SIR Model, Epidemics, and Social Media
Modelling of information diffusion on social networks with applications to WeChat:
https://www.sciencedirect.com/science/article/pii/S0378437117312785
In the context of network, similarities and differences exist between the diffusion of information and spreading of disease among population. Both processes exhibit some mechanisms of diffusion, and the spreading depends on local as well as global network structure. In the lecture, we discussed how the Galton–Watson process and SIR (Susceptible, Infectious, Removed) Model can be used to model the spreading of epidemics. The same model can also describe the diffusion of information in social medias, possibly with some modifications. In a paper published early this year, a group of information scientists modeled the information diffusion on WeChat, a Chinese multi-purpose messaging and social media mobile app released in 2011.
WeChat was one of the most popular mobile apps in the world with over 1 billion monthly active users. Like other social media apps such as Facebook and WhatsApp, WeChat allows the user to share content based on web pages and forward content shared by other users. The traces of user activities on online social networks such as WeChat give scientists new opportunities to understand the information diffusion process. In the paper, the researcher proposed two non-linear models to predict the spread of information on WeChat based on collected information cascade trees. The models were validated by the actual user data. The researcher discovered that a user with a large number of friends may have a limited influence on the diffusion on a particular web page she received because of her limited attention to each individual friend post.
Random Recursive Tree (RRT)
In the study, the information cascade tree the researcher used describes the spreading trajectory of a piece of information created or shared by a user. The researcher proposed two models, one for the topologies of the cascade trees, and one for the stochastic process of information diffusion. In the first model, the researcher adopted the Random Recursive Tree (RRT), a type of unordered non-planar tree labeled by distinct incrementing integers, to model the growth of cascade trees (shown in the figure above). This model quantified two fundamental properties of the cascade trees –the average path length and the degree variance in relation to the tree size, i.e., the number of nodes in the tree. The degree variance describes the variation of the number of neighbors, i.e., friends of the users in the network. The model allows the researcher to determine to what extend the information spreads “viral-like,” via “hop by hop propagation,” or “broadcast like,” via central hubs that linked to multiple users. The figures below show how the average path length and degree variance relate to the size of a network, in both the actual data and proposed model.
Average Path Length and Degree Variance versus the Size of a Network
The second stochastic Susceptible View Forward Removed (SVFR) model, a modified SIR model, depicts the user behavior, including creating, viewing, forwarding and ignoring a message in the information diffusion process. In the model, each node, i.e., user, can be in one of the four states: Susceptible, View, Forward, Removed. The four states are described as the following in the paper:
Susceptible (S)—the user has the potential to read a message/content, but has not yet read it,
View (V)—the user views the message,
Forward (F)—the user forwards the message,
Removed (R)—the user ignores the message either because (s)he does not want to read the message or has already viewed or forwarded the message.
The following state transition diagram describes the transition probability between each state. A user in the Susceptible state has a probability of β to view the message, and 1- β to ignore the message. After reading the message, she has a probability of γ to share or forward the message, and 1- γ to not share it. The SVFR model was able to capture the tree size distribution, the average path length and the degree variance of a tree in relation to the size of the tree.
SVFR Model
This research is highly related to the information cascade and network connectivity introduced in the class. The research shows how local structure, such as the number of friends of a user and her opinion on a message, can influence the information diffusion in the whole network. A user is able to see her friends’ actions upon an information, i.e., whether they shared it or not, but is not able to see their attitudes on the information. Then, based on her opinion on the information and the number of friends that shared the message, she needs to decide whether she wants to share or forward this message.
Furthermore, there are many similarities between the two models introduced in the paper and the models discussed in the lecture about epidemics. The Random Recursive Tree model is similar to the Galton–Watson process. In the branching process, when the basic reproductive number of an epidemic is lower than or equals to 1, i.e., the contacts are limited, or the probability of transmission is low, the disease will eventually die out. Similarity, in the RRT model, when the individuals in the network have limited interest in a piece of information, i.e, a low β or γ value, the message will eventually stop spreading. Both the RRT model and branching process describe the network diffusion globally.
On the other hand, the SVFR model in the paper and the SIR model for disease spreading both describe the diffusion on a local scale, i.e., how an individual node interacts with its neighbors. Although the SVFR model introduced two variables instead of just one, the underlying principle is the same as the general SIR model. There are differences as well. For example, while a message can spread via “central hubs” in a social media, this is unlikely in the spreading of an epidemic because of the physical contact involving a particular person is limited.