## 3 Ontology Modeling and Quality Measures
### Quality Metrics
#### 3.1.1 Structural Measures  
As ontology models are implemented using graphs, various authors have introduced metrics to measure the complexity and quality of the graph structure. The main aspects within structural metrics cover, among others, the balancing and density (richness) measures of the graph.  
Using the notation of the formal ontology model presented in [37], the ontology \(O\) is provided by  
\[O:=\left\{C,P,A,H^{C},prop,att\right\},\]  
where:  
* \(C\): the set of concepts;
* \(P\): the set of relations;
* \(A\): the set of properties;
* \(H^{C}\subseteq C\times C\): concept taxonomy;
* _prop_: \(P\to C\times C\), which relates concepts non-taxonomically, that is, every property may have domain and range axioms;
* _att_: \(A\to C\), which relates concepts with literal values.  
In the literature [37], the following quality measures are used for ontology evaluations:  
* _Relation Richness_: \(RR=\dfrac{|P|}{|SC|+|P|}\), where \(SC\) denotes the set of subclass relationships. An ontology that contains many relations other than class-subclass relations is richer than a taxonomy with only class-subclass relationships.
* _Property Richness_: \(AR=\dfrac{|att|}{|C|}\). It is assumed that as more properties are assigned to classes, more information can be conveyed in the ontology model.  
* _Inheritance Richness_: \(1R=\dfrac{\sum_{i\in C}|H^{C}(C_{1},C_{i})|}{|C|}\), which is defined as the average number of subclasses per class.
* _Class Richness_: This value shows the ratio of the number of classes that have instances and the total number of classes.
* _Average Population_: \(P=\dfrac{|I|}{|C|}\), where \(I\) is the number of instances; this shows the average number of instances for a class.
* _Cohesion_: This can be defined as the number of disjoint components of the ontology graph; a value of 1 signifies strong cohesion.
* _Importance of a Class_: This metric shows the importance of a class \(C_{i}\) as the number of instances that belong to the subtree rooted at \(C_{i}\) compared to the total number of instances.
* _Fullness_: The fullness of a class \(C_{i}\) is defined as the actual number of instances that belong to the subtree rooted at \(C_{i}\) compared to the expected number of instances under the subtree.
* _Inheritance Richness_: This shows the average number of subclasses per class in the subtree related to \(C_{i}\).
* _Connectivity_: This metric is defined for a class \(C_{i}\) as the number of instances of other classes that are connected to instances of \(C_{i}\).  
An alternative set of graph-oriented quality metrics has been proposed by [38], including the following heuristic elements:  
* _Number of objects_: the number of individuals belonging to a class;
* _Number of properties_: the number of predicates belonging to a class;
* _Number of children_: the number of subclasses;
* _Number of parents_: the number of superclasses;
* _Depth of inheritance tree_: the length of the longest path in the covered subgraph;
* _Centrality measure_: how far the depth of the class is from the average depth of all classes;
* _Density measure_: the number of related classes.  
As can be seen, these metrics are quantitative metrics against the schema of the ontology. The quality may relate to the integrity aspects of the ontology as well. A schema can be considered a low quality one when it contains a large number of conflicts [39]. This model distinguishes the following integrity measures:  
* property assertion conflicts;
* class assertion conflicts;
* statement assertion conflicts.  
Methods for checking integrity can be extended to the instance level as well [40].