# Feature Drift in Fake News Detection:
## 3 Methodology
### Feature Extraction  
We construct a propagation graph for each news item according to retweeting behavior, and then divide the features of the graphs into temporal features and structural features. The undirected propagation graph \(G_{q}=\{\mathcal{V}_{q},\mathcal{E}_{q}\}\) is constructed based on the retweeting relationship in the same event \(q\), where \(\mathcal{V}_{q}=\{v_{1},v_{2},\dots,v_{N_{q}}\}\) denotes the node set and the number of nodes is \(N_{q}\), and \(v_{i}\in\mathcal{V}_{q}\) denotes the \(i\)-th tweet that belongs to the event \(q\). \(\mathcal{E}_{q}\) denotes the edge set; if tweet \(v_{i}\) retweets tweet \(v_{j}\), a connected edge \((v_{i},v_{j})\in\mathcal{E}_{q}\) is established between nodes \(v_{i}\) and \(v_{j}\). **Temporal Features**. Temporal features [48] focus on the temporal patterns in the spread of news. In this work, we define the _creation time_ of tweet \(v_{i}\) as \(t_{i}\), which means that tweet \(v_{i}\) is posted at time \(t_{i}\). By utilizing this definition, we extract the temporal features from the propagation graphs as shown in Table 2. Detailed descriptions are as follows:  
* _Characteristic time (Ctime)_: For a propagation graph \(G_{q}\), we sort all _retweet times_ from small to large and obtain a time sequence \(R_{(1)},R_{(2)},\dots,R_{(l)},\dots,R_{(M)}\), where _retweet time_\(R_{(l)}\) denotes the difference in _creation time_ between two tweets that exist in a retweet relationship, \(l\) indicates the retweet time series index, and \(M\) denotes the number of retweets. Then, _Characteristic time_ is equal to \(R_{(|M*0.8|)}\), where \(\bigsqcup\) indicates that it is rounded down; to avoid an excessively long life cycle in the spread of news, we choose 0.8 as the cut-off, which can maintain most of the effective information. * _Max-degree time (Dtime)_: The _creation time_ of the tweet with the maximum degree (i.e., retweets) in the propagation graph. * _Response time (Rtime)_[48]: This indicates the timeliness of the responses, which is calculated by Equation (1): \[Rtime=\frac{1}{N_{q}}\sum_{i=1}^{N_{q}}2^{-(t_{i}-t_{0})}\] (1) where \(t_{0}\) denotes the _creation time_ of the source tweet. **Structural Features**. Structural features [49; 50; 51; 52; 53; 54] focus on the graph topological structure in the process of news propagation, and they well capture the complexity of news spreading. We extract the structural features from the diffusion dynamics of the propagation graph. All structural features are shown in Table 3 and detailed descriptions are as follows:  
* _Max degree (MaxD)_[49]: Maximum degree value in the propagation graph. * _Ratio of layer sizes (ROL)_[50]: Layer denotes the number of retweets since the source Twitter post. The ratio of layer sizes can be defined as \[ROL=\frac{n_{2}}{n_{1}}\] (2) where \(n_{2}\) and \(n_{1}\) denote the number of tweets whose distance is two and one from the source tweet, respectively. * _Average betweenness centrality (BCentr)_[49]: Average betweenness centrality of the propagation graph. * _Network diameter (Diameter)_[51]: The maximum distance between any two tweets in the propagation graph. * _Average degree (AveD)_[51]: The average value of degrees in the propagation graph. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Twitter** & **Gossipcop** \\ \hline True news & 205 & 6945 \\ Fake news & 205 & 3684 \\ Users & 173,487 & 739,166 \\ Tweets & 204,820 & 1,058,330 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistics of the datasets. * _Average shortest path (AvePath)_[51]: The average value of the distance of all node pairs in the propagation graph. * _Number of nodes (NoN)_[49]: Number of nodes in the propagation graph. * _Number of subgraphs (NoS)_[49]: Number of subgraphs in the propagation graph while deleting the source tweet. * _Tweet number in largest subgraph (MaxS)_[49]: The number of tweets in the largest subgraph. * _Structural heterogeneity (Hgeneity)_[52]: This indicates the connectivity of the propagation graph, which is calculated by Equation (3). \[