Complex behavior of individuals and collectives in a social system: An introduction to exploratory computational experimental methodology based on multi-agent modeling

Abstract

Although the multi-agent model has been used to analyze several economic and management problems, and the research results are regarded more profoundly, they all rely on certain scenarios. Once the scenarios are shifted to an unknown one, the results cannot be matched. In this paper, a new research method named exploratory computational experiment is introduced to resolve the problems coming from the social complex system, where individual’s behaviors are irrational, diverse, and complex, and collective behavior is dynamical, complex, and critical. Firstly, the foundation of the computational experiment is introduced, then several important problems, how individuals make the decision under complex environment, how collective behavior have emerged when different conflicts co-exist, and how to evaluate collect behaviors, are analyzed. To specify this new method, two examples of how to design a scientific mechanism to make the traffic system more effective and how is the evolution law of giant components in scale-free networks if the parameters are changed continuously. The results show that multi-agent modeling based on irrational behaviors controlled by individual dynamical game radius and memory length limited can describe the social problem more accurately, the exploratory computational experiment can give us more profound conclusions.

Introduction

The individual is complex because of his or her irrational behavior. Collective behavior can be simple or complex. This is because irrationality can be enlarged in the interaction process between individuals. Interaction between individuals could lead to system emergence, which produces different collectives with diverse behaviors (Hong et al., 2021; Baldassarri & Abascal, 2020; Mertikopoulos et al., 2010; Holger, 2014). The simplest behavior, such as the Heatbugs model, involves complex system behaviors and human behavior reflecting social, intelligent, autonomous, diverse, and random aspects.

Scientists have studied this interesting problem for several decades. A very broad range of complex adaptive systems has been studied. These include abstract ones (Boccaletti et al., 2006; Gupta et al., 2022; Niu et al., 2022)- such as the evolution of an economic system (Rand & Stummer, 2021; Schweitzer et al., 2009), complex adaptive systems, percolation (Albert et al., 2000; Brauer et al., 2010), emergence (Mertikopoulos et al., 2010; Bab & Brafman, 2008) and physical systems, such as city traffic design, management decisions (Hori et al., 2013; Benabdellah et al., 2020; Luis et al. 2021, Gaku Hashimoto, et al., 2015), spreading rumors (Bab, & Brafman, 2008), and insurance policy (Ma et al., 2019). They all have in common properties: one cannot hope to explain all outcomes mathematically, from a computation experiment viewpoint (Hori et al., 2013; Wang et al., 2017), or to offer a universal conclusion due to the complexity of the problem (Niu et al., 2022). Mainstream research methodology constructs a simulation model for a certain question, then applies the evolution law of different collective behaviors by simulating this model (Badano et al., 2010; Du et al., 2019; Ostrom, 2009). However, the simulation method seeks to explain why several known events happened and/or how they would change by supervised learning (Moumivand et al., 2021; Szolnoki et al., 2008; Laura et al. 2020). They do not conform to universal laws (Alan et al. 2020; Bollobás & Riordan., 2004; Pasztor et al., 2021). As far as the events that both possibly happen and have unhappened are considered, simulation is invalid because of the complexity and unknown. In addition, such simulation relies on certain strict conditions. Each simulation could be regarded as an independent Bernoulli test. In fact, we care what the law of a certain system while the conditions continuously change. As far as the unknown event is considered, some feasible and scientific research methods must be introduced in order to obtain an accurate result. Therefore, an exploratory computational experiment with non-supervision and driftlessness learning is focused on finding a certain law derived from certain conditions, which would satisfy the criteria of scientific research.

Social organization is a complex system because of random system structures and functions and furthermore, an individual’s behaviors are stochastic dynamic, diverse and complex, which can be described as a random probability to select a certain interaction among differential stochastic games, such as a stochastic adaptive game, stochastic quantum game, and so on (Wall, 2021; Xiong et al., 2021). This would affect a dynamic individual’s behavior changes due to a transitory state and the nature of the environment. In addition, an individual selects the right player to interact with according to a preferential attachment mechanism. As mentioned above, it is perhaps a fixed or random number controlled by several parameters of an individual’s degree, strength, or payoff, as shown in Fig. 1.

Fig. 1.

Diversity and Randomness of Individual Behavior

As shown in Fig. 1, it is conclude that the economic and management organization is a stochastic system, whereby not only the system’s structure and function are time-varying but are also due to the interactiveness of irrational behaviors. In addition, their rank can change randomly with time, which makes an unstructured decision-making process irresolvable by classical methods (Bai et al., 2016). Scientists care about what the system’ properties are and how the state is changed, comprising the following:

• How an arbitrary individual makes decisions under these complex conditions that consist of not only his or her interactive configuration and historic memories but also the system and the environment’s property;

• What property would emerge due to interaction between individuals? How do different collectives emerge because of interaction between individuals, what properties they have respectively, and how do they change over time?;

• How a phase transition of the system state happens and how the system could evolve over if some individuals change their behavior.

Obviously, these questions are vital to understanding, managing, and controlling this system. Unfortunately, contrasted with natural phenomena, social phenomena are highly complex. As mentioned above, social systems are general complex system individual interact together (Benabdellah et al., 2020; Ma et al., 2019; Niu et al., 2022; Rand & Stummer, 2021) and there are large enough number of individuals(e.g., 730,000 individuals are considered in NPS1(National Plan Scenario (1) model, at least 500 cars are considered in Transims system, 8.5 million of people are considered in Zika virus model, 1.4 million families are considered in Hallegatte model, et.al.) interact with each other (Hori et al., 2013; Ito et al., 2018; Ostrom, 2009), which make the question must be more complex. As a result, scientists cannot easily obtain profound conclusions by using classic reductionism methods. Although many scientists construct corresponding Multi-Agent models, modeling these dual complexities to prove several known theorems, they do not discover unknown laws that rely on occurring scenarios. As far as unknown ones are considered, the simulation method is invalid. So, a new method, the exploratory computational experiment, is introduced to study this complex adaptive system with large enough individuals interacting in this paper. The aim is to discover unknown situations and some interesting laws. Furthermore, for a social complex system, multi-agent modeling can describe the behavior for known events; if more known scenarios are analyzed, a basic law can be drawn. Then based on this multi-agent model and the corresponding basic law, the unknown event can be discussed on this model if the corresponding conditions have never been seen.

Irrational behavior occurs for several reasons, such as incomplete and/or imperfect information-driven by individual game radius and memory length, which is defined in Sect. 2; we look at the risk attitude in Sect. 3 and personal properties consisting of the learning ability level of intelligence, and adaptive ability. In addition, we examine selective ability to play off preferential attachment. These are all variables or parameters important to collective irrational behavior. Consequently, corresponding definitions are introduced in the paper. In this paper, the foundation of the computational experiment is introduced, then the problem and the research process of individual selection while facing different risks as well as the emergence of the irrational collective, has been discussed. Finally, this paper introduces some examples to prove that an exploratory computational experiment is feasible rational to analyze social problems. Several examples are introduced in this paper, traffic system, and evolution process of the giant components have been studied. It is conclude from traffic system that arbitrary individual must confirm to a simple rule and it is the simple behavior rule that makes the system’s behavior are more complex, especially makes traffic system is at critical state driven by scenarios more heavily. More important, environment would affect traffic system more sensitively if the system is of critical. The example of evolution process of the giant components reflects how collective behavior of homogenous individual evolves. In this example, we learn that collective behavior is always occurs sharply as giant component with homogenous behavior growth. Besides, diverse, random collective behaviors would be emerged in this process because of interaction between individuals. These examples reveal that exploration computational experiment is an important analysis method.

Several basic definitions

Game radius

An arbitrary individual interacts with others directly or indirectly, which produces a directed graph. For example, the maximum amount of a certain individual’s reachable distance is called his game radius, as shown in Fig. 2:

Fig. 2.

Game Radius

Definition 2.1

Suppose that graph $G$ is a graph corresponding to individual interactive topological configuration, then the maximum of reachable distance to arbitrary node $i$, is called his game radius $r_i$.

An arbitrary individual could adjust his local configuration such that $r_j(t)\ne r_i(t),\forall j\ne i,i,j\in \mathcal{A}$ always holds where $\mathcal{A}$ is all the individuals in a system and $r_i(s)\ne r_i(t),\forall s\ne t$, where $s,t\in \mathcal{T}$, where $\mathcal{T}$ is the set of time $t$.

Memory length

Generally, each individual cannot remember all historical events; the time steps coupled with maximum memory, called memory length, reflect his irrational decisions, shown as Fig. 3.

Fig. 3.

Memory length

Definition 2.2

For individual behavior equation.

$$X(t)=f(X(t-1),X(t-2),...,X(t-k))+w(t)$$ 1

the maximum $k$ that individual behavior can be affected by historical events is called memory length.

As far as individual $i$ is considered, $k_i\ne k_j,\forall i\ne j$ and $k_i(t)\ne k_i(s),\forall s\ne t$ always hold.

Definition 2.3

An individual always selects the fitness players to interact with so as to maximize his benefit according to.

$$\underset{\alpha }{max}\mathbb{P}\left(\pi \left({\alpha }^{j_i},{\alpha }^{k_i}\right)+{\varsigma }_{k_i}^{j_i}\ge \pi \left({\alpha }^{j_i},{\alpha }^{l_i}\right)+{\varsigma }_{l_i}^{j_i},\forall l_i\notin {\overline{\mathcal{N}}}^{j_i}(\omega )\right),t\gg t_c$$ 2

$$\underset{\omega }{max}\mathbb{P}\left(\pi \left({\alpha }^{j_i},{\alpha }^{k_{i^{{}^{\prime }}}}\right)+{\varsigma }_{k_{i^{{}^{\prime }}}}^{j_i}\ge \pi \left({\alpha }^{j_i},{\alpha }^{l_{i^{{}^{\prime }}}}\right)+{\varsigma }_{l_{i^{{}^{\prime }}}}^{j_i},\forall l_{i^{\prime }}\notin {\overline{\mathcal{N}}}^{j_i}(\omega )\right),t\gg t_c$$ 3

$$\underset{\omega }{max}\mathbb{P}\left(\pi \left({\alpha }^{j_i},{\alpha }^{k_{i^{{}^{\prime }}}}\right)+{\varsigma }_{k_{i^{{}^{\prime }}}}^{j_i}\le \pi \left({\alpha }^{j_i},{\alpha }^{l_{i^{{}^{\prime }}}}\right)+{\varsigma }_{l_{i^{{}^{\prime }}}}^{j_i},\forall l_{i^{{}^{\prime }}}\notin {\overline{\mathcal{N}}}^{j_i}(\omega )\right),t\gg t_c$$ 4

$$\underset{\omega }{max}\mathbb{P}\left(\pi \left({\alpha }^{j_i},{\alpha }^{N_i+1}\right)+{\varsigma }_{N_i+1}^{j_i}\ge \pi \left({\alpha }^{l_i},{\alpha }^{N_i+1}\right)+{\varsigma }_{N_i+1}^{l_i},\forall l_i\in i(\omega )\right),t\gg t_c$$ 5

$$\underset{\omega }{max}\mathbb{P}\left(\sum _{k_i\in {\mathcal{N}}^{j_i}}\left(\pi \left({\alpha }^{j_i},{\alpha }^{k_i}\right)+{\varsigma }_{k_j}^{j_i}\right)\le \sum _{l_j\in {\mathcal{N}}^{k_j}}\left(\pi \left({\alpha }^{l_j},{\alpha }^{k_j}\right)+{\varsigma }_{k_j}^{l_j}\right)\right),t\gg t_c$$ 6

where, $j_i$ is the $j^{\text{th}}$ individual of the $i^{\text{th}}$ sub-system, $\overline{\mathcal{N}}$ is the set of all neighbors of individual $j_i$, and $\mathcal{N}$ is the set of $\overline{\mathcal{N}}+j_i$. Equation (3)--(6) is called preferential attachment.

Equations (3)–(6) indicate that arbitrary rational individuals must select the maximum possible players who could bring the possible benefit compared to others.

Foundation of computational experiment

As a result, scientists have put forward a new method of Multi-Agent modeling by introducing an analysis method of a complex system to simulate these problems (Yogeswaranet al. 2012; Burggraef et al., 2019; Ito et al., 2018). They either construct a corresponding model to describe the problem and verify the known cases or analyze the state and evolution process of several unknown scenarios, which is called a computation simulation. However, the mission of scientific research is to discover unknown phenomena or their evolution process and to mine the corresponding law. Therefore, a new method should be introduced to implement this aim of science discovery.

The computational experiment is a boom-up modeling method, a revolutionary idea of reductionism. First, the individual’s behavior consists of behavior selection, and an interactive mechanism is constructed. Then, a Multi-Agent model is built to describe the system’s operation. Third, the index of collective behavior or system behavior is defined to verify whether the model is correct or not. There would be a need to revise the Multi-Agent model before the simulation results chime with the actual statistical data. Then, it is necessary to import the corresponding parameters according to the presupposed cases, then analyze the state, property, and corresponding evolutional process. In fact, if and only if the number of individuals in system is large enough, exploration computational experiment can play an important part on study (Bretas, A.M.C., Mendes, A., Jackson, M. et al., 2021).

Individual behavior

There are two kinds of individual behavior: behavior dynamics and individual interactive behavior. The former describes how an individual makes the decision, and the latter describes how one selects a person to interact with and how they interact with others. Individual dynamics usually be written to an order linear difference equation, a degenerated behavior dynamics of Eq. (1),

$$X(t+1)=AX(t)+B+\omega$$ 7

where $X(t)$ describe the probability vector that an individual selects certain behavior at time $t$. There are two problems that should be considered: how one individual selects a player to interact with and how she or he learns from the interactive process Khanzadi et al. (2019). The former reflects a mechanism named preferential attachment, see Eqs. (3)–(6), which means that this individual in some sense smart. As far as the latter is concerned, an individual could simulate, copy, learn, improve behavior that she or he thinks is good. Such behavior can be described as

$$\frac{{\dot{x}}_i}{x_i}=\beta (R_i-R)+\alpha (H_i-H)$$ 8

where $\beta$ reflects the learning ability and $\alpha$ is individual adaptive ability of behavior, $R_i$ is the behavior preference $i$ and $R$ is the average preference of all behaviors, $H_i=-log{x}_i$ is the self-information entropy of behavior $i$ and $H={\sum }_{n=1}^N{x}_{n}log{x}_n$ is the average information entropy of all behaviors. Obviously, this equation could describe how individuals learn other excellent strategies and how to improve their own.

Order parameters of collective behavior

Because collective behaviors are more stable and easier to observe, scientists prefer to use corresponding order parameters to verify whether the simulation model is correct or not. To do this, some order parameters should be introduced: such as changing frequency of individual strategies, structure and number of in-homogenous agents, the number of an agent’s categories, system properties, the nature of the environment (Wu et al., 2018). Besides, game radius, memory length, and other control parameters are also important. Furthermore, we need to focus on the configuration of giant components of homogenous behavior, which reflects system’s structure and function. For example, if the size of the maximum giant component equals or is larger than $(\lambda (p)+o(1))n$ (where $\lambda (p)$ is decided by selective probability $p$ and $exp(-\mathrm{\Theta }(1-p^2))\le \lambda (p)\le (1+5p/8)exp(-p/2)$ Bollobás and Riordan. (2004), collective behavior is robust. There should be other order parameters to describe collective behavior and system one, which would be redefined in specification research.

Process of computational experiment

Several processes should be done to analyze a certain problem by exploratory computational experiment. First, individual behavior should be defined as a stochastic differential equation and a series of constraints equations. Second, the Multi-Agent simulation model is constructed according to individual behavior. We do not stop revising the simulation model until the simulation results are confirmed with i.i.d. in relation to both the actual cases and behavior experiment results. The most important step is that scientists could imagine all possible cases, known as the difficult to observe or hardly appear at all, and then test them and obtain a corresponding result (Singh et al., 2021). Then we should judge whether the result is right or not. The last step is to make a conclusion for this problem by comparing all correct results of all possible scenarios.

According to verification tests, scientists would necessarily revise the simulation model if singular results have appeared in the analysis process but not in relation to the latter. One should record some singular results if they are not confirmed with the pre-set and continue to do the experiment. If a certain singular result repeatedly appears or does so with a relatively large probability, one should insist that this is an interesting result. If not, one should insist that it happens accidentally due to various reasons, and they should be found because they are important to the design mechanism. If several kinds of singular results appear with equal probability, one should insist that the system is critical and sensitive to initial systematic parameters. In this sense, the test failure is confirmed, and other analytic methods should be introduced (Singh et al., 2021).

In fact, each computational experiment can be regarded as a Bernoulli test because the scenario with certain conditions is a random variable holding with Normal distribution or Gaussian distribution. If and only if all scenarios that vary continuously are considered, and enough experiment results with corresponding scenarios are collected, the evolutionary law would be determined by reducing these Bernoulli sequences. However, if the system is critical, then theoretically, there is not a reliable conclusion because of the butterfly effect. In this case, an analytic method should be introduced by simplifying complex conditions to an integrated one under strict conditions. As a result, more profound conclusions would be drawn.

Analysis of computational experiment results

As mentioned above, a computational experiment aims to find the evolution law of a certain complex system and improve the system under all possible conditions. Therefore, an important question is arisen: how to draw a conclusion from experiment data of Bernoulli tests? The data for analysis comes from two aspects: what comes either from a social survey or a behavior experiment and what comes from a computational experiment. The latter is more important to a computational experiment because more interesting scientific laws could be discovered in this way. If the data satisfy a certain statistical distribution, it is easy to draw a conclusion by statistics; if the data is a stochastic process, then the evolution process is also easy to estimate using a corresponding analyzing method; if the data are a random pattern, then we insist that the system is critical. Even so, there always exist several singular data in the experiment. We should identify which are necessary under certain conditions and which are accidental and find the switching condition from the accidental event to a non-accidental one. By doing this, the conclusion is regarded as profound. Generally, a certain system always satisfies certain laws. In this sense, the collective behavior property and evolutionary law could be obtained by analyzing the distribution of certain order parameters that come from experimental results. The following sections propose several examples to explain how this computational experiment is used.

Individual behavior selection with bounded rationality on risk decision

Risk decision is a classical individual behavior selection problem, except for the variables mentioned in equation, one insists that individual decisions rely on the environment and vice versa. So, how to define the environment is so important to research outcomes that scientists could tear the vitals out of a subject (Majd & Hobson, 2020). One thinks that an environment comprises three parts: (1) the system environment that can be affected by arbitrary agents in the system; (2) the behavior set of other individuals within the game radius, which describes whom she or he affects and how they are affected; (3) his or her historical behaviors, that is, experiments relying on memory length which obviously can be innovated by learning. The corresponding decision process is specified in Fig. 4.

Fig. 4.

Risk decision process

Firure 4 describe the risk decision process, it is seen from Fig. 4 that the risk process is more complex. It is incorrect to try to draw a conclusion about this system according to classic tools, too. In this paper, a non-structural decision method, a computational experiment, is introduced to try to discover the evolution law about this complex adaptive system. The processes are listed as follows.

(1) Construct the dynamics equation of individuals’ behavior consisting of certain strategies with risk decision factors $x$, risk attitude $r$, and time $t$. A linear equation is always considered to make this research feasible and reasonable.

$$\begin{array}{cc}\hfill X{(t)}_{ħ(t)\times 1}=& A_{ħ(t)\times 1}^0+A_{ħ(t)\times \ell (t)}^{1}X{(t-1)}_{ħ(t)\times 1}+A_{ħ(t)\times \ell (t)}^{2}X{(t-2)}_{ħ(t)\times 1}\hfill \\ \hfill & +\dots +A_{ħ(t)\times \ell (t)}^{m}X{(t-\varpi )}_{ħ(t)\times 1}+\omega {(t)}_{ħ(t)\times 1}\hfill \\ \hfill \end{array}$$ 9

where, the rank of the matrix $A$ is a variable with time $t$, $m$ is the memory length. Obviously, this equation describes the dynamic behavior of an individual as a hidden exponential process if the time scale is not too large. Then the system function could be defined as

$$\begin{array}{c}Y{(t)}_{\wp \times 1}=C_{\wp (t)\times 1}+D_{\wp (t)\times ħ(t)}X{(t)}_{ħ(t)\times 1}+\vartheta {(t)}_{\wp (t)\times 1}\end{array}$$ 10

Suppose there are three categories of risk attitudes: risk preference, risk-neutral, and risk aversion, which are labeled $i,i=1,2,3$ respectively. Each would be activated at time $t$ with probability $p_{\ast }^{(i)}$, one insists that the risk attitude is deterministic, however, $p_{\ast }^{(i)}$ is always changed randomly, if

$$\begin{array}{cc}d{p}_1^{(i)}(t)=\hfill & f(p_1^{(i)}(t-1),p_1^{(i)}(t-2),...,p_1^{(i)}(t-m+1),\\ \hfill & p_2^{(\ast )}(t-1),p_3^{(\ast )}(t-1),...,p_l^{(\ast )}(t-1))dt+w(t)dz\end{array}$$ 11

hold, one thinks individuals are intelligent, social, and adaptive. Where $m$ is memory length, $l$ is a game radius (all individuals are labeled by a unique number), $w(t)$ is the noise, $\mathit{dz}$ is a Brownian movement.

In fact, risk attitudes are decided by several more important factors, such as the system properties and affective and cognitive parameters of decision-makers. System’s properties, uncertainty, complexity and stability, are most important to risk decision, which is decided on the term of $w(t)dz$. For example, people are always keen on selecting risk preference in stock market, or risk-neutral in product- produce of daily necessities, or risk aversion in catering industry under COVID-19. Furthermore, individual’s affective and cognitive driven by his historic knowledge and experience also affect the risk attitude. More abundant knowledge and experience one has, less degree of risk he would take; vice versa. In fact, one’s knowledge can be determined by what he or she learned from school and social, can be defined as his or her memory length; experiments comes from the interact with other individuals, which can be coupled with game radius. These two can be decided absolutely by parameter $m$ and $l$ of Eq. (11) and several properties of his nature of learning ability and adaptive ability that introduced in Eq. (8).

(2) Game radius and memory length of the arbitrary individual are random with time; furthermore, they are self-adaptive processes as shown in the following equation

$$p(l(t),m(t))=\alpha p(\pi (t-1),l(t-1),m(t-1))+\beta p(\pi (t-1),l(t-1))+v(t)$$ 12

where, $\pi$ is the individual payoff, $\alpha$ represents learning ability, $\beta$ is the adaptive ability. Similarly, $\alpha$ is also be looked at as the copying ability that one person copies the other’s strategy at time $t$, mutation ability that one changes the strategy of the opposite one at the transitory, cross transformation ability that one synthesizes different strategies coming from different neighbors (Björn et al. 2021). So, one can easily see that an individual’s learning ability is linked to adjusting her game radius and memory length by relying on Eq. (12). Adaptive ability $\beta$ means that individual must consider strategies the other ones within game radius, the property of system, and the environment, and must adjust the strategy adaptively. All individuals can get system noise $\widehat{w}(t)$ and his/her local noise $\tilde{w}(t)$, then produce the system noise $w(t)$ with a mapping $w(t)=\varrho (\widehat{w}(t),\tilde{w}(t))$.

(3) Each collective’s behavior is more complex than an individual one so that some order parameters are introduced to describe them: (1) the distribution character of a strategy due to an individual’s strategy and selected risk attitude, (2) collective payoff and its distribution, and the evolutionary law of these distributions and (3) the distribution of collective strategies coupled with an individual’s strategies, and the evolutionary law of these distributions. In fact, how to describe the collective behavior is so important that we take great effort to analyze them as below.

So, the process of making-decision is more complex. Furthermore, according to Eqs. (9)–(12), risk attitude is much important in this process. Unfortunately, risk attitude is totally random time-varied, as described in Eqs. (11)–(12), and considering the time-varying randomness of parameters of $\pi ,l,m$, and $\alpha$, $\beta$, $p$, coupled with the negative–positive feedback described as Eqs. (9)–(10), which makes the decision is hardly to make. Then, how risk attitude works in this process is what we should focus on, which will be discussed in following contend.

Individual risk decision behavior

Because the factors of risk decisions are multi-dimensional, multi-sourced, and time-varied, an individual finds it difficult to decide in order to ensure maximum payment (Castillo et al., 2021). Because arbitrary individual behavior relies not only on how many sources she or he has but also on what attitude she or he deals with risk. Furthermore, irrationality is one of the sources of risk attitude, which means that people generally do not select Pareto optimal solutions. Suppose we have completely rational people who always select a theoretical optimal strategy, rational people who select a theoretical optimal strategy with a certain probability, and completely irrational people who never select theoretic Pareto optimal strategy (Torres-Ruiz et al., 2018; Wang et al., 2019). The percentage of these three kinds of people is $w_1,w_2$ and $w_3$ respectively, with ${\sum }_{i=1}^3{w}_i=1$. Not only an individual adjusts the property of the game radius and the memory length, such that these corresponding parameter configurations vary with time, but also, she or he would shift their nature from being completely rational, rational, or incompletely rational people.

In the interactive process, we focus on two important facts: how to select the right player for the interaction and how to absorb others’ knowledge and improve and propagate it. The former consists of three parts: one selects a right individual to interact with, as expressed by Eqs. (2) and (3); an arbitrary individual abandons the player because he/she insists that the abandoned one perhaps provides a smaller payment than the other partners by searching all in his or her game radius, as described in Eq. (4); certain individuals could enter or quit this system by deriving from the system’s bounded fragmentation, as defined in Eqs. (5) and (6). The latter shows that an individual can learn average information directly and selectively. In a word, these behaviors do not only rely on the category and property of individuals but also rely on the nature of an individual and the properties of the local environment.

Set $p(r_{+})$, $p(r)$ and $p(r_{-})$ are the probability of risk attitude of risk preference $r_{+}$, risk-neutral $r$, and risk aversion $r_{-}$ respectively, $\sum p(r_{\bullet })=1$. The set individual selects $r_{+}$ coupled with a utility function $U(t)$; one insists that the selection is successful if $U(t)>U(t-1)$ otherwise, it fails. So,

$$\left\{\begin{array}{cc}p(r_{+}(t))=(1-\alpha )p(r_{+}(t-1))+\alpha ,\hfill & U(t)>U(t-1)\\ p(r_{+}(t))=(1-\alpha )p(r_{+}(t-1)),\hfill & U(t)<U(t-1)\end{array}\right)$$ 13

Similar, the probabilities of other selections should be

$$\left\{\begin{array}{cc}p(r(t))=(1-\alpha )p(r(t-1)),\hfill & U(t)\ge U(t-1)\\ p(r(t))=(1-\alpha )p(r(t-1)),\hfill & U(t)<U(t-1)\end{array}\right)$$ 14

$$\left\{\begin{array}{cc}p(r_{-}(t))=(1-\alpha )p(r_{-}(t-1)),\hfill & U(t)\ge U(t-1)\\ p(r_{-}(t))=(1-\alpha )p(r_{-}(t-1))+\alpha ,\hfill & U(t)<U(t-1)\end{array}\right)$$ 15

respectively. Equations (13), (14), and (15) describe how an individual decides what category of risk attitude she or he would be in and how to change their attitude in the interaction process between individuals. An individual could judge whose behavior could be learned and who could not and then adjust their risk attitude. The utility coupled with risk attitude should be written as

$$U_{r_{+}}=U(Y(1-\alpha ))+bR(1-{\mu }_{+})$$ 16

where, $Y$ is the income, $R(1-{\mu }_{+})$ is the payoff deriving from risk preference, and $1-{\mu }_{+}$ means the percent of the collective with risk preference. Similarly, the utility of risk-neutral can be written as

$$U_r=U(Y(1-\alpha ))+U\left(1-\frac{\left|\alpha -〈\alpha 〉\right|}{\alpha }\right)$$ 17

where, $〈\alpha 〉$ is the average learning probability of collective. The utility of risk aversion can be written as

$$U_{r_{-}}=U(Y(1-\alpha ))-bR(1-{\mu }_{-})$$ 18

where, $R(1-{\mu }_{-})$ is the payoff deriving from risk aversion. These behaviors defined by Eqs. (13)–(15) and (16)–(17) make an individual send their risk attitude to another linked directly or indirectly within their game radius, which affects them by learning.

Combing Eqs. (9)–(18), it is easy to see that the risk-decision of arbitrary Agent not only relies on his historic payoff coupled with corresponding strategies and the other Agents’ strategies within his/her game radius, but also relies on system properties with random time-varied and affective and cognitive to the system and environment that determined by Eqs. (11) and (12). Because most parameter and all variables are random with time-varied, classic method cannot deal with the corresponding complexity. However, we can constructed a corresponding Multi-Agent model to describe all agents complex behavior as well as the state of system and environment according to the steps and methods mentioned in Sect. 3, in simulation platform such as Repast, NetLogo, and so on.

Description of emerging collective behavior

Several directed order parameters, such as dynamic population number and size, dynamic collective utility, and system payoff, are introduced to describe collective behavior with different risk attitudes in the stochastic evolution process (Kranton et al., 2020; Xu et al., 2021).

In addition, the change frequency of individual strategies could be introduced to describe the stability of collective behavior, which reflects the volatility of collective behavior. Similarly, the change frequency of game radius and memory length describes the volatility of information coming from local configurations. The volatility and stability of collective behavior describe the collective uncertainty in turn.

Most importantly, giant connected components of homogenous individuals with common property could describe, in a spatial dimension, not only how many individuals select the same behavior and how they interact along a certain path, but also how different categories of individuals are distributed and whether certain behavior could clearly dominate if the corresponding giant component is its critical point. It is a more important order parameter of collect risk attitude.

As far as the risk decision is considered, the following order parameters of collective behavior should be considered: different risk attitudes’ probability distribution as well as the corresponding evolution process and the stochastic process of the population with the game radius of $1,2,\cdots$. Also important is memory length and the distribution of each category individual.

These order parameters describe collective behavior from homogenous individuals’ structure and strategies’ structure to graph structure of interaction between individuals respectively, from temporal to spatial, from static to dynamical, from local to global such that collective behavior is fully explicit.

Computation experiment

All possible scenarios should be considered because a system’s behavior is affected by environment and physical experiments. However, a more profound conclusion can be arrived at if the following questions are resolved: (1) Are effect factors fixed? What is the optimal behavior under certain conditions if the answer is yes? If the answer is no, what is the evolutionary law? (2) Is the recognition of risk fixed? If the answer is yes, what is the distribution of the risk attitude distribution? If the risk attitude is changed, how is the rule changed if the conditions are changed? (3) How is individual risk attitude affected by a collective state? Here, risk attitude is included. (4) How is risk attitude propagated in the collective? Here, preferential attachment, the ability to copy, crossover, variation, and risk attitude learning are included. (5) What conditions could lead to system optimization if the environment is changed?

To answer these questions, we designed scenarios coupled with corresponding environment parameters. Then we undertook a computation experiment in the Multi-Agent model repetitively for a certain scenario. We observed the data of order parameters and drew a conclusion from this scenario. The conclusion is given if and only if all scenarios are analyzed.

Emergent complexity of a collective due to individual interactivity

Interactivity and conflict between individuals

Individual behavior in society is both flexible and random, as is the interactive relationship between agents. In this sense, to describe its properties, one must assume that people can interact with others under a series of strictly limited conditions, as mentioned by Zheng (2014). Individuals in each Local-World have similar properties and are called homogenous agents; otherwise, they are called in-homogenous ones. Furthermore, the Super-Agent is defined as a collective of agents under certain strict interactive rules. Although the interactivities between agents are more complex, they can be simplified in two forms: cooperative game and non-cooperative game, as shown in Fig. 5.

Fig. 5.

Interaction between agents

Figure 5 indicates that there is a virtual agent, called Observer-Agent, who can observe the interaction between physical agents. Observer-Agent can stand in the Local-World or out of the Local-World, the property of the interaction between these agents he observes, that is, a non-cooperative game or a cooperative game, is decided by its location. $C$ represents the cooperative adaptive game and $N$ represents non-cooperative one. As far as the latter is concerned, the Observer-Agent would find each physical agent who takes part in this game keen to pursue to maximize the payment. However, as far as the former is concerned, she or he would state that each physical agent requests that his payment be distributed rationally among the coalition after he cooperates with others to maximize the implementation of the collective payment. In this example, the social, economic organization has a dissipative structure, i.e., this structure is dynamic when a large time-scale is invoked. This is not only reflected by its size-changing dynamically over time but also by the player selected both dynamically and adaptively.

According to the basic theory of a complex system, the conflict is defined as follows: there is much obvious conflict between in-homogenous agents in different Local-Worlds. However, as far as the conflict between homogenous agents in the same Local-World is considered, this is less obvious. It can be considered a non-cooperative game. Furthermore, whether the payment distributed to each agent is impartial or not, after they seek maximum payoff for the coalition by cooperating together, which could lead to conflict. No matter what game is considered, the interactive configuration is inevitable. Although these interactivities change randomly with time on a micro time scale, there always exists a relatively stable or gradient structure of collectives in the system on the macro time scale. Furthermore, if we focus on the multilevel system, the properties of collectives are more complex, and so are the conflicts (Bagrow et al., 2011; Sikder, 2020; Xu et al. 2021). Suppose that the objective function of agent $j_i$, and the $i^{\text{th}}$ agent in Local-World $j$, are denoted by $g^{j_i}(t,x,u)$, then the conflict could be mathematically described as:

$$\underset{u_{j_i}}{max}\sum _{j_i=1_i}^{n_i}{g}^{j_i}(t,x_{j_i},u_{j_i})$$ 19

$$\sum _{j_i=1_i}^{n_i}{v}^{(t)j_i}(t,x_{N_i}^{t\ast })=W^{(t)N_i}(t,x_{N_i}^{t\ast }),v^{(t)j_i}(t,x_{N_i}^{t\ast })\ge W^{(t)j_i}(t,x_{N_i}^{t\ast })$$ 20

where $j\ne j^{\prime },i=i^{\prime }$, $N_i$ is the number of agents in the sub-system $i$, $v^{(t)j_i}$ is the agent’s payment $j_i$ at time $t$, $W^{(t)N_i}$ is the payment of sub-system $i$, i.e., $\underset{u_{j_i}}{max}{\sum }_{j_i=1_i}^{n_i}{g}^{j_i}(t,x_{j_i},u_{j_i})$. If and only if (19)–(20) hold, can we say that the cooperative game would happen in the same sub-system. Furthermore, formula (20), which reflects the conflict between individual rationality and collective rationality, describes the essence of this conflict. On the other hand, if $i\ne i^{\prime },j\ne j^{\prime }$ is the interaction between these two agents, according to the hypothesis mentioned above called a non-cooperative game, it satisfies

$$\underset{u}{max}\left\{g^{j_i}[t,x_{j_i}(t),u_{j_i}(t)],g^{j_{i^{{}^{\prime \prime }}}}[t,x_{j_{i^{{}^{\prime \prime }}}}(t),u_{j_{i^{{}^{\prime \prime }}}}(t)]\right\},j\ne j^{\prime },i\ne i^{\prime }$$ 21

The conflict defined here reflects that agents fight for limited resources. If the time-scale is large, the conflicts between agents can be described as a mechanism of preferential attachment, consisting of two parts: select a right agent to play in a cooperative/non-cooperative game by maximizing connect probability, i.e., considering Eqs. (2)–(6), then determine what property and how important the interaction is by considering its local configuration, i.e., Eqs. (19)–(21).

Emergence of the process of collective conflict

The bottom-up modeling method and exploratory computational experiment analysis are introduced to describe the conflict by constructing a model from the agent’s behavior to systemic behavior due to the complexity of its emergence. In fact, much complex collective behavior happens due to the agent’s behavior—whether random, intelligent, social, or autonomous—even if the simplest individual’s behavior is considered in light of the fixed interactive rule (Xu 2019). To make the exploratory computational experiment model more accurate, a rational way would be to compare the experiment results with the physical state. If they hold with identical probability distribution, one could regard the model as right. If not, one would need to revise an agent’s behavior equation and the interactive equation so that the model is revised to match the property of the physical system, as described above. However, this comparison is affected if and only if the order parameters of collective behavior are defined.

Seen from a macroscopic scale, the proportion of homogenous strategy and the stability of the maximum proportion are all used to describe the collective conflict. Suppose that the probability of strategy $i$ at the moment $t$ is $P(i(t)),i\in \{1,2,...,m\},{\sum }_{i=1}^m{P}_i(t)=1$, where $m$ is the number of strategies, then the state and property of $P(i(t))$ can describe the size and stability of this strategy. Similarly, strategy distribution, individual distribution with homogenous strategy, the sensitivity of certain controls corresponding to a certain strategy, and the corresponding evolutionary law could all describe the property of collective conflict (Bai et al., 2020).

Diverse, random, intelligent, and social character makes individual behavior difficult to assess. So, let us first describe individual behavior. In this context, how one sets the parameter of corresponding Eqs. (1)–(10), (19)–(21) and how to solve them are vital to analysis. As already mentioned, several in-homogenous collective behaviors would emerge from the interactions (Lee et al., 2019). Because these behaviors are all multi-dimensional and incompatible, there must be some degree of conflict between these collectives, making collective behaviors more complex. Note that the payment of Eqs. (2)–(6) could degenerate to a degree (i.e., in-degree and out-degree or in-strength and out-strength) under certain conditions, so, the model could degenerate to classic research of collective behavior in complex networks supported by many research results (Giurge et al., 2020; Hori et al., 2013).

Suppose that there are 1,2,…, m dimensions of behaviors due to agents’ interaction, $n_1,n_2,...,n_m$ is the respective corresponding behavior number, then there exists ${\prod }_{i=1}^m{n}_i$ behaviors conflict together in the system. A computation experiment could obtain the property and evolutionary law of the behavior. As a result, one must focus not only on what strategy could be selected by a certain collective at a certain time but also on the distribution of the strategies and agents. So, the evolutionary process and their distribution of giant components connected to homogenous behavior would be observed, which describes the conflict between collectives (Lee et al., 2019).

How collective behavior changes

Several problems in a research process are crucial to scientists and the corresponding complex adaptive system management: how the collective behavior emerges and what property is reflected if several effect factors change (Lee et al., 2019). Collective behavior (See Eqs. (6)–(10)) is decided by individual behavior (See Eqs. (1)–(5)), interactive rules (i.e., including how to select a fitness player, how to cross copy and cross learn (See Eq. 5) and how to transfer corresponding knowledge to ones linked directly). Furthermore, individual behaviors are diverse, heterostructure, and time-varying (Agredo-Delgado et al., 2021; Lucas & Nordgren, 2020). So, the precise and profound law of collective behavior is much more difficult. In the experiment, all possible properties of the environment are analyzed as far as the same group of parameters is considered, and the parameters coupled with collective behavior are also identified according to Sect. 3. Then almost all different states corresponding to different systemic environments which happened in the evolutionary process are analyzed.

A certain collective selects a form of rational behavior to obtain more payment than any other behavior regarding collective rationality (Yuan et al., 2022). Suppose that several behaviors emerge in a system, then one insists that the collective selects one with a certain probability from these rational behaviors but not necessarily a fixed one. Of course, the rational selection process should comply with the preferential attachment rule in line with the order parameters of collective behavior. Due to the adaptive property of the system, the selected probability would change with the system state and environment property. In this sense, probability can be regarded as a stochastic process, and it could determine the rule of collective rationality selection. Except for preferential attachment, another mechanism, the learning and propagation process, is also important to determine the property of collective rationality and corresponding stability. It is a stochastic process for the same reason. Due to the difficulties linked to this kind of analysis, a more scientific inductive method should be introduced to summarize all computational experiment results.

Collective behavior could shift to an unexpected one under certain conditions, which could be implemented if precise computation experiments have been done. Because an arbitrary exploratory computational experiment is a Bernoulli trial, the result either conforms with certain invariable distribution with others or is singular, the latter is called phase transition of the complex adaptive system, and the corresponding conditions are more important to us. Other order parameters of collective behavior could analyze the change of collective behavior. Furthermore, the interaction between individuals makes a complex system close continuously to a critical point. In this state, collective behavior perhaps changes to the totally opposite behavior, even if a few individuals engage inconsistent behavior. Note that if the system is critical to the point of system symmetry breaking, then one would insist that collective rationality in the system should not exist (Shaukat et al., 2021). In fact, collective rationality is full of vulnerable factors if the system is critical, i.e., individual irrational behavior could lead to out-of-control collective rationality. As a result, collective rationality is very vulnerable and unstable so that it could be transferred to another transitory value, which makes the system reach a most innovative stage. However, it is crucial that the system always converges to this criticality, that is, a constant state. So, what is the critical state to be analyzed? To accomplish this, those vital agents, whose irrational behavior could lead to a system’s state phase transition and who are in the sets respective to the system’s kernel and bridge in topological configuration, should be focused on. To do this, collective rationality could be verified correctly by using Bernoulli s tests of the computational experiment.

Examples

Traffic system

Let us consider a classic model of traffic grid, there are number of cars run at the road, the drivers adjust the speed and race of his car according to his local traffic information and the state of the adjacent cars. This system enables the control of traffic lights and global variables, including the speed limit and the number of cars, and helps explore traffic dynamics. At each stage, cars attempt to move forward at their current speed. If their current speed is less than the speed limit and there is no car directly in front of them, they accelerate. If there is a slower car in front of them, they match the speed of the slower car and decelerate. Finally, if there is a red light or a stopped car in front of them, they stop.

Let’s consider the individual’s behavior, i.e., the speed and direction of the car, the driver always adjust the speed and direction of the car according to the traffic property. In this sense, the goals are both maximum speed and minimum loss, to implement them, the corresponding dynamic model should be

$$\left\{\begin{array}{c}{x}_i(t+\tau )=x_i(t)+\tau v_i(t)\\ {\varphi }_i=arg\left(\sum _{\{j\}}{e}^{i\theta }\right){\theta }_i(t+\tau )={\varphi }_i(t)+{\xi }_i(t)\\ \end{array}\right)$$ 22

In Eq. (22), $x_i(t)$ is the position of car $i$ at time $t$, $v_i(t)$ is the speed, $\tau$ is the response time. ${\theta }_i(t+\tau )\in [-\pi /2,\pi /2]$ is the angle of car $i$ at time $t+\tau$, which means which carriageways car $i$ will switch at time $t$. ${\varphi }_i$ is the mean value of angles of other cars within its game radius, which means that what arbitrary car steers is determined by the local information of state of other cars within its game radius. Furthermore, ${\varphi }_i$ updates according to the second formula of Eq. (22). ${\xi }_i$ is the behavior error. Obviously, Eq. (22) is the simplified formation of Eqs. (2)–(6), the interactive rule between individuals are much easier.

In classic simulation research, one could consider a certain scenario, set fitness parameters, run the model, observe how the traffic system changes, and then adjust corresponding controllable parameters to make the traffic system more rational. We randomly selected 200 cars in classic research, and the speed is limited to 100%. We also select ticks of per-cycles 12, 38, and 50, respectively. Where, ticks of per-cycles is the number of ticks that will elapse for each cycle, i.e., the average times of traffic lights change in each cycle. The results are as Fig. 6.where, the x-axis is the time step, second, the y-axis of (a) is the average speed of the car, the y-axis of (b) is number of stopped car in system, the axes of Figs. 7, 89. are the same as Fig. 6. One can see that the traffic system is more disorderly as ticks per-cycle increase. If ticks per-cycle equal 50, then the average speed of cars decreases quickly to 0, which leads to out-of-control traffic. Because the traffic system can be divided into an ordered state and a disordered state, several questions can be raised: how the traffic system changes within the parameters of continuity and change? Should there be a certain state, i.e., the criticality of a phased transition in the traffic system, such that the traffic system changes sharply and without warning? If so, what are the controllable and key factors, and what are the uncontrollable and key ones? To answer these questions, we utilize an exploratory computation experiment.

Fig. 6.

Dynamical effect from ticks to the traffic system

Fig. 7.

Dynamic effect from speed limited to the traffic system

Fig. 8.

Dynamic effect from the number of cars in the traffic system

Fig. 9.

The dynamic effect from ticks to traffic system at a critical state of traffic

Without a loss of generality, we select a parameter randomly to do computational experiments. We then know what the property of the system is by observing the experiment results in different scenarios and by inducing corresponding results. For example, suppose that ticks per-cycle 50 of the pseudo critical point is selected, the set speed limits are 100%, 80%, 70% and 60% respectively, ones could then observe how the speed affects the traffic.

Figure 7 indicates that a speed limit does not obviously impact a traffic system. Consequently, the speed limit is not a key factor in the traffic system. Then consider how the number of cars affects the traffic system. We do a computation experiment by setting numbers of the car are equal to 200, 150, and 100 respectively and setting the speed limit at 60%. After the experiment, one finds that the traffic system is ordered if the number of cars equals 150 and 100, but there is a disorder if there are 200 cars. So, a critical point should exist where the number of cars changes the traffic system from chaos to order. So, we set the number of cars at 180 and find the system is not stable. The experiment results are shown as Fig. 8.

From Fig. 8, one notes an interesting phenomenon by observing the experiment: in some scenarios, the traffic system operates smoothly. It fluctuates rhythmically if the car numbers are 100, 150, 175, 176, and 178, respectively. But it is out of control, i.e., it fluctuates non-rhythmically, if many cars stop on some roads and there are few cars on other roads when cars number between 180 and 200. We can hypothesize that there exists a critical point where the traffic system suddenly changes from order to disorder. When we slowly decrease the parameter by setting it at 178, 176, and 175, we find that the critical number of cars would be around 176–178. As a result, we discover the critical number of cars in an exploratory computational experiment.

Then an interesting question emerges: what property of the traffic system would there be if the traffic system is at a critical state, i.e., set the number of cars at 177 and the speed limit at 60%, and the ticks per-cycle change slowly. To draw a conclusion, we set the ticks per cycle at 25, 30, 40, and 50, respectively. Amazingly, we find that the ticks per cycle have a very limited impact on the traffic system. The experiment’s results are shown in Fig. 9.

According to the results of the traffic system exploratory computational experiment, the number of cars should be set to an appropriate scale to make the traffic system more scientifically valid. In addition, appropriate ticks in per-cycle and limit speed of cars should also be considered. More important, the environment, consisting of number of cars in system, ticks of per-cycles and limit speed of cars, have a strong impact on the state of the traffic system. Furthermore, this example shows not only that the complex adaptive system of traffic system is more random, diverse, dynamical and critical, but also that the system can be affected easily by environment if the system is at critical state.

Evolution process of the giant component in scale-free complex networks

As mentioned above, a connected giant component is always used to describe collective behavior. According to the interaction rule between individuals, a giant component connected with homogenous individuals could be produced according to the principle of birds of a feather flocking together, which reflects there are much number of individuals in the system. In fact, the numbers of people who believe an event is true would grow sharply if and only if the rank of this giant component connected is greater than or equal to a certain value, especially in the scale-free complex networks. The corresponding property is shown in Fig. 10

Fig. 10.

The evolutionary process of belief propagation in scale-free complex networks

Figure 10 shows the dynamic process of giant component evolution, where (a)–(c) refers to the intuitional topological structure of giant components respectively at 0.50, 1.00- and 2.00-time unit (per time unit is equal to 100 steps). In this process, one can understand how the giant components develop. (d) describes the relationship between fractions in the giant component and the connection of each individual, where the abscissa axis is the connection of each individual and the ordinate axis is the fraction of the individual in the giant component. Intuitively, a certain belief is slowly accepted at the outset. After a relatively long time, several individuals accept this belief. Once these individuals form a cycle (we call it an autocatalytic set), a real giant component emerges, and there is some information on belief feedback flows in this cycle. A system close to its criticality is shown as the vertical line in Fig. 10d. Then the number of people who believe that the issue is correct would grow sharply. In this way, we find the critical minimum effort level, no matter whether we would promote this opinion to all people in the system or prevent it from being propagated in the system.

To learn the property of the criticality and the phase transition process in detail, we focus on the state and corresponding development of the configuration of giant components in a small time scale. Considering the property of phase transition of belief system, we define a critical point to describe the great change. Due to the system’s complexity, the critical point is such a random number that scientists need to give an infimum and a supremum to constrain its range. So, we focus on the time-scale from infimum to the supremum of the critical point and observe the state and corresponding evolution process, which means that the details would need to be enlarged by using a microscope. We do exploratory computational experiments at a number of fixed parameters, observe the states and the evolutional process of giant components in each experiment, then obtain the results.

First, we do six exploratory computational experiments, and the results are shown in Fig. 11.

Fig. 11.

The evolutional process at criticality without singulars

The abscissa axis is time (logarithmic scale), and the ordinate axis is the fractions of the giant component (logarithmic scale). It is easy to see from Fig. 10 that (1) the fraction of the giant component changes continuously and smoothly, and (2) the figure presents the data from the simulation of the stochastic process, where the randomness comes from the stochastic time-varying property of belief system coupled with Eq. (2) and Condition A3). The stochastic process would converge into a stable attractor, (3) satisfying a certain statistical distribution law of $F=-11.32t^{-0.073}+11.56$, where $F$ is the fraction of giant component, and $t$ is time. Note that there is quite a big jump in the orange dots, which means that the belief system is at the criticality point such that the giant component of the belief system grows sharply. Whether the “big jumps” always occur or not should be what we focus on, which will be analyzed below. Because of randomness and non-equilibrium, there perhaps exists some singulars at the criticality in the process of belief system evolution. To prove it, we select another fixed-parameter closing to the orange dots and do four exploratory computational experiments repeatedly; the experiment results are shown in Fig. 12.

Fig. 12.

The evolutional process at criticality with singulars

We can see from Fig. 12 that there exists a singular in the experiment, which reflects its instability. After repeating this experiment several times, we identify several different dynamics to describe the random diversity of the dynamics if the belief system is critical. Then a question is raised: when should these dynamic equations be selected, and what are the corresponding selection conditions? Furthermore, we find that there are very significant differences between these dynamics equations, and there is not an identical statistical distribution, which reflects both that all possible events would happen and that there exists no stable dynamics if the belief system is in the criticality. In fact, there exists no “right” dynamics in this state, and we can select one randomly among those available to describe how a belief system evolves, as shown in Fig. 13, which means that the belief system is complex. Generally, the right intuition is not always correct, so how to judge the right dynamic is more important. This will be specified in the next paper.

Fig. 13.

Random diversity dynamics

In Fig. 13, the abscissa axis is time, and the ordinate axis is $X$ of Eq. (2). As seen in Fig. 13, the belief system could be bifurcated. It will develop if the pre-set trajectory has nothing to do with it, which forms an “S” curve. However, if a strong effect control is implemented, the belief system would turn to the opposite, as shown by the green line with “·–·–” in Fig. 13. If a weak effect control is added, the belief system always fluctuates around a certain state, seen as the “····” line in Fig. 13. Furthermore, we can see that the bifurcation happens at the critical point and also that the belief system would be changed randomly to an arbitrary evolution process among the three categories in Fig. 13. Consequently, there are no certain dynamics if the belief system is in criticality.

Conclusion

The complexity of collective behavior is not dependent on the complexity of individual behavior but dependent on a system’s structure, as well as the interactive configuration and the property of external disturbance. Because of analysis complexity, exploratory research of computation experiments was constructed to master the property and evolutionary law of the complex adaptive system. The personalized model coupled with certain problems could be studied in future research.

In fact, even with the simplest behavior of an individual, a system could emerge, showing more complex behavior of a collective because of interaction. The environment can be defined as an agent. In this sense, the interaction between this agent and the physical agents in a system could be defined as negative feedback from the environment to the system and positive feedback from the environment to the system. This is an important innovation of a computational experiment. A computational experiment is an efficient tool to deal with the emergence of a complex adaptive system with large enough individuals. In this process, scientists construct a corresponding behavior equation to describe different agents’ behavior due to his/her property. Then one revises the simulation model before the simulation results are subject to the same statistical distribution as the physical data. Then, one conducts the experiment under arbitrary possible scenes and obtains certain results. We arrive at the evolutionary law through induction if these results are subject to the same statistical distribution. If not, we must find that the results are subject to the same statistical distribution. In this sense, the conditions are important to discover scientifically if one can find a universal law. This is an important task in a computational experiment. The system is divided into several sub-systems due to the interactive properties between agents and the property of the environment and the collectives due to the system’s emergence. Collective behavior is decided by factors such as individual behavior, the environment, and several parameters of game radius and member length. Furthermore, collective behaviors are also described, including population number and size, change frequency of individual strategies, dynamic change of game radius and memory length, and configuration of the giant component.

Whether the collective behavior selected is rational or not is difficult to judge. One could judge whether the system is in equilibrium or not. If the system is in equilibrium, the emergence of collective behavior must be the optimal behavior. If the system is in non-equilibrium, collective behavior is critical, and we cannot find a stable optimal behavior. Finally, whether the system is at equilibrium or not is also judged by computational experiment.

Economical and management systems are general complex adaptive system, where environment controlling the state and properties of the system is affect by the system, which cannot be resolved by methods coming from reductionism both because the properties of randomness, diversity, time-varying and criticality and because the self-organizing, emerging, dynamic and evolving system of these systems. Exploration computational experiment is an unstructured analysis method to analyze the property beyond reductionism. So, this method would be an important and universal one to resolve the complex system that would be focused on in twenty-first century. Furthermore, this analysis method can be used in other sciences, such as physics, life science, chemistry, biology, social science, biology, and so on. It has wide application perspective.

Data availability statements

The author confirms that all data generated or analysed during this study are included in this published article.

Declarations

Conflict of interest

The author declares that they have no competing interests.