,47]. To effectively conduct this analysis, the variables z_i(t) are transformed, introducing a new representation of the velocity state v_i(t), as below:

$$z_i(t)={\displaystyle \frac{{\mathsf{\Delta }}_t}{\gamma (t)}}{v}_i(t)$$

(43)

where v_i(t) denotes the control input for individual i. With the change of variables in Equation (43), the dynamic of individual i can be defined by:

$$\left\{\begin{array}{l}{q}_i(t+1)=q_i(t)+\gamma (t)z_i(t),\hfill \\ z_i(t+1)=z_i(t)+\gamma (t)u_i(t),\hfill \end{array}\right.$$

(44)

where ${\Delta }_t{v}_i\left(t\right)$ is substituted with $\gamma \left(t\right)p_i\left(t\right)$, $t+{\Delta }_t\in {\mathbb{R}}_{\ge 0}$ is substituted with $t+1\in {\mathbb{Z}}_{\ge 0}$, $t\in {\mathbb{R}}_{\ge 0}$ is substituted with $t\in {\mathbb{Z}}_{\ge 0}$, with these changes applied to Equation (1). By combining the control methods of Equations (40) and (41) with the discrete time model of Equation (45), this results in the following:

$$\begin{array}{c}{q}_i(t+1)=q_i(t)+\gamma (t)z_i(t)\\ z_i(t+1)=z_i(t)+\gamma (t)\left\{-\nabla U\left(q_i(t)\right)-k_{di}{z}_i(t)\right.\\ \left.\hspace{1em}-\nabla {\mathsf{\Psi }}_G\left(z_i(t)\right)+k_4{\varphi }^T\left(q_i(t)\right){\hat{\mathsf{\Theta }}}_i(t+1)\right\}\end{array}$$

(45)

where ∇Ψ_G(p_i(t)) represents the gradient of the inconsistent function relative to z_i:

$$\nabla {\mathsf{\Psi }}_G\left(z_i(t)\right)=\sum _{j\in \mathcal{N}(i,q(t))}{a}_{ij}(q(t))\left(z_i(t)-z_j(t)\right)$$

4. Convergence Analysis

To investigate the convergence properties of Equations (38), (42) and (45), Ljung’s ODE approach is applied. The analysis technique, presented in the canonical form by Ljung, provides valuable insights into general recursive stochastic algorithms [44,46,47]. The convergence analysis presented demonstrates that the algorithms (46), (47), and (49) satisfy the regularity conditions C1-C11 given specific assumptions M1–M3, ensuring robust utilization of Ljung’s ODE approach for nonlinear observation processes. This provides a solid foundation for analyzing individual and collective behavior in complex social systems.

$$x(t)=x(t-1)+\gamma (t)Q(t;x(t-1),\phi (t))$$

(46)

and the observation process

$$\phi (t)=g(t;x(t-1),\phi (t-1),e(t))$$

(47)

To apply the ODE method for the nonlinear observation process outlined in (47), it is imperative to satisfy the specific regularity conditions detailed in Ljung’s work from 1975. These conditions ensure the reliability and accuracy of the analysis. A subset of the x space in (46), denoted as D_R, becomes the focus where these regularity conditions are upheld. Within this subset, the necessary criteria are met, allowing for a robust utilization of the ODE approach. By adhering to these conditions in a defined space, the analysis gains a solid foundation, enhancing the precision and validity of the results obtained from the ODE methodology.

C1: $\parallel (x,\phi ,e)\parallel <\mathrm{C}$ for all $\varphi ,e$ for all $x\in D_R$.

C2: For $x\in D_R$, $Q(t, x, \phi )$ is a function that is continuously differentiable with respect to x and $\phi$. In t, the derivatives are bounded for fixed values of $x$ and $\varphi$.

C3: $g(t; x, \phi , e)$ denotes differentiable continuously relative to $x\in D_R$.

C4: Define $\overline{\phi }(t,\overline{x})$ as

$$\overline{\phi }\left(t,\overline{x}\right)=g\left(t;\overline{x},\phi \left(t-1,\overline{x}\right),e\left(t\right)\right),\overline{\phi }(0,\overline{x})=0,$$

(48)

and suppose that $g (·)$ has the following properties:

$$\parallel \overline{\phi }(t, \overline{x})-\phi (t)\parallel <C \underset{n\le k\le t}{\max }\parallel \overline{x}-x(k)\parallel ,$$

if $\overline{\phi } (n,\overline{x})=\phi \left(n\right)$. This shows that a small change in x in (47) will not be increased to the higher amplitude of the observed $\phi$.

C5: Let ${\overline{\phi }}_1(t, \overline{x})$ and ${\overline{\phi }}_2(t, \overline{x})$ be the solution of (48), where ${\overline{\phi }}_1(s, \overline{x}) :={\overline{\phi }}_1^0$ and ${\overline{\phi }}_1(s, \overline{x}) :={\overline{\phi }}_2^0$. D_S can be defined as a set of all $\overline{x}$ as follows:

$$\parallel {\overline{\phi }}_1(t, \overline{x})-{\overline{\phi }}_2(t, \overline{x})\parallel <C ({\overline{\phi }}_1^0, {\overline{\phi }}_2^0){\lambda }^{t-s}\left(\overline{x}\right),$$

where $t>s$ and $\lambda \left(\overline{x}\right)<1$. This is the exponential stable area of (47).

C6: $\underset{t\to \infty }{\lim }EQ(t, \overline{x}, \overline{\phi }(t, \overline{x}))$ exists for $\overline{x}\in D_R$ and is represented by $f\left(\overline{x}\right)$. The expected value exceeds $\left\{e\right(·\left)\right\}$.

C7: The sequence of independent random variables is defined as $e(·)$.

C8: ${\sum }_{t=1}^{\mathrm{\infty }}\gamma \left(t\right)=\mathrm{\infty }$.

C9: ${\sum }_{t=1}^{\infty }\gamma \left(t\right)<\infty$ for some p.

C10: $\gamma (·)$ is a decreasing sequence.

C11: $\underset{t\to \infty }{\lim }sup[1/\gamma (t)-1/\gamma (t-1\left)\right]<\infty$. The “sup” operator, short for “supremum”, represents the least upper bound of a set.

In the application of algorithms, the introduction of projection is a common practice to fulfill the boundedness condition required by the ODE method [45,46]. Projection or saturation not only ensures that individual behavior stays within set limits but also satisfies the ODE approach’s boundedness requirement. Given that a single integrator model directs the movement of individuals, the control of their positioning is described as:

$$q_i(t+1)=q_i(t)+\gamma (t)p_i(t)$$

where p_i (t) represents the control mechanism. Subsequently, the standard saturation defined by [·]_D is as follows:

$$x(t)={[\mathsf{\Omega }(t)]}_D=\left\{\begin{array}{ll}\mathsf{\Omega }(t),& \mathsf{\Omega }(t)\in D\\ x(t-1),& \mathsf{\Omega }(t)\notin D\end{array}\right.$$

(49)

where D serves as the compact principal of DR, ensuring that the regularity condition is satisfied. In Equation (46), the left and right sides are denoted by x(t) and Ω(t), separately. The update in the projection algorithm occurs solely the updated numerical value is part of set D; if not, the prior state is maintained. This update mechanism ensures that the algorithm progresses effectively. Equation (46) can be transformed into a standard form based on the closed-loop system presented in Equation (45). In this standard form, the projection is integrated into the algorithm. In addition, the projection will gradually disappear in the average update direction. The disappearance of the projection means that individual cognition and behavior will change dynamically with uncertain environmental factors.

In complex social systems, the convergence and adaptability of algorithms is still a feasible research topic. To investigate this concept in more depth, the theoretical principles of Ljung and Soderstrom reasoning can be applied. This inference reveals that individuals or groups in complex social systems adjust their behavior in order to achieve collective goals. Then, to prove the regularity conditions C1–C11, assume the following:

M1: The individual and its neighbors gather a quantity s ≥ m of measurement values at the location ${\left\{p_k\right\}}_{k=1}^s$,

$${\displaystyle \sum _{k=1}^n\varphi (p_k){\varphi }^T(p_k)}\succ 0$$

where m is in Equation (15).

M2: Both the artificial potential force and the adjacency matrix are continuously differentiable for q, with their derivatives being bounded.

M3: To align with the algorithm (46), the projection algorithm (49) is utilized. In Equation (49), define D as a convex compact set determined by $D={\mathcal{M}}^{Ns}\times {\mathcal{M}}_p, where {\mathcal{M}}_p=[p_{min}, p_{max}{]}^{{2N}_s}$

Remark 4. 

M1 is considered the reward condition continuously in adaptive control. This suggests that individuals require continuous rewards or inputs to achieve dynamic cognitive processing and actions. According to (11) and Olfati-Saber, M2 can be satisfied. This reflects the adaptive ability of groups or individuals to adjust their behavior based on internal rules or external environments [38]. M3 satisfies the boundedness and constraint conditions of the ODE method. This represents the inherent limitations of individual cognitive processes and the spatial constraints of behavior in complex social systems.

Lemma 1. 

The reconciliation and estimation algorithms are converted to the standard form in (46) and (47), with the following lemma. The forms of (46) and (47) can be converted by algorithms (45) and (38), respectively, with the following definitions described as:

$$\begin{array}{l}q(t)=col(q_1 (t),\dots ,q_{N_s} (t))\in R^{2N_s},\\ z(t)=col(z_1 (t),\dots ,z_{N_s} (t))\in R^{2N_s},\\ x(t)={[q^T (t),z^T (t)]}^T\in R^{4N_s},\\ Q(t;x(t-1),\phi (t))=\left[\begin{array}{c}z\\ -\nabla U(q)-\left(\widehat{L}(q)+K_d\right)p-\nabla \widehat{C}(\phi ,q)\end{array}\right]\end{array}$$

(50)

According to the observation and sampling process in (47), the following can be obtained:

$$\begin{array}{ll}\phi \left(t\right)& =g\left(t; x\left(t-1\right),\phi \left(t-1\right),e\left(t\right)\right) \\ & =A\left(t; x\left(t-1\right)\right)\phi \left(t-1\right)+B\left(t; x\left(t-1\right)\right)e\left(t\right),\end{array}$$

(51)

This change reflects the continuous improvement in individual cognitive levels, as well as the continuous evolution and optimization of the group’s decision-making processes and behavior patterns.

Corollary 1. 

Algorithms (46), (47), and (49) must be examined in accordance with the regularity conditions C1–C11 [44] The subset D_R denotes an open, connected part of D_S. In Equation (49), D is represents as a compact subset of D_R, and it influences the trajectories of the associated ODE, which needs adhere to this definition.

$${\displaystyle \frac{d}{d\tau }}x\left(\tau \right)=f(x(\tau ))$$

(52)

where$f(x)=\underset{t\to \infty }{\lim }\to \mathbb{E}Q(t;x,\overline{\phi }(t,x)),$for τ > 0, The closed subset ${\overline{D}}_R$ of D_R is where the initial point in D can consistently remain. It is assumed that the domain of attraction D_A ⊃ D is possessed by the differential Equation (52), and that D_c is an invariant set. In that case, either:

x(t) → D_c, probability one is t → ∞,

(53)

or

x(t) → ∂D, probability one is t → ∞,

(54)

where ∂D can be defined as the boundary of D.

The trajectory of the differential equation in (39) needs to leave D in (49) to establish the conclusion in (54).

Remark 5. 

Suppose that the transformation recursive algorithm is considered after Lemma 1 is applied based on M1–M3 then the algorithm is constrained by the regularity conditions C1–C11, and $({\mathcal{M}}^{N_S}\backslash Z)\times {\mathcal{M}}_P\subset D\subset D_R$, where ${\mathcal{M}}_P=[p_{min}, p_{max}{]}^{{2N}_s}$ and Z are represented by:

$$Z=\left\{q\in {\mathcal{M}}^{N_s}\mid \sum _{j\in \left\{i\right\}\cup \mathcal{N}(i,q)}\varphi \left(q_j\right){\varphi }^{\mathrm{T}}\left(q_j\right)\nsucc 0,\forall i\in {\displaystyle \mathcal{l}}\right\}$$

(55)

Furthermore, f (x) in (52) of Corollary 1 can be obtained by

$$f(x)=\left[\begin{array}{c}p\\ -\nabla U(q)-\left(\widehat{L}(q)+K_d\right)p-\nabla C(q)\end{array}\right]$$

(56)

where $C\left(q\right) \in R\ge 0$ is defined as a cost function with respect to collective performance, as shown below:

$$C(q)=k_4{\displaystyle \sum _{i\in {\displaystyle \mathcal{l}}}\left[c_{\max }-c\left(q_i\right)\right]}$$

(57)

where $c_{max }$ represents the maximum of the potential field and is considered to have limits.

Proof. 

C1–C11 are validated as follows:

• C1: The assumptions regarding measurement noise in Equations (25) and (26), as well as Remark 2 in M1 and M3, can be satisfied.
• C2: Assuming that the radial basis functions in M2 and $\nabla \widehat{C}(\phi ,q)$ have smooth and bounded derivatives with respect to q achieves this.
• C3: In Equation (51), A and B are smooth within $D_R$ due to their dependence on smooth radial basis functions.
• C4: A similar rationale from Brus is introduced [48], $‖A(t; x){|}_{\tilde{x}}‖\le 1-\delta <1, \forall t$.
• C5: For a fixed $\overline{x}$.

$${\overline{\phi }}_i(t,\overline{x})=\prod _{k=s+1}^{t}A(k;\overline{x}){\overline{\phi }}_i(s,\overline{x})+\sum _{j=s+1}^t\left[\prod _{k=j+1}^{t}A(k;\overline{x})\right]B(j;\overline{x})e(j),\hspace{1em}i\in \{1,2\}.$$

Under M1 and M3 the following can be obtained:

$$‖{\overline{\phi }}_1(t,\overline{x})-{\overline{\phi }}_2(t,\overline{x})‖<{\lambda }^{t-s}(\overline{x})‖{\overline{\phi }}_1(s,\overline{x})-{\overline{\phi }}_2(s,\overline{x})‖,$$

• C6: The element of Q in (51) is considered to be a deterministic function of $x\in D_R$, except for $\nabla \widehat{C}(\phi (t), q)$. Due to M1 and (36), for a fixed q, it is described by:

$$\underset{t\to \mathrm{\infty }}{\lim }\mathbb{E}(\nabla \widehat{C}(\phi (t),q))=\nabla C(q),$$

from which C6 and (56) can be simultaneously verified

• C7: Because of the measurement noise assumption in (25), it can be satisfied.
• C8, C9, C10, C11: The conditions can be verified through the time-varying gain sequences shown in (42).

Finally, it is necessary for individuals to minimize and optimize overall performance costs [36,39]:

$$V(q(\tau ),p(\tau ))=C(q(\tau ))+U(q(\tau ))+{\displaystyle \frac{p^{\mathrm{T}}(\tau )p(\tau )}{2}}$$

(58)

where C(q) denotes a cost function related to the location of individual q, aimed at motivating the individual to track the maximum value of the potential field. This reflects the cognitive process of individuals focusing on the benefit-maximizing region. U(q) is a cost function associated with q, combining strategies for collective motion and obstacle avoidance, designed to maintain group cohesion during navigation. Additionally, this function includes the concept of artificial walls, which can be associated with the boundaries or norms guiding collective motion. The kinetic energy of the complex system is represented by the last term on the right side of (58), and it is treated as a separate cost function of p. This represent the costs or resources expended by an individual or group in pursuit of their goals. □

5. Simulation Results

In this work, a theoretical framework is developed to model the interactions of individuals in complex social systems. Each individual updates its understanding of the potential field with each iteration. This means that individuals adjust their cognition and behavior in response to new information and interactions in the social system. The initial velocity of the individual is zero, and the random location simulates the unpredictability and diversity of the social environment. The simulation results under different parameters and conditions are analyzed.

5.1. Standard Environment

Figure 2a shows the detailed parameters of the simulation in the initial environment. The number of individuals is set to 60, representing a moderate scale for observing individual actions and collective behaviors. The noise level parameter (W) is set to 1, simulating cognitive biases, information asymmetry, and sudden events in social systems. This parameter is dimensionless and serves as a relative measure of the randomness or uncertainty within the system. The equilibrium distance (D) is set to 0.6, representing the physical or psychological space maintained between individuals, which is crucial for social cohesion and personal comfort. This distance is measured in arbitrary units of distance, reflecting relative spacing within the simulation environment. Additionally, the simulation includes four potential fields, representing social resources, interests, identity, goals, etc. The peak value of these potential fields is set to 5, also measured in arbitrary units of potential value, indicating the strength or attractiveness of the fields.

Thus, to ensure clarity, arbitrary units are used to describe the parameters of distance and potential value in the simulation. This approach allows us to generalize the findings and apply them to various social systems without being constrained by specific measurement units.

As shown in Figure 2, the number of potential fields is set to 4, and the peak value of the potential field is set to 5. This configuration is chosen to balance complexity and clarity in observing individual and collective behavior within the social system. By setting the number of potential fields to four, the simulation ensures a manageable level of complexity, allowing for clear observation of how individuals interact with multiple attractive regions while avoiding an overly simplified or excessively intricate scenario. The peak value of the potential fields, set to five, represents a moderate level of attraction that is sufficient to influence individual behavior significantly without overpowering other factors in the simulation. This value ensures that individuals are drawn towards the potential fields, highlighting the interactions between individual decision-making processes and external influences. By maintaining a consistent peak value across all potential fields, the simulation provides a controlled environment to study the effects of other variables, such as noise intensity, individual density, and communication radius, on collective behavior. This approach allows for a focused examination of how individuals navigate towards areas of higher potential, reflecting real-world scenarios where individuals are attracted to areas with perceived higher benefits or resources. In summary, the selection of four potential fields with a peak value of five is designed to create a balanced and manageable simulation environment that effectively demonstrates the dynamics of collective behavior influenced by multiple attractive regions. This setup facilitates a clear analysis of individual and collective responses to potential fields, contributing to a deeper understanding of complex social systems.

In complex social systems, individuals often exhibit collective behaviors that arise from interactions within a diverse and dynamic environment. Figure 2b shows individuals gathering around the “yellow” potential field. This process demonstrates how collective interests, goals, or perspectives can draw people together, overcoming inherent cognitive biases or communication barriers. The attraction of individuals to a specific area reflects underlying commonalities in their interests or identities, which can be driven by cultural phenomena, social norms, or economic incentives. As individuals move towards the yellow potential field, their behavior exemplifies social identity formation, where the group coalesces around shared values or objectives. In Figure 2b, the collective behavior at iteration time t = 700 is also observed, where all individuals successfully converge to the potential field. This convergence is a reflection of herd behavior, where individuals align their actions with the group, often leading to conformity. Such behavior can be observed in various social contexts, from financial markets to social movements, where the actions of individuals are heavily influenced by the collective dynamics. The absence of oscillations near the maximum value indicates that the social group is in a critical equilibrium state, neither too rigid nor too loose. Instead, it represents a balanced state where the social group can maintain stability while being flexible enough to respond to external or internal changes. In fact, the behavior observed in Figure 2b exemplifies the complex interplay of individual actions and collective dynamics within social systems. The convergence towards the potential field highlights the emergent properties of these systems, where individual decisions, influenced by shared social, cultural, or economic factors, lead to the formation of cohesive group behavior. This equilibrium state is essential for the sustainability and adaptability of social systems.

Figure 2c shows the mean squared error (MSE) over iteration time, providing a quantifiable measure of the stability of the individuals’ search for the potential field. The sharp decline in the initial MSE indicates that individuals quickly adapt to the social environment and distributed learning, enhancing individual cognition, collective intelligence, and behavioral coordination. As iterations continue, the MSE value begins to stabilize, suggesting that further refinement of group behavior becomes more nuanced and incremental. After 700 iterations, individuals find a common social norm to abide by after a period of conflict, adjustment, and adaptation, thus achieving consensus or balance in the complex social system. This state indicates that distributed learning has overcome individual cognitive biases and environmental noise, reaching a stable convergence state. In conclusion, this understanding is vital for analyzing and predicting the behavior of social systems, whether in small communities or larger societal structures.

5.2. Noise Environment

Figure 3a,b show the collective behavior of individuals within a social system when the noise parameter is set to 20. This noise represents the disturbances and uncertainties individuals encounter during social interactions. These disturbances can take various forms, such as information asymmetry, false information, and cognitive biases. Information asymmetry occurs when there is an unequal distribution of information among individuals, leading to misunderstandings and misinformed decisions. False information, or misinformation, further complicates the social dynamics by spreading inaccurate or deceptive content that can mislead individuals. Cognitive biases, such as confirmation bias or anchoring, distort individuals’ perception and judgment, causing them to make irrational decisions based on incomplete or skewed data. These factors can lead to cognitive dissonance, a psychological state where individuals experience discomfort due to conflicting beliefs or information. Cognitive dissonance impairs individuals’ ability to process information logically and make rational decisions. As a result, the overall decision-making process within the group becomes less efficient and less stable.

The high noise level depicted in Figure 3a,b highlights the state of confusion and disruption that can ensue within a social system. This confusion can destabilize social stability, making it challenging for the group to maintain consistent adherence to social norms. Social norms, which are the shared expectations and rules that guide behavior within a community, rely on clear communication and mutual understanding to be effective. When noise interferes with these processes, individuals struggle to align their behaviors with the established norms. In addition, the formation of social cohesion is significantly affected by high levels of noise. Cohesion relies on shared understanding and coordinated action, both of which are undermined by noise. The resulting lack of cohesion can lead to fragmentation within the group, in which individuals are unable to interact or cooperate effectively.

Figure 3c illustrates the tendency of MSE to decline under conditions of severe noise interference. It reveals a direct correlation between an increase in noise parameters and a decrease in an individual’s ability to search potential fields effectively. This correlation indicates that group search efficiency is highly sensitive to noise, reflecting the vulnerability of collective behavior to false information and cognitive bias. Moreover, the results from Figure 3 highlight the need for strategies that enhance individual resilience to noise. Such strategies include fostering open and transparent communication, promoting critical thinking to counteract cognitive biases, and implementing systems to verify the accuracy of shared information. The simulation results not only demonstrate the impact of noise on individual and group performance but also underscore the importance of mitigating noise to ensure effective collaboration and decision making within social systems. In summary, by implementing strategies to manage noise, social systems can maintain stability, cohesion, and efficient functioning, even in the face of significant disturbances.

5.3. Density Environment

Figure 4 shows the correlation between individual density and collective action efficiency. In Figure 4a,b, when the group size reaches 180 individuals, the efficiency of collective search potential fields decreases with the increase in the number of individuals. This observation suggests that a higher individual density exacerbates social friction and competitive behaviors due to resource scarcity, thus hindering the group’s flexibility and responsiveness. As spatial density increases, several adverse effects emerge within the social system. Firstly, the social system may become overloaded, negatively impacting individual cognitive processing and decision-making efficiency. Overcrowding can lead to sensory overload and increased stress, making it difficult for individuals to process information effectively and make rational decisions, thus diminishing overall group performance and reducing the stability and convergence of social systems. In addition, an increase in density can exacerbate social frictions in the form of intensified interpersonal conflicts and competitive behaviors for limited resources. This competition can disrupt the harmony of the group, leading to inefficiencies and decreased collective productivity. The social dynamics become more strained, compromising the collaborative potential of the group.

To address these challenges and enhance collective efficiency, several strategies can be implemented. Optimizing group size is one such strategy, as limiting the size of the group to an optimal number can prevent overcrowding and reduce the negative impact of high density. Smaller, more manageable groups can maintain higher levels of efficiency and coordination. Additionally, strengthening communication channels is crucial in mitigating misunderstandings and conflicts within high-density settings. By improving communication channels, individuals can share information more, resolve disputes quickly, and align their actions towards common goals. This can involve using advanced communication technologies, establishing clear communication protocols, and promoting an open communication culture. Implementing distributed decision making can also alleviate the cognitive burden on individuals in high-density settings. By decentralizing decision-making processes, this reduces the pressure on any single individual and enables more efficient and effective group actions.

Figure 4c depicts the relationship between individual density and collective behavioral effectiveness within complex social systems. The sharp decline in MSE indicates an accelerated phase of distributed learning. During this phase, individuals rapidly optimize their behavior based on the interactive information and feedback provided by the system. However, the increasing number of individuals brings about significant challenges in coordination and communication. Beyond a certain threshold, the overall effectiveness of the team may plateau or even decline. This phenomenon is known as the diminishing returns of collective behavior [49]. The increase in individuals often leads to conflicting decisions and behaviors, which can impede the system’s ability to perform collective tasks efficiently. These conflicts can result in slower decision-making processes and reduced accuracy in task completion. In conclusion, while increasing the number of individuals in a system can initially lead to improved collective behavior through distributed learning, there is a critical point beyond which additional individuals may hinder performance due to coordination and communication challenges.

5.4. Without Communication Environment

Figure 5b emphasizes the crucial role of communication in collective behavior. When the communication radius is set to 0, the individual capability to efficiently search potential fields is diminished due to a lack of information sharing. Due to the lack of interaction and information sharing with neighboring individuals, individuals will detach from the group and act in isolation. This isolation not only impairs individual performance but also hinders the group’s ability to leverage collective intelligence. Collective intelligence represents the enhanced cognitive capacity and problem-solving potential that emerges when individuals within a group combine their skills and knowledge. Without the ability to communicate and share information, individuals are too isolated to tap into this powerful resource, resulting in a lack of cohesion within the group. Figure 5b highlights the necessity of fostering strong social networks within any complex system. The development and maintenance of these connections are crucial for building a cohesive and efficient team capable of navigating complex challenges together.

As shown in Figure 5c, in the initial phase, from 0 to 400 units of time, the MSE is high, indicating significant estimation errors among the individuals exploring the potential field. The rapid drop in MSE suggests a phase transition, where individuals quickly adapt to their local environment, gathering information and reducing uncertainty. This initial period represents a time of learning and adaptation. Without prior knowledge or communication, individuals must rely solely on their interactions with the environment. The high error rates reflect their lack of experience and information, but as they engage more with their surroundings, they develop localized strategies and heuristics that lead to significant improvements in their accuracy. In the later phase, from 400 to 700 units of time, the MSE continues to decrease but with less fluctuation. The reduced fluctuations in MSE suggest that the social system has reached a level of stability and robustness.

Overall, the graph of MSE over time for individuals exploring a potential field without a communication radius provides a comprehensive view of the dynamics within complex social systems. Initially, high errors and rapid adaptation highlight the emergent behavior as individuals independently develop strategies that collectively lead to a reduced MSE. Each individual adjusts and reorganizes in response to their environment, leading to nonlinear and unpredictable fluctuations. Finally, as the system stabilizes, individuals refine their strategies, resulting in a more robust and efficient exploration process. This interpretation illustrates how individual adaptation, emergent behavior, and self-organization contribute to the overall dynamics of the system, leading to greater stability and reduced error over time.

5.5. Multiple Potential Field Environment

As shown in Figure 6a, the amount of potential fields is set to 7, the peak value of the potential field in the bottom right corner is set to 10, and the peak value of the other potential fields is set to 5. In Figure 6b, it is observed that individuals are naturally attracted to the area of the highest peak of the potential field. The potential fields with the highest peaks are represented by brighter colors. Initially, individuals are scattered but show a tendency to move towards regions with higher potential, especially the potential field with a peak value of 10 in the lower right corner, which exerts a stronger attractiveness compared to other areas with lower peak values of 5. This means that individuals respond to environmental information and seek out areas that maximize their personal benefits. However, individuals do not directly gravitate towards the region with the maximum potential field peak. Instead, they move continuously towards the nearest potential field before progressively aggregating into the area with the maximum potential. Firstly, individuals move towards the nearest potential field, reflecting local interactions and decision-making processes. Over time, as they gather more information and adjust their strategies, they progressively converge into the area with the maximum potential. This gradual aggregation demonstrates the emergent behavior of complex social systems, where local interactions and individual decisions lead to the formation of larger, organized patterns. Cognitively, this indicates that individuals process information about their environment and make decisions based on a combination of proximity, perceived value, and accessibility. This reflects the human tendency to be drawn towards immediate or easily accessible benefits before moving on to more significant but less accessible benefits.

As shown in Figure 6c, the overall trend shows a steep initial decline in MSE, indicating rapid adaptation and learning, followed by a more gradual decrease and stabilization in the later phase. Due to the complexity of the setting and the influence of other potential fields, the path to the potential field with the maximum peak value is not direct. This complexity can cause fluctuations in MSE, reflecting the individuals’ navigation, learning, and strategy adjustment processes. As individuals continue to interact and adjust their behavior, the MSE shows a gradual decrease and stable trend. This reflects the process in social systems where, over time, social norms, shared knowledge, and collective strategies emerge, producing more stable and predictable behavior. In summary, in social systems, collective behavior is shaped by the local interactions between individuals and the complex environment. Despite the complexity of the environment leading to indirect paths, the continuous movement towards the potential field with the maximum peak value demonstrates the adaptiveness and emergence of social systems. This emphasizes the importance of distributed learning, as individuals and groups continuously optimize their behavior in a dynamic and interconnected environment.

5.6. Multiple Negative Potential Field Environment

As shown in Figure 7a, the amount of potential fields is set to 4, and the peak value of the potential field is set to −5. As can be seen in Figure 7b, the peak of the negative potential field generates a strong repulsive force, which impedes the individuals from completing the global task of exploring the potential field. All individuals gather together to form a grid-structured swarm. The formation of grid structures highlights the ability of individuals within complex social systems to self-organize into structured patterns. This self-organization is a feature of complex social systems where local interactions and simple rules can lead to the emergence of collective behavior. By constantly adjusting their position and maintaining a structured formation, individuals optimize their behavior to navigate the environment created by the repulsive potential fields. This structured formation demonstrates the inherent robustness and flexibility of complex social systems, allowing them to adapt to and navigate dangerous conditions effectively.

Moreover, individuals exhibit emergent behavior by approaching each other while maintaining a safe distance. This phenomenon can be interpreted as a form of local interaction, where individuals are influenced both by the repulsive forces of the potential field and by the proximity of their peers. This dual influence ensures that individuals do not merely react to the negative potential but also take into account the positions and movements of those around them. The balance between attraction to peers and repulsion from negative fields enables the group to maintain cohesion and avoid overcrowding, thus optimizing their collective movement and interaction. This emergent behavior highlights the adaptability of individuals in responding to both local and global influences in their environment. It shows how individuals collectively adjust their strategies to optimize their performance and achieve a balance between competing forces. Furthermore, the observed behavior emphasizes the importance of distributed learning and collective adaptation in complex social systems. Distributed learning refers to the process where individuals learn and adapt based on local information and interactions rather than relying on a central authority. This decentralized approach allows for greater flexibility and faster adaptation to changes in the environment.

Figure 7c depicts a convergence process, indicating that when individuals are subjected to the repulsive forces of a negative potential field, they experience a period of adjustment characterized by trial and error. Initially, the high MSE reflects a lack of coordination and the difficulties individuals face in optimizing their positions relative to others. Over time, individuals learn and adjust their strategies. The decline in MSE indicates that the behavior of individuals continues to be optimized, and they increasingly adhere to the rules of balancing proximity and safe distance. This adaptive behavior demonstrates a decentralized form of intelligence. The observed pattern suggests the resilience and flexibility of individuals in overcoming the initial disruptions caused by negative potential fields. Based on the gradual reduction in the MSE, they demonstrate an ability not only to survive but also to form grid-like teams in a complex and potentially dangerous environment. This convergence process can be attributed to the self-organizing properties and emergent behaviors within the social system, where individuals utilize local interactions and feedback mechanisms to enhance the group robustness and adaptability. In summary, Figure 7 reveals the process from initial chaos to a well-coordinated state, emphasizing the role of collective intelligence and local interactions in navigating and stabilizing within a negative potential field.

6. Conclusions

The simulations reveal interdisciplinary insights into the collective behavior of individuals within various social environments. In standard conditions, individuals exhibit rapid adaptation and convergence towards common goals, reflecting social identity formation and collective behavior. High noise levels disrupt social cohesion and efficient decision making, highlighting the need for strategies to enhance resilience against misinformation and cognitive biases. Increased density leads to social friction and decreased efficiency, underscoring the importance of optimizing group size and improving communication channels. The absence of communication significantly hampers collective intelligence, demonstrating the crucial role of information sharing. In environments with multiple potential fields, individuals progressively adapt, showing the importance of distributed learning and emergent behavior in complex systems. Lastly, negative potential fields drive individuals to form structured patterns, emphasizing self-organization and adaptability in adverse conditions. These findings collectively enhance the understanding of how social dynamics and environmental factors influence collective behavior, offering valuable implications for both theoretical research and practical applications.

The research offers a comprehensive analysis of how individual interactions and distributed learning contribute to emergent behaviors and collective adaptation, providing novel insights into the mechanisms underlying social cohesion and collective intelligence. They provide valuable implications for practical applications, such as urban planning, resource management, policy making, and the design of resilient social systems. Future research should continue to explore these dynamics, incorporating more complex interactions and real-world scenarios to further refine the understanding of collective behavior in social systems.

The theoretical future research on collective behavior in social systems should enhance simulation models by incorporating complex and diverse agent interactions, such as emotional influences and varying communication strengths. This includes examining heterogeneous agents to understand how diversity affects adaptability. Investigating dynamic noise levels and environmental feedback loops, where agents’ actions modify their environment, will provide insights into adaptive behavior over time. Additionally, understanding the impact of different communication networks, information quality, and misinformation is crucial for comprehending their influence on social cohesion and decision making. Research should also explore scalability and density effects by varying the number of agents to observe emergent behaviors and efficiencies, developing strategies to optimize agent density for practical applications. Applying these models to real-world phenomena, such as crowd behavior or information spread, can guide policymakers in promoting social stability. Interdisciplinary approaches, integrating psychological and sociological theories, will enrich research and enhance model applicability.