Regret Psychology-Driven Information Propagation and Behavioral Adoption in Complex Social Networks

Abstract

In recent years, information propagation on social networks has attracted extensive attention, with psychological characteristics of individuals exerting a significant influence on the diffusion process. Our study investigates the role of regret psychology and its impact on information spreading and behavioral adoption. We categorize individuals into regretful and non-regretful groups and introduce regret intensity together with the proportion of regretful individuals as dynamic variables. Based on this, we construct a two-layer interactive model consisting of a psychological layer and a behavioral layer. Then we establish the behavioral adoption model for the heterogeneous population and study the propagation characteristics of the regretful individuals on social networks. Furthermore, we derive the propagation dynamics using edge-based compartmental theory to examine the transmission mechanism. Numerical simulations, which coincide nicely with our theoretical analyses, reveal the crossover phenomena in phase transitions: as the regret threshold increases, adoption dynamics shift from second-order continuous to first-order discontinuous transitions. More importantly, for a given propagation probability, there exists an optimal regret threshold that maximizes the final adoption size. These findings highlight the crucial role of regret psychology in reshaping the propagation mechanism and provide a new theoretical perspective for understanding symmetry transformations and group heterogeneity in social contagion dynamics.

1. Introduction

Social networks, as a key infrastructure of the information era, have reshaped patterns of human social interaction through complex connections among nodes [1,2,3]. The network structure not only accelerates the flow of information but also profoundly influences the formation and diffusion of group behaviors [4,5,6]. Information propagation and behavioral adoption are mutually reinforcing processes. Individuals’ adoption of new behaviors, such as technological innovation, green consumption, or environmental protection concepts, relies heavily on effective information transmission within social networks [7,8,9]. With the rapid development of Internet technology and online social platforms, information dissemination has become a major mechanism shaping public attitudes and behavioral choices [10,11]. The diversification of communication channels, including news applications, social media, and short video platforms, has greatly amplified the social contagion effect [12,13]. Factors such as information exposure frequency, the reputation of the information source, and individuals’ initial attitudes jointly determine the breadth of information diffusion and the depth of behavioral adoption [14,15,16]. Prior studies have shown that individuals typically experience a dynamic “exposure–evaluation–reinforcement–adoption” chain, where the adoption process exhibits non-Markovian characteristics, driven by accumulated effects of historical exposure, information redundancy, and cognitive adaptation [17,18,19,20]. Therefore, psychological factors become a crucial variable in the dynamics of information propagation [21].

Human decision-making essentially results from the joint effects of cognition and emotion, among which regret psychology plays a particularly significant role [22,23,24]. Regret arises as a cognitive dissonance emotion triggered by counterfactual thinking when actual outcomes deviate from expectations. Investors may regret prematurely selling rising stocks, while consumers may question impulsive purchases, illustrating the persistent “post-decision ripple” of regret. More importantly, in the age of social media, such regret emotions can spread through social relations as a form of psychological contagion, thereby reshaping others’ risk preferences and behavioral choices. For instance, exposure to negative messages such as “adverse reactions after vaccination” may evoke regret and reduce adoption willingness, whereas the regret of unvaccinated individuals may promote protective behaviors among others. This bidirectional psychological feedback mechanism implies that regret can either suppress or facilitate behavioral diffusion. Previous studies have demonstrated that factors such as network topology [25,26], individual interaction tendencies [27], and information memory effects [28,29] play significant roles in determining the efficiency of information propagation. Threshold models, as a representative paradigm for describing non-Markovian diffusion, have been extensively employed to characterize the mechanisms of behavioral adoption in complex networks [30,31,32]. However, most existing models still regard psychological states as static parameters, making it difficult to accurately capture the dynamic evolution of regret emotions and their cross-layer transmission effects [33,34,35,36].

The dynamic evolution of regret psychology essentially reflects the transition from symmetry to asymmetry within social information propagation systems. When individuals’ psychological feedback tends to be consistent, the group spreading behavior remains symmetric and stable. However, as regret effects accumulate and amplify the differences in group responses, the system’s symmetry is broken, leading to abrupt changes in the propagation state. This dynamic symmetry–asymmetry transformation induced by psychological feedback provides an important clue for understanding the nonlinear mechanisms of social contagion. In addition, recent advances in active particle systems have provided powerful paradigms for describing collective behavioral dynamics [37,38,39,40]. These approaches, inspired by Herbert Simon’s theory of the Artificial World, model each agent as an active particle whose motion and decision rules are influenced by internal states and social interactions. This perspective offers a kinetic description of cognitive and emotional feedback within groups. Therefore, it is essential to develop a new class of information propagation models that integrate psychological feedback effects to reveal the intrinsic interplay between cognitive emotion and behavioral adoption in social networks. Such a model can uncover how psychological feedback triggers symmetry breaking and dynamic restoration in social contagion, elucidate the coupling relationship between the psychological and behavioral layers, and provide a new symmetry-based theoretical perspective for interpreting the nonlinear evolution of group behaviors in real social networks.

Inspired by the aforementioned social issues and based on the aforementioned research gaps, our paper focuses on the transmission characteristics of regretful individuals in social networks and the mechanism of their influence on behavior adoption. Our study innovatively takes regret intensity and the proportion of regretful individuals as dynamic variables, constructing a two-layer interactive model that includes the psychological layer and the behavioral layer. In the psychological layer, the accumulation and decay patterns of regret emotions are quantified, and in the behavioral layer, the critical conditions for triggering behavioral reversal by the regret threshold are defined. Firstly, we divide the population into regret and no-regret groups and establish regret/no-regret behavioral adoption threshold functions to illustrate the propagation characteristics of regret psychology. The regret threshold is proposed in the functions to account for the regret property. Then, we derive the dynamic propagation equations using edge compartmental theory based on the regret threshold to analyze the information propagation process. Finally, ER and SF networks are used to simulate the information propagation of heterogeneous populations.

The central structure of this paper is as follows. In Section 2, a social propagation model with regret/no-regret adoption threshold models is constructed. Section 3 proposes the edge compartmental theory based on the regret threshold. Section 4 compares the experimental results obtained from simulated experiments and theoretical derivations. Finally, some conclusions are drawn in Section 5.

2. Model Description

To explore the effect of regret behavior on the information propagation mechanism, we construct a social network model with N nodes, where each node represents a user and the edge between two nodes represents the closeness between users. The degree distribution of the network follows $P\left(k\right)$, where k is the degree of a node.

We utilize the SAR (Susceptible-Adopted-Recovered) model [14] to classify nodes on the network into susceptible, adopted, and recovered states, denoted as S-state, A-state, and R-state. The SAR model always is used to illustrate the information propagation mechanism in social networks.

Agent S (the S-state node): receives information from its neighboring node with a success probability $\lambda$. The cumulative successful received information pieces is denoted as m. Once an information is successfully received, the received information cumulative count m increases by 1, i.e., $m\to m+1$. An S-state node cannot adopt the behavior. Initially, there is no information propagation in the social network, i.e., $m=0$ for each node. Each edge can successfully transmit the same piece of information only once.

Agent A (the A-state node): adopts behavior and transmits information to all neighboring S nodes and then changes to the R-state with probability $\gamma$.

Agent R (the R-state nodes): neither receives nor transmits information.

Most nodes are the S-state at the beginning of information propagation. Only a small number of seed nodes are selected as A-state, meaning that they will transmit information to their neighbor nodes. When no A-state nodes are left in the social network, the information propagation process will have finished.

In order to describe the effect of regret behavior on information propagation, we divide the population into two groups: the no-regret population and the regret population. Then we propose two behavioral adoption threshold functions for the two group populations to characterize the propagation characteristic, as shown in Figure 1b. For the no-regret population, the behavioral adoption threshold function is calculated as follows:

$$h_n(x,\epsilon )=\left\{\begin{array}{c}sin{\displaystyle \frac{\pi x}{\epsilon }},0<x\le \epsilon ,\\ 0,\epsilon <x<1.\end{array}\right.$$

(1)

For the regret population, the behavioral adoption threshold function is calculated as follows:

$$h_r(x,\epsilon )=\left\{\begin{array}{c}sin{\displaystyle \frac{\pi x}{\epsilon }},0<x\le \epsilon ,\\ sin{\displaystyle \frac{\pi (x-\epsilon )}{2(1-\epsilon ),}}\epsilon <x<1,\end{array}\right.$$

(2)

where $\epsilon$ represents the regret threshold, x is denoted as the proportion of the aggregate received information pieces m by the S-state node to its degree k, i.e., $x={\displaystyle \frac{m}{k}}$. And $h_r(x,\epsilon )$ (or $h_n(x,\epsilon ))$ is the non-regret (or regret) individual behavioral adoption probability. The adoption function of the no-regret group, $h_n(x,\epsilon )$, shows a unimodal characteristic. It monotonically increases as a sine function when the information exposure ratio x is less than or equal to $\epsilon /2$, and then drops sharply to zero at the point $\epsilon$, reflecting the phenomenon of resistance to information saturation among rational decision makers. The regret group’s function, $h_r(x,\epsilon )$, has a two-stage feature, initially responding in the same way as the non-regret group, but when $x>\epsilon$, the adoption probability rises again, forming a “wait-and-see - follow-the-rally” behavior pattern driven by counterfactual thinking. In the discrete simulation process, since $m\in \{0,1,\dots ,k\}$, $h(x,\epsilon )$ is evaluated at $x={\displaystyle \frac{m}{k}}$ and updated at each integer step m, where k is the degree of node. Linear interpolation is used for intermediate values to guarantee a smooth transition between discrete states.

Figure 1. The information propagation model (a) and the regret/no-regret behavior adoption models (b). $\epsilon$ is the regret threshold. The adoption probability of the no-regret groups embodies an initially rising and subsequently declining line. The adoption probability of the regret groups represents a rising line again because of the regret property.

In real life, the no-regret group shows more consistent adoption willingness before and after the behavior. As the proportion of A-state neighbors around them increases, their interest gradually grows, but their interest also weakens when this proportion drops significantly. As shown in Figure 1, the adoption probability of the no-regret group first monotonically increases with the increase in x, and then decreases. The adoption willingness of the regret group is influenced by multiple factors and is prone to behavioral reversals due to sudden information interference. In the initial state, like the no-regret group, people’s interest in information behavior gradually increases, but as the proportion of A-state neighbors around them increases, their interest subsequently decreases. However, under the influence of regret psychology, this group undergoes a behavioral reversal and begins to regain interest in information and behavior. For example, in the real-life stock market, people buy stocks and make profits. Investors cash out when the stock price reaches the expected rate of return and refuse to take subsequent fluctuation risks. However, the regret group will let go once they suffer losses because the stock price drops, but if the stock price rebounds, they will regret it and reevaluate the market trend and re-enter the market. The adoption threshold functions (1) and (2) originate from the behavioral feedback mechanism of regret and are mathematically formulated to capture the sequential phases of “hesitation-reversal -re-adoption”. The sine function naturally represents smooth and periodic variations within the interval [0, 1], providing both a bounded domain and a nonlinear growth-decay pattern that mirrors the cyclical responses observed in real human behavior. Furthermore, its continuity embodies the recurrent cognitive effects induced by regret psychology and reflects the bounded rationality reported in psychological experiments. Therefore, the adoption threshold functions (1) and (2) are both theoretically justified and practically meaningful.

The information propagation process is summarized as follows: In the initial state, we randomly select a p proportion of the regret population and a q proportion of the no-regret population, where $p+q=1$. Then we randomly selected ${\rho }_0$ proportion of nodes as the A-state nodes (seeds). All other nodes default to S-state. The A-nodes transmit information to their neighbor nodes. The S-state node successful receives the information from its A-state neighbor node with probability $\lambda$. Upon successfully receipt of the information, the S-state node increases the accumulated information pieces by one (i.e., $m\to m+1$), and it will no longer accept information from the same A-state neighbor node due to non-redundant information propagation. Moreover, the S-state node with a regretful (no-regret) psychology will adopt the behavior and change to the A-state with the probability function $h_r(x,\epsilon )$ (or $h_n(x,\epsilon )$). The A-state nodes change to the R-state with a specific probability $\gamma$ after successfully transmitting information. The propagation ends when no A-state nodes are left in the social network.

3. Theoretical Analysis

We utilize theoretical analysis to explore the information propagation on social networks under the heterogeneous populations. Taking into account edge-based compartmental theory [41], we model the dynamic propagation process and analyze the transmission mechanism. It is assumed that node i is in the cavity state [42], which allows it to acquire information from neighboring nodes but prevents it from transmitting information to neighboring nodes.

Following the classical edge-based compartmental framework proposed by Wei Wang [14], we consider a social contagion process incorporating regret psychology. This baseline model provides the probabilistic foundation for describing information transmission across edges. To avoid redundancy, only key relationships and definitions are retained. The baseline model equations serve as the starting point for our psychological extension and are not repeated in full. In this study, we extend the above framework by incorporating a dynamic evolutionary mechanism of regret psychology into the threshold-based adoption process.

We define the probability that node i does not receive a message from a neighbor node before time t as $\theta \left(t\right)$. Therefore, by time t, the probability that node i receives m pieces of information from neighboring nodes is as follows:

$$\varphi (k_i,m,t)=\left(\begin{array}{c}{k}_i\\ m\end{array}\right)\theta {\left(t\right)}^{k_i-m}{\left[1-\theta \left(t\right)\right]}^m,$$

(3)

where $k_i$ is the degree of node i.

If S-state node i is prone to regret, it remains in a susceptible state with a probability ${\displaystyle \prod _{a=0}^m}[1-h_r({\displaystyle \frac{a}{k}},\epsilon \left)\right]$ after receiving m pieces of information. Thus, its probability of receiving m pieces of information and maintaining the susceptible state by time t is as follows:

$$\begin{array}{cc}\hfill s_r(k_i,m,t)& =\sum _{m=0}^{k_i}\varphi (k_i,m,t)\prod _{a=0}^m[1-h_r({\displaystyle \frac{a}{k_i}},\epsilon \left)\right]\hfill \\ \hfill & =\sum _{m=0}^{⌊\epsilon k_i⌋}\varphi (k_i,m,t)\prod _{a=0}^m(1-sin{\displaystyle \frac{\pi a}{\epsilon k_i}})\hfill \\ \hfill & +\sum _{m=⌈\epsilon k_i⌉}^{k_i}\varphi (k_i,m,t)\prod _{a=0}^{⌈\epsilon k_i⌉}(1-sin{\displaystyle \frac{\pi a}{\epsilon k_i}})\prod _{a=⌈\epsilon k_i⌉}^m(1-sin{\displaystyle \frac{\pi (\frac{a}{k_i}-\epsilon )}{2-2\epsilon }}).\hfill \end{array}$$

(4)

If S-state node i is a no-regret population, it remains in a susceptible state with a probability ${\displaystyle \prod _{a=0}^m}[1-h_n({\displaystyle \frac{a}{k}},\epsilon \left)\right]$ after receiving m pieces of information. Thus, its probability of receiving m pieces of information and maintaining the susceptible state by time t is as follows:

$$\begin{array}{cc}\hfill s_n(k_i,m,t)& =\sum _{m=0}^{k_i}\varphi (k_i,m,t)\prod _{a=0}^m[1-h_n({\displaystyle \frac{a}{k}},\epsilon \left)\right]\hfill \\ \hfill & =\sum _{m=0}^{⌊\epsilon k_i⌋}\varphi (k_i,m,t)\prod _{a=0}^m(1-sin{\displaystyle \frac{\pi a}{\epsilon k_i}})+\sum _{m=⌈\epsilon k_i⌉}^{k_i}\varphi (k_i,m,t)\prod _{a=0}^{⌈\epsilon k_i⌉}(1-sin{\displaystyle \frac{\pi a}{\epsilon k_i}}).\hfill \end{array}$$

(5)

Therefore, the probability of S-state node i receiving m pieces of information and maintaining the susceptible state before time t can be expressed as follows:

$$s(k_i,t)=(1-{\rho }_0)\left[p{s}_r(k_i,m,t)+q{s}_n(k_i,m,t)\right].$$

(6)

Considering the degree distribution of the social network, the proportion of susceptible state nodes in the social network at time t can be obtained as follows:

$$\begin{array}{cc}\hfill S\left(t\right)& =\sum _{k}P\left(k\right)s(k,t)\hfill \\ \hfill & =(1-{\rho }_0)[p\sum _{k}P\left(k\right)s_r(k,m,t)+q\sum _{k}P\left(k\right)s_n(k,m,t)].\hfill \end{array}$$

(7)

In order to calculate $S\left(t\right)$, we explore $\theta \left(t\right)$. Since there are three states of nodes in the network, which are susceptible state, adopted state, and recovered state, $\theta \left(t\right)$ can be decomposed as follows:

$$\theta \left(t\right)={\xi }_S\left(t\right)+{\xi }_A\left(t\right)+{\xi }_R\left(t\right),$$

(8)

where ${\xi }_S\left(t\right)$, ${\xi }_A\left(t\right)$, and ${\xi }_R\left(t\right)$, respectively, represent the probability that the neighbor node j of the S-state node i is in the S-state, A-state, and R-state and have not transmitted information to node i by time t.

Considering that node i is in the cavity state, its neighboring node j can only receive information from other neighboring nodes except node i. By time t, the probability that node j with degree $k_j$ receives cumulatively n pieces of information from its neighbor nodes is calculated as follows:

$$\varphi (k_j-1,n,t)=\left(\begin{array}{c}{k}_j-1\\ n\end{array}\right)\theta {\left(t\right)}^{k_j-n-1}{\left[1-\theta \left(t\right)\right]}^n.$$

(9)

If S-state node j is regretful, the probability that node j receives n pieces of information and remains in a susceptible state is ${\displaystyle \prod _{j=0}^n}[1-h_r({\displaystyle \frac{j}{k}},\epsilon \left)\right]$. Similarly, we can conclude that the probability that node j is in the susceptible state at time t after receiving n pieces of information is as follows:

$$\begin{array}{cc}\hfill {\Theta }_r(k_i,n,t)& =\sum _{n=0}^{k_j-1}\varphi (k_j-1,n,t)\prod _{a=0}^n[1-h_r({\displaystyle \frac{a}{k_j}},\epsilon \left)\right]\hfill \\ \hfill & =\sum _{n=0}^{⌊\epsilon k_j⌋}\varphi (k_j-1,n,t)\prod _{a=0}^n(1-sin{\displaystyle \frac{\pi a}{\epsilon k_j}})\hfill \\ \hfill & +\sum _{n=⌈\epsilon k_j⌉}^{k_j-1}\varphi (k_j-1,n,t)\prod _{a=0}^{⌈\epsilon k_j⌉}(1-sin{\displaystyle \frac{\pi a}{\epsilon k_j}})\prod _{a=⌈\epsilon k_j⌉}^n(1-sin{\displaystyle \frac{\pi (\frac{a}{k_i}-\epsilon )}{2-2\epsilon }}).\hfill \end{array}$$

(10)

Otherwise, S-state node j is no-regret. At time t, the probability that the node j receives n pieces of information and still remains in the susceptible state is calculated as follows:

$$\begin{array}{cc}\hfill {\Theta }_n(k_j,t)& =\sum _{n=0}^{k_j-1}\varphi (k_j-1,n,t)\prod _{a=0}^n[1-h_n({\displaystyle \frac{a}{k_j}},\epsilon \left)\right]\hfill \\ \hfill & =\sum _{n=0}^{⌊\epsilon k_j⌋}\varphi (k_j-1,n,t)\prod _{a=0}^n(1-sin{\displaystyle \frac{\pi a}{\epsilon k_j}})+\sum _{n=⌈\epsilon k_j⌉}^{k_j-1}\varphi (k_j-1,n,t)\prod _{a=0}^{⌈\epsilon k_j⌉}(1-sin{\displaystyle \frac{\pi a}{\epsilon k_j}}).\hfill \end{array}$$

(11)

Therefore, the probability of S-state node j receiving n pieces of information and remaining in the susceptible state before time t is calculated as follows:

$$\Theta (k_j,t)=p{\Theta }_r(k_j,t)+q{\Theta }_n(k_j,t).$$

(12)

By integrating the regret-dependent threshold functions into the edge-based compartmental framework, these new expressions explicitly capture the coupling between the psychological regret dynamics and the behavioral layer adoption. All other formulations identical to the standard framework are omitted here for brevity and are summarized in Appendix A. After linearizing the system around its initial state, the outbreak threshold conditions can be obtained. The proposed theoretical extension thus bridges traditional threshold models and dynamic psychological feedback, providing a new perspective for analyzing social contagion.

4. Results and Discussions

To confirm the findings of the proposed model and the corresponding theoretical analyses, we used the Erdös-Rényi (ER) network [43] and the Scale-Free (SF) network [44] to simulate the information propagation process. During the simulation process, to avoid stepwise discontinuities caused by discrete sampling, we introduced a linear interpolation method to ensure smooth transitions of the propagation curves in the numerical space. In a complete experiment, the propagation probability $\lambda$ was sampled at 100 evenly spaced points within the interval [0, 1]. For each $\lambda$, the regret threshold $\epsilon$ was also sampled at 100 points within [0, 1], resulting in 10,000 numerical combinations for simulation and verification. Furthermore, we repeated 1000 independent simulation experiments to verify the consistency between the theoretical analysis and numerical results, ensuring that the observed outcomes were not incidental.

We set the network size as N = 10,000 and the average degree of the social network as $⟨k⟩=10$, respectively. That is, the number of people in each social network is 10,000. Setting the recovery probability $\gamma =1$, i.e., the adopted state nodes will change to the recovered state after successfully transmitting information [14]. In the initial state, only a very small proportion of nodes (${\rho }_0=0.0001$) are selected as seed nodes (A state).

When the propagation process tends to a stationary state, there will be no adopted state nodes in the network. Thus, we can identify the final propagation range and the final adoption size by observing the proportion of recovered state nodes when the information propagation is steady. At the critical point of the propagation process, the information propagation and the final behavior adoption outbreak. In order to determine the critical value of the propagation process intuitively, we introduce the relative variance for numerical verification:

$$\chi ={\displaystyle \frac{⟨\left(R(\infty )\right)-{⟨R(\infty )⟩}^2⟩}{{⟨R\left(\infty \right)⟩}^2}},$$

(13)

where $⟨\cdots ⟩$ represents the ensemble average. The critical point of the final propagation range is displayed by the peak of $v_R$.

4.1. The Information Propagation on ER Social Network

First, we explore the information propagation characteristics of the regret population on an ER network, where the node degree distribution of the ER network follows the Poisson distribution, i.e., $p\left(k\right)=e^{-k}{⟨k⟩}^k/k!$. Dotted lines and symbols denote theoretical solutions and numerical simulations, respectively.

Figure 2 explores the relationship between the final propagation range $R(\infty )$ and the propagation probability $\lambda$ on the ER network with different regret thresholds. The proportion parameters of the regret population are $p=0.2$, 0.5, and 0.8 in Figure 2(a1), Figure 2(b1), and Figure 2(c1), respectively. From Figure 2(a1–c1), we can see that as the propagation probability increases, the propagation range shows a first increase and subsequently tends to stabilize. Furthermore, as the regret threshold rises, the propagation range increases to global adoption. The propagation curves present a continuous second-order phase transition when $\epsilon =0.2/0.5$ and a discontinuous first-order phase transition when $\epsilon$ increases to 0.8. Figure 2(a2)–(c2) shows that the phase transition phenomenon will occur and reach global behavioral adoption when the propagation probability reaches the critical value. Figure 2(a2)–(c2) shows that the phase transition phenomenon will occur and reach global behavioral adoption when the propagation probability reaches the critical value. In addition, the proportion of the regret population has a little influence on information propagation in this network topology and parameters. Figure 2 shows that the theoretical prediction agrees well with the simulation.

Figure 2. The illustration of the relationship between the final adoption size $R(\infty )$ and the propagation probability $\lambda$. The various proportion parameters of the regret population are $p=0.2$, 0.5, and 0.8 in subgraphs (a1), (b1), and (c1), respectively. The symbols and dotted lines represent the simulated and theoretical values, respectively. In addition, the relative variances of the theoretical analyses are depicted in subgraphs (a2), (b2), and (c2), respectively, along with the critical points of (a1–c1).

Then we explore the effects of regret threshold parameters $\epsilon$ on $R(\infty )$ with different propagation probabilities, $\lambda$, in Figure 3. The various regret population proportion parameters are $p=0.2$, 0.5, and 0.8 in Figure 3(a1), Figure 3(b1), and Figure 3(c1), respectively. We can see that $R(\infty )$ varies non-monotonically with $\epsilon$ for the given $\lambda$. With regret threshold $\epsilon$ increases, $R(\infty )$ rises initially and subsequently declines towards zero. The reason is that at the small value of $\lambda$, the regretful behavior dominates the information propagation process. When $\epsilon$ is small, individuals have a chance of regret and gradually increase their interest in the information. So more S-state individuals will receive information and behavioral adoption, leading to the steady increase in $R(\infty )$. However, as $\epsilon$ increases, it will decline for interest on the information and the chance of regret, which drops $R(\infty )$ gradually. Moreover, we can obtain an optimal value of $\epsilon$ and a maximum value of $R(\infty )$.

Figure 3. The effects of regret thresholds $\epsilon$ on the final adoption size $R(\infty )$ under different propagation probability $\lambda$. The various regret population proportion parameters are $p=0.2$, 0.5, and 0.8 in subgraphs (a1), (b1), and (c1), respectively. The relative variances of the theoretical analyses are depicted in subgraphs (a2), (b2), and (c2), respectively, along with the critical points of (a1–c1).

Figure 4 investigates the joint effect of the transmission probability $\lambda$ and the regret threshold ratio $\epsilon$ on the final spread range in the ER network. In Figure 4a–d, the regret population ratio parameter p is 0, 0.2, 0.8, and 1, respectively. In Figure 4a–c, the parameter plane $(\lambda ,\epsilon )$ can be divided into four regions. In region I, $\epsilon$ is very small, and the information hardly spreads in the population, with no population adopting the behavior. This occurs because $\epsilon$ hinders information diffusion throughout the population. In region II, as $\lambda$ increases, the information spreads to a limited extent but does not achieve global adoption. In region III, as $\lambda$ increases, the information spread gradually enters global, and as $\epsilon$ increases, the growth pattern exhibits a second-order continuous phase transition. In region IV, as $\epsilon$ increases, the growth pattern transitions from a second-order continuous phase transition to a first-order discontinuous phase transition. More interestingly, for a given $\lambda$, the final spread range varies non-monotonically with $\epsilon$. When $\lambda$ is low, $R(\infty )$ initially rises and then falls, analogous to the behavior in Figure 3. In this case, due to the low information transmission probability, regret behavior dominates. As the information transmission probability increases, $R(\infty )$ rises monotonically. In this case, non-regret behavior dominates. In Figure 4d, in addition to the similar phenomena in Figure 4a–c, it can be clearly observed that there is a mutation in region I that is different from the other subplots. When $\epsilon$ is very small, $R(\infty )$ increases with $\lambda$ to the global behavior adoption state. In Figure 4d, all individuals are susceptible to regret. When the unit transmission probability $\lambda$ is relatively large, although the adoption probability is small, agents are highly likely to experience regret. The regret characteristic promotes information spread. Therefore, the information can spread and the behavior can be globally adopted throughout the group. As $\epsilon$ increases, the threshold function exhibits a two-stage pattern—an initial rise followed by a decline, and then a secondary sinusoidal increase. This “rise–fall–rise” curve shape introduces a hesitation zone in the system, during which individuals temporarily lose their willingness to adopt the behavior.

Figure 4. The joint effects of propagation probability $\lambda$ and regret threshold $\epsilon$ on the final propagation range $R(\infty )$ for the ER network. The regret population proportion parameters are $p=0$, $p=0.2$, $p=0.8$, and $p=1$ in subgraphs (a), (b), (c), and (d) respectively.

4.2. The Information Propagation on SF Network

Next, we explore the propagation characteristics of the regret population on the SF network. The node degree distribution on the SF network follows a power-law distribution $p\left(k\right)=\varsigma k^{-v}$, where $\varsigma =1/{\sum }_k{k}^{-v}$. The parameter v denotes the degree index of the SF network, and the smaller v is, the greater degree of heterogeneity of the SF network. In addition, the minimum and maximum degrees of the SF network are $k_{min}=4,$ $k_{max}=100$, respectively.

Figure 5 depicts the relationship between $\lambda$ and $R(\infty )$ on the SF network with degree distribution heterogeneity. As the propagation probability increases, $R(\infty )$ gradually increases and reaches global adoption. In Figure 5a, a decrease in degree heterogeneity (with a large value of v) will facilitate the information propagation with a slight regret threshold. The degree of heterogeneity effect on information propagation weakens while the regret threshold becomes large. Comparing the subgraphs in Figure 5(a1)–(c1), the increase in the regret threshold prevents the information propagation and facilitates the change in the phase transition pattern from the second-order continuity to the first-order discontinuity. Figure 5(a2)–(c2) correspond to the critical propagation probabilities of Figure 5(a1)–(c1). The simulated values (symbols) fit well with the theoretical values (lines).

Figure 5. The effects of $\lambda$ and $R(\infty )$ on the SF network, where $\epsilon =0.2,0.5,0.8$ in subgraphs (a1), (b1), and (c1), respectively. Subgraphs (a2), (b2), and (c2) represent the relative variances of the theoretical analyses and the critical points of (a1), (b1), and (c1), respectively. The symbols and dotted lines represent the simulated and theoretical values, respectively. The other parameter is $p=0.5$.

Then we explore the effects of regret threshold parameters $\epsilon$ on $R(\infty )$ with degree distribution heterogeneity v in Figure 6. The various degree distribution exponents are $v=2.1$, 3, and 4 in each subgraph. Comparing Figure 6(a1)–(c1), as regret threshold $\epsilon$ increases, $R(\infty )$ rises initially and subsequently declines towards zero for the given $\lambda$. Therefore, we can obtain an optimal value of $\epsilon$ and a maximum value of $R(\infty )$ and v. The reason is that at the small value of $\lambda$, the regretful behavior dominates the information propagation process. When $\epsilon$ is small, people are more inclined to regret and gradually increase their interest in the information. So more S-state individuals will receive information and behavioral adoption, leading to the steady increase in $R(\infty )$. However, as $\epsilon$ increments, people don’t regret quickly and gradually decline their interest in information and the chance of regret, gradually dropping $R(\infty )$. In addition, the increment of degree distribution heterogeneity (with a smaller value of v) will restrain information propagation and behavioral adoption and promote the increment pattern change from the second-order continuous phase transition to the first-order discontinuous.

Figure 6. The effect of $ϵ$ and $\lambda$ on the final propagation range $R(\infty )$ for the SF network. The degree distribution heterogeneity $v=2.1,3,4$ in subgraphs (a1), (b1), and (c1), respectively. The symbols and lines represent the simulated and theoretical values, respectively. The relative variances of the theoretical analyses are depicted in subgraphs (a2), (b2), and (c2), respectively, along with the critical points of (a1–c1). The other parameters are $p=0.5$.

Figure 7 investigates the joint effect of $\lambda$ and the regret threshold parameter $\epsilon$ on the final propagation range in the SF network. Figure 7a–c correspond to $v=2.1,3,4$, respectively. The regret population proportion parameter is $p=0.5$. In Figure 7a–c, the parameter plane $(\lambda ,\epsilon )$ can be divided into three regions. There is no global behavioral adoption outbreak in region I. That is because at a low value of $\epsilon$, an individual has little interest in information and behavioral propagation. The final propagation range shows a second-order continuous phase transition pattern in region II and a first-order discontinuous phase transition in region III. More exciting, the final propagation range varies non-monotonically with $\epsilon$ for a given $\lambda$. With a low $\lambda$, $R(\infty )$ rises initially and subsequently declines. The reason is similar to Figure 6. In this case, regretful behavior dominates the information propagation process due to the lower information propagation probability. $R(\infty )$ rises monotonically to global behavioral adoption as information propagation probability increases. In this case, the non-regret behavior dominates the information propagation process. Comparing Figure 7a–c, when v and $\epsilon$ are small, such as in Figure 7a, the increase in behavioral adoption population is slow, continuous, and long as the propagation probability increases.

Figure 7. The joint effect of $\lambda$ and threshold $\epsilon$ on the final propagation range $R(\infty )$ for the SF network. The first to third columns’ subgraphs correspond to $v=2.1$, 3, and 4 in subgraphs (a), (b), and (c), respectively. The other parameters is $p=0.5$.

5. Conclusions

This study explores the information propagation mechanism of regret psychology on social networks, where we randomly select a p proportion of the regret population and the rest of the population as the no-regret population. A social network consisting of N nodes is established. Then, we propose two behavioral adoption threshold functions to describe the behavioral characteristics of the regretful and non-regretful populations. The non-regretful group’s interest in information and behavior changes sequentially, increasing and then dropping. The regretful group has the chance of regret and gradually increases their interest again in information and behavioral adoption. We propose an edge compartmental theory based on the regret behavioral adoption threshold model to theoretically analyze the characteristics of the regretful group in information propagation. In addition, we simulate the information propagation process with the regret threshold using ER and SF networks. We find the crossover phenomena of phase transition through theoretical analysis and simulation results. More importantly, we find an optimal regret threshold for the given information propagation probability at which a maximum final adoption size can be obtained.

In this study, we quantitatively modeled and simulated the characteristics of regret psychology in the communication process, thereby explaining its underlying mechanisms in information propagation and behavioral adoption. By incorporating a dynamic evolutionary mechanism of regret psychology into the conventional threshold framework, our work not only compensates for the limitation of existing information propagation models that treat psychological states as static parameters but also uncovers the cross-layer regulatory influence of regret emotions on the coupled dynamics of information diffusion and behavioral adoption. This integration establishes a novel theoretical paradigm for understanding the intrinsic propagation patterns of real-world social networks. In marketing, enterprises prefer that individuals believe and take action accordingly. Therefore, based on Figure 3 and Figure 6, by increasing the product exposure intensity and triggering regret stimuli, the effectiveness of marketing activities can be maximized. For false information, it is desired that individuals do not believe or adopt the misleading content. Thus, by reducing the information exposure intensity and controlling the proportion of regretful groups and the regret threshold, impulsive adoption behaviors can be suppressed. In the future, we plan to collaborate with enterprises to verify the group adoption behavior model using real data, providing a reasonable basis for the subsequent development of strategies.