=Paper=
{{Paper
|id=Vol-1244/GViP-paper6
|storemode=property
|title=Interactive Visualization of Formal Concept Lattices
|pdfUrl=https://ceur-ws.org/Vol-1244/GViP-paper6.pdf
|volume=Vol-1244
|dblpUrl=https://dblp.org/rec/conf/diagrams/Pattison14
}}
==Interactive Visualization of Formal Concept Lattices==
https://ceur-ws.org/Vol-1244/GViP-paper6.pdf
Interactive Visualisation of Formal Concept
Lattices
Tim Pattison
tim.pattison@defence.gov.au
Defence Science & Technology Organisation
West Ave, Edinburgh
South Australia 5111
Abstract. Formal Concept Analysis (FCA) is suitable for use within
organisations at different levels of maturity in information management.
It takes as input a bigraph, into which both structured and unstructured
data can be readily transformed, and produces a multiple-inheritance
type hierarchy suitable for formal knowledge representation. Accord-
ingly, FCA has been widely applied in areas such as information re-
trieval, knowledge discovery and knowledge representation. The multiple-
inheritance hierarchy produced by FCA is a complete lattice which can
be represented as a labelled, directed, acyclic graph (DAG). We adopt a
visual analytic approach to FCA by combining computational analysis
with interactive visualisation. Scaling FCA to the interactive analysis of
large data sets poses two fundamental challenges: the time required to
enumerate the vertices, arcs and labels of the lattice DAG; and the diffi-
culty of meaningful and responsive user interaction with a large digraph.
This paper briefly describes three software prototypes which address as-
pects of these scalability challenges.
1 Introduction
Formal Concept Analysis (FCA) [1] derives a multiple-inheritance type hierarchy
from a formal context. A formal context is a bigraph, consisting of a set of object
vertices, a set of attribute vertices, and edges specified by a binary relation
between these two sets. The “types” derived by FCA correspond to maximal
bicliques in this bigraph, and are known as formal concepts. Each formal concept
consists of a set of objects, called its extent, and a set of attributes, called
its intent, which are fully interconnected and jointly maximal: neither objects
nor attributes can be added while preserving full interconnection. The set of
formal concepts, when partially ordered by set inclusion on their extents, forms
a complete lattice. This lattice can be efficiently represented as a single-source,
single-sink, labelled, directed acyclic graph (DAG), whose vertices are formal
concepts, and whose adjacency relation is the transitive reduction [2] of the
ordering relation.
The resultant multiple-inheritance hierarchy of formal concepts constitutes
a useful generalisation of a hierarchy for applications such as the storage and
78
retrieval of data objects using keywords or tags, the representation of a Descrip-
tion Logic subsumption hierarchy [3], or the partial ordering of closed frequent
item sets in association mining. Accordingly, FCA has been widely applied in
such disparate fields as information retrieval, knowledge discovery and knowledge
representation [4].
Formal Concept Analysis is an analytic technique suitable for organisations at
different levels of maturity in information management. For those who primarily
retrieve and read unstructured text, the context bigraph is equivalent to the
“bag of words” representation common to a number of statistical techniques for
natural language processing, such as Latent Semantic Analysis [5]. For those
analysing user-tagged data, including various forms of social media, the tags
are attributes associated with the media objects of interest. FCA constitutes
a form of association mining for structured data such as the membership of
people in organisations or communities of interest. For organisations aspiring to
automated reasoning, the output of FCA is an empirically-derived subsumption
hierarchy suitable for use in Description Logics.
“Visual analytics is the science of analytical reasoning facilitated by inter-
active visual interfaces,” which, inter alia, “seeks to marry techniques from in-
formation visualisation with techniques from computational transformation and
analysis of data” [6]. We adopt a visual analytic approach to FCA by combining
computational analysis with interactive visualisation. Scalability is a key chal-
lenge for visual analytics. Algorithms must scale to large data sets, visualisations
must make efficient and intelligible use of screen real-estate, and both must be
responsive for interactive use. The number of formal concepts derived from a
formal context is bounded above by an exponential function of the number of
objects and attributes in that context. Consequently, two fundamental challenges
confront those who wish to scale FCA to the interactive analysis of large data
sets: the time required to enumerate the vertices, arcs and labels of the lattice
DAG; and the difficulty of meaningful and responsive user interaction with a
large lattice digraph.
This paper is organised as follows. Section 2 provides a graph-theoretic in-
troduction to Formal Concept Analysis. Section 3 undertakes a brief survey
of techniques aimed at improving the scalability of FCA for more responsive
visualisation and interaction. Section 4 briefly presents three techniques and as-
sociated software prototypes through which the Defence Science and Technology
Organisation has addressed selected aspects of FCA scalability.
2 Graph-theoretic introduction to FCA
A formal context (G, M, I) is a bipartite graph, or bigraph, with object vertex
set G, attribute vertex set M , and undirected edge set I ⊆ G × M . Each edge
is adjacent to one object and one attribute vertex. Each object and attribute
vertex has a unique label which derives from the domain of application. For an
information retrieval domain, for example, the object labels may be document
titles and the attribute labels keywords. Figure 1a shows an example formal
79
context, in which the object and attribute vertices have numerical and alphabetic
labels respectively.
G
H
10
1
I 8 F H I
10
5 1
G
6
5 E J
J
12 12 8 6
F
E
(a) Bigraph (b) Hasse diagram
Fig. 1: Bigraph and Hasse diagram (v.i.) for example formal context.
A sub-context (G0 , M 0 , I 0 ) of the formal context (G, M, I) is a bigraph con-
sisting of a subset G0 ⊆ G of its objects, a subset M 0 ⊆ M of its attributes, and
the subset I 0 = I ∩ (G0 × M 0 ) of its edges adjacent to those object and attribute
vertices.
2.1 Formal concepts
A formal concept consists of a set of objects, called the extent, and a set of
attributes, called the intent, which form a maximal biclique in the context bi-
graph. For example, ({5, 12}, {F, G}) is a formal concept in the formal context
of Figure 1a, since both attributes in its intent {F, G} are connected to both
objects in its extent {5, 12}, and no other objects or attributes can be added
while preserving full inter-connection. Two concepts are said to be comparable
iff the extent of one is a subset of the extent of the other.
2.2 Concept lattice DAG
FCA produces a set of formal concepts which are, or can be, partially-ordered
by extent set inclusion. This partially-ordered set forms a complete lattice [7],
which includes inter alia a unique maximum element, called the supremum, and
a unique minimum element, called the infimum.
This complete lattice can be represented as a DAG, in which each vertex rep-
resents a formal concept, and each arc connects a lower neighbour in the partial
order to its upper neighbour. This DAG has a single source vertex, correspond-
ing to the infimum of the lattice, and a single sink vertex, corresponding to the
supremum. Two concepts are comparable iff there is a directed path between
their corresponding vertices in the DAG.
80
2.3 Hasse diagram
A Hasse diagram is a layered drawing of this DAG in which the vertical compo-
nent of each arc is upwards on the page. This convention aids the interpretation
of the partial ordering and obviates the need to explicitly indicate the direction
of each arc. The source [sink]1 vertex appears at the bottom [top] of the dia-
gram, and all other vertices are assigned to intervening layers. Figure 1b shows
the Hasse diagram resulting from FCA of the formal context shown in Figure 1a.
Each concept bears an attribute label in the Hasse diagram, and is said to be
an attribute concept, iff its extent is the set of objects adjacent to that attribute
in the context bigraph. Similarly, each concept bears an object label, and is
said to be an object concept, iff its intent is the set of attributes adjacent to
that object. For example, the top vertex in Figure 1b is an attribute concept
for attribute G and an object concept for object 10. Attribute [object] labels are
placed above [below] the labelled concept.
A concept inherits the attributes [objects] appearing as labels on comparable
concepts above [below] it in the Hasse diagram. The vertex having attribute label
set {F} and object label set {5} in Figure 1b corresponds to the concept having
extent {5, 12} and intent {F, G}. In addition to its own object and attribute
labels, it inherits attribute G from its upper neighbour and object 12 from its
lower neighbour.
3 Layout, visualisation and interaction
This section provides a brief survey of techniques aimed at improving the scal-
ability of FCA for more responsive visualisation and interaction.
3.1 Reducing DAG size
The most obvious approach to improving Hasse diagram layout, visualisation
and interaction, is to reduce the number of formal concepts. Querying and filter-
ing the input context to remove objects and attributes which are not of interest
will expedite concept enumeration and reduce the number of labels on the Hasse
diagram, but is not guaranteed to reduce the number of concepts [8]. Another
means of achieving this objective is to impose a threshold on extent set cardinal-
ity, so that screen real-estate and user attention are reserved for formal concepts
which represent suitably large subsets of the objects in the formal context. The
partial order amongst these frequent closed item sets is referred to as an iceberg
lattice. Algorithms exist [9] which exploit the monotonicity of the constraint on
extent cardinality to expedite enumeration of the formal concepts.
1
Square brackets are used throughout this paper to indicate that a sentence is true
both when read without the bracketed terms and when read with each bracketed
term substituted for the term which precedes it.
81
3.2 Layout of Hasse diagram
Standard algorithms [10, 11] and genetic variants (see e.g. [12]) exist for assigning
the vertices of a DAG to layers and ordering them within each layer to improve
aesthetic criteria such as edge crossings. In the present case of a lattice digraph,
layer assignment is constrained by the maximum path length of a vertex from the
source, and to the sink, vertex. The assignment of vertices to layers in the Hasse
diagram is typically under-constrained by the partial order, so that both layer
assignment and horizontal order within a layer can be permuted when adjusting
the graph layout to optimise aesthetic criteria. This graph layout problem has
combinatorial complexity [11].
3.3 Interactive visualisation
Usability testing of FCA applied to the management of email has demonstrated
that users can successfully interpret Hasse diagrams [13]. However, the combi-
natorial explosion of concepts with increasing size of the formal context poses
challenges for the layout and visualisation of, as well as interaction with, the
lattice digraph. On-demand construction and layout of the entire lattice digraph
cannot be achieved in interactive timescales for large lattices, so that either
prior or user-guided construction and layout is required to support responsive
interaction2 . For contexts of even moderate size, the potentially large number of
resultant vertices and arcs compete for limited screen real estate and challenge
user comprehension.
To help the user manage this problem of scale, interactive exploration, as
opposed to static presentation, of the Hasse diagram is essential. Information
visualisation techniques such as pan and zoom, focus-plus-context, details-on-
demand and structural navigation [14] can support user interaction with, and
comprehension of, large graphs [15]. For example, geometric zooming or distor-
tion of the Hasse diagram [16, 17] can help allocate more screen real-estate to
an area of interest. Alternatively, structural navigation of the lattice digraph
can be facilitated by presentation of the immediate graph neighbourhood of the
current vertex [16, 18, 19], possibly combined with an overview showing where
that vertex resides in the full lattice. A third option is an interactive version of
nested Hasse diagrams [1, 16], in which each vertex serves as a container within
which to display the Hasse diagram for the same object set and (some of) the
remaining attributes. In many applications, however, it is not clear a priori how
best to group the attributes, or how to order the groups for nesting.
3.4 Discovering or imposing tree structure
A range of mature visualisation and interaction techniques exist for tree, as
opposed to lattice, data structures [20–22]. Operating system interfaces for the
structural navigation of directory hierarchies are ubiquitous, and user intuition is
2
User-guided construction of the DAG is addressed in section 4.3.
82
accordingly well established [16]. This intuition can be exploited for visualisation
of the concept lattice digraph, provided that a tree structure can be discovered
in, or imposed on, the graph.
Any spanning tree of the lattice digraph, which is rooted at the source or sink
vertex, would arguably serve this purpose. Melo et al. [23] investigate various
criteria by which a single parent can be chosen for each concept. Whereas any
given vertex will typically lie on multiple directed paths from the source [to the
sink] of the lattice digraph, the corresponding path in a spanning tree is unique.
To make it easier to purposefully locate a vertex of interest, or more likely that
such a vertex might be encountered during less goal-directed user exploration,
each vertex, along with the sub-lattice of which it is the supremum, can be
replicated on demand under each of its parent vertices [16, 24].
Another approach to imposing tree structure on a graph to facilitate user in-
teraction is hierarchical clustering or partitioning of its vertex set [15]. Hierarchi-
cal clustering involves the recursive application of a graph clustering algorithm to
the clusters (sub-graphs) it identifies. Graph clustering involves optimising some
measure of cluster quality, such as modularity, which takes into account factors
such as the number, or total weight, of intra- versus inter-cluster links. Whilst
the global optimisation of modularity is NP complete, sub-optimal solutions can
be computed for large graphs in responsive timeframes [25].
A range of techniques and tools exist for browsing hierarchically clustered
graphs. Amongst these are structural zooming on inclusion layouts [22], and
the GrouseFlocks environment [26] which supports the use and modification of
multiple hierarchical clusterings on the same graph.
3.5 Demand for enhanced tool support
There is a clear trend in operating system interfaces towards tagging and query-
ing rather than navigation of a hierarchical file system. Users typically require
assistance in recalling or constructing a set of tags with which to retrieve a suit-
ably small set of objects which contains the object(s) of interest. This trend will
drive a demand for well-designed user interfaces through which, like trees before
them, multiple-inheritance hierarchies become intuitive with use. The concep-
tually simple generalisation of a tree to allow a vertex to have multiple parents
poses significant challenges for user navigation. More generally, scalable visual-
isation and interaction of multiple-inheritance hierarchies, and in particular of
the DAG produced by FCA, remains an open challenge.
4 Three FCA prototypes
This section briefly presents three software prototypes which the Defence Science
and Technology Organisation has developed to address aspects of this scalability
challenge.
83
4.1 Hierarchical parallel decomposition
Carve [27] supports interactive analysis of large formal contexts by discovering
and exploiting hierarchical structure which is present in some contexts. That
hierarchical structure is used to expedite and enhance both the layout of, and
user interaction with, the concept lattice. The Carve algorithm [28] discovers
a hierarchical decomposition of amenable contexts and of the corresponding
lattice DAG. It produces a tree, representing a partial parallel decomposition
of the DAG [11], along with the DAG itself. The decomposition tree for the
example context in Figure 2a is shown in Figure 2b. Carve uses this tree both
to divide and conquer the computational problem of laying out the DAG as a
Hasse diagram, and as a coordinated view to facilitate user interaction with the
DAG.
Each vertex of the tree returned by the Carve algorithm corresponds to the
lattice digraph for a sub-context identified during hierarchical decomposition of
the formal context. This tree can be drawn using an inclusion layout, in which
each vertex of the tree is represented as a container within which the containers
representing descendant tree vertices are nested. In Figure 2c, these nested con-
tainers are shown as coloured boxes whose colour is that of the corresponding
vertex in the decomposition tree in Figure 2b. Each leaf vertex of this tree serves
as a container for the Hasse diagram of the corresponding trivial or otherwise
indivisible sub-lattice digraph.
��
�
� �
� �
�
�
�
� �
�
��
�
�
� �
�
�� �
�
�
� �
(a) Context bigraph (b) Decomposition tree (c) Hasse diagram
Fig. 2: Example context bigraph, the corresponding decomposition tree, and the
modified Hasse diagram with discovered sub-lattices within nested containers.
The sink [source] vertex of the sub-lattice digraph corresponds to a concept
in the global context (G, M, I) iff it has an attribute [object] label. Sink [source]
vertices which are concepts are shown in Figure 2c as circles with black [white]
fill, while those which are not are represented as point junctions of the arcs
from their lower [to their upper] neighbours. Such junctions can be seen at the
top and bottom of the grey container in Figure 2c. Each sink [source] vertex is
84
connected by an arc to its counterpart in the parent container. In the case where
the former vertex is not a concept, it serves as a collection [distribution] point
for a “trunk” arc to [from] its counterpart. These trunk arcs reduce clutter by
condensing multiple arcs into a single line.
4.2 Structural navigation
The SORTeD prototype supports the retrieval of documents (objects) from a
corpus based on queries over the terms (attributes) they contain. User queries are
constrained to term combinations which occur in the corpus, and are generalised
by removing, or specialised by adding, terms to navigate to comparable concepts.
Unlike previous interfaces for structural navigation of the DAG [16, 18, 19], those
comparable concepts are not constrained to be neighbours of the current concept.
Depicted in Figure 3, the user interface offers valid terms to add or remove from
the query. The computational challenge is to compute the set of all concepts
comparable to the query concept to facilitate interactive use. Despite structurally
navigating the DAG through a keyhole view, the user may be unaware of the
DAG’s existence, relying instead on intuition established through long-term use
of conventional information retrieval interfaces.
In SORTeD, FCA provides a mechanism for literal search over the corpus,
with the user interface assisting the construction and refinement of conjunctive
Boolean queries. The search results are ranked using Latent Semantic Analysis
(LSA) according to their cosine similarity to the search terms [5], and the search
terms are ranked according to their cosine similarity to the result set. The latter
ranking is less conventional, indicating the comparative relevance of the search
terms to the result set, from which the user may judge whether the result set is
likely to satisfy their requirements. In addition to offering “valid” search terms
which co-occur with the existing search terms to assist the user to refine the
query, the interface also offers, ranks and visually distinguishes terms which
are only semantically related to the result set. Selecting one of these initiates
a literal query in which the selected term is substituted for the existing set of
query terms.
Fig. 3: SORTeD interface for information retrieval combining FCA and LSA.
85
4.3 User-guided FCA
The DAnCe prototype improves the scalability of FCA for interactive use by
allowing the user to steer the analysis towards areas of interest, and to halt
construction of the DAG as soon as their analytic objectives are satisfied. The
resultant DAG will be more task-focused, and, depending on the application,
may be considerably smaller, than the lattice DAG for the original context.
We have developed the DAnCe prototype to explore this possibility, modi-
fying a top-down algorithm for concept enumeration to respond to user control
and guidance. Instead of autonomous enumeration of all concepts followed by
batch-mode construction of the DAG and corresponding Hasse diagram, DAnCe
allows the user interactive control over the process of concept enumeration and
provides dynamic, incremental update of the Hasse diagram. In addition to being
able to start, stop, restart and step through the process of concept enumeration,
the user can: select a concept of interest and prioritise the enumeration of com-
parable concepts which are below it in the lattice; or select multiple concepts
and prioritise the generation of the concepts which correspond to the set inter-
sections of their intents and extents. Each vertex and arc is displayed either as
soon as it is discovered, or in a batch-mode update of the Hasse diagram after a
specified number of steps of the enumeration algorithm.
Visualisation challenges faced by DAnCe include: ensuring intelligible layout
of the partially-constructed diagram and maintaining the user’s mental model
while vertices and arcs are added; and modifying the labelling scheme described
in Section 2.3 to allow consistent interpretation when some DAG vertices and
edges have yet to be discovered. DAnCe maintains the complete Hasse diagram
for the partial order amongst the concepts generated to date. Figure 4 shows
mock-ups of this Hasse diagram for an example formal context consisting of
people and their physical attributes. Figure 4a depicts the state of the Hasse
diagram first presented to the user. By this stage, all of the attribute and object
concepts have been generated by the concept enumeration algorithm, labelled,
and laid out to establish the framework for insertion of subsequent concepts.
Figure 4b shows the result of the user selecting in this diagram the attribute
concepts for “Beard” and “Moustache”, which are highlighted in response with
small grey halos. This multiple selection triggers the calculation of the extent
and intent intersections for the selected concepts. The former corresponds to the
infimum, which is accordingly highlighted with a green halo; the latter corre-
sponds to a new concept, which is consequently inserted into the Hasse diagram
and highlighted with a purple halo. Since this extent intersection has path length
2 from the supremum, a new row has been inserted to accommodate concepts
with path length 3, and the selected concepts demoted to it. The extent inter-
section is inserted into row 2 at ordinal position 2 of 5; this position is based on
the horizontal barycentre of its associated layer 1 ancestors (atoms) and layer 5
descendants (co-atoms), which are predominantly to the left of the centreline.
86
Male Yellow Hair Brown Eyes Female
Black Hair Brown Skin Red Hair
Male Yellow Hair Brown Eyes Female
Kyle
White Hair Beard Moustache Brown Hair
White Hair Beard Moustache Brown Hair Black Hair Brown Skin Red Hair
Kyle
Andy Justin Jon Jake Joshua Megan Sarah William Tyler Daniel
Andy Justin Jon Jake Joshua Megan Sarah William Tyler Daniel
(a) Initial diagram (b) Multiple selection
Fig. 4: The initial Hasse diagram and the result of multiple selection.
5 Summary
In this paper I have described, in graph-theoretic terms, the analytical technique
of FCA, which is applicable to a range of clients of the Defence Science and
Technology Organisation (DSTO). The scalability challenges of construction and
interactive visualisation of the resultant DAG have been described, along with
three prototype tools developed by DSTO to address aspects of these challenges.
Acknowledgement The author gratefully acknowledges the contributions of
Aaron Ceglar and Derek Weber to the development of the prototypes and screen-
shots described in this paper. Graphviz was used in the preparation of most graph
drawings.
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