Modeling and threshold algorithm of rumor propagation system under random switching perspective

Abstract

The present invention relates to the technical field of rumor propagation prediction, and discloses a rumor propagation system modeling method under a random switching perspective, which includes the following steps: classifying the population in the social network rumor model, drawing a relationship flow chart of each population, constructing a differential equation expression of the rumor model, finding the rumor-free equilibrium point and rumor-containing equilibrium point of the model based on whether the rumor is popular, calculating the basic reproduction number, and further explaining the switching model. The present invention first proposes a hypothesis: assuming that there is a social network composed of N individuals, the individuals can be regarded as nodes in the network, and the relationships between individuals can be regarded as edges between network nodes. In the entire network, the switching model is further explained using a nonlinear coupling relationship, adding a Markov chain to the contact infection rate, and adding anti-rumor agents to the traditional model, which is more in line with practical situations.

Description

Technical Field

[0001] The present invention relates to the technical field of rumor propagation prediction, and in particular to a rumor propagation system modeling method under random switching perspectives. Background Art

[0002] With the rapid development of new media, people have more channels for accessing information, and social networks have become increasingly complex. Through WeChat official accounts, Douyin (TikTok), and Weibo, people easily access a vast amount of fragmented information, often mixed with pseudoscience and false information. The spread of such information can negatively impact social order. Therefore, effectively predicting rumor spread trends can help governments and relevant social platforms take countermeasures and protect public resources. Rumor propagation models are based on infectious disease models. Since the pioneering work of Kermack and McKendrick, mathematical models have become a crucial tool for understanding the spread and control of infectious diseases. For SIR bins, they divide the population of a region into three bins for a given infectious disease: susceptible, infected, and removed. The form of rumor propagation is similar to that of disease propagation. In 1964, Daley-Kendal proposed a classic DK rumor propagation model; in 1973, Maki-Thomson proposed the MT rumor propagation model, which revealed the basic laws of rumor propagation while also pointing out the difference between rumor propagation and disease propagation; in 2017, Li et al. proposed the SIRS infectious disease model with Markov switching; in 2012, Zhao et al. combined the network topology of rumor propagation and studied the SIQR rumor propagation model with a suspicion mechanism; in 2020, Zhuang et al. studied the rumor propagation stage, the two stages before and after the coexistence of rumors and counter-rumors, and compared the simulation of the model with the data of real rumor cases.

[0003] Based on this situation, and to address the omissions of the previous model, the IMDS rumor propagation model was proposed. This model breaks down the population into four categories: ignorant, misinformation spreaders, debunkers, and stiflers. This paper focuses on adding debunkers to the traditional model to achieve a more realistic scenario. Furthermore, we explored the nonlinear coupling relationship between rumor transmission among different groups of people, adding a Markov chain to the contact transmission rate. This model, when applied to social networks, provides a more realistic picture of rumor propagation. Summary of the Invention

[0004] The purpose of this invention is to provide a rumor propagation system modeling method under a random switching perspective, and to provide a theoretical basis for preventing and controlling rumors through a rumor propagation model that is closer to the actual situation.

[0005] To achieve the above object, the present invention provides the following technical solutions:

[0006] A rumor propagation system modeling method under random perspective switching includes the following steps:

[0007] S1: Classify the people in the social network rumor model and draw a relationship flow chart for each group of people.

[0008] S2: Construct the differential equation expression of the rumor model.

[0009] S3: Depending on whether the rumor is popular, find the model's equilibrium point with and without rumors.

[0010] S4: Calculate the basic reproduction number and further explain the switching model.

[0011] Preferably, each type of people in S1 is represented by a node, so there are four types of people, divided into: the ignorant (I), the group who do not know about the rumors but are easily affected; the rumor spreaders (M), the group who hear the rumors and spread them; the anti-rumor people (D), the group who hear the rumor information and fight back against the rumor information based on their own experience and knowledge; the silent people (S), the group who are no longer interested in the information. The dynamic transfer relationship between different groups of people is represented by links with arrows (i.e., lines between nodes), and an overall relationship diagram is obtained.

[0012] Preferably, the IMDS model can construct the following kinetic equation:

[0013]

[0014] In the equation, ∧ is the number of people moving into the ignorant node per unit time, μ is the migration rate of each type of node; α represents the probability that an ignorant person becomes a rumor spreader when seeing online rumor information; β represents the probability that an ignorant person becomes an anti-rumor spreader when seeing online rumor information; the subscript r(t) of α and β is a Markov chain that can take values in the finite state space M = {1,2,...,N}; γ represents the probability that an ignorant person becomes an anti-rumor spreader when seeing online anti-rumor information; f1(I,M) and f2(I,D) are general functions that can be expressed in specific forms; represents the probability that a spreader becomes a silent person; represents the probability that a spreader becomes a silent person.

[0015] Preferably, the switching of the social network of the IMDS model is a time-continuous Markov chain, the social network takes values in the finite state space M = {1, 2, ..., N}, and the transition probability P can be expressed as q e,e' Indicates that state e and state e' represent the state of the system at time t and time t+1 respectively. When the state at the next moment is different from the state at this moment, q e,e' ≥0, when the state at the next moment is the same as the state at this moment, q e,e' =-∑e'≠eq e,e' .

[0016]

[0017] Preferably, in the search for the equilibrium point, the right side of the differential equation is set to zero. According to the solution, three equilibrium points can be obtained, namely, the equilibrium point E0 = (I0, 0, 0, 0) with only ignorant people, the equilibrium point E1 = (I1, 0, D1, S1) without rumor spreaders, and the equilibrium point E2 = (I2, S2, D2, S2) with persistent rumors. The coordinate values within the equilibrium points are all constants, and the threshold R0 is calculated. By controlling the parameter strategy to make R0 < 1, the number of rumor spreaders and anti-rumor spreaders tends to zero, and the rumor information will automatically disappear after a period of time; if R0 > 1, the rumor information will continue to be popular.

[0018] Compared with the prior art, the present invention has the following beneficial effects:

[0019] This rumor propagation system modeling method from a random switching perspective uses a threshold algorithm to classify the people in the social network rumor model and draw a relationship flow chart for each group; constructs a differential equation expression for the rumor model; finds the model's rumor-free equilibrium point and rumor-containing equilibrium point based on whether the rumor is popular; calculates the basic reproduction number, further explains the switching model, and adds anti-rumor agents to the traditional model to conform to more realistic situations. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0021] Figure 1 This is a schematic diagram of the overall process of the rumor propagation system modeling method according to an embodiment of the present invention;

[0022] Figure 2 This is a flow chart of the conversion relationship among groups of people in the rumor propagation system modeling method of the present invention. DETAILED DESCRIPTION

[0023] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0024] Example:

[0025] See also Figure 1-2 The rumor propagation system modeling method under random switching perspective provided by the present invention includes the following steps:

[0026] S1: Classify the people in the social network rumor model and draw a relationship flow chart of each group. First, make a hypothesis. Suppose there is a social network composed of N individuals. Individuals can be regarded as nodes in the network, and the relationship between individuals can be regarded as edges between network nodes. In the entire network, people can be divided into four categories: the group that does not know the rumor information but is easily affected is called the ignorant (I); the group of people who believe in rumors and become rumor spreaders (M); the group of people who have the ability to judge the truth of rumors or already know the truth and come forward to refute the rumors are called anti-rumor groups (D); the group of people who lose interest in rumors over time and do not participate in the discussion are called silent people (S).

[0027] Based on this assumption, we can get a flow chart of the conversion relationship between different groups of people in the rumor spreading organization, as shown in the following figure: Figure 1 shown.

[0028] Parameter description:

[0029] ∧: the number of people moving into the ignorant node per unit time;

[0030] μ: the migration rate of each type of node;

[0031] α: When an ignorant person sees online rumor information, he or she becomes a rumor spreader with a probability of α;

[0032] β: The probability that an ignorant person will become an anti-rumor advocate with a probability of β when seeing online rumor information;

[0033] r(t): is a Markov chain that can take values in a finite state space M = {1, 2, ..., N};

[0034] γ: The probability that an ignorant person will become a rumor counter-reactor with a probability of γ when seeing anti-rumor information on the Internet;

[0035] η: the probability that a rumor spreader (M) becomes a silent person (S);

[0036] δ: the probability that a rumor counter-spreader (D) becomes a silent one (S);

[0037] f1(I,M): represents the contact coefficient of I and M, and the function f1(I,M)≤I·C1(M);

[0038] f2(I,D): represents the contact coefficient of I and D, and the function f2(I,D)≤I·C2(D);

[0039] C1(M) and C2(D) are continuous functions, and their derivatives are not less than zero.

[0040] It should be noted that α is a contact infection rate in the finite state space M. Due to the irreducibility of the Markov chain, the probability distribution of the contact infection rate in each space M is fixed, and ∑M=1.

[0041] S2: Construct the differential equation expression of the rumor model. All nodes of the rumor propagation model are positive numbers, and all parameters are positive. The amount of outflow from the node is represented by a "-" sign, and the amount of inflow into the node is represented by a "+" sign. The following differential dynamic equation can be constructed:

[0042]

[0043] S3, according to whether the rumor is popular, find the model's equilibrium point with and without rumors.

[0044] Through calculation, we know that there are three equilibrium points in the system of this model, namely, the rumor-free equilibrium point The equilibrium point where rumor counterattacks continue to exist is E1 (I1, 0, D1, S1), and the equilibrium point where rumor spreaders continue to exist is E2 = (I2, S2, D2, S2).

[0045] The first three equations do not contain the silent operator (S), so we discuss the first three equations of the differential equation system.

[0046]

[0047] The Jacobian matrix of the above formula is:

[0048]

[0049] In order to consider the influence of various parameters on the stability of each equilibrium point, the following assumptions are made:

[0050]

[0051]

[0052]

[0053]

[0054]

[0055] Theorem 1: When conditions (H1) to (H2) are satisfied, the equilibrium point E0 is locally asymptotically stable.

[0056] Proof at the equilibrium point The Jacobian matrix at is:

[0057]

[0058] The characteristic equation at the equilibrium point E0 can be written as:

[0059]

[0060] Therefore, the characteristic equation has three characteristic roots:

[0061] λ1=-μ

[0062]

[0063]

[0064] Since μ is a positive real number, λ1<0, if condition (H1) is met, That is, λ2<0

[0065] If condition (H2) is met, That is, λ3<0

[0066] At this time, all the characteristic roots of j(E1) have negative real parts. According to stability theory, it is locally asymptotically stable at the equilibrium point E0.

[0067] Theorem 2: When conditions (H3) to (H5) are satisfied, the equilibrium point E1 is locally asymptotically stable.

[0068] Prove that E1(I1,0,D1,S1) is the equilibrium point without rumor spreaders (M). The relationship between the ignorant, rumor counter-spreaders, and silent people is expressed as:

[0069]

[0070]

[0071] S1=δD1.

[0072] The Jacobian matrix at E1(I1,0,D1,S1) is:

[0073]

[0074] The characteristic equation at the equilibrium point E1 can be written as:

[0075] (α r(t) C1(I1)-(η+μ)-λ)

[0076] [λ 2 +(γC2'(I1)D1+2μ+δ-γC2(I1))λ+(δ+μ)γC2'(I1)D1-μγC2(I1)-μδ+μ 2 ]=0

[0077] The characteristic equation has a characteristic root:

[0078] λ1=∑ e∈M α e C1(I1)-(η+μ), the other two characteristic roots λ2, λ3 satisfy the equation:

[0079]

[0080] If condition (H3) is met, ∑ e∈M π e α e C1(I1)-(η+μ)<0, that is, λ1<0. If conditions (H4) ~ (H5) are established, γC2'(I1)D1+2μ+δ-γC2(I1)>0, (δ+μ)γC2'(I1)D1-μγC2(I1)-μδ+μ 2 >0, that is, λ2>0,λ3<0. At this time, all the characteristic roots of j(E1) have negative real parts. According to stability theory, the equilibrium point E1 is locally asymptotically stable.

[0081] E2=(I2,M2,D2,S2) is the equilibrium point where rumors persist. The relationship between the ignorant, rumor spreaders, rumor counter-spreaders, and silent people is expressed as:

[0082]

[0083]

[0084]

[0085]

[0086] When the function f is a deterministic function, the local asymptotic stability conditions of the equilibrium point E2 can be calculated through the stability criterion.

[0087] S4: Calculate the basic reproduction number and further explain the switching model. In order to consider the impact of various parameters on the global stability of the rumor-free equilibrium point, the following assumptions are made:

[0088]

[0089]

[0090] Taking the logarithm of the second equation:

[0091] Integrating the above inequality, we can obtain from Birkhoff's ergodic theorem:

[0092]

[0093] If (H6) is satisfied, then the right side of the above inequality satisfies That is, limM(t)=0 under any initial conditions.

[0094] For the third equation:

[0095]

[0096] Since (H6) satisfies, there exists M(t)<ε

[0097]

[0098] According to the comparison theorem:

[0099]

[0100] If (H7) is satisfied, then the right side of the above equation satisfies Then the system (1) under any initial conditions:

[0101] If hypotheses (H6) to H7) are met, the number of silent and ignorant people can be calculated as follows: Therefore, the basic reproduction number of the model is:

[0102] When R0<1, we do not need to take any human measures and the rumor information will automatically die out; when R0>1, the rumor spreaders may continue to exist. At this time, the spread of rumors can be controlled by reducing the contact infection rate α or the migration rate η of the rumor spreaders.

[0103] The present invention first proposes a hypothesis: assuming that there is a social network composed of N individuals, the individuals can be regarded as nodes in the network, and the relationships between individuals can be regarded as edges between network nodes. In the entire network, the switching model is further explained, using a nonlinear coupling relationship, the contact infection rate plus the Markov chain, and the anti-rumor agent is added on the basis of the traditional model, which is more in line with the actual situation.

[0104] While embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions, and variations may be made to these embodiments without departing from the principles and spirit of the invention, and that the scope of the invention is defined by the appended claims and their equivalents.

Claims

1. A method for modeling a rumor propagation system under random switching perspectives, characterized by: The following steps are involved: S1: Classify the groups of people in the social network rumor model and draw a relationship flow chart for each group of people; S2: Construct the differential equation expression of the rumor model; S3: Find the model’s equilibrium point with and without rumors, depending on whether the rumor is popular. S4: Determine the basic reproduction number and further explain the switching model; In S2, the IMDS model constructs the following kinetic equations: In the equation, ∧ is the number of people moving into the ignorant node per unit time, μ is the migration rate of each type of node; α represents the probability that an ignorant person will become a rumor spreader when seeing online rumor information; β represents the probability that an ignorant person will become a rumor counter-spreader when seeing online rumor information; the subscript r(t) of α and β is a Markov chain that can take values in the finite state space M = {1,2,...,N}; γ represents the probability that an ignorant person will become a rumor counter-spreader when seeing online rumor counter-spreader information; f1(I,M) and f2(I,D) are general functions that can be expressed in specific forms; η represents the probability that a spreader will become a silent person; δ represents the probability that a counter-spreader will become a silent person; μ represents the migration rate of each type of node; In S4, in order to consider the impact of various parameters on the global stability of the rumor-free equilibrium point, the following assumptions are made: Taking the logarithm of the second equation: Integrating the above inequality, we can obtain from Birkhoff's ergodic theorem: If (H6) is satisfied, then the right side of the above inequality satisfies ∑ e ∈ M π e (α e C1(I)-(η+μ)), that is, limM(t)=0 under any initial value conditions; For the third equation: Since (H6) satisfies, there exists M(t)<ε According to the comparison theorem: If (H7) is satisfied, then the right side of the above equation satisfies Then the system under any initial conditions: If hypotheses (H6) to (H7) are met, the number of silent and ignorant people can be calculated as follows: Therefore, the basic reproduction number of the model is: In the equation, C1 and C2 are continuous functions, and their derivatives are not less than zero; f1(I,M) represents the contact coefficient of I and M, and the function f1(I,M)≤I·C1(M); f2(I,D) represents the contact coefficient of I and D, and the function f2(I,D)≤I·C2(D).

2. The rumor propagation system modeling method under random switching perspective according to claim 1 is characterized by: Each type of people in S1 is represented by a node, so there are four types of people, divided into: the ignorant (I), the group who do not know about the rumors but are easily affected; the rumor spreaders (M), the group who hear the rumors and spread them; the anti-rumor people (D), the group who hear the rumor information and fight back against the rumor information based on their own experience and knowledge; the silent people (S), the group who are no longer interested in the information. The dynamic transfer relationship between different groups of people is represented by links with arrows (i.e., lines between nodes), and an overall relationship diagram is obtained.

3. The rumor propagation system modeling method under random switching perspective according to claim 1 is characterized by: The switching of the social network in the IMDS model is a right-continuous Markov chain over time. The social network takes values in the finite state space M = {1, 2, ..., N}. The transition probability P can be represented by qe,e'. State e and state e' represent the state of the system at time t and time t+1 respectively. When the state at the next moment is different from the state at this moment, qe,e' ≥ 0. When the state at the next moment is the same as the state at this moment, qe,e' = -∑e' ≠eqe,e' 4. The rumor propagation system modeling method under random switching perspective according to claim 1 is characterized by: To find the equilibrium point, let the right side of the differential equation be zero. According to the solution, three equilibrium points can be obtained, namely the equilibrium point E0 = (I0, 0, 0, 0) with only ignorant people, the equilibrium point E1 = (I1, 0, D1, S1) without rumor spreaders, and the equilibrium point E2 = (I2, S2, D2, S2) with persistent rumors. The coordinate values within the equilibrium points are all constants. The threshold R0 is calculated and given by controlling the parameter strategy to make R0 < 1. The number of rumor spreaders and anti-rumor spreaders will tend to zero, and the rumor information will automatically disappear after a period of time; if R0 > 1, the rumor information will continue to be popular.

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