=Paper=
{{Paper
|id=Vol-2536/om2019_Tpaper3
|storemode=property
|title=Identifying Mappings among Knowledge Graphs by Formal Concept Analysis
|pdfUrl=https://ceur-ws.org/Vol-2536/om2019_LTpaper3.pdf
|volume=Vol-2536
|authors=Guowei Chen,Songmao Zhang
|dblpUrl=https://dblp.org/rec/conf/semweb/ChenZ19
}}
==Identifying Mappings among Knowledge Graphs by Formal Concept Analysis==
https://ceur-ws.org/Vol-2536/om2019_LTpaper3.pdf
Identifying Mappings among Knowledge Graphs
by Formal Concept Analysis
Guowei Chen1,2 and Songmao Zhang2
1
University of Chinese Academy of Sciences, Beijing, P.R. China
2
Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese
Academy of Sciences, Beijing, P.R. China
chenguowei17@mails.ucas.ac.cn, smzhang@math.ac.cn
Abstract. Formal Concept Analysis (FCA) is a well-developed math-
ematical model for clustering individuals and structuring concepts. In
one of our previous studies, we proposed to incrementally match classes
and properties across complex biomedical ontologies based on FCA. We
intend to apply the approach to matching knowledge graphs (KGs) and
this paper reports a preliminary result. Compared with ontologies which
model the schema knowledge of classes, KGs are much larger and fo-
cus on instances and their properties. We build three token-based for-
mal contexts for classes, properties, and instances to describe how their
names/labels share lexical tokens, and from the concept lattices com-
puted, lexical mappings can be extracted across KGs. An evaluation on
the 9 matching tasks of OAEI Knowledge Graph Track shows that our
system obtains the highest recall in class, property, instance, and over-
all matching over the seven systems participated in the track in OAEI
2018. Additionally, our system is able to identify cases when one entity
in a KG does not have any correspondence in another KG. Based on the
lexical instance mappings, we further construct a property-based formal
context to identify commonalities among properties in a structural way,
which indicates a promising direction for taking full advantage of the
knowledge within KGs.
Keywords: knowledge graph ·formal concept analysis ·ontology match-
ing
1 Introduction
Ontologies serve as the foundation of the Semantic Web by defining basic classes
and their structures that constitute various domain knowledge, thus can be used
to semantically annotate the Web resources. Ontology matching (OM) tech-
niques [1] have been developed to detect the correspondence among diverse yet
overlapping ontologies so that search engines and applications can understand
the equivalence on the Web as well as mismatches. Since Google invented the
Copyright © 2019 for this paper by its authors. Use permitted under Creative Com-
mons License Attribution 4.0 International (CC BY 4.0).
2 G. Chen and S. Zhang
notion of Knowledge Graph (KG) and made its own system in 2002, and with the
prevailing of the TransE series algorithms [2,3] for embedding KGs in a numerical
way, the Semantic Web has evolved into the KG time. Soon the OM community
realized the inevitable of identifying semantic connections among KGs. Started
in 2018, the annual OAEI competition 3 presents a KG track where 9 KGs in
the category of Games, Comins, and TV&Books, respectively, yield a total of
9 pairwise matching tasks [5,6]. Seven OM systems were able to participate in
the KG track in 2018, including the well-known AML [7], LogMap family [8],
POMAP++ [9], Holontology [10], and DOME [11].
By design, both ontologies and KGs have classes, properties and instances.
Ontologies primarily model the schema knowledge of classes whereas KGs are
much larger and mostly describe instances and their properties. This means that
techniques for mapping KGs focus more on instance matching [12]. In one of our
previous studies [18,19,20], we proposed the FCA-Map system that incrementally
matches classes and properties across complex biomedical ontologies based on
Formal Concept Analysis (FCA). FCA is a well-developed mathematical model
for clustering individuals and structuring concepts [14]. The purpose of FCA-
Map is to push the envelop of the FCA formalism in exploring as much knowledge
as possible within ontologies, including class names, subclass relations, part-
whole relations, disjointedness, and other logical axioms. In this paper, we intend
to apply the approach to matching knowledge graphs and a preliminary result
is reported.
Concretely, based on the rationale of lexical matching in FCA-Map, we con-
struct three token-based formal contexts for classes, properties, and instances,
respectively, to describe how their names/labels share lexical tokens. The derived
formal concept lattices represent the clustering of classes/properties/instances
by names, and thus lexical mappings can be extracted across KGs. An evalua-
tion on the OAEI KG Track shows that, when compared with the seven OAEI
2018 participants, our system obtains the highest recall and comes second in
F-measure in terms of average performances on 9 tasks. In addition, our system
can identify most of the null mappings provided in the OAEI gold standard
for entities that do not have any correspondence in another KG. Based on the
lexical mappings, we further build a structural formal context to describe how
properties across KGs have common in linking the same instances. The map-
pings identified solely by structural matching indicate a promising direction for
taking full advantage of the knowledge within KGs.
Although FCA has been applied to modeling KGs [13], to the best of our
knowledge, this is a first attempt to identify the correspondence among KGs by
a FCA-based approach. In Section 2 of the paper, we will present the lexical
matching part and its evaluation on the OAEI KG Track. A first step of struc-
tural matching is described in Section 3, and our on-going work is discussed in
Section 4 at last.
3
http://oaei.ontologymatching.org/
Identifying Mappings among KGs by FCA 3
2 Identifying lexical mappings between KGs
FCA is a principled approach of deriving a concept hierarchy from a collection
of objects and their attributes. The fundamental notions are formal context and
formal concept, and the former is defined as a binary table K := (G, M, I),
where G is a set of objects as rows, M a set of attributes as columns, and I
a binary relation between G and M in which (g, m) ∈ I reads object g has
attribute m , generally represented by “×” in the table cell. A formal concept
of context K is a pair (A, B) consisting of a subset of objects A ⊆ G and a
subset of attributes B ⊆ M such that B equals all the attributes common to
objects in A and at the same time, A equals the set of objects that have all
the attributes in B. The subconcept-superconcept relation can be defined as:
(A1 , B1 ) ≤ (A2 , B2 ) :⇔ A1 ⊆ A2 (⇔ B1 ⊇ B2 ), leading to a lattice structure of
formal concepts.
For the instances in two KGs, we use the following example to illustrate the
construction of token-based formal context, the derivation of concept lattice and
the extraction of instance mappings. The similar process applies to the classes
and properties in two KGs.
Example 1. Given two KGs memory-beta (MB), stexpanded (STEX) from OAEI
2018, the left of Fig. 1 shows some instances and their label strings. Note that
one string can be shared by instances across KGs, as listed on the right of Fig. 1.
We extract names and labels of all instances in the two KGs and separate the
tokens in them through normalization techniques [17]. As shown in Fig. 2 on
the left, the token-based formal context is constructed with each string as an
object, each token as an attribute, and the cell in the context marked when the
string contains the token. The gray area in the table presents a formal concept
indicating the duality between its objects and attributes, i.e., the subset of tokens
are identified to co-exist solely in the two strings.
From the token-based formal context, formal concepts and their lattice struc-
ture can be derived automatically, as shown on the right of Fig. 2, where
each node represents a formal concept and the line denotes the subconcept-
superconcept relation from the lower to the upper node 4 . For identifying map-
pings, we pay attention to formal concepts that contain exactly two strings
relevant to instances across KGs. Take for example the gray node on the right
of Fig. 2 which corresponds to the gray area in the context on the left. Four
instance mappings can be extracted from this formal concept:
⟨MB:USS_Fredrickson, STEX:USS_Fredrickson⟩
⟨MB:USS_Fredrickson_(NCC-42111), STEX:USS_Fredrickson_(NCC-42111)⟩
⟨MB:USS_Fredrickson, STEX:USS_Fredrickson_(NCC-42111)⟩
⟨MB:USS_Fredrickson_(NCC-42111), STEX:USS_Fredrickson⟩
The first two are exact matches and the latter partial matches.
4
For the sake of efficiency, we use the Galois Sub-Hierarchy (GSH) [15] which preserves
solely the necessary elements of the lattice and implement the Hermes[16] algorithm
for computing the lattice.
4 G. Chen and S. Zhang
"NCC-42111"
rdf
USS_Fredrickson_(NCC-42111) s:la
bel
USS_Fredri…
rdfs
:lab
el NCC-42111
"USS Fredrickson (NCC-42111)"
String Instances
rdfs MB:USS_Fredrickson
USS_Fredrickson :lab
el uss fredrickson
rdfs
STEX:USS_Fredrickson
:lab
el USS_Fredrickson_(NCC-42111) MB:USS_Fredrickson_(NCC-42111)
uss fredrickson (ncc-42111)
STEX:USS_Fredrickson_(NCC-42111)
"USS Fredrickson"
ncc-42111 MB:NCC-42111
rdfs fredrickson system MB:Fredrickson_system
Fredrickson_system :lab
el
rdfs
:lab USS_Fredrickson
el
"Fredrickson system"
instances of MB
instances of STEX
Fig. 1. Left: An RDF graph representation of part of two KGs in Example 1. Right:
Strings and the instances (can be across KGs) having them as labels.
uss fredrickson, uss fredrickson (ncc-42111), ncc-42111, fredrickson system
fredrickson
uss fredrickson, uss fredrickson (ncc-42111), fredrickson system
system
fredrickson
42111
fredrickson system
ncc
uss
uss fredrickson (ncc-42111), ncc-42111 fredrickson, system
ncc, 42111
uss fredrickson ×× uss fredrickson, uss fredrickson (ncc-42111)
uss, fredrickson
uss fredrickson (ncc-42111) × × × ×
ncc-42111 ×× uss fredrickson (ncc-42111)
uss, fredrickson, ncc, 42111
fredrickson system × ×
uss, fredrickson, ncc, 42111, system
Fig. 2. Left: The token-based formal context for instances in Example 1. Right: The
derived formal concept lattice.
There are 9 knowledge graphs in the OAEI KG Track, as listed in Table 1,
and on its corresponding 9 KG matching tasks, we evaluate our FCA-based
lexical matching approach. The results are shown in Fig. 3 according to the gold
standard5 and evaluation tool6 provided by OAEI 2018. One can see that our
approach is able to achieve high performances in recall, and the quality of class
mappings is better than that of property mappings which is then better than
instance mappings while at the same time the number of mappings identified for
class, property and instance increases.
A comparison with the seven OAEI 2018 KG Track participants is listed in
Table 2. Again, our approach favors recall and ranks the first in average over 9
tasks for class, property, instance and overall matching. Moreover, our approach
obtains the second best F-measures in all matching types, indicating that a bal-
5
https://github.com/sven-h/dbkwik/tree/master/e_gold_mapping_interwiki/
gold
6
http://oaei.ontologymatching.org/2018/results/knowledgegraph/kg_track_
eval.zip
Identifying Mappings among KGs by FCA 5
Table 1. An overview of 9 knowledge graphs of the OAEI KG Track
KG Category #Class #Property #Instance
RuneScape Wiki (runescape) Games 106 1,998 200,605
Old School RuneScape Wiki (oldschoolrunescape) Games 53 488 38,563
DarkScape Wiki (darkscape) Games 65 686 19,623
Marvel Database (marvel) Comics 2 99 56,464
Hey Kids Comics Wiki (heykidscomins) Comics 181 1,925 158,234
DC Database (dc) Comics 5 177 128,495
Memory Alpha (memory-alpha) TV 0 326 63,240
Star Trek Expanded Universe (expanded) TV 3 201 17,659
Memory Beta (memory-beta) Books 11 413 63,223
darkscape~oldschoolrunescape 10 4 runescape~darkscape 10 4 runescape~oldschoolrunescape 10 4
1 2 1 1 8
Precision
4
F-measure
Recall 1.5 6
#mappings
3
0.5 1 0.5 0.5 4
2
0.5 1 2
0 0 0 0 0 0
Class Property Instance Class Property Instance Class Property Instance
heykidscomics~dc 10 4 marvel~dc~results marvel~heykidscomics 10 4
1 8 1 4000 1 3
6 3000
2
0.5 4 0.5 2000 0.5
1
2 1000
0 0 0 0 0 0
Class Property Instance Class Property Instance Class Property Instance
4
memory-alpha~memory-beta 10 memory-alpha~stexpanded memory-beta~stexpanded
1 2 1 1 6000
4000
1.5
3000 4000
0.5 1 0.5 0.5
2000
2000
0.5 1000
0 0 0 0 0 0
Class Property Instance Class Property Instance Class Property Instance
Fig. 3. The results of FCA-based KG matching. Charts in the same row are about
the same category, i.e., Games, Comics, and TV&Books. In each chart, the bars show
precision, F-measure and recall of each task, whereas the lines show the number of
mappings identified by our approach.
6 G. Chen and S. Zhang
ance can be achieved between quality and quantity. Overall, the DOME system
[11] stands out by having the best precision and F-measure in both property
matching and instance matching for most cases, followed by Holontology [10]
which ranks the first in overall precision.
Table 2. Comparing with OAEI 2018 KG Track participants by average performance
over 9 matching tasks, where # stands for the number of tasks that the system is able to
generate non-empty alignments, and Size the average number of generated mappings.
Class Property Instance overall
System #
Size Prec. F-m. Rec. Size Prec. F-m. Rec. Size Prec. F-m. Rec. Size Prec. F-m. Rec.
AML 5 11.6 0.85 0.64 0.51 0.0 0.00 0.00 0.00 82380.9 0.16 0.23 0.38 102471.1 0.19 0.23 0.31
POMAP++ 9 15.1 0.79 0.74 0.69 0.0 0.00 0.00 0.00 0.0 0.00 0.00 0.00 16.9 0.79 0.14 0.08
Holontology 9 16.8 0.80 0.83 0.87 0.0 0.00 0.00 0.00 0.0 0.00 0.00 0.00 18.8 0.80 0.17 0.10
DOME 9 16.0 0.73 0.73 0.73 207.3 0.86 0.84 0.81 15688.7 0.61 0.61 0.61 15912.0 0.68 0.68 0.67
LogMap 7 21.7 0.66 0.77 0.91 0.0 0.00 0.00 0.00 97081.4 0.08 0.14 0.81 97104.8 0.09 0.16 0.64
LogMapBio 9 22.1 0.68 0.81 1.00 0.0 0.00 0.00 0.00 0.0 0.00 0.00 0.00 24.1 0.68 0.19 0.11
LogMapLt 6 22.0 0.61 0.72 0.87 0.0 0.00 0.00 0.00 82388.3 0.39 0.52 0.76 88893.1 0.42 0.49 0.60
Our System 9 22.7 0.68 0.81 1.00 250.9 0.64 0.74 0.86 25903.9 0.39 0.55 0.95 26177.4 0.45 0.61 0.93
Table 3. Null mappings identified by our system, where Gold stands for the number
of null mappings in the gold standard.
Class Property Instance
old
old
ld
KG matching task
go
ng
ng
ld
ld
ld
in
ti
ti
go
go
go
ld
ld
ld
t
Go
Go
Go
No
No
No
In
In
In
darkscape�oldschoolrunescape 7 6 22 6 6 455 38 34 25,032
runescape�darkscape 5 5 38 10 10 1,339 13 3 107,941
runescape�oldschoolrunescape 4 3 53 8 8 1,611 37 11 115,061
heykidscomics�dc 13 12 123 10 8 1,512 53 40 156,744
marvel�dc 3 3 0 12 11 143 65 56 164,543
marvel�heykidscomics 10 4 128 10 8 1,517 42 38 160,706
memory-alpha�memory-beta 11 11 1 10 7 511 49 42 92,334
memory-alpha�stexpanded 3 3 1 11 11 339 60 57 69,823
memory-beta�stexpanded 14 14 0 12 11 369 55 51 67,848
The gold standard of OAEI KG Track contains not only 1:1 mappings but
also cases where one entity in a KG is matched to “null” in the other KG.
They represent the uniqueness of classes, properties and instances to one knowl-
edge base with respect to another, which is complementary to 1:1 and complex
mappings in revealing the whole picture of the relationship between two sys-
tems. We call them null mappings, and the OAEI evaluation takes them into
account solely for calculating false positives in 1:1 mappings. By taking advan-
Identifying Mappings among KGs by FCA 7
tage of the inherent feature of the FCA formalism, our system is able to iden-
tify such null mappings. When a formal concept in the derived lattice contains
strings solely from one entity in a KG, the corresponding entity contributes to
a null mapping. As shown in Table 3, there are 571 null mappings in the gold
standard and our system has successfully detected 473 of them, accounting for
83%, as exemplified by ⟨darkscape:Room, oldschoolrunescape:null⟩ for class null
mapping, ⟨marvel:null, dc:runtime⟩ for property, and ⟨memory-beta:Victoria,
stexpanded:null⟩ for instance. At the same time, a large number of null map-
pings identified are not in the gold standard, and their validity needs further
investigation as the gold standard is only partial as reported by OAEI.
3 Identifying structural mappings between KGs
We call the obtained lexical mappings anchors, based on which we can build
formal contexts from the structural knowledge in KGs so as to extract addi-
tional mappings. A KG can be seen as an RDF graph where the vertex generally
represents a class or an instance and the edge a property from one instance
to another, or a type relation from an instance to a class. For given two KGs,
a property-based formal context is constructed by taking properties from two
KGs as objects, and pairing the lexical instance anchors across KGs as attributes.
When a property is used to link two instances in an anchor pair, the correspond-
ing cell in the formal context is marked. After the lattice is derived, if a formal
concept contains solely two properties from two KGs, respectively, they can be
extracted as a structural mapping. Again, in the following we use an example to
illustrate the matching process.
Example 2. Given two KGs memory-alpha (MA), memory-beta (MB) from OAEI
2018, a part of their (subject, predicate, object) (SPO triples) are listed in Table 4.
Table 4. Some SPO triples from two KGs MA and MB.
subject predicate object
MA:Rules_of_Acquisition_(episode) MA:wsstoryby MA:Hilary_J._Bader
MA:Rules_of_Acquisition_(episode) MA:wsteleplayby MA:Ira_Steven_Behr
MA:Battle_Lines_(episode) MA:wsstoryby MA:Hilary_J._Bader
MA:Battle_Lines_(episode) MA:wsteleplayby MA:Richard_Danus
MA:Paradise_Lost_(episode) MA:wsteleplayby MA:Robert_Hewitt_Wolfe
MB:Rules_of_Acquisition_(episode) MB:story MB:Hilary_J._Bader
MB:Rules_of_Acquisition_(episode) MB:teleplay MB:Ira_Steven_Behr
MB:The_Nagus MB:teleplay MB:Ira_Steven_Behr
MB:Battle_Lines_(episode) MB:story MB:Hilary_J._Bader
MB:Paradise_Lost_(episode) MB:teleplay MB:Robert_Hewitt_Wolfe
8 G. Chen and S. Zhang
Some lexical instance anchors between MA and MB are as follow:
a = ⟨MA:Battle_Lines_(episode), MB:Battle_Lines_(episode)⟩
b = ⟨MA:Hilary_J._Bader, MB:Hilary_J._Bader⟩
c = ⟨MA:Ira_Steven_Behr, MB:Ira_Steven_Behr⟩
d = ⟨MA:Paradise_Lost_(episode), MB:Paradise_Lost_(episode)⟩
e = ⟨MA:Rules_of_Acquisition_(episode), MB:Rules_of_Acquisition_(episode)⟩
f = ⟨MA:Richard_Danus, MB:Richard_Danus⟩
g = ⟨MA:Robert_Hewitt_Wolfe, MB:Robert_Hewitt_Wolfe⟩
h = ⟨MA:The_Nagus, MB:The_Nagus⟩.
MA:wsteleplayby MA:wsstoryby MB:teleplay MB:story
(a, f )
(d, g)
(h, c)
(a, b)
(e, c)
(e, b)
MA:wsteleplayby
MA:wsstoryby
MB:teleplay
MB:story
(d,g) (e,c) MB:teleplay
MA:wsteleplayby × × × (d,g) (e,c) (h,c)
(e,b) (a,b)
MB:teleplay × × × MA:wsteleplayby
(d,g) (e,c) (a,f)
MA:wsstoryby × ×
MB:story × × (d,g) (e,c) (e,b) (a,b) (a,f) (h,c)
Fig. 4. Left: The structural formal context for properties in Example 2. Right: The
derived formal concept lattice.
Table 5. The property mappings solely identified structurally between two KGs MA
and MB.
Property mapping
⟨MA:relative, MB:otherRelatives⟩
Those in the gold standard ⟨MA:wsteleplayby, MB:teleplay⟩
⟨MA:wsstoryby, MB:story⟩
⟨MA:prev, MB:before⟩
Those not in the gold standard ⟨MA:next, MB:after⟩
⟨MA:relative, MB:grandparents⟩
⟨MA:abreadby, MB:narrator⟩
The constructed property-based formal context is presented on the left in
Fig. 4 and the lattice derived on the right. As shown by the gray area, a property
mapping ⟨MA:wsteleplayby, MB:teleplay⟩ is identified by structural knowl-
edge rather than by names. For the matching task between KGs MA and MB,
7 property mappings are detected solely by the structural matching, as listed
in Table 5, of which 2 are true positives. Note that the OAEI 2018 KG gold
standard is declared to be only partial, and the lower part of Table 5 shows
promising candidates. With these additional structural mappings, the precision,
F-measure and recall for the property task have all increased compared with the
lexical matching step, as shown by Fig. 5.
Identifying Mappings among KGs by FCA 9
memory-alpha~memory-beta
1 140
Precision
0.9 F-measure
Recall 120
0.8 #mappings
0.7 100
0.6
80
0.5
60
0.4
0.3 40
0.2
20
0.1
0 0
Lexical Structural Lexical Structural
Fig. 5. Evaluation of the additional structural mappings between properties of two
KGs MA and MB.
On the other hand, the structural property matching does not affect the
performance of the other 8 tasks, either because the mappings found are not
in the gold standard or none mappings are found at all. Note that as shown
by Fig. 3, these 8 property tasks have already obtained a higher performance
compared with the MA-MB task at the lexical matching step. To further improve,
comprehensive ways shall be explored to augment the structural formal contexts
with extended knowledge in KGs.
4 Discussion and conclusions
This paper reports an on-going study of constructing multiple FCA structures
for the purpose of matching knowledge graphs. Its lexical matching part already
receives the best recall and the second best F-measure in class, property, in-
stance, and overall matching for the OAEI 2018 KG Track tasks, revealing the
advantage of our FCA-based approach. Moreover, our system has identified 83%
of null mappings provided in the OAEI gold standard. All these come from the
inherent capability of FCA formalism in detecting commonalities among individ-
uals and accordingly forming concepts and classifying them in a lattice structure.
For the structural matching, we have realized a property-based lattice from the
knowledge of property linking one instance to another in KGs. Obviously, fur-
ther an instance-based lattice shall be computed similarly to identify structural
instance mappings. Moreover, the knowledge of instance belonging to class in
KGs can be used as well to explore commonalities among instances. As a matter
of fact, we are developing an iterative framework so as to perform class, prop-
erty, and instance matching in an augmented way until no further matches can
be found.
Our previous system FCA-Map is for matching ontologies and thus targets
classes. Although there are classes in the OAEI KGs, they are much fewer than
instances and properties, and basically none schema knowledge is specified. This
10 G. Chen and S. Zhang
says that the structural matching part in FCA-Map cannot be applied directly,
and alternative types of formal contexts are being designed targeting instances
and properties. In addition to matching, FCA-Map includes a structural valida-
tion step to eliminate wrong mappings based on the disjoint axioms in ontologies.
When there is no such knowledge in KGs, we shall develop alternative validation
strategies so as to ensure the quality of mappings and prevent the mismatches
from propagating in the iterative framework.
What is worth noting is that the systems participated in OAEI 2018 are
basically ontology matching systems and not specifically tailored for knowledge
graph matching. Therefore it is understandable that the performance can be
unsatisfactory for some tasks. Nevertheless, systems like DOME still managed to
outperform. DOME uses the doc2vec approach to train vector representations for
ontology classes and instances based on large texts, so that the similarity among
entities can be computed according to the distance of vectors. Such numerical
ways of embedding KG entities into a high-dimensional, continuous space are
called representation learning, which have already been adopted for matching
ontologies, as in [21,22,23]. To compare our FCA-based approach with these
works will be of interest, not only by conducting comparative experiments but
also exploring the possible combining ways.
Acknowledgements. This work has been supported by the National Key Research
and Development Program of China under grant 2016YFB1000902 and the Natural
Science Foundation of China under No. 61621003.
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