Research on rumor and anti rumor propagation models based on quantum superposition states

Abstract

Building an effective rumor propagation model is the key to blocking rumor propagation. This article is based on the quantum superposition theory, analyzing the propagation decision of unknown individuals from the perspectives of consciousness and behavior, determining their time-varying propagation state, characterizing the propagation process of online rumors, and constructing a rumor propagation model to solve the basic reproduction number. It analyzes the local stability of equilibrium points without rumor propagation and equilibrium points with rumor propagation. The experimental results better fit the rule of change of rumor propagation group state, and verify the effects of the average rumor belief level in society and the degree of social management of rumors on the scale of rumor propagation, providing theoretical support for suppressing rumor propagation.

Introduction

The Internet is characterized by openness, sharing, freedom, and real-time communication. While it facilitates access to and sharing of information, it also significantly accelerates the emergence and spread of online rumors. Rumors exhibit diverse types, uncertain durations, rapid and widespread propagation, and dynamically evolving derivative risks, all of which complicate governance efforts. The spread of rumors not only misleads the public but also threatens social stability.

The study of rumor spread mostly originates from infectious disease models. In 1926, Kermack and McKendrick constructed the SIR¹(Susceptible Infected Recovered) infectious disease model. In 1964, Daley, D J, compared the process of rumor spread with the infectious process of diseases, and proposed the SIR model of rumor spread. Many scholars have established dynamic models such as the SEIR²model, SIRS³model, SIDR⁴model, SC1C2IR⁵model, etc., based on this model. However, the disease process is a passive contagious process, while rumor spread is an active behavior. The rumor spread model based on the infectious disease model does not consider the subjective initiative of individuals. Many scholars have also referred to or integrated theories from other fields to explain the laws of rumor spread, such as the wildfire spread model⁶, the swarm insect immune strategy⁷, the principle of elastic collision of billiard balls⁸, the mechanics of attraction model⁹, the drug diffusion principle¹⁰, and thermodynamic formulas¹¹.

In recent years, the application of quantum theory in the field of communication has attracted extensive attention from the academic community. Scholars have explored the correlation between quantum theory and information propagation from different dimensions: Tran Minh C¹² focused on the study of the temporal characteristics of information propagation in quantum systems; Li Penghua et al.¹³ innovatively proposed a model of Elman neural network based on the structure of quantum gates, which significantly improves the learning performance of the network; Wenjuan Zhang et al.¹⁴ and Yu Zhao et al.¹⁵, on the other hand, from the perspectives of quantum entanglement theory and quantum genetic algorithm, respectively, deeply explained the intrinsic mechanism of cross-cultural information propagation, and further proposed a model of information propagation blocking. With the depth of research, the application value of quantum theory in the field of network information propagation has gradually emerged, especially in the study of network rumor propagation, which provides new theoretical and methodological support. In terms of theoretical research, Zhang Yibing et al.¹⁷ introduced the epistemological method of quantum mechanics into the study of network information, revealing the mechanism of propagation, aggregation and amplification of information in the network through the quantum properties of microscopic particles, which provides a new perspective for understanding the dynamic process of rumor propagation. In terms of model construction, Wang Ying et al.¹⁶ combined quantum game theory and constructed a network information propagation model based on quantum game from the perspective of the information propagator, with comprehensive consideration of the mobility and trust mechanism; Yan Fei et al.¹⁸ proposed a continuous time quantum walk information propagation model (CTQW-IPM), which estimates the individual criticality through the probability distribution of quantum observation, effectively solving the problem of the problem of many iterations and complex parameters of traditional social network simulation models, providing a more efficient modeling tool for rumor propagation simulation. In addition, the information propagation blocking model designed by Zhao Yu et al.¹⁵ based on quantum genetic algorithm provides technical support for the research of rumor propagation intervention strategy by virtue of its fast processing capability. However, despite the progress made in existing research, the potential applications of core quantum properties such as superposition in the context of online rumor propagation have not yet been fully explored. This field remains a vast area for further investigation and deepening.

While traditional models usually reduce individual states to a single dimension (e.g., “spreading rumors” or “not spreading rumors”), quantum states can cover multiple dimensions (e.g., cognitive and behavioral dimensions) at the same time. Based on the above analysis, this study analyzes the complex psychological and behavioral characteristics of individuals in rumor propagation and constructs a model of rumor and anti-rumor propagation based on quantum superposition theory. By representing individual states as a superposition of basic states at the cognitive and behavioral levels, analyzing the state transfer probability and the “collapse” law, describing the complexity and dynamics of individual decision-making, the model reveals the intrinsic mechanism and law of rumor propagation. In addition, the model simulates and compares the rumor and counter-rumor propagation process and the influence of parameter changes by using simulated data and real networks, which provides a theoretical basis for preventing and reducing rumor propagation.

Section Quantum superposition theory introduces the fundamental principles of quantum superposition states, Sect. Model establishment develops the theoretical model, and Sect. Theoretical Analysis performs a theoretical analysis. Finally, Sect. Simulation presents simulation experiments to discuss the results. The model more effectively captures the influencing factors and dynamic processes of individual decision-making, offering both a theoretical foundation and practical insights for the governance of online rumors.

Quantum superposition theory

In order to quantitatively describe the state of microscopic particles, a wave function is introduced and represented by $\psi$in quantum mechanics. The following are three basic expressions of the principle of quantum superposition states¹⁹.

Statement 1 For a specified quantum system, if there is a set of complete set state functions $\left\{ {{\varphi _i}} \right\},i=1,2, \ldots ,t$, any state $\left| \psi \right\rangle$ in the system can be formed by a linear combination (superposition) of these configuration functions, that is,

$$\left| \psi \right\rangle =\sum\limits_{{i=1}}^{t} {{c_i}\left| {{\varphi _i}} \right\rangle }$$

(1)

.

Among them, ${c_i}$ is a set of constants (which can be complex), ${\varphi _i}$ represents a complete set of state functions refers to a group of functions that can fully describe all possible states of a quantum system, and any state $\left| \psi \right\rangle$ satisfies the normalization condition: $\left\langle \psi \right.\left| \psi \right\rangle =1$.

All possible states of this quantum system constitute a mathematical linear space (vector space), known as the Hilbert space of the quantum system.

Statement 2 If $\left\{ {{\varphi _i}} \right\},i=1,2, \ldots ,t$ is a state (wave function) that the system can achieve, then any linear superposition of them (1) always represents a state that the system can achieve.

Statement 3 When the physical system is in the superposition state Eq. (1), it can be considered that the probability of the system being in the $\left| {{\varphi _i}} \right\rangle$ quantum state is ${\left| {{c_i}} \right|^2}$.

Model establishment

When exposed to rumor information, individuals engage in cognitive processing at the conscious level to assess the credibility of the rumor and make behavioral decisions regarding whether to forward it. Consequently, individuals may exhibit one of four possible states: (1) believing and spreading the rumor, (2) believing but not spreading the rumor, (3) not believing yet spreading the rumor, and (4) not believing and not spreading the rumor.

Consciousness layer

At the conscious level, an individual’s judgment determines whether they believe the rumors to which they are exposed. Key factors influencing this judgment include user intimacy, rumor credibility, and the individual’s ability to discern information.

Intimacy level between users ${\alpha _1}$

The level of intimacy between users plays a significant role in determining their belief in received rumors. Research indicates that in information propagation, individuals are more inclined to accept and forward information from close friends. Moreover, the likelihood of believing such information increases with the number of close friends sharing it²⁰. Additionally, a higher level of intimacy further enhances the probability of belief.

Based on this, the intimacy level ${\alpha _1}$ between users is defined as follows,

$${\alpha _1}=\frac{{\left| {u \cap v} \right|}}{{\left| {u \cup v} \right|}}$$

(2)

.

where $\left| {u \cap v} \right|$ represents the number of friends that individuals u and v share, $\left| {u \cup v} \right|$ represents the total number of friends that individuals u and v each have, $0 \leqslant {\alpha _1} \leqslant 1$.

The credibility of rumors ${\alpha _2}$

The credibility of a rumor is a critical determinant of an individual’s belief in its content. Research has demonstrated a positive correlation between the number of comments endorsing a rumor and its perceived credibility²¹.Based on this, the definition of rumor credibility ${\alpha _2}$ is given as follows,

$${\alpha _2}=\frac{{\sum {{C_b}} }}{{\sum C }}$$

(3)

Where ${C_b}$ represents the number of individuals expressing a belief attitude in the comment, and C represents all individuals participating in this rumor comment,$0 \leqslant {\alpha _2} \leqslant 1$.

User discrimination ability ${\alpha _3}$

The ability of users to discern the rumors they encounter significantly influences their likelihood of believing such rumors³². Their discernment ability is influenced not only by their level of knowledge but also by the influence of neighboring nodes. Based on this, the user discrimination ability ${\alpha _3}$ is defined as follows:

$${\alpha _3}=\frac{{Num\left( {R\_adj(u,t)} \right)}}{{Num\left( {R\_adj(u,t)} \right)+Num\left( {T\_adj(u,t)} \right)}}$$

(4)

where $Num\left( {R\_adj(u,t)} \right)$ represents the number of neighboring nodes who believe the rumor, and $Num\left( {T\_adj(u,t)} \right)$represents the number of neighboring nodes who do not believe the rumor,$0 \leqslant {\alpha _3} \leqslant 1$.

Add and average the values of ${\alpha _1}$, ${\alpha _2}$, and ${\alpha _3}$ for an individual. If the average value is greater than 0.5, it is more inclined to believe in rumors; otherwise, it is not believed in rumors.

Behavioral layer

At the behavioral level, an individual’s decision to spread rumors is influenced by factors such as their past behavior, current interest in the rumor, and other contextual variables.

Previous historical behavior of user ${\beta _1}$

A significant correlation exists between an individual’s past forwarding behavior and their current and future propagation choices³³. Using deep learning, topic mining is conducted on an individual’s past forwarded information to extract keyword sets. Simultaneously, topic analysis is applied to the information currently received by the user to derive its keyword set. The following definition of historical behavior is given:

$${\beta _1}=\frac{{\left| {k\_pre(i) \cap k\_now(i)} \right|}}{{\left| {k\_now(i)} \right|}}$$

(5)

where $k\_pre(i)$ is the set of subject keywords that individual i has previously disseminated information on, and $k\_now(i)$ is the set of subject keywords i that individual currently receives, $0 \leqslant {\beta _1} \leqslant 1$.

Current interest in rumors ${\beta _2}$

Changes in an individual’s interest in information also influence their decision to forward it. As individuals repeatedly forward rumors, they may critically evaluate the false or contradictory aspects, leading to a gradual decline in interest among those who initially chose to spread the rumors. Once this interest falls below a certain threshold, individuals will cease forwarding the rumors. Scholars have suggested that rumors are highly appealing initially, especially when their validity remains unverified^5,22. However, as individuals repeatedly share them in social interactions, the rumors’ attractiveness diminishes over time. Based on the above analysis, the definition of interest ${\beta _2}$ in rumors at the current moment is given:

$${\beta _2}(t)={1 \mathord{\left/ {\vphantom {1 {{e^{c*t}}}}} \right. \kern-0pt} {{e^{c*t}}}}$$

(6)

where c is the decay constant, t is a time variable measured in days. The initial interest value of an individual receiving a rumor is 1, which gradually decreases over time, thereby affecting the individual’s choice of rumor propagation.

Take the average of the influencing factors ${\beta _1}$ and ${\beta _2}$. If the average value is greater than 0.5, it is more inclined to spread rumors. Conversely, if it is less than 0.5, it is more inclined to not spread.

A rumor propagation model based on quantum superposition state

In the traditional SIR (Susceptible-Infectious-Recovered) compartmental model, the state of an individual is typically singular and deterministic. However, in the context of rumor propagation, an individual’s behavior (to spread or not to spread) and their belief (to believe or not to believe) are complex and intricately intertwined. The quantum superposition method can naturally describe the superposition of these states. Therefore, this study draws on the quantum superposition approach to characterize the hesitancy, ambivalence, or uncertainty in an individual’s psychological state when confronted with rumors.

Basic States

The previous analysis shows that the present study is modeled by a two-quantum system that simultaneously describes the state changes of an individual with respect to both rumor awareness and behavioral dimensions. The observable states mathematically represented as $\left| 0 \right\rangle$ and $\left| 1 \right\rangle$. According to the double quantum bit combination, the rumor ignoramus can be converted into four states after being exposed to the rumor information, namely, Rumor Spreader, Anti-Rumor Spreader, Bystander, Insensitive person. The detailed mapping of these state transitions is shown in Table 1.is described systematically in the following section.

Table 1 Individual state and quantum ground state mapping table.

When an individual is exposed to rumor information, if the level of user intimacy, the rumor’s inherent influence, and the individual’s discernment ability are all low, the conscious-level decision is disbelief, represented by the quantum bit $\left| 0 \right\rangle$. Simultaneously, if the individual’s historical forwarding behavior shows low correlation with the topic, their interest in the topic is minimal, and the behavioral-level decision is not to propagate, the quantum bit is denoted as $\left| 0 \right\rangle$. This group neither believes nor spreads the rumor. As shown in Table 1, they are referred to as Insensitive individuals in the rumor propagation process, denoted as R, with the corresponding quantum state $\left| {00} \right\rangle$. The other three states can be obtained by the same analysis.

In conclusion, when an ignorant individual ${\text{S}}$ is exposed to rumor information, it may exhibit four basic states, such as $\left| {11} \right\rangle$, $\left| {10} \right\rangle$, $\left| {01} \right\rangle$, and $\left| {00} \right\rangle$. Specifically, as shown in Fig. 1.

Fig. 1

Individual decision-making and state change graph.

Formulaic individual state

According to the description of Quantum Superposition Theory, the wave function of individual equation can be expressed as:

$${\Psi _i}={c_{i1}}{\Psi _{\left| {11} \right\rangle }}+{c_{i2}}{\Psi _{\left| {10} \right\rangle }}+{c_{i3}}{\Psi _{\left| {01} \right\rangle }}+{c_{i4}}{\Psi _{\left| {00} \right\rangle }}$$

(7)

,

where the wave function of individual ${\Psi _i}$ is used to represent the current state of the individual, and $\left\{ {{c_{i1}},{c_{i2}},{c_{i3}},{c_{i4}}} \right\}$ is a set of constants. According to the principle of quantum superposition states, for an individual, $\left\{ {{\Psi _{\left| {11} \right\rangle }},{\Psi _{\left| {10} \right\rangle }},{\Psi _{\left| {01} \right\rangle }},{\Psi _{\left| {00} \right\rangle }}} \right\}$ is a set of complete set state functions, and any state of the linear combination (superposition) of these configuration functions is a state that the individual i may exist in at a certain moment.

States transition

When individuals are exposed to rumor-related information, they may simultaneously exhibit tendencies to both spread and not spread the rumor, as well as to believe and not believe it. This model captures the uncertainty and probabilistic nature of individual behavior through the representation of probability amplitudes and superposition states. Assuming that the probability of an individual’s consciousness choosing “Belief” is denoted as $\alpha$, and the probability of “Disbelief” is denoted as $\left( {1 - \alpha } \right)$; Similarly, the probability of an individual’s behavior layer choosing “Spread” is denoted as $\beta$, while “Non Spread” is denoted as $\left( {1 - \beta } \right)$. And individuals may show different behavioral tendencies at different time points. When the probability of individual selection changes, the node state of the individual at the next moment may also change, the change values are denoted by $\Delta$ respectively.

The phenomenon of measurement collapse in quantum mechanics suggests that a system will randomly collapse to a certain deterministic state when observed. Analogous to rumor spreading, an individual may end up exhibiting a specific behavior (e.g., spreading a rumor or opposing a rumor) after receiving information due to factors such as external environment, social pressure, or personal preference. The state transition diagram of the network rumor propagation model based on quantum superposition states is shown in Fig. 2.

Fig. 2

State transition diagram of rumor propagation model.

The dynamic equation for rumor propagation is as follows:

$$\begin{gathered} \frac{{dS\left( t \right)}}{{dt}}= - \alpha \beta S\left( t \right){I_r}\left( t \right) - \alpha \left( {1 - \beta } \right)S\left( t \right)B\left( t \right) - \left( {1 - \alpha } \right)\beta S\left( t \right){I_t}\left( t \right) \hfill \\ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{} \end{array}}&{} \end{array} - \left( {1 - \alpha } \right)\left( {1 - \beta } \right)S\left( t \right)R\left( t \right) \hfill \\ \frac{{d{I_r}\left( t \right)}}{{dt}}=\alpha \beta S\left( t \right){I_r}\left( t \right)+\frac{{\Delta {\alpha _2}}}{3}\beta \left[ {{I_t}\left( t \right) - {I_r}\left( t \right)} \right] - \alpha \frac{{\Delta {\beta _2}}}{2}{I_r}\left( t \right) \hfill \\ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{} \end{array}}&{} \end{array} - \frac{{\Delta {\alpha _2}}}{3}\frac{{\Delta {\beta _2}}}{2}{I_r}\left( t \right) \hfill \\ \frac{{d{I_t}\left( t \right)}}{{dt}}=\left( {1 - \alpha } \right)\beta S\left( t \right){I_t}\left( t \right)+\frac{{\Delta {\alpha _2}}}{3}\beta \left[ {{I_r}\left( t \right) - {I_t}\left( t \right)} \right] \hfill \\ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{} \end{array}}&{} \end{array} - \frac{{\Delta {\alpha _2}}}{3}\frac{{\Delta {\beta _2}}}{2}{I_t}\left( t \right) - \alpha \frac{{\Delta {\beta _2}}}{2}{I_t}\left( t \right) \hfill \\ \frac{{dB\left( t \right)}}{{dt}}=\alpha \left( {1 - \beta } \right)S\left( t \right)B\left( t \right)+\alpha \frac{{\Delta {\beta _2}}}{2}{I_r}\left( t \right)+\frac{{\Delta {\alpha _2}}}{3}\frac{{\Delta {\beta _2}}}{2}{I_t}\left( t \right) \hfill \\ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{} \end{array}}&{} \end{array} - \frac{{\Delta {\alpha _2}}}{3}\beta B\left( t \right) \hfill \\ \frac{{dR\left( t \right)}}{{dt}}=\left( {1 - \alpha } \right)\left( {1 - \beta } \right)S\left( t \right)R\left( t \right)+\frac{{\Delta {\alpha _2}}}{3}\frac{{\Delta {\beta _2}}}{2}{I_r}\left( t \right)+\frac{{\Delta {\alpha _2}}}{3}\beta B\left( t \right) \hfill \\ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{} \end{array}}&{} \end{array}+\alpha \frac{{\Delta {\beta _2}}}{2}{I_t}\left( t \right) \hfill \\ \end{gathered}$$

(8)

,

where $S\left( t \right)+{I_r}\left( t \right)+{I_t}\left( t \right)+B\left( t \right)+R\left( t \right)=N(t)$.

States collapse

When an individual is “measured” or influenced by external factors, their superposition state will collapse to a certain determined state based on the probability amplitude. This mechanism can effectively simulate the behavioral changes of individuals after receiving new information or being influenced by society.According to the superposition principle Statement 3, it can be assumed that individual state ${\Psi _i}$ will collapse to state ${\Psi _{\left| {ab} \right\rangle }}$ with probability, where the value of a or b is 0 or 1. Based on the definition of individual state, the probability distribution is assumed to be as follows:

$$\left\{ \begin{gathered} {\left| {{c_{i1}}} \right|^2}=\alpha \cdot \beta \hfill \\ {\left| {{c_{i2}}} \right|^2}=\alpha \cdot \left( {1 - \beta } \right) \hfill \\ {\left| {{c_{i3}}} \right|^2}=\left( {1 - \alpha } \right)\cdot \beta \hfill \\ {\left| {{c_{i4}}} \right|^2}=\left( {1 - \alpha } \right)\cdot \left( {1 - \beta } \right) \hfill \\ \end{gathered} \right.$$

(9)

,

Based on the above assumptions, the state of individual is

$${\Psi _i}=\left\{ \begin{gathered} {\Psi _{\left| {11} \right\rangle }},if \alpha ,\beta \geqslant 0.5 \hfill \\ {\Psi _{\left| {10} \right\rangle }},if \alpha \geqslant 0.5,0 \leqslant \beta <0.5 \hfill \\ {\Psi _{\left| {01} \right\rangle }},if 0 \leqslant \alpha <0.5,\beta \geqslant 0.5 \hfill \\ {\Psi _{\left| {00} \right\rangle }},if 0 \leqslant \alpha ,\beta <0.5 \hfill \\ \end{gathered} \right.$$

(10)

In this way, the probability values of each state at that moment can be determined through state transitions, and the final state of the individual can be predicted through “measurement”.

Theoretical analysis

In infectious disease dynamics, the basic reproductive number (${R_0}$) is a critical concept, representing the average number of secondary infections caused by a single infected individual in a fully susceptible population during the early stages of an outbreak²³. Typically, ${R_0}=1$ serves as a threshold parameter for determining whether a disease will eventually die out.

Based on this, this article defines the basic reproduction number ${R_0}$ as the expected number of new generation rumor spreaders generated by a single rumor spreader, and uses ${R_0}$as an important indicator to judge the ability of rumor propagation²⁴. Therefore, this section will conduct theoretical analysis on the basic reproduction number ${R_0}$ of the rumor propagation system, the stability of the equilibrium point without rumors, and the stability of the equilibrium point with rumors.

The basic reproduction number

There are three types of spreaders in the rumor and anti-rumor propagation model based on quantum superposition states proposed in this article, namely $B(t)$, ${I_r}(t)$, and ${I_t}(t)$.

To distinguish between infectious compartments and non-infectious compartments, the system is partitioned into ${\rm X}$ and ${\text{Y}}$vectors by applying the next-generation matrix method²⁵. Here, the ${\rm X}$ vector typically represents the infectious compartments, which include all compartments directly associated with the spread of rumors. The ${\text{Y}}$ vector, on the other hand, generally denotes the non-infectious compartments, encompassing all compartments not directly involved in the propagation of rumors.

$${\rm X}=\left[ {\begin{array}{*{20}{c}} {B(t)} \\ {\begin{array}{*{20}{c}} {{I_r}(t)} \\ {{I_t}(t)} \end{array}} \end{array}} \right] {\text{Y=}}\left[ {\begin{array}{*{20}{c}} {S(t)} \\ {R(t)} \end{array}} \right]$$

(11)

Write the expression of ${\rm X}$ regarding F and V based on ${\rm X}$ and ${\text{Y}}$, and the roles of the F matrix and the V matrix are to describe the occurrence of new infections and the transition processes of infected individuals, respectively.

$$\begin{gathered} {\rm X}=\left[ {\begin{array}{*{20}{c}} {B(t)} \\ {\begin{array}{*{20}{c}} {{I_r}(t)} \\ {{I_t}(t)} \end{array}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {\alpha (1 - \beta )SB} \\ {\begin{array}{*{20}{c}} {\alpha \beta S{I_r}} \\ {(1 - \alpha )\beta S{I_t}} \end{array}} \end{array}} \right] - \left[ {\begin{array}{*{20}{c}} {\frac{{\Delta {\alpha _2}}}{3}\beta B - \alpha \frac{{\Delta {\beta _2}}}{2}{I_r} - \frac{{\Delta {\alpha _2}}}{3}\frac{{\Delta {\beta _2}}}{2}{I_t}} \\ { - \frac{{\Delta {\alpha _2}}}{3}\beta ({I_t} - {I_r})+\alpha \frac{{\Delta {\beta _2}}}{2}{I_r}+\frac{{\Delta {\alpha _2}}}{3}\frac{{\Delta {\beta _2}}}{2}{I_r}} \\ { - \frac{{\Delta {\alpha _2}}}{3}\beta ({I_r} - {I_t})+\frac{{\Delta {\alpha _2}}}{3}\frac{{\Delta {\beta _2}}}{2}{I_t}+\alpha \frac{{\Delta {\beta _2}}}{2}{I_t}} \end{array}} \right] \hfill \\ ={F_{12}}(B,{I_r},{I_t}) - {V_{12}}(B,{I_r},{I_t}) \hfill \\ \end{gathered}$$

(12)

Calculate the Jacobian matrix for ${F_{12}}(B,{I_r},{I_t})$ and ${V_{12}}(B,{I_r},{I_t})$, respectively,

$$\begin{gathered} {\text{F}}=Jacobian({F_{12}}(B,{I_r},{I_t}))=\left[ {\begin{array}{*{20}{c}} {\alpha (1 - \beta )S}&0&0 \\ 0&{\alpha \beta S}&0 \\ 0&0&{(1 - \alpha )\beta S} \end{array}} \right] \hfill \\ {\text{V}}=Jacobian({V_{12}}(B,{I_r},{I_t})) \hfill \\ ~~~~=\left[ {\begin{array}{*{20}{c}} {\frac{{\Delta {\alpha _2}}}{3}\beta }&{ - \alpha \frac{{\Delta {\beta _2}}}{2}}&{ - \frac{{\Delta {\alpha _2}}}{3}\frac{{\Delta {\beta _2}}}{2}} \\ 0&{\frac{{\Delta {\alpha _2}}}{3}\beta +\alpha \frac{{\Delta {\beta _2}}}{2}+\frac{{\Delta {\alpha _2}}}{3}\frac{{\Delta {\beta _2}}}{2}}&{ - \frac{{\Delta {\alpha _2}}}{3}\beta } \\ 0&{ - \frac{{\Delta {\alpha _2}}}{3}\beta }&{\frac{{\Delta {\alpha _2}}}{3}\beta +\alpha \frac{{\Delta {\beta _2}}}{2}+\frac{{\Delta {\alpha _2}}}{3}\frac{{\Delta {\beta _2}}}{2}} \end{array}} \right] \hfill \\ \end{gathered}$$

(13)

By observing the above two equations, it can be found that there is similarity in the characteristics of the two types of rumor propagation user groups. To simplify the solving process, the process solutions of ${I_r}$ and ${I_t}$ can be merged. Therefore, the basic reproduction number

$${R_0}=\frac{\beta }{{\Delta {\beta _2}}}$$

(14)

corresponding to the rumor propagation model.

Equilibrium point in the absence of rumors

The model considers a stable group size without any additional input-output ratio, so the equilibrium point without rumors is ${P_0}=\left( {1,0,0,0,0} \right)$, and the Jacobian matrix of that point is.

$$\begin{aligned}{{J}_{0}}({{P}_{0}})&={{\left[ \begin{matrix} \beta S-\Delta {{\beta }_{2}} & 0 \\ \frac{\Delta {{\beta }_{2}}}{2} & \alpha (1-\beta )S-\frac{\Delta {{\alpha }_{2}}}{3}\beta \\ \end{matrix} \right]}_{(1,0,0,0,0)}}\\&=\left[ \begin{matrix} \beta -\Delta {{\beta }_{2}} & 0 \\ \frac{\Delta {{\beta }_{2}}}{2} & \alpha (1-\beta )-\frac{\Delta {{\alpha }_{2}}}{3}\beta \\ \end{matrix} \right]\end{aligned}$$

(15)

Therefore, the eigenvalues of the solution to this formula are

$$\left\{ {\begin{array}{*{20}{c}} {{\lambda _1}=\beta - \Delta {\beta _2}} \\ {{\lambda _2}=\alpha (1 - \beta ) - \frac{{\Delta {\alpha _2}}}{3}\beta <0} \end{array}} \right.$$

(16)

.

Due to ${R_0}<1$, i.e. $\beta <\Delta {\beta _2}$, we obtain ${\lambda _1}<0$. According to the South Hurwitz²⁶ criterion, when ${R_0}<1$, the rumor free equilibrium point ${P_0}=\left( {1,0,0,0,0} \right)$ is locally asymptotically stable; When ${R_0} \geqslant 1$, ${P_0}$ is unstable.

The equilibrium point in the spread of rumors

Next, we set the differential equation to zero,

$$\left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} { - \alpha \beta S\left( t \right){I_r}\left( t \right) - \alpha \left( {1 - \beta } \right)S\left( t \right)B\left( t \right) - \left( {1 - \alpha } \right)\beta S\left( t \right){I_t}\left( t \right) - \left( {1 - \alpha } \right)\left( {1 - \beta } \right)S\left( t \right)R\left( t \right)=0} \\ {\alpha \beta S\left( t \right){I_r}\left( t \right)+\frac{{\Delta {\alpha _2}}}{3}\beta \left[ {{I_t}\left( t \right) - {I_r}\left( t \right)} \right] - \alpha \frac{{\Delta {\beta _2}}}{2}{I_r}\left( t \right) - \frac{{\Delta {\alpha _2}}}{3}\frac{{\Delta {\beta _2}}}{2}{I_r}\left( t \right)=0} \end{array}} \\ {\left( {1 - \alpha } \right)\beta S\left( t \right){I_t}\left( t \right)+\frac{{\Delta {\alpha _2}}}{3}\beta \left[ {{I_r}\left( t \right) - {I_t}\left( t \right)} \right] - \frac{{\Delta {\alpha _2}}}{3}\frac{{\Delta {\beta _2}}}{2}{I_t}\left( t \right) - \alpha \frac{{\Delta {\beta _2}}}{2}{I_t}\left( t \right)=0} \\ {\alpha \left( {1 - \beta } \right)S\left( t \right)B\left( t \right)+\alpha \frac{{\Delta {\beta _2}}}{2}{I_r}\left( t \right)+\frac{{\Delta {\alpha _2}}}{3}\frac{{\Delta {\beta _2}}}{2}{I_t}\left( t \right) - \frac{{\Delta {\alpha _2}}}{3}\beta B\left( t \right)=0} \\ {\left( {1 - \alpha } \right)\left( {1 - \beta } \right)S\left( t \right)R\left( t \right)+\frac{{\Delta {\alpha _2}}}{3}\frac{{\Delta {\beta _2}}}{2}{I_r}\left( t \right)+\frac{{\Delta {\alpha _2}}}{3}\beta B\left( t \right)+\alpha \frac{{\Delta {\beta _2}}}{2}{I_t}\left( t \right)=0} \end{array}} \right.$$

(17)

,

We can find the equilibrium point ${P_1}=({S^ * },I_{r}^{ * },I_{t}^{ * },{B^ * },{R^ * })=(\frac{{\Delta {\beta _2}}}{\beta },\frac{{{I^ * }}}{2},\frac{{{I^ * }}}{2},{B^ * },{R^ * })$ for the spread of rumors, where

$$\begin{gathered} {B^ * }=\frac{{3\beta \Delta {\beta _2}}}{{2\left( {\Delta {\alpha _2}{\beta ^2} - 3\alpha (1 - \beta )\Delta {\beta _2}} \right)}}{I^*} \hfill \\ {R^ * }=\frac{\beta }{{2(1 - \alpha )(1 - \beta )}}\left( { - 1 - \frac{{\Delta {\alpha _2}{\beta ^2}}}{{\Delta {\alpha _2}{\beta ^2} - 3\alpha (1 - \beta )\Delta {\beta _2}}}} \right){I^*} \hfill \\ \end{gathered}$$

(18)

.

Then the Jacobian matrix of the point is,

$$\begin{gathered} {J_1}({P_1})={\left[ {\begin{array}{*{20}{c}} { - \beta I - \alpha (1 - \beta )B - (1 - \alpha )(1 - \beta )R}&{ - \beta S}&{ - \alpha (1 - \beta )S} \\ {\beta I}&{\beta S - \Delta {\beta _2}}&0 \\ {\alpha (1 - \beta )B}&{\frac{{\Delta {\beta _2}}}{2}}&{\alpha (1 - \beta )S - \frac{{\Delta {\alpha _2}}}{3}\beta } \end{array}} \right]_{({S^ * },I_{r}^{ * },I_{t}^{ * },{B^ * },{R^ * })}} \hfill \\ ^{{}}\begin{array}{*{20}{c}} {}&{} \end{array}=\left[ {\begin{array}{*{20}{c}} { - \beta {I^*} - \alpha (1 - \beta ){B^*} - (1 - \alpha )(1 - \beta ){R^*}}&{ - \Delta {\beta _2}}&{ - \frac{{\alpha (1 - \beta )\Delta {\beta _2}}}{\beta }} \\ {\beta {I^*}}&0&0 \\ {\alpha (1 - \beta ){B^*}}&{\frac{{\Delta {\beta _2}}}{2}}&{\frac{{\alpha (1 - \beta )\Delta {\beta _2}}}{\beta } - \frac{{\Delta {\alpha _2}}}{3}\beta } \end{array}} \right] \hfill \\ \end{gathered}$$

(19)

.

Therefore, the eigenvalues of the solution to this formula are

$$\left\{ {\begin{array}{*{20}{c}} {{\lambda _1}= - \beta {I^*}<0} \\ \begin{gathered} {\lambda _2}+{\lambda _3}= - \Delta {\beta _2}+\frac{{\alpha (1 - \beta )\Delta {\beta _2}}}{\beta } - \frac{{\Delta {\alpha _2}}}{3}\beta <0 \hfill \\ {\lambda _2}{\lambda _3}= - \Delta {\beta _2}\left( {\frac{{\alpha (1 - \beta )\Delta {\beta _2}}}{\beta } - \frac{{\Delta {\alpha _2}}}{3}\beta } \right)+\frac{{\Delta {\beta _2}}}{2} \times \frac{{\alpha (1 - \beta )\Delta {\beta _2}}}{\beta }>0 \hfill \\ \end{gathered} \end{array}} \right.$$

(20)

.

From the eigenvalue relationship of the solution, it can be seen that all three eigenvalues are negative and follow the 3rd order Routh Hurwitz criterion. Therefore, when ${R_0} \geqslant 1$, the rumor propagation equilibrium point ${E_1}=({S^ * },I_{r}^{ * },I_{t}^{ * },{B^ * },{R^ * })$ of the model is stable.

Simulation

This chapter is divided into three main sections. The first section validates and compares the model using both constructed networks (e.g., BA network and small-world network) and real-world networks (e.g., Facebook and Twitter social networks). The second section analyzes and compares the similarities and differences between the rumor propagation model and the SIR infectious disease model. The third section examines the changes in key model parameters to investigate the dynamics of rumor propagation group states.

Model validation

Network description

BA Network: The BA network²⁷ is a stochastic connectivity model based on preferential attachment, a fundamental concept in complex network theory. Its core principle is that the likelihood of a node forming new connections is proportional to its existing degree. The BA network exhibits scale-free properties, small-world characteristics, and high clustering coefficients.This article forms a 500 scale BA network based on the generation rules of the BA network for subsequent simulation experiments.

Small-World Networks: Small-world networks²⁸ are a unique class of complex network structures where most nodes are not directly connected, yet any two nodes can be linked through a small number of steps. Their defining features include relatively short average path lengths and high clustering coefficients.This article forms a 500 scale small-world network based on the generation rules of the small-world network for subsequent simulation experiments.

Facebook Social Network: This dataset²⁹ comprises “circles” (or friend lists) from Facebook, collected from survey participants via a dedicated Facebook application. It includes node characteristics (profiles), circles, and ego networks. The detailed network structure is presented in Table 1.

Twitter Social Network: This dataset²⁹ includes “circles” (or lists) from Twitter, crawled from public sources. It consists of node characteristics (profiles), circles, and ego networks. The network structure is detailed in Table 2.

Table 2 Dataset statistics.

.

Fig. 3

Demonstration of network structure (BA network (top left), Small World Network (top right), Facebook social network (bottom left), Twitter social network (bottom right)).

Experimental process

Based on the model described above, the pseudo-code defining the functions for belief in the rumor (consciousness layer), propagation of the rumor (behavior layer), and network state updates is presented below:

Functions 1: whether to believe the rumor (consciousness layer)

Functions 2: whether to spread the rumor (behavior layer)

Functions 3: the network state update

In this study, within the four generated networks, a node is randomly selected as the rumor spreader, designated as the rumor source, while the states of all other nodes are initially set as unknown. The numbers of insensitive individuals and bystanders are set to zero, implying that all nodes are susceptible to the influence of the rumor. For the sake of observation, the time required for each individual to spread the rumor once is considered equal and defined as a unit time step. Based on the functions defined during the experiment, a series of comprehensive iterations are conducted to observe the scale of rumor propagation within the random networks, analyze the dynamic principles of rumor propagation, and investigate the impact of individual behaviors on network stability and the final structure of information propagation.

Experimental results

As illustrated in Fig. 4, the blue, orange, green, red, and purple lines represent the changes in the number of Rumor Spreaders, Anti-Rumor Spreaders, Bystanders, Insensitive Individuals, and Ignorant Individuals, respectively. In Fig. 4, the number of Rumor Spreaders and Anti-Rumor Spreaders exhibits a rapid initial increase, followed by a sharp decline after reaching a peak, eventually approaching zero. Prior to the emergence of official information, individuals cannot distinguish between rumors and anti-rumors, maintaining their own perspectives, which may shift over time. Consequently, the state change curves for Rumor Spreaders and Anti-Rumor Spreaders are similar. Individuals who believe the rumor but choose not to spread it are classified as Bystanders. The number of Bystanders increases as rumors and anti-rumors spread, driven by the transformation of Ignorant individuals upon initial exposure to rumors, as well as the conversion of Rumor Spreaders and Anti-Rumor Spreaders into Bystanders due to changes in their neighbors’ states and their own interest levels. Simultaneously, the number of Insensitive Persons—users who lose interest in believing or spreading the message—increases as the number of Ignorant Individuals, Rumor Spreaders, and Anti-Rumor Spreaders declines, eventually stabilizing. This phenomenon mirrors the rapid initial spread of a disease during an outbreak, followed by a decline in infection rates as preventive measures or treatments are implemented.

Fig. 4

BA network simulation modeling state change.

When a user spreads received rumor information to their network, the recipient often exhibits heightened interest due to initial exposure, leading to further propagation. However, as the information propagates repeatedly, the individual’s interest diminishes, or their surrounding contacts withdraw from the propagation process, causing the individual to cease forwarding the rumor. Thus, Fig. 4 effectively captures the dynamics of user states during rumor propagation in the absence of control measures or official counter-information.

Additionally, the number of Rumor Spreaders and Anti-Rumor Spreaders exhibits a fluctuating trend, likely due to the challenges in distinguishing the source or authenticity of rumors and false information, leading to repeated cycles in the propagation process. In BA networks, rumor and anti-rumor information propagate rapidly around hub nodes, resulting in sharp peaks in the graph. This phenomenon can be attributed to the scale-free nature of BA networks, characterized by a small number of highly connected nodes.

Fig. 5

Small-world network simulation modeling state changes.

As illustrated in Fig. 5, the trends represented by the lines align with those observed in Fig. 4. The simulation results for small-world networks and BA networks are comparable, with similar trends observed in the numbers of Rumor Spreaders, Anti-Rumor Spreaders, Bystanders, Insensitive Individuals, and Ignorant Individuals as rumor and anti-rumor information propagate. In small-world networks, the propagation of rumors and anti-rumors is notably rapid, attributed to their high clustering coefficient and short average path lengths, which facilitate faster information propagation. In this model, the number of individuals in all categories decreases, particularly Ignorant and Insensitive Individuals. This indicates that rumors and anti-rumors in small-world networks propagate less efficiently and are less likely to trigger widespread panic or social instability.

However, it is observed that the time required for the number of spreaders to peak is shortest in small-world networks, followed by BA networks. This can be explained by the network structure: compared to BA networks, small-world networks have shorter average path lengths, enabling faster information propagation across nodes and quicker peak attainment.

Fig. 6

Facebook network simulation modeling state changes.

As illustrated in Fig. 6, the trends represented by the lines align with those observed in Fig. 4. The number of Ignorant Individuals declines rapidly after an initial peak, while other groups show a gradual increase at later stages. This indicates that on Facebook, the propagation of rumors and anti-rumors predominantly occurs in the early stages before gradually diminishing. Analyzing the changes in crowd states for rumor and anti-rumor propagation, multiple peaks are observed, suggesting rapid information propagation at specific time points. This phenomenon may arise from the highly community-structured nature of Facebook networks, where information spreads rapidly within specific friend circles before propagating across the broader network.

Fig. 7

Twitter network simulation modeling state changes.

As illustrated in Fig. 7, the trends represented by the lines align with those observed in Fig. 4. Similar to Facebook, the propagation of rumors and anti-rumors on the Twitter network is more active initially before gradually slowing down. The number of Rumor Spreaders and Anti-Rumor Spreaders initially increases before declining, whereas the number of Bystanders and Insensitive Individuals continues to rise. This highlights the time-sensitive nature of rumor and anti-rumor propagation on Twitter, as well as the heightened susceptibility of communicators to rumors. The Twitter network exhibits rapid information propagation with pronounced peaks, as information can be retweeted quickly. This phenomenon may arise from the presence of highly connected communities in the Twitter network, similar to Facebook, facilitating rapid and widespread information propagation.

The rapid increase in the number of Ignorant and Insensitive Individuals across all four networks—BA networks, Small-World networks, Facebook social networks, and Twitter social networks—demonstrates the rapid spread of rumors and the initial susceptibility of most individuals. Rumor Spreaders and Anti-Rumor Spreaders were observed in all four networks, serving as the primary agents of rumor propagation. The networks exhibit variations in the number of Rumor Spreaders and Anti-Rumor Spreaders, reflecting differences in their influence and communication strategies. Additionally, the speed of rumor propagation varies across networks. For instance, rumors spread faster in Facebook and Twitter networks but slower in BA networks. This variation is influenced by the network structure. The quantitative relationship between Rumor Spreaders and Anti-Rumor Spreaders also differs across networks. In Facebook and Twitter networks, Rumor Spreaders dominate, whereas in BA networks, Anti-Rumor Spreaders play a more significant role.

Comparative analysis

In the absence of official counter-information, individuals cannot distinguish between rumors and anti-rumors. Therefore, rumors and anti-rumors are treated as two opposing perspectives on the same information, and these two types of information spreaders can be regarded as belonging to the same category, referred to as “spreaders,” and can thus be analyzed collectively. On the other hand, bystanders and insensitive individuals, both of whom are unwilling to engage in further communication and no longer influence the spread of rumors, can be categorized as “recovered.” By employing this statistical approach, a comparison can be made with the traditional SIR model, thereby validating the effectiveness of the quantum-based method.

Fig. 8

State change diagram for the presence of only one propagator(BA network (top left), Small World Network (top right), Facebook social network (bottom left), Twitter social network (bottom right)).

Fig. 9

SIR model state change diagram.

Figures 8 and 9 illustrate the comparison between the traditional SIR model for infectious diseases and the quantum superposition-based rumor propagation model proposed in this study. In Fig. 3, the blue, orange, and green lines represent the change curves for the number of spresders (the sum of Rumor Spreaders and Anti-Rumor Spreaders), recoverers (the sum of Insensitive Individuals and Bystanders), and susceptibles (Ignorants), respectively. In Fig. 4, the purple, red, and green lines depict the change curves for the number of susceptibles, propagators, and recoverers in the SIR or SIRS model, respectively. By analyzing the trends, peaks, and intersection points of these two models, significant similarities are observed, validating the effectiveness of the proposed rumor propagation model.

However, compared to the traditional SIR model, the proposed model incorporates more granular considerations in simulating individual behavior. It not only accounts for individual decision-making at both behavioral and conscious levels but also more accurately captures the dynamic process of state transitions. Additionally, the model integrates realistic factors, including the historical behavior of individuals, fluctuations in rumor interest, and changes in rumor influence, enabling it to better approximate real-world scenarios and achieve a more accurate fit.

In summary, the quantum superposition-based rumor propagation model proposed in this study retains the core features of the traditional SIR model while significantly enhancing its accuracy in modeling real-world rumor propagation. By incorporating the complexity of individual decision-making, the influence of historical behaviors, and dynamic considerations of rumor interest and influence, it provides a more effective theoretical tool and analytical framework for studying and managing online rumor propagation.

Parameter adjustment

In this section, we adjust the thresholds for the consciousness and behavior layers in the quantum superposition-based rumor propagation model, examine the impact of varying thresholds on the model, and analyze its practical implications.

Consciousness layer

The model stipulates that an individual’s belief in a rumor is determined by averaging their user intimacy ${\alpha _1}$, rumor credibility ${\alpha _2}$, and discernment ability ${\alpha _3}$. If the average exceeds 0.5, the individual is more likely to believe the rumor; otherwise, they are less likely to believe it. Thus, this value $\alpha$ represents the average rumor belief level of society, reflecting the general public’s trust in unconfirmed information or rumors within a specific time frame and social context.

The average rumor belief level of a society is influenced by factors such as social culture, education level, information transparency, and crisis events. Different cultural backgrounds and education levels shape varying attitudes toward rumors, while increased information transparency and the occurrence of crisis events significantly influence rumor propagation and acceptance.

Next, the average rumor belief level of society is adjusted to examine the impact of this parameter on the scale of rumor propagation.

Fig. 10

Parameter tuning of social average rumor belief level (BA network (1), Small World Network (2), Facebook social network (3), Twitter social network (4)).

In Fig. 10, the blue, green, and red lines represent the social average rumor belief levels of 0.4, 0.5, and 0.6, respectively. The solid lines depict the changes in the number of Rumor Spreaders, while the dashed lines represent the changes in the number of Anti-Rumor Spreaders. From the four figures, it is evident that as the average rumor belief level of society increases, the number of Rumor Spreaders and Anti-Rumor Spreaders decreases. This phenomenon may result from the public’s enhanced ability to discern information, leading to greater caution when encountering information of uncertain veracity. The increased “threshold” for believing rumors and counter-rumors reflects society’s heightened immunity to rumors. Individuals are less likely to be swayed by unverified information and are more inclined to adopt a wait-and-see approach or disregard information of uncertain authenticity. This wait-and-see or dismissive attitude may arise from past experiences with frequent rumor propagation and a general skepticism toward information authenticity in an era of information overload. Consequently, society has become less susceptible to rumors and has reduced indiscriminate propagation, reflecting improved social information literacy and more mature information consumption practices.

The average rumor belief level is crucial for shaping public policy, maintaining social stability, and guiding public education efforts. It is not static but fluctuates with dynamic changes in the social environment and information propagation. Therefore, governments and media must continuously adapt their strategies to these changes, ensuring public access to accurate information and mitigating the harm caused by rumors.

Behavioral layer

The model stipulates that an individual’s decision to spread rumors is determined by averaging their historical behavior ${\beta _1}$ and current interest ${\beta _2}$ in rumors. If the average exceeds 0.5, the individual is more likely to spread rumors; otherwise, they are less likely to do so. Thus, this value $\beta$ represents society’s permissiveness toward rumors, reflecting the propensity of individuals to spread unverified information.

The degree of societal management of rumors is influenced by factors such as individual information literacy, social trust, media reporting tendencies, legal regulations, and psychosocial conditions. Individual information literacy determines the ability to identify and counteract rumors, while social trust influences judgments about information sources. Media reporting tendencies can either amplify or suppress rumor propagation, and legal regulations govern information propagation at an institutional level. Additionally, societal psychological states, particularly during crises or periods of uncertainty, significantly impact sensitivity to rumors and the willingness to disseminate them.

Next, the degree of societal management of rumors is adjusted to examine the impact of this parameter on the scale of rumor propagation.

Fig. 11

Parameter tuning of the degree of social management of rumors (BA network (1), Small World Network (2), Facebook social network (3), Twitter social network (4)).

In Fig. 11, the blue, green, and red lines represent the degree of social management of rumors set to 0.4, 0.5, and 0.6, respectively. The solid lines depict changes in the number of Rumor Spreaders, while the dashed lines represent changes in the number of Anti-Rumor Spreaders. From the four figures, it is evident that fluctuations in the number of Rumor Spreaders and Anti-Rumor Spreaders decrease as the degree of societal management of rumors increases. This reflects the enhanced effectiveness of societal information regulation and rumor control mechanisms. Stricter societal management of rumors has suppressed rumor propagation, while the propagation of counter-rumor information has become more organized, reducing the tendency to blindly follow rumors. This management enhances the stability of the information environment, fostering greater public composure in response to uncertain information and reducing fluctuations in communication behavior. Consequently, society has developed a more mature response to rumors, and the dynamics of information propagation have stabilized, reflecting improvements in social information governance and the optimization of the information ecosystem.

It not only reflects society’s general attitude toward information authenticity but also relates to maintaining social order and ensuring public safety. A low degree of governance may signal vulnerabilities in the social information environment and public distrust of authoritative information. Therefore, efforts are needed to enhance public information discernment through education, media responsibility, and legal regulation, thereby increasing social resilience and safeguarding harmony and stability.

Conclusion

Based on the principle of quantum superposition states, this study analyzes the conscious and behavioral choices individuals make when exposed to rumor information. Users are categorized into four possible states: Rumor Spreaders $\left| {11} \right\rangle$, Bystanders $\left| {10} \right\rangle$, Anti-Rumor Spreaders $\left| {01} \right\rangle$, and Insensitive Individuals $\left| {00} \right\rangle$. The individual wave function state superposition equation is formulated, and individual states probabilistically collapse into different conditions. Finally, the temporal changes in individual states are statistically analyzed, yielding a state transition diagram of rumor propagation based on quantum superposition theory.

This study innovatively applies the principle of quantum state superposition to rumor propagation, describing the decision-making process of individuals when exposed to rumor information and proposing a quantum superposition-based rumor propagation model. First, by analyzing individual propagation decisions at both conscious and behavioral levels using quantum superposition theory, quantum superposition states are employed to characterize the uncertainty and complexity of individuals’ decision-making during online rumor propagation. Second, the basic reproduction number, non-rumor equilibrium point, and rumor existence equilibrium point of the proposed model are derived, and their local stability is proven. Finally, the BA network, small-world network, Facebook social network, and Twitter social network are used to validate the results and compare them with the classical SIR model, confirming the rationality and validity of the proposed rumor propagation model. Additionally, analysis of changes in the average rumor belief level and the degree of societal rumor management reveals that higher average rumor belief levels, education levels, and information transparency increase the likelihood of individuals believing rumors. Higher levels of societal rumor management correlate with increased individual information literacy, stricter rumor control, suppressed rumor propagation, and more organized propagation of counter-rumor information, reducing the tendency to blindly follow rumors.

This study employs quantum states to describe individual behavior in rumor propagation, capturing the complexity and uncertainty of psychological and behavioral responses. Quantum states enable individuals to simultaneously exist in multiple possible states (e.g., believing, skeptical, opposing, or indifferent), with actual behavior reflecting a random collapse into one of these states. This approach better captures the dynamics and diversity of individual behaviors in rumor propagation, offering an innovative theoretical framework for studying complex human behaviors.

Moreover, societal attitudes and behaviors toward rumors constitute a complex social phenomenon, requiring collaborative efforts from multiple stakeholders for effective management and control. The media plays a crucial role in shaping public attitudes toward rumors, and responsible reporting combined with timely and accurate rumor debunking can significantly reduce rumor belief levels. Furthermore, enhancing legal and institutional frameworks is essential for curbing rumor propagation. Legal measures targeting rumor mongers and spreaders can enhance societal capacity to prevent and control rumors.

This study carries out statistical analysis by simulating the individual decision-making process. Future research will combine large-scale data with individual behavioral data (e.g., likes, retweets, etc.) to further identify rumor propagation trends and predict the evolution of related public opinion, thus providing support for timely management. Meanwhile, the average rumor propagation time is used as the unit propagation step in this study, and the factor of time will be considered in subsequent studies on rumor and anti-rumor information in the study. In addition, understanding the change of information propagation state in quantum networks is important for improving channel capacity and transmission efficiency³⁰. By describing the process of superposition and collapse of quantum states during information reception, the model proposed in this study is expected to provide a new theoretical basis for optimizing the information propagation path and enhancing the efficiency of quantum networks³¹.