## 2 Materials and Methods
### Metrics for the Evaluation of PROforma Models
#### 2.2.3 Partitionability Metrics  
Partitionability is used for referring to the relationship of subcomponents to the overall model. Within this category are the metrics separability, sequentiality, structuredness, depth, and the new metric model depth.  
* _Separability_, \(\Pi\): it tries to capture how far certain parts of the model can be considered in isolation. An increase in the value of this metric might imply a simpler model. It is computed as the ratio of cut vertices to tasks. A cut vertex (or articulation point) is a node whose deletion separates the graph into several components. \[\Pi(G)=\frac{|\{n\in N/n\text{ is a cut vertex}\}|}{|N|-2}\]
* _Sequentiality_, \(\Xi\): it measures how sequential is a plan. This metric relates to the fact that sequences of nodes are the most simple components in a graph. It is calculated as the ratio of the maximum possible number of scheduling constraints between non-connector tasks to the total number of scheduling constraints. \[\Xi(G)=\frac{|A\cap(NCT\times NCT)|}{|A|}\]
* _Structuredness_, \(\phi\): it measures how far a process model is made of nesting blocks of matching join and split connectors. For this metric, it is necessary to obtain the reduced process graph applying the graph reduction rules defined by [PERSON] [11]. Structuredness is computed as one minus the number of tasks in the reduced process graph, \(|N^{\prime}|\), divided by the number of tasks in the original process graph. The structuredness value for a structured graph is 1. \[\phi=1-\frac{|N^{\prime}|}{|N|}\]
* _Depth_, \(\Lambda\): it is related to the maximum nesting of structured blocks in a graph. It is computed as the maximum depth of all nodes, where the depth of a node \(\lambda(n)\) is calculated as the minimum of the in-depth and out-depth of the node. The in-depth \(\lambda_{in}(n)\) refers to the maximum number of split connectors that must be traversed in a path reaching the node from the start node, minus the number of join connectors in the same path. The out-depth \(\lambda_{out}(c)\) is defined analogously with respect to the end node. \[\Lambda(G)=max\{\lambda(n)/n\in N\}\]
* _Model depth_, Y: it computes the maximum nesting of a task in the hierarchy of plans. Starting at the top-level plan, where it would be initialized to 1, each time the process logic traverses a plan, it would be increased by one. Therefore, it can be defined as the maximum number of plans that it is necessary to descend to reach a task. We define the model depth of a task \(t\) recursively as follows (notice that in PROforma, plans are a type of tasks). \[modelDepth(t)=1,\ t\in top\_level\ plan\] \[modelDepth(t)= 1+modelDepth(p),\] \[t\
ot\in top\_level\ plan,\ t\in p,\ p\in P\] \[\text{Y}=maximum(modelDepth(t))\ \forall t\] Note that this latter metric is different from the depth metric that considers the nesting of a task in a graph with respect to the split/join connections traversed. In contrast, model depth measures the nesting of a task considering the hierarchy of graphs. Although it is possible to compute the model depth of any plan, we have only considered the model depth metric of the top-level plan (full model).