I-SFI model of propagation dynamic based on user’s interest intensity and considering birth and death rate

ABSTRACT

Everyone has a different level of interest in a trending topic posted on social media, which may affect user’s behaviour. In order to find out the way it affects the process of information transmission, we construct an interest intensity-based susceptible-forwarding-immune$\left(\mathit{I}-\mathit{SFI}\right)$ propagation dynamic model and two parameters birth rate$\mathit{A}$ and death rate$\mathit{\mu }$ are introduced to represent the users who newly join the group of disseminated information and the users who leave this population. And we give different birth rates to people with various levels of interest, which helps us to determine the interest intensity of potential users to a certain extent. We use the forwarding data of the real topic on Chinese Sina-microblog for data fitting, which can accurately parameterize the model and quantify the impact of interest intensity. And sensitivity analyses also give some strategies for increasing the impact of information dissemination process from the perspective of interest intensity.

KEYWORDS:

• Dynamic model
• Information propagation
• Interest intensity

1. Introduction

In the information network era, each of us is exposed to all kinds of messages posted on the Internet every day. Different people have different levels of interest in the same topic, which affects whether a user selects to retweet the post he has viewed. If we consider forwarding as the act of participating in the information dissemination process, then our level of interest in the topic will affect the forwarding rate and thus the whole communication process. For the same piece of information on a specific theme released on the Internet, due to the difference on users’ interest intensity, the forwarding rate will vary. Hence, studying how interest intensity affects information dissemination in social media supports the necessity for platforms to recommend information on specific topics to people with different interest profiles, and also provides ideas for improving the efficiency of information dissemination.

When an original post owner publishes a blog on the web, some people are very interested in the topic of the blog posted, some find it less interesting, and others are somewhere in between them in terms of interest. So we divide the susceptible users into three groups, respectively strong-interest susceptible users, medium-interest susceptible users and weak-interest susceptible users. After reading the message these three groups of users will retweet at different forwarding probabilities. At the same time, there will be some people with different interest intensity become new susceptible users, and some people who belong to the group will leave at a certain speed. The whole process is shown in Figure 1.

Figure 1. A schematic diagram depicting the process of information dissemination among users with different degrees of interest intensity.

It is always a popular challenge to build models describing the information dissemination process to explore the mechanism of different types of messages such as news and rumours. There are many new and relevant studies in this area. For instance, Xiao et al. [1] proposed a quantitative rumour & rumouranti-rumour mutual influence model, using a model of unsupervised GAN to enhance homomorphism data in the sample space and introducing evolutionary game theory. Zhang et al. [2] studied a predator-prey model with constant delay in both predator and prey equations and applied the proposed model to the underlying relationship between the existing rumour propagating through social media and the related authoritative information containing the truth broadcast. Li et al. [3] proposed a comprehensive macromodel based on the tripartite cognitive game and the rumour and antirumour and stimulate-rumour information dissemination model.

Since the process of information transmission is similar to the spread of infectious disease in the population, many infectious disease epidemiological models are applied to profile information propagation. For instance, the susceptible-infected$\mathit{\hspace{0.17em}}$(SI) model [4,5], the susceptible-infected-recovered (SIR) epidemic model [6,7], the susceptible-exposed-infected-recovered (SEIR) epidemic model [8,9] and the susceptible-exposed-susceptible (SIS) epidemic model [10,11] all have been applied. In order to make the model more in line with the real situation of information dissemination and show the characteristics of the information transmission, many scholars introduce new influence factors into their models or use different ways to categorize the users involved in information dissemination. Zhao et al. [12] develop a new rumor spreading model, susceptible-infected-hibernator-removed (SIHR) model by taking account of the mutual effect of forgetting and remembering mechanisms and adding a new kind of people-Hibernators. On the basis of prior studies, Xia et al. [13] established susceptible – exposed – infected – removed (SEIR) model with hesitating mechanism by considering the attractiveness and fuzziness of the content of rumors. Cheng et al. [14] propose a modified ISR rumor spreading model by taking into account the activity and infectivity of nodes and propagation environment is introduced. In 2019, Yin et al. [15] constructed susceptible-forwarding-immune$($SFI$)$ propagation dynamic model to adapt to the information dissemination mechanism of Sina. Then in the basis of SFI model, they introduce key factors such as opinion-delay, emotional and cross-transmission behaviours to establish OD-SFI model [16], E-SFI model [17] and CT-SFI model [18] separately.

In 2005, Iamnitchi et al. [19] mentioned that users in the social network will form groups of common interest naturally, which indicates that interest is a significant influencing factor in the process of information dissemination. In 2014, Ciobanu et al. [20,21] first presented a basic interest-based dissemination algorithm and a kind of algorithm named ONESIDE was put forward to indicate that network nodes tend to combine together due to interests. And Ciobanu et al. [22] proposed an interest-based dissemination framework for opportunistic networks entitled Interest Spaces in 2015. Then in 2017 Zhao et al. [23] conducted a further research, they focused on how the individual interests’ changing behaviour impacts the dynamics of information propagation. In 2018, Gan et al. [24] analysed the throughput of data dissemination at the level of users’ interests and concluded that users’ interests have the ability to drastically improve upon existing throughput scaling’s established. In 2021, based on the conclusions of previous studies [25,26] that the information dissemination in the real world is affected by many social factors, such as the interest of the topic, Li et al. [27] proposed a new public opinion evolution HK – SEIR model which combines the opinion fusion HK and the epidemic transmission SEIR models, and the topic interest degree was added to the model. It is known from previous studies that the interest intensity is an important influencing factor in the communication process, and considering the interest factor in the model is a decision that is more in line with the actual transmission process. Thus we introduce interest intensity into the SFI [15] model to build new model to investigate impact mechanisms.

Most dynamic models of information dissemination assume that the population size is consistent, they would consider birth rate equal to death rate or even not take birth rate and death rate into account. However, in the real situation, potential users on the communication platform join the group, and those who in the group remove for reasons that they actively cancel or are banned, etc. The population size of the group during transmission is constantly changing, so it is necessary to introduce two parameters in the model, birth rate and death rate [28], indicating the joining of potential users and the leaving of users in the cluster, respectively.

To the best of our knowledge, there is no dynamic model that includes the factor of user interest intensity and takes into account the dynamics of the research group’s size. Considering that we develop the interest intensity-based susceptible-forwarding-immune (I-SFI) propagation dynamic model and use forwarding data to parameterize the model.

This article is organized as follows: in model description, we formulate the $\mathit{I}-\mathit{SFI}$ model and analyse it in details; in data fitting, we perform the data fitting analysis based on real data from Chinese Sina-microblog to verify the feasibility of model; in parameter sensitivity analyses, discussions on the basis of sensitivity analyses are introduced which can help us propose better strategies for information dissemination and increase the visibility of your message.

2. Model description

On the basis of SFI model framework, we develop our interest intensity-based susceptible-forwarding-immune$\left(\mathit{I}-\mathit{SFI}\right)$ propagation dynamic model as shown in Figure 2. In which only the accessible people in the process of information dissemination and the diffusion of information generated by the forwarding behavior of individuals are considered.

Figure 2. A schematic describing the information dissemination process considering the intensity of interest and the dynamics of the study population.

Previously, other models based on SFI-model have been proposed, such as the delayed opinion susceptibility immunity (OD-SFI) [16] model which emphasizes on the influence of opinion leaders’ opinions on the information dissemination process. the emotion-based susceptible-forwarding-immune (E-SFI) [17] model that focuses on the impacts of different emotional categories and the emotional choices of user communities, the cross‐transmission susceptible-forwarding-immune (CT-SFI) model [18] that concentrates on the cross-transmission effects, and so on. As distinct from them, our $\mathit{I}-\mathit{SFI}$ model is more concerned with the impact of differences in interest intensity on the propagation process.

In this model, we stratify the population$\mathit{N}$ into five categories. Each individual only belongs to a unique state at any given time and since the dynamics of the population size is taken into account, the value of $\mathit{N}$ does not remain constant. The definitions for the five categories are listed in Table 1.

Table 1. Five groups divided according to three states (susceptible, forwarding and immune) and interest intensity.

Group	Interpretation
$S_1$	In which people are strong-interest susceptible users. They are not exposed to the information, but have access and are susceptible to forwarding behavior.
$S_2$	In which people are medium-interest susceptible users. They are not exposed to the information, but have access and are susceptible to forwarding behavior.
$S_3$	In which people are weak-interest susceptible users. They are not exposed to the information, but have access and are susceptible to forwarding behavior.
$\mathit{F}$	In which users have already retweeted and they still have the ability to influence the individuals in those three susceptible states.
$\mathit{I}$	In which individuals consist of two parts: those who have already forwarded will not forward again and no longer have the ability to influence other susceptible users, and those susceptible users with varying levels of interest in the message remain not forwarding after exposure.

Table 2 illustrates all the parameters and corresponding interpretations in our$\mathit{I}-\mathit{SFI}$ model, which is listed as below.

Table 2. Parameters definition.

Parameter	Interpretation
$\mathit{\beta }$	The average exposure rate that a susceptible user can be exposed to the information, which is the same for susceptible users with varying levels of interest.
$p_1$	The average probability that a strong-interest susceptible user will forward the information after being exposed to it.
$p_2$	The average probability that a medium-interest susceptible user will forward the information after being exposed to it.
$p_3$	The average probability that a weak-interest susceptible user will forward the information after being exposed to it.
$A_1$	The average entering rate to the group at which an individual newly become the strong-interest susceptible user.
$A_2$	The average entering rate to the group at which an individual newly become the medium-interest susceptible user.
$A_3$	The average entering rate to the group at which an individual newly become the weak-interest susceptible user.
$\mathit{\mu }$	The average removing rate of individuals within the population break away from the group, which is the same for the five categories.
$\mathit{\alpha }$	The average rate related to the behavior law of group users at which a user in the forwarding state become inactive to forwarding.

Our $\mathit{I}-\mathit{SFI}$ model takes the form of the following systems of differential equations:

(1) $\left\{\begin{array}{c}\frac{d{S}_1\left(\mathit{t}\right)}{\mathit{dt}}=A_1-\mathit{\beta }{S}_1\left(\mathit{t}\right)\mathit{F}\left(\mathit{t}\right)-\mathit{\mu }{S}_1\left(\mathit{t}\right)\\ \frac{d{S}_2\left(\mathit{t}\right)}{\mathit{dt}}=A_2-\mathit{\beta }{S}_2\left(\mathit{t}\right)\mathit{F}\left(\mathit{t}\right)-\mathit{\mu }{S}_2\left(\mathit{t}\right)\\ \frac{d{S}_3\left(\mathit{t}\right)}{\mathit{dt}}=A_3-\mathit{\beta }{S}_3\left(\mathit{t}\right)\mathit{F}\left(\mathit{t}\right)-\mathit{\mu }{S}_3\left(\mathit{t}\right)\\ \frac{\mathit{dF}\left(\mathit{t}\right)}{\mathit{dt}}=p_1\mathit{\beta }{S}_1\left(\mathit{t}\right)\mathit{F}\left(\mathit{t}\right)+p_2\mathit{\beta }{S}_2\left(\mathit{t}\right)\mathit{F}\left(\mathit{t}\right)+p_3\mathit{\beta }{S}_3\left(\mathit{t}\right)\mathit{F}\left(\mathit{t}\right)\\ -\mathit{\alpha F}\left(\mathit{t}\right)-\mathit{\mu F}\left(\mathit{t}\right)\\ \frac{\mathit{dI}\left(\mathit{t}\right)}{\mathit{dt}}=\left(1-p_1\right)\mathit{\beta }{S}_1\left(\mathit{t}\right)\mathit{F}\left(\mathit{t}\right)+\left(1-p_2\right)\mathit{\beta }{S}_2\left(\mathit{t}\right)\mathit{F}\left(\mathit{t}\right)\\ +\left(1-p_3\right)\mathit{\beta }{S}_3\left(\mathit{t}\right)\mathit{F}\left(\mathit{t}\right)+\mathit{\alpha F}\left(\mathit{t}\right)-\mathit{\mu I}\left(\mathit{t}\right)\end{array}\right.$(1)

In this dynamic system, there are $\mathit{\beta }{S}_1\left(\mathit{t}\right)\mathit{F}\left(\mathit{t}\right)$ strong-interest susceptible users contacting with forwarding users and the number of whom choose to forward the information and then become the forwarding users is $p_1\mathit{\beta }{S}_1\left(\mathit{t}\right)\mathit{F}\left(\mathit{t}\right)$, so the number of users who remain not forwarding after being exposed to the message and thus removing to immune state is $\left(1-p_1\right)\mathit{\beta }{S}_1\left(\mathit{t}\right)\mathit{F}\left(\mathit{t}\right)$. In an analogous manner, weak-interest susceptible users and medium-interest susceptible users remove to forwarding state and immune state through the same process described above at different forwarding rate $p_2$ and $p_3$ respectively. Meanwhile the numbers of strong-interest susceptible users, weak-interest susceptible users and medium-interest susceptible users increase by $A_1$, $A_2$ and $A_3$ respectively. As time goes by, $\mathit{\alpha F}\left(\mathit{t}\right)$ individuals in the forwarding state no longer have the ability to influence others and then transfer to the immune state. Throughout the whole process, individuals in each state are breaking away from the group at the same removing rate $\mathit{\mu }$. Due to considering the dynamic changes of the study population, the model is closer to the real information transmission process.

For the single information propagation dynamics model, the cumulative forwarded population $\mathit{C}\left(\mathit{t}\right)$ and current forwarded population $\mathit{F}\left(\mathit{t}\right)$ are two significant quantities for the model analysis, which can be acquired from the web platform. Here, we can also obtain the value of cumulative forwarded population$\mathit{C}\left(\mathit{t}\right)$ through the differential equation as below:

(2) $\frac{\mathit{dC}\left(\mathit{t}\right)}{\mathit{dt}}=p_1\mathit{\beta }{S}_1\left(\mathit{t}\right)\mathit{F}\left(\mathit{t}\right)+p_2\mathit{\beta }{S}_2\left(\mathit{t}\right)\mathit{F}\left(\mathit{t}\right)+p_3\mathit{\beta }{S}_3\left(\mathit{t}\right)\mathit{F}\left(\mathit{t}\right)$(2)

In the initial stage of information release, current forwarded population $\mathit{F}\left(\mathit{t}\right)$ will grow with time until it reaches its maximum value $F_{\mathit{max}}$. As the timeliness of information fades away, cumulative forwarded population $\mathit{C}\left(\mathit{t}\right)$ will cease to increase and thus reach a stable value $C_s$ which indicates the final size of users forwarding the information.

In infectious disease models, ${\mathrm{\Re }}_o$ refers to the basic reproduction number or basic reproduction number. It is one of the most important parameters to describe the rate and extent of transmission of an infectious disease. ${\mathrm{\Re }}_o$ is the average number of other people an infected person can infect in a population without immune protection. If ${\mathrm{\Re }}_o$ is greater than 1, this means that each infected person can infect more than one person on average and therefore the disease will continue to spread. If ${\mathrm{\Re }}_o$ is less than 1, the disease will gradually disappear because each infected person can only infect less than one person on average.

Based on the definition of reproduction number ${\mathrm{\Re }}_o$ which is regarded as a significant indicator to measure whether a disease can break out in the epidemic model, we put forward the interest intensity reproduction ratio ${\mathrm{\Re }}_o$to estimate the possibility for an explosion of public opinion. It represents the value of susceptible users with varying levels of interest intensity who are affected by a forwarding individual to remove to forwarding state in the early phase of information dissemination. We can derive ${\mathrm{\Re }}_o$ from the following calculation procedure:

(3) ${\mathrm{\Re }}_o=\frac{p_1\mathit{\beta }{S}_1+p_2\mathit{\beta }{S}_2+p_3\mathit{\beta }{S}_3}{\mathit{\mu }+\mathit{\alpha }}$(3)

3. Data fitting

3.1. Data description

To analyse our model by using the real event, we choose topics about the COVID-19 and the Russian-Ukrainian war, which are widely discussed and has gained a lot of attention and repercussions on the social platform. For some reasons, such as the outbreak of the COVID-19, which affects the daily lives of per individual to different degrees, users’ interest intensity in this topic will also vary; differences in people’s interest in politically related topics will also cause them to show different levels of interest in war topics. The real data we use for data fitting is from the Chinese Sina-microblog. We capture the number of retweets on the topic at different moments in time, and the interval for collecting forwarding numbers is set to 10 minutes. And in order not to affect the consistency of data when users do not pay attention to information on the Internet during the late at night, we need to eliminate some distracting external factors. Thus we delete the forwarded data in the night time when the users are sleeping, and fit with the pure real data.

3.2. Parameter estimation

Here we select two topics for data fitting, where topic 1 is # Will the epidemic situation outbreak massively? # and Topic 2 is # Ukraine’s president says he is not afraid to negotiate with Russia. #. We collected retweets for topic 1 in the time period from 1 August 2021, 16:50 to 8 August 2021, 17:12, containing 6437 data, and for topic 2 in the time period from 25 February 2022, 10:15 to 12 March 2022, 11:42, containing 7634 data. After processing the data, we obtained the cumulative number of retweets for both topics during these periods.

The LS method is used for estimation of model parameters and initial susceptible population. Least squares is a classical data fitting method whose goal is to find a function that minimizes the sum of squares of errors with respect to the actual data points. It is remarkable in that it provides an efficient mathematical approach to data analysis and model fitting problems. With this method, the correlation of the data and the prediction results can be estimated quickly, while the reliability of the prediction results can be assessed. Parameters can be set as Θ: $\left(\mathit{\beta },p_1,p_2,p_3,A_1,A_2,A_3,\mathit{\mu },\mathit{\alpha }\right)$ and the calculation process of LS error function is listed as below:

(4) $\mathit{LS}=\sum _{\mathit{k}=0}^T{\left|f_C\left(\mathit{k},\mathrm{\Theta }\right)-C_k\right|}^2$(4)

In which $C_k$ denotes the real cumulative forwarding population, $f_C\left(\mathit{k},\mathrm{\Theta }\right)$ denote the corresponding numerical calculation according to the parameter vector for $C_k$, and k = 1, 2, … is the sampling time. We chose $C_k$ for model calibration because it reflects how many people saw and spread this message, and can be used to measure the impact and spread of the message on social media. Most importantly, it is a result that can be calculated from real data.

As illustrated in Figures 3 and 4, we fit the real data mentioned above and the fitting results roughly match with the actual situation which proves the feasibility and accuracy of the model to a large extent. In the graphs, the horizontal coordinate indicates the time, the vertical coordinate indicates with the cumulative forwarding population, the yellow star indicates the actual cumulative forwarded population considering the interest intensity, and the red line is the estimated cumulative forwarded population calculated according to the formula.

Figure 3. Data fitting results to the $\mathit{I}-\mathit{SFI}$ model of the topic 1, showing the change in the actual cumulative forwarded population (yellow star) and the estimated cumulative forwarded population (red line) as time changes.

Figure 4. Data fitting results to the $\mathit{I}-\mathit{SFI}$ model of the topic 2, showing the change in the actual cumulative forwarded population (yellow star) and the estimated cumulative forwarded population (red line) as time changes.

Table 3 is the estimated result of parameters fitted by the real data. In this result, we can see that during the birth and growth in this information dissemination total circle group, the average entering rate to the group which is called$\mathit{A}$ in this model is high. This is due to the fact that the communication platform is large enough to have a great number of potential users who can become susceptible users. As the degree of interest decreases from high to low, the different birth rates of the population $A_1$, $A_2$, $A_3$ and the average probabilities that susceptible users will forward the information $p_1$, $p_2$, $p_3$ also reduce. Besides, the average removing rate of whom break away from the group which is called $\mathit{\mu }$ in this model is slightly low. On the contrary, we can see that the average rate representing a user become inactive to forward the information which is called $\mathit{\alpha }$ is relatively high. So when an original post owner publishes a blog on the web, people who have strong-interest to this topic are more likely to forward it, and the growth rate of the number of strong-interest susceptible users is the fastest which means that most of the people who have not entered into the group have strong-interest to the topic.

Table 3. Parameter results to the $\mathit{I}-\mathit{SFI}$ model of the two topics.

	$\mathit{\beta }$	${\mathit{p}}_1$	${\mathit{p}}_2$	${\mathit{p}}_3$	${\mathit{A}}_1$	${\mathit{A}}_2$
Topic 1 Topic 2	2.8111×10^−5 1.7429×10^−4	0.2035 0.0204	0.1683 0.0178	0.0838 0.0091	883.8027 311.0628	579.8274 253.0508
	${\mathit{A}}_3$	$\mathit{\mu }$	$\mathit{\alpha }$	${\mathit{S}}_{10}$	${\mathit{S}}_{20}$	${\mathit{S}}_{30}$
Topic 1 Topic 2	568.5857 228.6956	0.0075 0.0024	0.9770 0.0572	1.1784×10^5 1.2961×10^5	7.7310×10^4 1.0544×10^5	7.5811×10^4 9.5290×10^4

When each part of this dynamic model has not yet changed, it is an initial state in steady. The stable state of the I- SFI model can be listed as $\mathit{dF}\left(\mathit{t}\right)/\mathit{dt}=0.$Thus the initial value of susceptible users with three levels of interest can be calculated like: $S_{10}=A_1/\mathit{\mu };S_{20}=A_2/\mathit{\mu };S_{30}=A_3/\mathit{\mu }.$

In addition, the values of $\mathit{\alpha }$ and $\mathit{\mu }$, which represent negative effects in our model, are both positive, so we can also use this experimental result to confirm a psychological communication phenomenon, which is called ‘the sleeper effect’ firstly proposed by Hovland, a psychologist [29]. Well, we can also see in these two real events, values $p_1$, $p_2$and $p_3$ representing forwarding rates of different intensities are all a little low. It can be inferred that Sina-Microblog users are not strongly interested in forwarding these information. However, differences in forwarding rates between susceptible individuals with different levels of interest can still be seen.

For the topics that is used for our data fitting # Will the epidemic situation outbreak massively? # and # Ukraine’s president says he is not afraid to negotiate with Russia. #, obviously different users will have different levels of interest in them. Some people are very concerned about the outbreak because it will affect their daily travel, career, etc., so they have a strong interest intensity in the topic. Some people are less negatively affected by COVID-19 so they show little interest in this topic. Over time, some people may have gotten used to COVID-19, so they are not very interested in this news either. Similarly, some people who are very concerned about politics and the international situation will show a strong interest in topic 2. Some people do not like to talk about politics-related topics, so they will also lack interest in Topic 2. Since there are enough channels to get information about the COVID-19 and the situation in Russia and Ukraine, the forwarding rates of susceptible population in different levels of interest are generally low, but we still can see the differences and the relationship between the interest intensity and the forwarding rate: the stronger the interest intensity, the higher the forwarding rate. Probably the implications of the COVID-19 for people are indelible, we can learn from the value of $A_1,A_2,A_3$ that most people who are not in this group are strongly interested in this topic. And compared with topic 1, people who are not in this group are significantly less interested in topic 2 than topic 1. It is not difficult to find that features of these topics can fit with our model, and through data fitting and parameter estimation, we are also able to access the interests of users who are not in the group to some extent.

3.3. Comparison

The $\mathit{I}-\mathit{SFI}$ model constructed in this paper is a modification of the SFI model based on the intensity of interest, while taking into account the two influence factors such as birth rate and death rate. In order to reflect the necessity of considering these three variables on the basis of the original SFI model, to illustrate the improvement of the new model compared to the old one, and to assess whether the addition of these three variables can enhance the ability of the model to describe the dissemination of information on social platforms, we have conducted comparative experiments.

Similarly, we first use the same two topics (topic 1 and topic 2 mentioned above) for data fitting to the SFI model. The fitting results are depicted in Figures 5 and 6.

Figure 5. Data fitting results to the I-SFI model and the SFI model of the topic 1, showing the change in the actual cumulative forwarded population (yellow star), the estimated cumulative forwarded population (red line and blue line) as time changes.

Figure 6. Data fitting results to the I-SFI model and the SFI model of the topic 2, showing the change in the actual cumulative forwarded population (yellow star) and the estimated cumulative forwarded population (blue line and red line) as time changes.

By comparing the fitting results of the two different models, it is not difficult to find that the fitting results of the SFI model did not match the actual situation as well as the $\mathit{I}-\mathit{SFI}$ model. And to make the comparison more intuitive, we chose to calculate the mean absolute percentage error (MAPE) between the fitting results (the estimated cumulative forwarded population) of the two models in different events and the true values (the actual cumulative forwarded population). MAPE is commonly used to measure the average percentage error of a model’s predictions relative to the true value and is a percentage indicator. It can be calculated by this formula:

(5) $\mathit{MAPE}=\frac{1}{n}\sum _{\mathit{i}=1}^n\left|\frac{y_i-{\hat{y}}_{\mathit{i}}}{y_i}\right|\ast 100$(5)

Where $\mathit{n}$ is the total number of samples. $y_i$ is the true value of samples, in this case the actual cumulative forwarded population. $\widehat{y_i}$ is the predicted value of the model for samples, in this case the estimated cumulative forwarded population. After the calculation, the values of MAPE are listed in Table 4.

Table 4. MAPE values to two models of the two topics.

	$\mathit{I}-\mathit{SFI}$	$\mathit{SFI}$
Topic 1 Topic 2	6.8187 4.2217	9.0623 7.7633

Obviously, for both different topics, the $\mathit{I}-\mathit{SFI}$model has lower MAPE values than the SFI model. This indicates that the $\mathit{I}-\mathit{SFI}$ model will have less inaccuracy compared to the SFI model when describing and predicting the process of information dissemination on social platforms. It also reflects that it is necessary to consider the influencing factors of interest intensity, birth rate and death rate in the model to improve the accuracy and make it more optimized.

4. Parameter sensitivity analyses

In order to evaluate the influence of different parameters for the information transmission process, we use the partial rank correlation coefficients (PRCCs) to analyse how the nine parameters$(A_1,A_2,A_3,p_1,p_2,p_3,\mathit{\alpha },\mathit{\beta },\mathit{\mu })$ affect the three indicators$\left(C_{\mathit{S},}{F}_{\mathit{max}},{\mathrm{\Re }}_o\right)$ in the $\mathit{I}-\mathit{SFI}$model respectively. For each input parameters, we use 1000 samples to evaluate the sensitivity. Figure 7 presents the results of the analysis using scatter plots and histograms. In the bar chart, if the value of PRCCs is positive, it means that the parameter value shows positive correlation with the corresponding index, and if the value of PRCCs is negative, it means that the parameter value shows negative correlation with the corresponding index, and the larger the absolute value of PRCCs, the greater the influence of correlation. The scatter is an aid to judge the influence of correlation, the more scattered the points in the graph, the weaker the correlation, and the opposite, the stronger the correlation. The trend of the points in the scatter plot can also be used to judge the positive and negative correlations. We can see that $p_1,p_2$and$\mathit{\beta }$ has strong positive influence on $C_s,F_{\mathit{max}}$and${\mathrm{\Re }}_o$, while $\mathit{\alpha }$ has a great negative effect on $C_s,F_{\mathit{max}}$and${\mathrm{\Re }}_o$. Compared with $p_1$ and $p_2$, $p_3$ has less impacts on the three indicators but cannot be ignored. $\mathit{\mu }$ is negatively correlated with the three indexes and has little effect on them especially for ${\mathrm{\Re }}_o$. At the same time, $A_1,A_2$and$A_3$ have almost no effect on $C_s,F_{\mathit{max}}$and${\mathrm{\Re }}_o$.

Figure 7. The results of PRCC sensitivity analysis on parameters of $C_S$,$F_{\mathit{max}}$ and ${\mathrm{\Re }}_o$, in which the correlation of the parameters $A_1$, $A_2$, $A_3$, $p_1$, $p_2$, $p_3$, $\mathit{\alpha }$, $\mathit{\beta }$ and $\mathit{\mu }$ with the indices $C_s$, $F_{\mathit{max}}$ and ${\mathrm{\Re }}_o$ are shown.

With the known improvement in the performance of the model after taking into account the interest intensity, birth rate and death rate, we can find through sensitivity analysis that the birth rate and death rate have very little effect on the model metrics, which suggests, to some extent, that while the practice of taking birth and death rates into account in the model enables the model to simulate a more realistic scenario of information dissemination, it does not have a significant impact on the performance of the model. The result also shows that in a real network information spreading situation, although the dynamic change of the number of people in the groups of propagation always exists, it does not have a decisive effect on the process of information transmission, whereas $p_1$, $p_2$,$\mathit{\alpha }$ and $\mathit{\beta }$ are key factors in determining outbreak of the event and the scale of information dissemination.

As $p_1,p_2$ and $p_3$ are of great importance to reflect the different levels of interest intensity, and $\mathit{\beta }$ is an essential parameter representing the density of the communication network, so we conduct a more detailed experiment on the important factors $p_1,p_2,p_3$ and $\mathit{\beta }$ to show how they affect the change of the three indices, whose results can also be verified with the results of PRCC sensitivity analysis. In this experiment, we change the value of only one key factor $p_1,p_2,p_3$ or $\mathit{\beta }$ at a time, and the other parameters are fixed at their best-fit values. Figure 8 displays important factors’ effects on current forwarded population and cumulative forwarded population respectively, where $\mathit{F}$ denotes the current forwarded population, $\mathit{C}$ denotes the cumulative forwarded population, and the three different types of lines represent the changes in the current and cumulative forwarded populations over time when the key factors are different values and the other parameters are fixed best-fit values. Obviously, with the increase of different forwarding data $p_1,p_2,p_3$ and the average exposure rate $\mathit{\beta }$, the current forwarded population reaches the peak earlier and the maximum value also rises. Meanwhile the cumulative forwarded population will move to a much larger stable state at an early time. However, when the forwarding data of susceptible users with different interest levels increased by the same value, the positive effect of $p_1$ on current forwarded population and cumulative forwarded population increase most significantly. The positive impact that the average exposure rate $\mathit{\beta }$ brings to current forwarded population and cumulative forwarded population is also of great importance and cannot be overlooked.

Figure 8. The effect of key factors on instantaneous forwarding population and cumulative forwarded population: (a) only $p_1$ changes; (b) only $p_2$ changes; (c) only $p_3$ changes; (d) only $\mathit{\beta }$ changes.

In accordance with the two experiments based on the data of real event on Chinese Sina-microblog, if we want more people to participate in the process of information transmission through forwarding behaviour, first we can select the information diffusion source with higher network density at the beginning of releasing the information to obtain a lager average exposure rate $\mathit{\beta }$. And then when we post a blog, we can choose bloggers of the type related to the content of the blog as the post owner which means most of their fans are interested in the information or the platform target the information to users who are interested in such topics, through which more strong-interest susceptible users will participate in the process of information dissemination and a higher forwarding rate is obtained.

5. Conclusions

In this paper, we construct an interest intensity-based susceptible-forwarding-immune$\left(\mathit{I}-\mathit{SFI}\right)$ propagation dynamic model under the consideration for differences in users’ interest levels and the dynamics of population size. Based on the forwarding data of real event on Chinese Sina-microblog, we perform data fitting and sensitivity analysis on parameters. It is found that people will be more likely to forward the information that they feel interested in, which attach great importance to improving the efficiency of information dissemination. And the system takes people moving in and out into account, which makes our model more fit the real information transmission process. Since we make the birth rates of susceptible users with different interest intensity vary, so to a certain extent we can also determine the interest intensity of users outside the study population by the different birth rates of the three groups.

But the average entering rate$\mathit{\hspace{0.17em}}\mathit{A}$ and the average removing rate$\mathit{\hspace{0.17em}}\mathit{\mu }$ generate changes of the network structure at real time, how the changing network structure affects the information dissemination process needs to be further investigated. Meanwhile, exploring better data collection methods to break through the limitations of social platform data collection is also worth exploring. It is also important to filter false postings before applying the model to prevent misjudgements caused by the extremes of information dissemination caused by false postings.

Acknowledgments

The work was supported by the National Key Research and Development Program (No. 2021YFF0901700); State Key Laboratory of Media Convergence and Communication, Communication University of China; the Fundamental Research Funds for the Central Universities; the High-quality and Cutting-edge Disciplines Construction Project for Universities in Beijing (Internet Information, Communication University of China).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the The National Natural Science Foundation of China [No. 62372418]; The Beijing Natural Science Foundation [No. 4232015].