# DR-SCIR Public Opinion Propagation Model with Direct Immunity and Social Reinforcement Effect

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## Abstract

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## 1. Introduction

## 2. SCIR Public Opinion Propagation Model

_{IR}indicates the probability that the communicator does not believe the public opinion for some reason, thus turns to be immune, that is, the repost immunity rate; and ${P}_{CR}$ indicates the probability that the hesitant contacts the information, and then converts to immune directly without propagating, that is the indirect immunity rate.

## 3. DR-SCIR Public Opinion Propagation Model

#### 3.1. Direct Immune

#### 3.2. Social Reinforcement Effect

#### 3.3. DR-SCIR Network Public Opinion Propagation Model

- (i)
- Assume that the total number of users remains constant during the spread of public opinion, corresponding to the network is the total number of nodes N remains unchanged, that is, at any time t, there will be:$$S(t)+C(t)+I(t)+R(t)=N$$
- (ii)
- In the process of public opinion propagation, the topology of the entire social network remains the same, that is, the friend relationship between users in the social network remains constant.
- (iii)
- Users treat the information from different friends equally, that is, under the effect of social reinforcement effect, the weight of the impact of each time they receive the information is the same.

#### 3.4. Transition Probability

- (i)
- When the unknown state S contacts the propagation state I, there will be three transition states for S: the first one is to transform to the hesitant state C with probability ${P}_{SC}$, the second one is to transform to the propagation state I with probability ${P}_{SI}$, and the last one is to transform to the immune state R with probability ${P}_{SR}$.
- (ii)
- When the hesitation state C contacts the propagation state I, one part will transforms to the propagation state I with probability ${P}_{CI}$, and the other part will transforms to the immune state R with probability ${P}_{CR}$.
- (iii)
- The propagation state I will transforms to the immune state R with probability ${P}_{IR}$.
- (iv)
- The immune state R will transform to the hesitant state C with probability ${P}_{RC}$ under the function of social reinforcement effect.

- (i)
- Suppose node j is in state S at time t; the relationship between propagation probabilities is given by$${P}_{SS}^{j}+{P}_{SC}^{j}+{P}_{SI}^{j}+{P}_{SR}^{j}=1$$Considering ${n}_{1}={n}_{1}(t)$ as the number of nodes at state I in the neighborhood of node j at time t, we can write the above probabilities as follows,$$\left\{\begin{array}{c}{P}_{SC}^{j}={(\mathsf{\Delta}t\cdot {P}_{SC})}^{{n}_{1}}\\ {P}_{SI}^{j}={(\mathsf{\Delta}t\cdot {P}_{SI})}^{{n}_{1}}\\ {P}_{SR}^{j}={(\mathsf{\Delta}t\cdot {P}_{SR})}^{{n}_{1}}\end{array}\right.$$Thus,$${P}_{SS}^{j}=1-[{(\mathsf{\Delta}t\cdot {P}_{SC})}^{{n}_{1}}+{(\mathsf{\Delta}t\cdot {P}_{SI})}^{{n}_{1}}+{(\mathsf{\Delta}t\cdot {P}_{SR})}^{{n}_{1}}]$$Supposing node j has k edges and ${n}_{1}$ is a random variable obeying the binomial distribution,$$\prod ({n}_{1},t)}=\left(\begin{array}{c}k\\ {n}_{1}\end{array}\right)\phi {(k,t)}^{{n}_{1}}+{(1-\phi (k,t))}^{k-{n}_{1}$$$$\phi (k,t)={\displaystyle \sum _{{k}_{1}}p({k}_{1}|k)}p({I}_{{k}_{1}}|{S}_{k})$$$p({k}_{1}|k)$ is a degree correlation function, which represents the conditional probability that a node with degree k is adjacent to a node with degree k
_{1}.$p({I}_{{k}_{1}}|{S}_{k})$ represents the probability that a node with k_{1}edges is in the propagation state I under the condition that it is connected to a susceptible node with degree k.Use ${p}^{I}({k}_{1},t)$ to represent the density of nodes in state I with degree of k_{1}at time t, and then $\phi (k,t)$ is approximately:$$\phi (k,t)={\displaystyle \sum _{{k}_{1}}p({k}_{1}|k)}{p}^{I}({k}_{1},t)$$Then, the average probability $\overline{{p}_{SC}}(k,t)$ that the S state node with degree k becomes state C within the time period $[t,t+\mathsf{\Delta}t]$ is:$$\overline{{p}_{SC}}(k,t)={\displaystyle \sum _{{n}_{1=0}}^{k}\left(\begin{array}{c}k\\ {n}_{1}\end{array}\right)\phi {(\mathsf{\Delta}t\cdot {P}_{SC})}^{{n}_{1}}\phi {(k,t)}^{{n}_{1}}{(1-\phi (k,t))}^{k-{n}_{1}}}$$Substituting Equation (12) into Equation (13), there is:$$\overline{{p}_{SC}}(k,t)={(\mathsf{\Delta}t\cdot {P}_{SC}\cdot {\displaystyle \sum _{{k}_{1}}p({k}_{1}|k)}{p}^{I}({k}_{1},t))}^{k}$$Similarly, the average probability $\overline{{p}_{SI}}(k,t)$ that the S state node with degree k becomes state I within the time period $[t,t+\mathsf{\Delta}t]$ is:$$\overline{{p}_{SI}}(k,t)={(\mathsf{\Delta}t\cdot {P}_{SI}\cdot {\displaystyle \sum _{{k}_{1}}p({k}_{1}|k)}{p}^{I}({k}_{1},t))}^{k}$$Similarly, the average probability $\overline{{p}_{SR}}(k,t)$ that the S state node with degree k becomes state R within the time period $[t,t+\mathsf{\Delta}t]$ is:$$\overline{{p}_{SR}}(k,t)={(\mathsf{\Delta}t\cdot {P}_{SR}\cdot {\displaystyle \sum _{{k}_{1}}p({k}_{1}|k)}{p}^{I}({k}_{1},t))}^{k}$$Thus, the average probability $\overline{{p}_{SS}}(k,t)$ that the S state node with degree k maintains its state in the time period $[t,t+\mathsf{\Delta}t]$ is:$$\begin{array}{c}\overline{{p}_{SS}}(k,t)=1-\overline{{p}_{SC}}(k,t)-\overline{{p}_{SI}}(k,t)-\overline{{p}_{SR}}(k,t)\\ =1-(\mathsf{\Delta}t\cdot {P}_{SC}\cdot {\displaystyle \sum _{{k}_{1}}p({k}_{1}|k)}{p}^{I}({k}_{1},t){)}^{k}\\ -(\mathsf{\Delta}t\cdot {P}_{SI}\cdot {\displaystyle \sum _{{k}_{1}}p({k}_{1}|k)}{p}^{I}({k}_{1},t){)}^{k}\\ -(\mathsf{\Delta}t\cdot {P}_{SR}\cdot {\displaystyle \sum _{{k}_{1}}p({k}_{1}|k)}{p}^{I}({k}_{1},t){)}^{k}\end{array}$$ - (ii)
- Suppose node j is in state C at time t; the relationship between propagation probabilities is given by$${P}_{CC}^{j}+{P}_{CI}^{j}+{P}_{CR}^{j}=1$$Specifically, there are$$\left\{\begin{array}{c}\overline{{p}_{CI}}(k,t)={P}_{CI}^{j}=\mathsf{\Delta}t\cdot {P}_{CI}\\ \overline{{p}_{CR}}(k,t)={P}_{CR}^{j}=\mathsf{\Delta}t\cdot {P}_{CR}\end{array}\right.$$Thus, the average probability $\overline{{p}_{CC}}(k,t)$ that the C state node with degree k maintains its state in the time period $[t,t+\mathsf{\Delta}t]$ is:$$\overline{{p}_{CC}}(k,t)=1-\mathsf{\Delta}t\cdot {P}_{CI}-\mathsf{\Delta}t\cdot {P}_{CR}$$
- (iii)
- Suppose node j is in state I at time t; the relationship between propagation probabilities are as followings,$${P}_{II}^{j}+{P}_{IR}^{j}=1$$$$\overline{{p}_{IR}}(k,t)={P}_{IR}^{j}=\mathsf{\Delta}t\cdot {P}_{IR}$$Thus, the average probability $\overline{{p}_{II}}(k,t)$ that the I state node with degree k maintains its state in the time period $[t,t+\mathsf{\Delta}t]$ is:$$\overline{{p}_{II}}(k,t)=1-\mathsf{\Delta}t\cdot {P}_{IR}$$
- (iv)
- Suppose node j is in state I at time t; the relationship between propagation probabilities is given by,$${P}_{RR}^{j}+{P}_{RC}^{j}=1$$Considering ${n}_{2}={n}_{2}(t)$ as the number of nodes at state I in the neighborhood of node j at time t,$${P}_{RR}^{j}={(1-\mathsf{\Delta}t\cdot {P}_{RC})}^{{n}_{2}}$$Supposing node j has k edges and ${n}_{2}$ is a random variable obeying the binomial distribution,$$\prod ({n}_{2},t)}=\left(\begin{array}{c}k\\ {n}_{2}\end{array}\right)\phi {(k,t)}^{{n}_{2}}+{(1-\phi (k,t))}^{k-{n}_{2}$$$$\phi (k,t)={\displaystyle \sum _{{k}_{1}}p({k}_{1}|k)}p({I}_{{k}_{1}}|{R}_{k})$$$p({I}_{{k}_{1}}|{R}_{k})$ represents the probability that a node with k
_{1}edges is in the propagation state I under the condition that it is connected to an immune node with degree k.Use ${p}^{I}({k}_{1},t)$ to represent the density of nodes in state I with degree of k_{1}at time t; then, $\phi (k,t)$ is approximately:$$\phi (k,t)={\displaystyle \sum _{{k}_{1}}p({k}_{1}|k)}{p}^{I}({k}_{1},t)$$Then, the average probability $\overline{{p}_{RC}}(k,t)$ that the R state node with degree k becomes state C within the time period $[t,t+\mathsf{\Delta}t]$ is:$$\overline{{p}_{RC}}(k,t)={\displaystyle \sum _{{n}_{2=0}}^{k}\left(\begin{array}{c}k\\ {n}_{2}\end{array}\right)\phi {(\mathsf{\Delta}t\cdot {P}_{rc})}^{{n}_{2}}\phi {(k,t)}^{{n}_{2}}{(1-\phi (k,t))}^{k-{n}_{2}}}$$Substituting Equation (28) into Equation (29), there is:$$\overline{{p}_{RC}}(k,t)={(\mathsf{\Delta}t\cdot {P}_{RC}\cdot {\displaystyle \sum _{{k}_{1}}p({k}_{1}|k){p}^{I}({k}_{1},t)})}^{k}$$Thus, the average probability $\overline{{p}_{RR}}(k,t)$ that the R state node with degree k maintains its state in the time period $[t,t+\mathsf{\Delta}t]$ is:$$\overline{{p}_{RR}}(k,t)=1-\overline{{p}_{RC}}(k,t)=1-{(\mathsf{\Delta}t\cdot {P}_{RC}\cdot {\displaystyle \sum _{{k}_{1}}p({k}_{1}|k){p}^{I}({k}_{1},t)})}^{k}$$

## 4. Simulation Analysis

#### 4.1. Simulation Dataset

#### 4.2. Impact of Propagation Probability Upper Limit T on Propagation

_{CR}= 0.2, P

_{SC}= 0.7, P

_{CI}= 0.5, P

_{IR}= 0.2, P

_{SI}= 0.2, P

_{SR}= 0.1, $\alpha =0.001$, and b = 10. The influence of different values of the upper limit T of the propagation probability on the density of various nodes is shown in Figure 4.

#### 4.3. Impact of Positive Social Reinforcement Effect Factor b on Propagation

#### 4.4. Impact of Negative Social Reinforcement Effect Factor b on Propagation

#### 4.5. Impact of Direct Immunization ${P}_{SR}$ on Propagation

#### 4.6. Comparative Analysis of SIR Model, SCIR Model and DR-SCIR Model

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Schematic diagram of three networks: (

**a**) random network; (

**b**) NW network; and (

**c**) BA network.

**Figure 4.**The number of four states nodes against time t with varying upper limit T: (

**a**) density change of S state node; (

**b**) density change of C state node; (

**c**) density change of I state node; and (d) density change of R state node. The final propagation scope with T = 0.01, 0.05, 0.1, 0.2, and 0.3, respectively. Other parameters are the same as the default settings.

**Figure 5.**The number of four states nodes against time t with varying positive social reinforcement effect factor b: (

**a**) density change of S state node; (

**b**) density change of C state node; (

**c**) density change of I state node; and (

**d**) density change of R state node. The final propagation scope with b = 0, 1, 2, 5, and 10, respectively. Other parameters are the same as the default settings.

**Figure 6.**The number of four states nodes against time t with varying negative social reinforcement effect factor b: (

**a**) density change of S state node; (

**b**) density change of C state node; (

**c**) density change of I state node; and (

**d**) density change of R state node. The final propagation scope with b = 0, –1, –2, –5, and –10 respectively. Other parameters are the same as the default settings.

**Figure 7.**The number of four states nodes against time t with varying P

_{SR}: (

**a**) density change of S state node; (

**b**) density change of C state node; (

**c**) density change of I state node; and (

**d**) density change of R state node. The final propagation scope with P

_{SR}= 0, 0.1, 0.2, 0.3, and 0.5, respectively. Other parameters are the same as the default settings.

**Figure 8.**The number of four states nodes against time t with different models: (

**a**) density change of S state node; (

**b**) density change of C state node; (

**c**) density change of I state node; and (

**d**) density change of R state node. The final propagation scope with SIR model, SCIR model, DR-SCIR model with negative b, and DR-SCIR model with positive b, respectively. Other parameters are the same as the default settings.

Name | Node | Edge | Average Degree | Max Degree | Min Degree | Average Path Length | Average Clustering Coefficient |
---|---|---|---|---|---|---|---|

Random | 1000 | 499,345 | 499.354 | 548 | 440 | 1.501 | 1.501 |

NW | 1000 | 105,013 | 105.013 | 126 | 86 | 1.895 | 0.105 |

BA | 1000 | 7981 | 7.981 | 230 | 8 | 3.233 | 0.029 |

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## Share and Cite

**MDPI and ACS Style**

Li, W.; Guo, T.; Wang, Y.; Chen, B.
DR-SCIR Public Opinion Propagation Model with Direct Immunity and Social Reinforcement Effect. *Symmetry* **2020**, *12*, 584.
https://doi.org/10.3390/sym12040584

**AMA Style**

Li W, Guo T, Wang Y, Chen B.
DR-SCIR Public Opinion Propagation Model with Direct Immunity and Social Reinforcement Effect. *Symmetry*. 2020; 12(4):584.
https://doi.org/10.3390/sym12040584

**Chicago/Turabian Style**

Li, Weidong, Tianyi Guo, Yunming Wang, and Bo Chen.
2020. "DR-SCIR Public Opinion Propagation Model with Direct Immunity and Social Reinforcement Effect" *Symmetry* 12, no. 4: 584.
https://doi.org/10.3390/sym12040584