# Realistic Aspects of Simulation Models for Fake News Epidemics over Social Networks
## 3 Model
### Network Construction  
We model our social network as a graph. Every node of the graph represents an agent, and edges convey a relationship between its two endpoints. A key problem in this scenario is to generate a synthetic network that resembles real OSNs like Facebook or Twitter. Generating graphs with given properties is a well-known problem: network models are the fundamental pieces to study complex networks [47; 48; 49]. However, these mathematical models are solely driven by the topology of a network and do not take into account features and similarities among nodes, which instead have a key role in shaping the connections in an OSN. In this work, we build our network model via an approach similar to [50]. We generate our networks according to the following properties:  
* Geographical proximity: A user is likely to be connected with another user if they live nearby, e.g., in the same city. The coordinates of a node are assigned by sampling \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Distribution** & **Description** \\ \hline State & \(\{S,B,FC\}\) & State of the node \\ Vulnerability & \(\mathcal{N}_{0}^{1}\)(0.5, 0.2\({}^{2}\)) & Tendency to follow opinions \\ Sharing rate & \(\mathcal{N}_{0}^{1}\)(0.5, 0.2\({}^{2}\)) & Tendency to share the news \\ Recovery rate & \(\mathcal{N}_{0}^{1}\)(0.2, 0.2\({}^{2}\)) & Tendency to do fact-checking \\ Interest attributes & \(\mathcal{N}_{-1}^{1}\)(0, 0.4\({}^{2}\)) & Connect nodes based on their interests \\ Coordinates & \(\mathcal{U}\)(0, 1) & Geographical position of the node \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Attributes of common and influencer nodes. from a uniform distribution in the square \([0,1]^{2}\). This procedure follows the random geometric graph formalism [51] and ensures that geographical clusters appear. We evaluate the distance between two nodes using the Euclidean distance formula, normalized such that the maximum possible distance is 1. In our simulations, we created an edge between two nodes if their attribute distance was less than a threshold equal to 0.03. * Attributes' proximity: Each node has a set of five \"interest\" attributes, distributed according to a truncated Gaussian distribution; we employ these parameters to model connections between agents based on their interests. This helps create connections and clusters in the attribute domain, rather than limiting connections to geographical proximity criteria. The distance between two sets of interest attributes is evaluated using the Euclidean distance formula, normalized such that the maximum possible distance is 1. In our simulations, an edge between two nodes is created if their distance is less than a threshold equal to 0.03. * Randomness: To introduce some randomness in the network generation process, an edge satisfying the above geographical and attribute proximity criteria is removed from the graph with a given probability \(p_{rem}=0.5\). The above thresholds reproduce the higher likelihood of connections among OSN agents that are geographically close or have similar interests. In addition to connectivity properties, every node has three parameters that affect the behavior of the corresponding user in the simulation: the vulnerability (defined as the tendency to follow opinions: if higher, it makes it easier for a user to change its state when receiving fake news or its debunking); the sharing rate (defined as the probability to share fake news or debunking piece); and the recovery rate (defined as the probability to do fact-checking: if higher, it makes the node more inclined to verify a fake news item before re-sharing it). We assigned these parameters at random to each node, according to a truncated Gaussian distribution with the parameters reported in Table 2. The distributions were truncated between 0 and 1. We build the network starting from common nodes and creating edges according to the above rules.