## 3 Preliminaries
### Problem Statement  
Let us say we have \(S\) source tweets containing the fake news link, \(\mathbb{S}=\{s_{i}\}(1\leq i\leq S)\).  
**Fake News.** Two key factors are used to define fake news: authenticity and intent [2]. First, fake news contains claims that can be verified by users as wrong information. Second, fake news is created with malicious intentions to mislead readers. Based on these two key features, the definition of fake news can be divided into two categories: narrow and broad. News needs to satisfy both key characteristics in order to meet the narrow definition of fake news. On the other hand, a broader definition of fake news focuses on the content's authenticity or intent. In this paper, we have adopted the broad definition to involve more data instances, such as inadvertently created false news content or biased news articles for political propaganda purposes.  
**Cascade.** An information cascade can be viewed as a diffusion topology, which is depicted in the tree's structure. Each node in the tree represents one step of information propagation. For example, we refer to the post's diffusion tree generated by the source tweet \(s_{i}\) referencing the URL and all its retweets as the cascade.  
**Observed Cascade.** For each source tweet \(s_{i}\), the observed cascade is recorded as the set of early spreaders \(u\) within the observation time window \(T\), i.e., \(\mathbf{C}_{t=T}^{s_{i}}=\{u_{1},u_{2},\ldots,u_{n_{T}^{s_{i}}}\}\), in which \(n_{T}^{s_{i}}\) is the number of users propagating the source tweet \(s_{i}\) within the observation time window \(T\).  
**Popularity of Post.** If the tweet is retweeted, then the retweet will become a child node of the source tweet. Therefore, we can define the size of a post cascade in social media as the number of users involved in the retweeting process, which is the post's total number (including source tweets and retweets). We quantify the popularity of a post using the cascade size, as defined in [8; 14].  
**Future Popularity Prediction.** Given an observation time \(T\) and a source tweet \(s\) with the news link, we have the observed cascade \(\mathbf{C}_{T}^{s}\) and underlying network \(\mathbf{G}_{T}^{s}=(\mathbf{V}_{T}^{s},\mathbf{E}_{T}^{s})\), in which \(\mathbf{V}_{T}^{s}\) is the set of users associated with \(s\) within the observation time window \(T\), and \(\mathbf{E}_{T}^{s}\subset\mathbf{V}_{T}^{s}\times\mathbf{V}_{T}^{s}\) is the set of relationships between all users. For each user in the cascade, the profile and the historical tweets are retrieved as well. In this work, we aim to predict the natural logarithm of the final popularity \(log(\mathbf{Y}_{t_{p}=\infty}^{s}+1)\) of the source tweet \(s\). Figure 9 shows that when approaching seven days after publications, the cascade saturates. Thus, in this work, we consider seven days as a good approximation to the final popularity; i.e., we choose seven days as the prediction time, \(t_{p}\).  
Figure 8: Diffusion network and social network among users. Yellow lines depict the diffusion paths among users. Blue lines depict the users’ social networks. The direction of the arrow is the direction of information flow.