Empirical Findings on Ontology Metrics

D. Rodŕıgueza,b∗, M.A. Siciliaa, E. Garćıaa
aDepartment of Computer Science

University of Alcalá, Ctra. Barcelona, Km. 31.6
28871 Alcalá de Henares, Madrid, Spain

bDepartment of Computing and Communication Technologies
Oxford Brookes University

Wheatley Campus, Oxford OX33 1HX, UK
{daniel.rodriguezg,msicilia,elena.garciab}@uah.es

Abstract

Ontologies are becoming the preferred way of representing, dealing
and reasoning with large volumes of information in several domains.
In consequence, the creation, evaluation and maintenance of ontolo-
gies has become an engineering process that needs to be managed and
measured using sound and reliable methods. As part of any ontol-
ogy engineering or revision process, metrics can play a role helping
in identifying possible problems or incorrect use of ontology elements,
along with providing a kind of quality assessment that complements
reviews requiring expert inspection. However, in spite of the fact that
there are a number of ontology metric proposals described in the liter-
ature, there is a lack of empirical studies that provide a basis for their
interpretation. This paper reports a systematic exploration of exist-
ing ontology metrics from a large set of ontologies extracted from the
Swoogle search engine. The Ontorank value used inside the Swoogle to
rank search results is used as a contrast for some existing metrics. The
results show that the existing proposed metrics evaluated in general
do not seem to be indicators of that ranking, but they can be helpful
to identify particular kinds of ontologies.

Keywords. Ontology, metrics, Swoogle, Ontorank

∗Corresponding author

1



1 Introduction

Ontologies [4] are explicit representations of domain concepts and their re-
lationships. More formally, an ontology defines the conceptualisations of a
problem domain and a set of constraints on how terms can be combined to
model the domain, using a formal language. Ontologies are nowadays used
for different purposes in different systems, and as a consequence, they have
become artifacts in the development of software.

In the Software Engineering domain, metrics play an important role in
design, development and identifying software defects of future maintenance
problems. Like it happens in the software engineering tasks, generally the
construction of ontologies follows an iterative and incremental approach and
a number of metrics has been proposed to measure different aspects such as
reliability, reusability, cohesion, etc. (see for example [14]). However, un-
like it happens in software engineering domain, there is a lack of empirical
studies that validate how the different metrics are capable of judging qual-
ity properties and their interpretation. Inquiry on the relation of quality
indicators and metrics is in consequence an important open challenge for a
more systematic approach to developing ontology-based software systems.
The problem with ontology metrics is that ontologies are very heterogeneous
in their structure, objectives and level of formality. In consequence, there is
a need to carry out exploratory studies that could eventually result in in-
sights about the metrics applicable to different kinds of ontologies and their
interpretation for each of them.

This paper reports on on such an exploratory study, carried out by mea-
suring a large set of OWL ontologies obtained from the Swoogle reposi-
tory together with a measure of their popularity using Ontorank [2]. The
study has been carried out by implementing and using an open source soft-
ware framework for computing ontology metrics expressed in the Ontology
Web Language (OWL)1, that was already used in a previous preliminary
study [9]. OWL has been chosen as the language is a W3C recommendation
and has rapidly become the standard language for expressing formal ontolo-
gies. The results of the study point out that current metrics are not directly
interpretable in a general way, but there seems to be different categories of
ontologies that could be discernible from the value of the metrics.

The rest of the paper is structured as follows. Section 2 covers the
background on ontology metrics and some related work. Next, Section 3
describes the statistical analyses performed together with some findings.

1
http://www.w3.org/TR/owl-guide/

2



Finally, Section 4 concludes the paper and outlines future research work.

2 Background

Ontologies have been a resource in the field of artificial intelligence since
the 70’s. However since the inception of the Semantic Web, ontologies have
become the principal recourse to integrate and deal with online information,
and a new set of standards to formally represent them has been developed
and gained widespread adoption. The Ontology Web Language (OWL) is
one of such standards that belongs to a family of knowledge representa-
tion languages created for the Semantic Web [1]. The OWL has reached
the status of W3C (World Wide Web Consortium) recommendation. From
a technical point of view OWL extends the RDF (Resource Description
Framework) and RDF-S (RDF Schema) allowing us to integrate a variety of
applications using XML as interchange syntax.

The OWL has evolved since its inception in 2004 to a new specification in
2009 (OWL2) with different sub-languages depending on the expressiveness
and reasoning capabilities provided. There are however common elements
and OWL ontologies are composed of (i) classes that can be nested as sets of
individuals, (ii) individuals as instances of classes, i.e., objects of the domain
and (iii) properties as binary relations between individuals. It is also possible
to specify property domains, cardinality ranges and reasoning on ontologies.
From these elements a number of authors have proposed metrics to measure
the quality of ontologies.

Several authors have proposed metrics related to ontologies or adapted
from Object-oriented software. For example, Yao et al. [17] proposed a set
on metrics to measure the cohesion in ontologies such as the number of root
classes, number of leaf classes and average depth of inheritance tree of all
leaf nodes. Authors validated the metrics both theoretically and empirically.
Their empirical validation consisted of comparing statistically the results of
the metrics and subjective evaluation of eighteen evaluators.

Tatir et al. [13] propose three orthogonal dimensions to evaluate the
quality of an ontology. The first dimension is quality of the ontology schema
representing the real world. The second dimension evaluates the content of
the ontology, i.e, how well populated is the ontology and if it reflects the
real world. Finally, the third dimension evaluates if the of instances and
relations agree with the schema. Schema metrics proposed by the authors
include relationship, attribute and inheritance richness. Instance metrics
are divided into (i) knowledge base metrics such as class richness, average

3



population and cohesion and (ii) class metrics which include importance,
fullness, inheritance richness, relationship richness connectivity and read-
ability. Authors analysed two general purpose ontologies SWETO and TAP
as well as the GlycO (Glycomics Ontology) ontology.

Vrandečić and Sure [15] propose a framework for designing ontologies
though a process of normalisation and considering stable metrics, i.e., met-
rics considering the open world assumption.

Metrics and metric frameworks can help with processes or methodologies
for the design of ontologies or ontological tools. A well-known methodology
for the evaluation of ontological adequacy of taxonomic relationships is On-
toClean defined by Guarino and Welty [5]. Another process is the OntoMet-
ric process defined by Lozano-Tello and Gómez-Pérez [7]. OntoMetric is a
process to select ontologies using a broad range of metrics, not only those
that can be collected form the actual ontologies.

A suite a of ontology metrics to measure design complexity has been re-
cently proposed [18], reporting an analysis of desirable properties for metrics
described by Weyuker, and providing results of an empirical evaluation in
which the metrics proposed are shown to be discriminating ontologies that
are considered by the authors to be of different complexity.

In another direction, Ding et al. [3, 2] have developed Swoogle, a se-
mantic search engine in which the popularity of a semantic web document is
defined using the Ontorank measure adapted from the well-known PageRank
algorithm [12]. That ranking metric is an external criteria that can be used
to contrast ontology metrics. The problem with the ranking is that its inter-
pretation as a quality measure is not evident and has not been empirically
studied. However, as other quality measures are not available for ontologies,
it represents an interesting opportunity to contrast existing metrics.

We also developed an open source framework to extract the OWL on-
tology metrics called OntoMetrics. The OntoMetrics is an open source Java
implementation that makes use the Java libraries of Protégé2 an ontology
editor that provides an application programmers interface (API) for loading,
saving and manipulating OWL and RDF files. In this paper we have use
this framework to collect metrics from a large number of ontologies together
with their popularity measured using Ontorank.

2
http://protege.stanford.edu/

4



3 Empirical Analysis of Ontology Data

In this section, the metrics gathered are described and then the results
of the statistical analysis is reported. Only basic metrics related to the
main elements of ontologies have been implemented and used. Particularly,
metrics dealing with classes inferred (i.e., defined classes) or axioms were
not considered as a majority of the ontologies available for the study lacked
these kinds of constructs.

3.1 Dataset

In order to empirically analyse the relationships between the previously de-
fined metrics (Section 2), we downloaded around 1,500 ontologies defined in
OWL using Swoogle3, which is a search engine for the Semantic Web. A
crawler was developed ad hoc that submits a blank query to Swoogle and
then navigates the pages downloading automatically the OWL files. Only
ontologies expressed in OWL were considered as they represent the majority
of the available ones, and allows for an homogeneous analysis.

Using the previously commented Ontometrics framework, we collected
the following metrics:

• No. of Classes (noc) is a simple count of the number of classes (|C|)
contained in the ontology.

• No. of Instances (noi) is a simple count of the number of instances
(|I|) contained in the ontology.

• No. of Properties (nop) is a simple count of the number of properties
(|p|) contained in the ontology.

• No. of Root Classes metric (norc) [17] corresponds to the number of
root classes (without superclasses) explicitly defined. Let C the classes
in an ontology: norc = |Ci|, ¬∃Cj | Ci * Cj. The range of this metric
is from 1 to |C|. There is at least one root and the larger the number
of root classes, the more diverse the ontology will be.

• No. of Leaf Classes metric (nolc) [17] is the sum of all leaf classes,
i.e., those without subclasses, in an ontology. Let C the set of classes
in an ontology: nolc = |Ci|, ¬∃Cj | Cj * Ci. A a root class without
out inheritance can also be considered a leaf class, the range of this
metrics is from 1 to |C|.

3
http://swoogle.umbc.edu/

5



• Average Population metric (ap) [13] measures the average distribution
of instances across all classes. Formally, it is defined as: ap =

|I|
|C|

According the authors, this metric is supposed to be used in conjunc-
tion with the Class Richness metric as an indication if there is enough
information in the ontology.

• Class Richness metric (cr) [13] is the ratio between the number of
classes that have instances (|C′|)divided by the total number of classes.
This metric provides an indication of how many instances are really
related to classes defined in the schema.

• Explicit Depth of Subsumption Hierarchy (dosh). This metric cor-
responds to the maximum length of hierarchical relationships in the
ontology explicitly defined, i.e., longer path from a root class.

• Relationship Richness metric (rr) [13] is defined as the ratio of the
number of relationships divided by the sum of the number of subclasses
plus the number of relationships: rr =

|P|
|SC|+|P|, where |P | is the

number of relationships and SC is the number of subclasses (or the
number of inheritance relationships). According to the authors, this
metric reflects the diversity of relations and placement of relations in
the ontology. An ontology that contains many relationships other than
class-subclass relations (values close to 1) is richer than a taxonomy
with only class-subclass relationships (values close to zero).

• Inheritance Richness metric (ir) [13] is defined as the average number
of subclasses per class. Formally, ir is defined as:

ir =

∑
Ci∈C |H

c(C1, Ci)|
|C|

where Hc(C1, Ci)| is the number of subclasses (C1) for a class (Ci)
and the divisor (C) is the total number of classes. According to the
authors, this metric represents the distribution of information across
different levels of the ontology inheritance tree and serves as an indi-
cator of how well knowledge is grouped into different categories and
subcategories in the ontology. Values close to zero indicate flat or hor-
izontal ontologies representing perhaps more general knowledge while
large values represent vertical ontologies describing detailed knowledge
of a domain.

6



• Ontorank [2, 3] is an adaptation from Google’s PageRank [12] to em-
ulate user’s navigation behaviour at document level of granularity.
OntoRank for a semantic document a is defined as

OntoRank(a) = wPR(a) +
∑

x∈OTC(a)

wPR(x)

where OTC(x) is the set of semantic documents that transitively a
imports or extends, and wPR(a) is defined as: and

wPR(a) = (1 − d) + d
∑

linkto(x,a)

wPR(x) · f(x, a)∑
link(x,−,y) f(x, y)

f(x, a) =
∑

link(x,l,a)

weight(l)

where link(a, l, b) represents a link from a to b semantic documents
with semantic tag l, linkto(x, a) set of documents directly linked to
a, weight(l) is a user specified navigation preference on semantic links
with tag l, d is the probability of navigating from one document to
another.

The Ontorank was directly provided by Swoogle, and represents an in-
dicator of quality analogous to the PageRank [11] algorithm used in the
Google search engine.

Table 1 summarizes the main metrics reported on the literature, classified
by the kind of elements considered. Table 5 in Appendix A shows an excerpt
from the data collected.

3.2 Statistical Analysis

As stated previously, we downloaded a large number of ontologies from
Swoogle from which we could extract the metrics previously defined. The
total number of ontologies from which we could collect all metrics was 1,413.
Table 2 shows the descriptive statistics for the metrics collected.

The first interesting observation is that the average, median and third
quartile values for most metrics are very low (see Table 2 and Figure 1),
i.e., the vast majority of ontologies are small and quite flat (the ontology

7



Table 1: Summary of Ontology Metrics used in this Work
Metric Acronym

No. of Classes noc
No. of Instances noi
No. of Properties nop
No. of Root Classes norc
Number of Leaf Classes nolc
Average Population ap
Class Richness cr
Explicit Depth of Subsumption Hierarchy dosh
Inheritance Richness ir
Relationship Richness Metric rr
Ontorank or

Table 2: Descriptive Statistics
Avg Med V ar StdDev Min Max Range Q3 StdSkw StdKrt

or 2.74 0.85 968.5 31.12 0.64 962.31 961.67 1 379 5300.1
ap 1.34 1 8.83 2.97 0 67 67 1.20 170.44 1469.95
cr 0.54 0.75 0.20 0.44 0 1 1 1 -1.84 -13.9

dosh 2.54 1 5.75 2.4 1 30 29 3 40.13 113.57
ir 0.34 0 0.17 0.42 0 1.83 1.83 0.75 9.97 -9.31

noc 36.11 6 6166.3 78.53 1 945 944 34 80.44 293.25
noi 28.13 6 9522.8 97.59 0 991 991 9 94.85 328.43
nolc 27.46 5 3815.8 61.77 0 823 823 27 90.87 367.80
nop 24 0 2997.8 54.75 0 952 952 23 107.11 690.53
norc 6.69 5 127.37 11.29 0 194 194 6 131.7 857.18

rr 2.78 1 13.57 3.68 0 71 71 5 156.8 1326.26

8



0 33.5 67

 508

 75  69
 28

 11

 143

 579

0 0.5 1 1 15.5 30

ap cr dosh

 4  3  0  0  1

0 0.92 1.83 1 473 945 0 35.5 71

ir noc rr

Figure 1: Histograms

hierarchy is in average very small). An exception is the ClassRichness (cr)
metric with a ’u’ shape which may be a reflection of two different ways of
designing or using an ontology, either the instances are kept separately of
the classes that define them or both classes and instances are stored together
with the ontology.

Also in general, the distributions of the metrics are not following any
identifiable distribution. Values of skewness and kurtosis in Table 2 allow
us to discard normal distributions for all of them.

The few large values are statistically outliers as shown in Figure 2 by
using the red colour. When analysing actual ontologies with large values,
these correspond to special ontologies such as those in the Open Biomedi-
cal Ontologies (OBO) repository. For example, the NCI Thesaurus (ncithe-
saurus.owl) and the disease ontology.owl are both extremely large ontologies
with a special purpose. There are reasons for this that could be hypothesized
to come from some differences on the languages and approaches to engineer
ontologies in the biomedical domain [6].

Table 3 shows the Spearman rank correlation coefficient between the
different metrics (also graphically in Figure 3). It can be observed that the
OntoRank metric is not correlated with any other metric, and therefore,
it seems that the popularity of an ontology cannot be identified by the
set of metrics used in this study. We have analysed this relationship with

9



Figure 2: Box-and-whisker Plots

Table 3: Spearman Rank Correlation Coefficients
cr dosh ir noc noi nolc nop norc rr

ap 0.6878 -0.4387 -0.4735 -0.3157 0.6261 -0.2800 -0.2754 0.1959 0.4837
cr -0.6995 -0.7151 -0.5607 0.2789 -0.5105 -0.6500 0.0787 0.7250

dosh 0.9776 0.8073 0.1370 0.7460 0.7959 0.0998 -0.7443
ir 0.7853 0.0848 0.7148 0.7709 0.0288 -0.7526

noc 0.3173 0.9812 0.6583 0.4275 -0.4580
noi 0.3369 0.2525 0.5574 0.1828
nolc 0.6154 0.4613 -0.3822
nop 0.2487 -0.5948
norc 0.2400

other data mining approaches using several Weka’s [16] classifiers such as
regression or classification trees to confirm this effect.

The relationship between the rest of the metrics were contrasted using
factor analysis. Table 4 shows the resulting factor loadings using princi-
pal component analysis with standarised values as an extraction method in
which 3 components were extracted.

The first component is characterized by high values in dosh, ir, noc
that roughly measure depth and breadth of the subsumption hierarchy, i.e.,
metrics related with the schema of the ontology. The second component is
characterized by a high correlation with the ap and noi so it is more related
with the content of the ontology rather than the schema. and properties
and also with relationship richness that relates both of them. In the third
component, the largest values are also related to the ap and noi.

There are some threats to the validity of this study. First of all, we
analysed single files, but Semantic Web documents can import or extend
other documents in order to be able of working together. Coupling [10] or
cohesion [8] metrics should be also be included. Furthermore, the metrics

10



Figure 3: Graphical representation of the Spearman Rank Correlations

Table 4: Factorial Analysis of the Ontological Metrics
Factor 1 Factor 2 Factor 3

ap -0.0786494 0.551809 -0.681197
cr -0.727543 0.33865 0.167987

dosh 0.855587 -0.205325 -0.164372
ir 0.825037 -0.29855 -0.300246
noc 0.835051 0.129089 0.290564
noi 0.344157 0.653781 -0.458953
nolc 0.797677 0.165676 0.337331
nop 0.676127 0.259735 0.0893104
norc 0.437119 0.61869 0.338713
rr -0.351663 0.566569 0.2992

11



computed were extracted from the OWL files, with no associated reasoning
process. This entails that the metrics do not consider elements that would
be potentially inferred from reasoners, e.g. automatic classifications or sub-
sumption relationships that are not explicit. While this may seem a major
limitation, the fact that most ontologies do not contain axioms or defined
classes (classes expressed through logical necessary and sufficient conditions)
suggests that explicit elements are enough for the empirical assessment of
metrics in the current state of development of ontologies.

4 Conclusions and Future Work

This paper addressed the empirical analysis of a large number of ontologies
through a collection of metrics in order to propose interpretations for the
metrics that can be subject to hypothesis contrast and eventually lead to a
practical applicability of these metrics. The OntoRank was used as a con-
trast as the only available assessment mechanism that can be automatically
computed without requiring expert assessment of the ontologies. It should
be noted that such kind of indicators cannot substitute but complement ex-
pert of logical assessments as those that have been already reported in the
literature.

To do so, we developed an open source tool, called Ontometrics, used
to collect the set of metrics presented in this paper together with their
Ontoranking value provided by Swoogle. The results showed that there is
no correlation between the popularity of an ontology and the metrics used
in this work. There are some peculiarities for example most ontologies are
quite flat and small, perhaps reflecting that most of the ontologies are not
mature. Some exceptions can be found in the OBO foundry4 repository
which contains mature ontologies from the biomedical domain. In this case,
although it is difficult to establish concrete thresholds to define when an
ontology needs to be improved, such metrics helped in identifying ontologies
for special purposes such as thesaurus.

Future work will be directed to further develop our metrics framework
with new metrics and update the framework to the new OWL API5. There
is also room for further analysis of the metrics, for example, analysing how
ontologies evolve by carrying out longitudinal studies accounting for the
evolution of ontologies.

4
http://www.obofoundry.org/

5
http://owlapi.sourceforge.net/

12



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15