# Realistic Aspects of Simulation Models for Fake News Epidemics over Social Networks
## 3 Model
### Time Dynamics  
We considered two time dynamical aspects in our simulation: the connection of an agent to the network and the engagement of an agent.  
Several simulation models assume that agents connect to the networks simultaneously at fixed time intervals or \"slots\". While this may be a reasonable model as a first approximation, it does not convey realistic OSN interaction patterns.  
Conversely, in our model, we let each agent decide independently when to connect to the network. Therefore, agents are not always available to spread contents and do not change state instantly upon the transmission of fake news or debunking messages. We modeled agent inter-access time as having an exponential distribution, with a different average value that depends on the agent. This makes it possible to differentiate between common users and bots. In particular, we set the average time between subsequent OSN accesses for bots to be:  
\[\frac{1}{\lambda_{bot}}=\frac{4}{\lambda_{common}}\,, \tag{1}\]  
or four times that of common users, where \(\lambda_{common}=16\) (minutes\({}^{-1}\)). Conversely, in the baseline case without time dynamics, all nodes access the network at the same, a fixed rate of once every 16 minutes.  
When accessing the OSN, we assume that every agent reads its own feed, or \"wall\". In this feed, the agent will find the news received while it was offline. The feed may contain messages that induce a state change in the agent (e.g., from susceptible to believer, or from believer to fact-checker). We also assumed the first node was infected at time \(t=0\).  
To model the typical loss of interest in a news item over time and also to model the different spreading effectiveness (or \"virality\") of different types of news, we introduced an engagement coefficient, whose variation over time obeys the differential equation:  
\[\frac{\partial E}{\partial t}=-\gamma E\,, \tag{2}\]  
where \(\gamma\) is the decay constant. The solution to (2) is:  
\[E(t)=E_{0}e^{-\gamma t}\,, \tag{3}\]  
and models an exponential interest decay, where the engagement \(E_{0}\) at time \(t=0\) controls the initial virality of the news and the constant engagement decay factor \(\gamma\) controls how fast the population loses interest in a news item. The factor \(E_{0}\) allows us to model the different level of initial engagement expected for different types of messages. For example, in our simulations, we set \(E_{0}^{(f)}=1\) for fake news and \(E_{0}^{(d)}=0.1E_{0}^{(f)}\) for debunking messages. This gives debunking messages a 10-fold lower probability to change the state of a user from believer or susceptible to fact-checker, with respect to the probability of transitioning from susceptible to believer.  
In our setting, this means that a user believing in a fake news item will likely be attracted by similar pieces confirming the fake news, rather than by debunking, a behavior know as confirmation bias [53]. In the same way, a susceptible agent will tend to pay more attention to fake news, rather than to common news items. This models the fact that the majority of news regarding scientific studies tends not to reach and influence people [11].  
In addition, we set \(\gamma=2/T_{sim}\), where \(T_{sim}\) is the simulation time; in this way, at the end of the simulation, fake news still induces some small, non-zero engagement. By setting this value, we can simulate the lifetime of the fake news, from its maximum engagement power to the lowest one. For a given time \(t\), \(E(t)\) is a multiplicative factor that affects the probability of successfully infecting an individual at time \(t\).