## 5 Experimental Study
### Complexity Optimization  
We aimed to investigate several measures of complexity and to make a comparison across the event logs and types of cycles folding. In this study, we considered four measures, such as average degree, entropy, and two additional ones that we empirically derived in Section 4.2. Measures (9) and (10) are structural. They indicate the relative size of the model, i.e., the ratio of the number of elements in the model and the number of possible or theoretical ones. The first is the number of edges presented in the model divided by the number of edges of a corresponding complete directed graph. This way, we can judge how close the graph structure is to having all pairs of relations, which complicates its understanding. A complete graph does not imply loops, so, in this formula, we do not account for them in the number of edges even though they present in the model. The last \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Monitoring** & **Nurse** & **Physician** \\  & **Process** & **Workflow** & **Workflow** \\ \hline \multicolumn{2}{c}{Num. of cases} & 272 & 165 & 43 \\ \hline \multirow{2}{*}{Event classes} & Clinical & Lab tests and Follow-up & Appointments \\  & Non-clinical & Triage duties & COVID-19 treatment \\ \hline \multicolumn{2}{c}{Num. of unique events} & 18 & 19 & 29 \\ \hline \multicolumn{2}{c}{Total num. of events} & 35,611 & 1042 & 1077 \\ \hline \multirow{2}{*}{Case length} & Max & 674 & 33 & 61 \\ \cline{2-4}  & Min & 3 & 1 & 1 \\ \cline{2-4}  & Mean & 131 & 6 & 25 \\ \hline \multicolumn{2}{c}{Record duration} & 355 days & 454 days & 377 days \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Datasets summary. measure, (10), is the equally weighted sum of the activities and transitions ratios. \"Start\" and \"end\" events are not included in the set of nodes for this measure because they do not present in the log activities and are just auxiliary. Accordingly, in- and upcoming relations for the initial and terminal nodes are not included in the set of edges. We plotted landscapes for each of the complexity measures considered. The measures visualize the complexity value and its relationships with activity and transition rates that are basic options in our algorithm to regulate the process model completeness. Process models in the area near the rate limits are mostly useless due to either very high complexity (with very high \(r_{a}\) and \(r_{t}\)) or the reduction of almost all significant activities in the model (with very low \(r_{a}\) and \(r_{t}\)). We also revealed that there are no meta-states for the event log of nurse workflow. So, only the results for two event logs of the monitoring program and physician workflow are discussed further. A summary of cycles found in the models is given in Table 3. It should be pointed out that \"start\" and \"end\" events and relations associated with them are accounted for in the number of model elements. We also want to highlight that the maximum and the minimum numbers of cycles and meta-states are not always for the boundary levels, and the mean values are rounded down. We also provide and investigate fitness and complexity landscapes across the considered measures and event logs. At first sight, they may seem monotonic. However, intricate patterns in behaviors of real processes involve irregularities in the landscapes that are amplified with the presence of meta-states in the model. Nevertheless, we can still address the optimization problem within such conditions. The results of process model optimization are shown in Figure 8, where a red marker is plotted to indicate optimal rates. One can see that all complexity measures can facilitate decreasing the transition rate (as the optimal value is very close to zero when it concerns the transition rate), which directly affects the ability to comprehend a model effortlessly. Meanwhile, \(H\) and \(K_{n}\) allowed the maximum activity rate to be optimal in all cases. This is not appropriate if there are too many activities and highly varying process behaviors.