=Paper=
{{Paper
|id=Vol-2123/paper6
|storemode=property
|title=How Hierarchies of Concept Graphs Can Facilitate the Interpretation
of RCA Lattices?
|pdfUrl=https://ceur-ws.org/Vol-2123/paper6.pdf
|volume=Vol-2123
|authors=Sébastien Ferré,Peggy Cellier
|dblpUrl=https://dblp.org/rec/conf/cla/FerreC18
}}
==How Hierarchies of Concept Graphs Can Facilitate the Interpretation
of RCA Lattices?
==
https://ceur-ws.org/Vol-2123/paper6.pdf
How Hierarchies of Concept Graphs Can
Facilitate the Interpretation of RCA Lattices??
Sébastien Ferré and Peggy Cellier
Univ Rennes, CNRS, INSA, IRISA
Campus de Beaulieu, 35042 Rennes cedex, France
ferre@irisa.fr, peggy.cellier@irisa.fr
Abstract. Relational Concept Analysis (RCA) has been introduced in
order to allow concept analysis on multi-relational data. It significantly
widens the field of application of Formal Concept Analysis (FCA), and
it produces richer concept intents that are similar to concept definitions
in Description Logics (DL). However, reading and interpreting RCA con-
cept lattices is notoriously difficult. Nica et al have proposed to represent
RCA intents by cpo-patterns in the special case of sequence structures.
We propose an equivalent representation of a family of RCA concept lat-
tices in the form of a hierarchy of concept graphs. Each concept belongs
to one concept graph, and each concept graph exhibits the relationships
between several concepts. A concept graph is generally transversal to
several lattices, and therefore highlights the relationships between dif-
ferent types of objects. We show the benefits of our approach on several
use cases from the RCA litterature.
Keywords: Formal Concept Analysis, Relational Concept Analysis, Data Min-
ing, Concept Graph
1 Introduction
Many domains produce multi-relational data. For example, in the health domain,
one can have patients taking drugs, drugs giving some symptoms and interact-
ing with other drugs, doctors taking care of patients and prescribing drugs to
patients, etc. In order to extract knowledge from that kind of data, many data
mining techniques, such as Formal Concept Analysis (FCA) [8], require the flat-
tening of multi-relational data but this results in loss of structural information,
and a more difficult interpretation of discovered patterns. It is therefore desirable
to have direct methods for multi-relational mining [2]. Several generalizations
of FCA have been proposed to handle relational data: Power Context Fami-
lies [10], Relational and Logical Concept Analysis [7], Relational Concept Anal-
ysis (RCA) [11], and Graph-FCA [5]. RCA has so far been the most frequently
used approach with applications in health [11,9] or model driven engineering [4].
?
This research is supported by ANR project PEGASE (ANR-16-CE23-0011-08).
c paper author(s), 2018. Proceedings volume published and copyrighted by its editors.
Paper published in Dmitry I. Ignatov, Lhouari Nourine (Eds.): CLA 2018, pp. 69–80,
Department of Computer Science, Palacký University Olomouc, 2018. Copying
permitted only for private and academic purposes.
70 Sébastien Ferré and Peggy Cellier
An important issue for the effective use of RCA is the interpretation of its
outputs. Indeed, RCA produces not one but several concept lattices, and the
intent of each concept may depend on the intent of other relationally-related
concepts, recursively. To the best of our knowledge, there has been only one pro-
posal to automatically extract and graphically represent the relational patterns
that are buried into concept intents: Nica’s cpo-patterns [9]. However, it has sev-
eral restrictions. First, it is defined only for sequential data. Second, it generates
cpo-patterns only for the concepts of one chosen lattice. Third, it generates a
cpo-pattern for each concept of the chosen lattice, missing potential factoriza-
tions between patterns and thus interesting information between patterns.
In this paper, we propose a novel and generic graphical representation of RCA
outputs that emphasizes the relational patterns. We call it hierarchy of concept
graphs. It has the following good properties. First, it makes no assumption on
the context family, and can therefore handle all kinds of graph structures, not
only sequences. Second, it is at the same time a complete and non-redundant
representation of the family of concept lattices, and does not require to choose
one concept lattice as starting point. Third, it offers a better balance in the
display between generalization ordering (lattice edges), and relationships (rela-
tional attributes). Fourth, it clusters concepts into concept graphs, and hence
produce a coarser-grained representation. Fifth, it can be efficiently computed
from the concept lattices, in linear time.
Section 2 discusses related work. Section 3 shortly recalls the main definitions
of RCA. Section 4 introduces our representation of RCA outputs as a hierarchy
of concept graphs, illustrates it on a reference example of RCA, and discusses its
properties. Section 5 evaluates our approach on a few use cases, and discusses the
impact of representation choices. Section 6 concludes and draws perspectives.
2 Related Work
Several generalizations of FCA have been proposed to handle relational data.
Power Context Families [10] has a formal context for each relation arity, i.e.
a context of objects, a context of couples of objects, a context of triples of
objects, etc. A concept lattice is computed for each context, independently of
other contexts. The resulting concepts are used as a vocabulary of types and
relations to build concept graphs that are similar to Conceptual Graphs [12,1].
Relational and Logical Concept Analysis [7] takes as input a power context
family limited to unary and binary relations but extended to complex logical
descriptions. It generates a single concept lattice where concept intents combine
both unary and binary descriptors, and where the labeling of the concept lattice
is extended with relationships between concepts. Relational Concept Analysis
(RCA) [11] takes as input a power context family limited to unary and binary
relations. In practice, the unary context is split in several unary contexts, one
for each type of object. RCA generates a concept lattice for each type of object,
where concept intents are sets of classical attributes and relational attributes.
The latter represent relationships to other concepts in the concept lattice family.
Hierarchies of Concept Graphs in RCA 71
Graph-FCA [5] takes as input a power context family without restriction on
arities. It generates a set of graph patterns where each node represents a unary
concept, each pair of nodes represents a binary concept, etc. For each concept
arity, the set of all concepts forms a concept lattice.
The above shows that there are two kinds of representations of the results:
concept lattices and concept graphs. They complement each other: concept lat-
tices emphasizes the generalization ordering between concepts, while concept
graphs emphasize the relationship patterns between objects in data. In RCA,
the native representation is made of concept lattices, and the relationship pat-
terns are only indirectly accessible through relational attributes. Recently, Nica
et al [9] have proposed a solution to combine concept lattices with graph pat-
terns. However it is not a general solution for RCA because of the limitations
already discussed in the introduction.
3 Relational Concept Analysis (RCA)
We here recall the definitions of context family and lattice family in RCA. We
therefore focus on the input and output of RCA, and we ignore the methodology
and algorithms that are used to compute the concept lattices from the context
family. Indeed we are here concerned with the graphical representation of RCA
lattices rather than on their computation. A detailed presentation of RCA is
available in previous papers, in particular [11]. The input data of RCA is called
a relational context family (RCF). In words, it is a collection of formal contexts,
one for each kind of objects, together with a collection of binary relations going
from the objects of one context to the objects of the same or another context.
K , R ) where:
Definition 1. A Relational Context Family (RCF) is a pair (K
– K = {Ki }i=1..n is a set of contexts Ki = (Oi , Ai , Ii ), and
– R = {rk }k=1..m is a set of relations rk where rk ⊆ dom(rk ) × ran(rk ), and
dom(rk ), ran(rk ) ∈ {Oi }i=1..n are respectively the domain and range of rk .
As a running example, we reuse the RCF defined in [11] about pharma-
covigilance of AIDS patients and drugs: ({Kp , Kd }, {takes, itb, iw }). Context Kp
describes 4 patients in terms of age, gender, and observed Adverse Drug Reac-
tions (ADR) (14 attributes). Context Kd describes 6 drugs in terms of active
molecule, and expected ADR (16 attributes). Relation takes relates patients to
the drugs they have taken. Relation itb (“is taken by”) is the inverse of takes.
Relation iw (“interacts with”) relates couples of drugs that interact with each
other (it is symmetric).
Given an RCF, the output of RCA is a collection of concept lattices, one
for each context of the RCF. The relations of the RCF are taken into account
by repeatedly applying a mechanism of relational scaling on each context and
its concept lattice, until convergence is reached. This leads to the introduction
of relational attributes that express relational constraints, and contribute to the
formation of concepts (see [11] for more details).
72 Sébastien Ferré and Peggy Cellier
takes:d10
p6 takes:d14
HairLoss
Oedema
p4 takes:d12
Adult
p7 Fatigue takes:d2
p14 takes:d21
takes:d3
takes:d6
female p13 Bleeding p12
Hives p5 takes:d16
HeartFailure
p2 p8 takes:d18 Male
p9 Nausea
BreathDisorder
takes:d9 Vomiting
Headache takes:d0
p3 takes:d13 p0 Senior p11 p10 takes:d15
Lane Farley Shana Trudy
p1
Fig. 1. The relational concept lattice of patients Lp (reduced labelling)
Definition 2. Let (KK , R) be a RCF. The Relational Concept Lattice Family
(RCLF) is a set of concept lattices L = {Li }i=1..n , one for each context Ki .
Each concept ci in Li is a pair (X, Y ) where:
– X ⊆ Oi is the extent of the concept, and
– Y is the intent of the concept, and contains attributes in Ai and relational
attributes in the form ρ r : cj where ρ ∈ {∃, ∀∃, ...} is a scaling operator,
r ∈ R , dom(r) = Oi , ran(r) = Oj , and cj ∈ Lj .
In this paper, we only consider existential scaling (ρ = ∃) even though our
approach is applicable to other scaling operators. Figure 1 shows the concept
lattice Lp of patients, and Figure 2 the concept lattice Ld of drugs. Both are
represented with reduced labelling, i.e. each object/attribute appears only once.
Object labels are placed below the concept, while attribute labels are placed
above and on the right of the concept. Each relational attribute ∃r : cj is dis-
played as r : cj .
The reading and interpretation of RCA lattices is notoriously difficult. The
main reason is probably that reading the intent of a concept requires not only to
traverse the lattice upward, as in FCA, but also to follow relationships to other
concepts through the relational attributes. For example, concept p7 groups the
adult patients who have fatigue, and take a drug in concept d10 and a drug in
concept d14. Concept d10 groups the drugs for which diarrhea is expected, and
Hierarchies of Concept Graphs in RCA 73
itb:p7
d7
itb:p2
itb:p5 itb:p12
itb:p8
Fatigue
d23 iw:d12 d10 Diarrhoea d22 itb:p9 d14 d19
Vomiting
Rash
d11 d8 d4 Nausea
itb:p10
d16 d21 d20 d18
d12 d2 Headache
LiverDamage Actinomycin
itb:p0 itb:p3 BreathDisorder
d5 itb:p11
HairLoss
d17 iw:d3 d15
iw:d13
Dactinomycin
Rifonavir Raltegravir Efavirenz HeartFailure
Tenofovir Maraviroc
d6 d3 iw:d9 d0 d13 iw:d6 d9
Viread Kaletra Isentress Sustiva Selzentry
d1
Fig. 2. The relational concept lattice of drugs Ld (reduced labelling)
which are taken by patients in concept p7. Loops in the exploration of the intent,
like in that example, lead to circular definitions of concepts, and contribute to
the difficulty of interpretation.
4 Hierarchies of Concept Graphs
Our objective is to facilitate the reading of the intent of RCA concepts, in order
to facilitate their interpretation. The first idea is to display the different lattices
side-by-side, and to materialize each relational attribute r : c2 on a concept c1 as
r
a relational edge, i.e. a labeled and directed edge c1 − → c2 . However, the graph-
ical representation becomes denser and even less readable; and its structure is
dominated by the lattice structures at the cost of elongated relational edges. A
better balance between lattice edges and relational edges is desirable. The second
idea is to identify relational structures as subsets of interrelated concepts from
different lattices, and to use them as building blocks in the graphical representa-
tion. We propose to define those relational structures as the Strongly Connected
Components (SCC) [3] of the dependency graph between concepts. The intuition
behind that dependency graph is that the intent of a concept depends on its
ancestors in the lattice, and also on the target concepts of relational attributes.
Dependency Graph. We first define the notion of concept dependency and then
the dependency graph of a RCLF.
74 Sébastien Ferré and Peggy Cellier
Definition 3. Let L be a RCLF and c1 , c2 be two concepts in L . Concept c1
depends on concept c2 , denoted by c1 → c2 , if:
– c2 is a parent concept of c1 (c1 ≤ c2 ),
– or c1 is labeled by a relational attribute ρ r : c2 .
Definition 4. Let L be a RCLF. The dependency graph of L is the directed
graph GL = (V, E) where:
– V is the set of all concepts of all lattices in L except bottom concepts,
– and E = {(c1 , c2 ) | c1 , c2 ∈ V and c1 → c2 }.
SCCs as Concept Graphs. From there, a SCC of GL is a maximal set of concepts
(possibly from several lattices) where each concept has a dependency path to all
other concepts in the SCC for the definition of its intent. SCCs are used to define
concept graphs, which are the building blocks of our graphical representation.
Definition 5. A concept graph is the subgraph of the lattice family (enriched
with relational edges) that is induced by a SCC of GL . It therefore mixes concepts
from several lattices, and both lattice edges and relational edges.
In Figure 3, each rounded box contains a concept graph (G1-G6). Nodes
are concepts from the two RCA lattices in Figures 1 and 2 (same id, same
labels). Relational edges (arrows) replace the relational attributes to graphically
represent the relational dependencies. Lattice edges that cross concept graph
boundaries are displayed as dotted lines to keep the graph light, and to emphasize
the concept graphs over the global lattice structures. It is notable in this example
that no relational edge crosses concept graph boundaries. This is because in the
context family each relation either has an inverse relation (e.g., takes and itb)
or is a symmetric relation (e.g., iw ).
Furthermore, it is known that the SCCs of a graph form a directed acyclic
graph, where SCC1 → SCC2 if any concept in SCC1 depends on any concept
in SCC2 . The concept graphs of a RCLF can therefore be organized into a hi-
erarchy of concept graphs. For example, concept graph G2 is a child of concept
graph G1 in Figure 3 because several concepts in G2 have lattice edges (dotted
lines) to concepts in G1: e.g., from p4 to p6, or from d12 to d14. Those hierarchi-
cal relationships can be seen as a complex version of the lattice edges, combining
several lattice edges across different lattices.
Interpretation. We here give a short interpretation of the hierarchy of concept
graphs in Figure 3. G1 represents the most general pattern between patients and
drugs. It shows that all patients take drugs expected to give fatigue and diarrhea,
and that all drugs are taken by an adult with fatigue. G2-4 are specializations
of G1. For example, G4 specializes G1 to patients with bleeding. G2 specializes
G1 to patients with hairloss and oedema, which all take drugs giving vomiting
and rash, which are taken by a patient with hives, and by a female patient.
G5 and G6 represent patterns that are specific to individual patients and drugs,
p6 G1 d7
Adult Diarrhea Fatigue
p7 Fatigue d10 d14
G2 d23 G3 d22 d19 G4
lattice edges across concept graphs
HairLoss Bleeding
Vomiting Rash Nausea
Oedema
p4 d11 d8 p14 d4 p8 d16 d18
male Headache
Hives
female HeartFailure
p2 p5 d12 p12 p9 d2 d21
Nausea
d20
G6
BreathDisorder BreathDisorder
Vomiting Raltegravir Actinomycin
HairLoss
LiverDamage p10 d0 d15
G5 p13 d5 d17
iw iw Isentress
Trudy Dactinomycin
HeartFailure
Headache Tenofovir Rifonavir Maraviroc
Senior iw Efavirenz
p0 p11 d6 d3 d9 d13 p3
iw
Farley Shana Viread Kaletra Selzentry Sustiva Lane
Hierarchies of Concept Graphs in RCA
75
patients to drugs, itb from drugs to patients, iw between drugs), dotted lines represent
Fig. 3. Hierarchy of concept graphs: arrows represent relational attributes (takes from
76 Sébastien Ferré and Peggy Cellier
and are therefore less interesting from a data-mining perspective. G5 shows that
some patients (p13) take drugs (d6 and d3) that cause liver dammage (d5) and
are in interaction with drug Sustiva (d13) which is taken by patient Lane (p3).
Discussion. The hierarchy of concept graphs has a number of good theoretical
properties. First, it is a complete representation because it keeps all concepts
and edges from the concept lattices. It is also parcimonious in that it does not
duplicate any concept, edge, or label. Second, it is more readable because it
displays relation attributes as relation edges, and because its layout offers a
better balance between lattice edges and relational edges. Moreover, when the
many concepts are clustered in a small number of concept graphs, the RCLF can
be read at a higher level of granularity. Third, it is efficient to compute because
the SCCs can be extracted in time linear with the size of GL , and hence in the
cumulated size of lattices in L .
5 Use Cases
In this section we present two use cases in order to compare graph concepts with
results of Graph-FCA and cpo-patterns from [9]. The first one describes the royal
family. The second use case is about flu patients and medical examinations.
5.1 Royal Family: Genealogical Data
G1 p0
male female
G2 p1 G3 p3
G4
chi nt
p10 ld pare p6
p8 p9 p4
G5
Fig. 4. Hierarchy of concept graphs about the royal family (G5 not detailed)
The first use case is the one used for Graph-FCA [6]. It describes a
subset of the British royal family: Charles, Diana, William, Harry, Kate,
Hierarchies of Concept Graphs in RCA 77
George, and Charlotte. The power context family uses two attributes (male
and female), and two relations (parent and its inverse child). Note that the
relations have the same object type as domain and range: people. RCA
produces one lattice containing 19 concepts. Figure 4 shows the hierarchy of
concept graphs obtained from that lattice with our approach. Concepts are
clustered in 5 concept graphs. G5 is not detailed because it contains very
specific concepts, and hence does not bring new knowledge. The hierarchy of
concept graphs enables to reach the following interpretations for each concept:
p0 people p8 male parents (fathers)
p1 male people (men) p9 female parents (mothers)
p3 female people (women) p6 people with a father and mother (children)
p10 people with a child (parents) p4 male children (sons)
The concept of “daughters” has a single instance (Charlotte), and is found in
G5, a specialization of G4. The relation from p10 (parents) to p4 (sons) shows
that, in the context, every parent has a son but not necessarily a daughter.
The relations froms p6 to p8 and p9 shows that, in the context, every child
who has a known father also has a known mother, and reciprocally. Concept
graph G4 exhibits the relational pattern of a nuclear family, relating children to
their father and mother as parents, with the specificity in this context that all
parents have a son.
Comparing those results to Graph-FCA, it is interesting to note that the
graph patterns of Graph-FCA are equivalent to the RCA concept graphs, up to a
few representation changes. Graph-FCA patterns only represent relational edges,
not lattice edges. The generalization ordering between concepts and patterns is
therefore not explicitly represented. In Graph-FCA the use of inverse relations
is implicit so that the child relation is redundant with the parent relation. Note
that Graph-FCA also defines n-ary concepts such as “couple” or “sibling”. The
equivalence with Graph-FCA on this example must not be generalized. In fact,
it does not hold on the running example about patients and drugs.
5.2 Medical Histories and Comparison to cpo-Patterns
The second use case is the one used for cpo-Patterns [9]. It describes flu pa-
tients through their symptoms, their viral tests and their medical examinations.
The specificity of that dataset is the sequentiality of the data. For instance for
patient p1 we know that a viral test on 28/09 is preceded by a medical ex-
amination on 26/09 which is also preceded by another medical examination on
25/09. The power context family uses 6 symptom attributes (COUGHmoderate,
FEVERmoderate, ?moderate, COUGHhigh, FEVERhigh, ?high) and two rela-
tions: RME-ipb-ME (sequential relation between medical examinations), RVT-
ipb-ME (sequential relation from viral tests to medical examinations). It de-
scribes five viral tests and ten medical examinations. RCA produces two lattices.
The viral test lattice contains 12 concepts and the medical examination lattice
contains 18 concepts.
Figure 5 shows the hierarchy of concept graphs obtained from those lattices
with our approach. We note that it is a special case, indeed each concept is a
78 Sébastien Ferré and Peggy Cellier
1 0
?moderate
?high COUGHmoderate
7 14 2 8
FEVERhigh
6 17 13
COUGHhigh
4 10 16 15
24 9 19 29
23 21 5 12
Fig. 5. Hierarchy of concept graphs about medical histories. Each concept is a concept
graph on its own. Greyed concepts are about viral tests, others are about medical
examinations. Only concepts with support greater than one are shown.
concept graph by its own. It is due to the fact that the relation ”is-preceded-
by” has no inverse relation, and forms no cycle because of its sequential nature.
It is thus impossible to find more than one concept in a strongly connected
component. The greyed concepts are the concepts from the viral test lattice,
other concepts come from the medical examination lattice. In the graph only
concepts with support greater than one are shown. We can read in the hierarchy
that all viral tests (top concept 0) are preceded by a medical examination (top
concept 1). That relational pattern has two specialisations. The first one where
the symptom during the examination is moderate cough (concepts 2 and 8). The
second one where there is a high symptom (concepts 7 and 14). We can also note
that parts of sequential patterns are shared by several concepts. For instance,
concept 24 and 10 are preceded by a medical examination with high cough
(concept 4). In fact, concept 9 specializes concept 10 by inserting between the
viral test and the high cough (concept 4) two additional medical examinations
(concepts 21 and 23), which are themselves specialisations of concept 24. It
highlights the overlaps between sequential patterns.
We have also conducted experiments when considering the inverse relation
of ”is-preceded-by” (ipb), i.e. ”is-followed-by” (ifb). The power context family
is thus extended with two relations: RME-ifb-ME and RME-ifb-VT. RCA still
produces two lattices but the hierarchy of concept graphs is different. Indeed,
eleven concept graphs are extracted. Each of them contains several concepts and
only one viral test concept. Figure 6 shows an excerpt of the hierarchy of concept
graphs with those inverse relations. For the sake of readability and compactness,
we modified the representation of concept graphs in two ways: (a) only the most
Hierarchies of Concept Graphs in RCA 79
ipb
G1
ifb
G2 G3
G4 ?moderate G5
COUGHmoderate
15 ?high
?high COUGHhigh
31 13 4 10
16
FEVERhigh
?high
G9 G7 G6
Fig. 6. Hierarchy of concept graphs about medical histories with inverse relation ifb.
Only concept graphs whose concepts have their support greater than one are shown.
specific concepts of a concept graph are kept, and (b) the full intent of those
concepts is shown, instead of the reduced intent, so that each concept graph
can be read in isolation. Eight concept graphs among the eleven are shown and
only two of them (G4 and G5) are detailled in the figure. Concept graph G4
can be read as ”a viral test (concept 13) preceded by a medical examination
with moderate cough (concept 15) and a medical examination with high fever
(concept 16) and both of them are preceded by a medical examination with a
high symptom (concept 31)”.
The interesting result is that the eleven concept graphs match exactly the
eleven cpo-patterns extracted by [9]. However, there are some differences in the
display of the patterns. Indeed, in order to compute the strongly connected
components, the inverse relations have to be added and they appear in the
result. For instance between concepts 13 and 15 there are two arrows, one in each
direction, because the relation is-followed-by (ifb) is the inverse relation of is-
preceded-by (ipb). In the same vein, ifb and ipb are transitive relations, and thus
some arrows are redundant. For example, the arrows between concepts 13 and
31 can be deduced from the paths through concepts 15, and 16 by transitivity.
In [9], the representation was specialized for sequential data, and so that kind
of redanduncies were avoided. On the contrary, our approach is general and
allows to take into account any kind of relations without any assumption on
them. In order to avoid those redundancies the description language of power
context families should be modified in order to add a way to specify relation
properties (e.g., is transitive, is symmetric, has an inverse), and then it should
be taken into account when computing and displaying the concept graphs. The
same can be said for the redundancy on attributes. Indeed, when looking at
the intent of concept 15, we can note the redundancy between ”?moderate” and
80 Sébastien Ferré and Peggy Cellier
”COUGHmoderate”. It is due to the conceptual scaling used on the symptom
attributes. By taking into account the hierarchy between attributes, the display
of the concept graphs can be simplified without loosing information.
6 Conclusion and Perspectives
We have proposed a novel and general representation of RCA concept lattices,
called hierarchy of concept graphs, in order to facilitate their interpretation. The
key idea is to exhibit relational patterns by having a better balance in the display
between lattice edges and relational edges. Each concept graph clusters a set of
concepts (from different lattices) whose intents are mutually dependent, and
exhibits a relational pattern. Concept graphs are organized into a hierarchy so
that generalization ordering between concepts is lifted to concept graphs. As
future work, we plan to study the impact of relation properties (e.g., inverse,
transitivity) on the trade-off between the number of concept graphs and the size
of each concept graph. It will also be necessary to develop tools for the dynamic
visualization of large hierarchies of concept graphs, à la Conexp1 .
References
1. Chein, M., Mugnier, M.L.: Graph-based knowledge representation: computational
foundations of conceptual graphs. Advanced Information and Knowledge Process-
ing, Springer (2008)
2. Džeroski, S.: Relational Data Mining, pp. 887–911. Springer US, Boston, MA
(2010), https://doi.org/10.1007/978-0-387-09823-4 46
3. Even, S.: Graph algorithms. Cambridge University Press (2011)
4. Falleri, J.R., Arévalo, G., Huchard, M., Nebut, C.: Use of model driven engineering
in building generic fca/rca tools. In: CLA. vol. 7, pp. 225–236 (2007)
5. Ferré, S.: A proposal for extending formal concept analysis to knowledge graphs.
In: Baixeries, J., Sacarea, C., Ojeda-Aciego, M. (eds.) Int. Conf. Formal Concept
Analysis (ICFCA). pp. 271–286. LNCS 9113, Springer (2015)
6. Ferré, S., Cellier, P.: Graph-FCA in practice. In: Haemmerlé, O., et al. (eds.) Int.
Conf. Conceptual Structures (ICCS). pp. 107–121. LNCS 9717, Springer (2016)
7. Ferré, S., Ridoux, O., Sigonneau, B.: Arbitrary relations in formal concept analysis
and logical information systems. In: ICCS (2005)
8. Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations.
Springer-Verlag New York. (1999)
9. Nica, C., Braud, A., Dolques, X., Huchard, M., Le Ber, F.: Extracting hierarchies
of closed partially-ordered patterns using relational concept analysis. In: Int. Conf.
Conceptual Structures. pp. 17–30. Springer (2016)
10. Prediger, S.: Simple concept graphs: A logic approach. In: Int. Conf. Conceptual
Structures. pp. 225–239. LNCS 1453 (Aug 1998)
11. Rouane-Hacene, M., Huchard, M., Napoli, A., Valtchev, P.: Relational concept
analysis: mining concept lattices from multi-relational data. Annals of Mathematics
and Artificial Intelligence 67(1), 81–108 (2013)
12. Sowa, J.: Conceptual structures. Information processing in man and machine.
Addison-Wesley, Reading, US (1984)
1
http://conexp.sourceforge.net/