## 3 Materials and Methods
### Virtual Environment
#### 3.3.1 Influence Simulation  
The primary objective of most social bots is to influence the real world. One way they do this is by gaining the trust of other users and persuading them to adopt their perspectives. Figure 2: Framework. The construction of the virtual environment starts by using real-world data, analyzing bot influence and user engagement, and reconstructing social networks. Then, a profile editor with a substitute detector that aligns with a black-box adversary detector and an adversarially trained generator is developed. This allows for user clustering and preselection of potential targets. The action selection and target selection agents are trained subsequently, which help guide a new account to act as either a fake account or an influencer within the virtual environment, simulating the creation of a social bot. The image representing the detector comes from BotRGCN [24]. Previous studies have used the ICM model to simulate how the bot's influence spreads. In this model, the social network is represented as a directed graph. An edge from user \(u\) to user \(v\) indicates that the information can spread from \(u\) to \(v\), i.e., \(v\) follows \(u\). It is assumed that all users were deactivated at the beginning. The bot needs to interact (retweet, mention, or follow) with a user several times to activate, i.e., to influence the user. The number of interactions required depends on the number of followers of the bot and the user. However, our analysis results do not support this assumption. The following network \(G_{F}\) restored from Twibot-22 (see Section 4.1.1 for more information) contains 612,329 normal users and 81,432 bots. The normal users have followed the bots 153,763 times in total. The bot has not interacted with the user 96,180 times, indicating that the majority of the bot's influence does not come from direct interactions with other users. We think that the primary reason these followers are following the bots is because of the bots' tweets. To demonstrate this, \(k\) seed bots with the slowest follower growth rate were chosen, and the similarity between their tweets and those of other bots was compared. The \(k\) seed bots can be represented as  
\[ \begin{split}|Seed_{F}|=k\quad\&\\ Seed_{F}\subset V^{\prime}_{F}&\quad s.t.\forall u \in V^{\prime}_{F}\setminus Seed_{F}\forall v\in Seed_{F},\\ &\quad\frac{N_{Follower}(u)}{N_{Tweet}(u)}\geq\frac{N_{Follower} (v)}{N_{Tweet}(v)}\end{split}  \tag{1}\]  
where \(V^{\prime}_{F}\) are the bots in \(V_{F}\), and \(N_{Follower}(u)\) and \(N_{Tweet}(u)\) are the number of followers and tweets of user \(u\). With \(k\) set to 1000, the similarity is assessed by calculating the Euclidean distance between the tweet embeddings generated using the RoBERTa model:  
\[\mathcal{S}_{F}(u)=dist(E(u),\frac{\sum\limits_{v\in Seed_{F}}E(v)}{|Seed_{F}|}) \tag{2}\]  
where \(E(u)\) is the tweet embedding of \(u\). The tweet information of 64,741 bots is provided. Based on their similarity, the bots are grouped into 13 clusters of 5000 bots. Figure 2(a) presents the relationship between the mean follower growth rate and the similarity. It is evident that bots that are less similar to the seed bots generally have higher follower growth rates. The assumption of the probability of a user following back after interaction is also doubtful. Existing studies believe that this probability should be proportional to the number of the bot's followers \(N_{Follower}(b)\) and be inversely proportional to the number of the user's followers \(N_{Follower}(u)\). \[q\times(\frac{1+N_{Follower}(b)}{1+N_{Follower}(u)}) \tag{3}\]  
where \(q\) is a hyperparameter set to 0.3 [10]. However, the possibility of the target user following back on different interactions is different. After excluding the users who have been interacted with in multiple ways, statistics show that 15.4% of the followed users followed back, 3.3% of the retweeted users followed back, and 2.8% of the mentioned users followed back.