FORMALIZING A MODEL TO REPRESENT AND VISUALIZE

CONCEPT SPACES IN E-LEARNING ENVIRONMENTS

Antonina Dattolo

Dipartimento di Matematica e Informatica, Universit`a di Udine, Udine, Italy

Flaminia L. Luccio

Dipartimento di Informatica, Universit`a Ca’ Foscari Venezia, Venezia, Italy

Keywords:

Adaptive educational hypermedia, concept maps, zz-structures, graph theory, e-learning.

Abstract:

Zz-structures offer graph-centric views capable of representing contextual interconnections among different

information. In this paper we use these structures in order to represent and visualize concept spaces in e-

learning environments, and we present their formal analytic description in terms of graph theory. In particular,

we focus our attention on the formal description of two views (H and I views), and we extend these notions

to a number n > 2 of dimensions. We also apply both this formal description, and the particular properties of

zz-structures, to an example in the Web-based education ﬁeld.

1 INTRODUCTION

Adaptive Educational Hypermedia (AEH) (Cristea

et al., 2006) seek to apply the personalized possibil-

ities of Adaptive Hypermedia (Brusilovsky, 2001) to

the domain of education, thereby granting learners a

lesson individually tailored to them. A fundamental

part of these systems is the concept space (Dagger

et al., 2005): this provides an ontology of the subject

matter including the concepts and their relationships

to one another.

The purpose of concept mapping is not the production

of a map representing in absolute terms the relation-

ships between concepts, but the production of a visual

layout, which can make that speciﬁc issue clearer.

Concept spaces are traditionally visualized using a

concept map diagram, a downward-branching, hier-

archical tree structure. In mathematical terms, a con-

cept space map is a directed acyclic graph, a general-

ization of a tree structure, where certain sub-trees can

be shared by different parts of the tree.

Concept maps have got the double advantage of vi-

sually representing an information map and linking it

to useful material contained in a database. Learners

have a referring map to which they can come back to

review previous steps, and, mostly, learn how to orga-

nize information so “it makes sense” for them.

Unfortunately, traditional concept maps (Freire and

Rodriguez, 2005) are inadequate to capture and vi-

sualize very large collections of interrelated infor-

mation. Many of the more innovative tree visual-

ization techniques are not well suited to represent

concept maps: for example Shneiderman’s Treemaps

(Shneiderman, 1992) and Kleiberg’s Botanical trees

(Kleiberg et al., 2001) cannot easily differentiate be-

tween relationship types; other models (e. g. (Cassidy

et al., 2006), based on hyperbolic geometry, or (Suk-

somboon et al., 2007), based on S-nodes are not able

to dynamically switch from a view to another one. It

is often not possible to view the entire concept space

on-screen without zooming out so far that the concept

and relationship labels are no longer readable. Sim-

ilarly, the large number of relationships improve the

difﬁculty of understanding the structure of the con-

cept space.

In particular, in the e-learning ﬁeld, there are many

reasons to deﬁne opportune structure models for stor-

ing and visualizing concept maps:

• They allow the system to be adaptive: current ap-

proaches and tools (see WebCT, Moodle, etc.) are

not adaptive, as they neither support a comprehen-

sive analysis of users’ needs, demands and oppor-

tunities, nor they support a semantic analysis of

texts.

• They provide interoperability between different

adaptive systems: this feature becomes not only

desirable but also necessary, as it enables the re-

use of previouslycreated material without the cost

339

Dattolo A. and L. Luccio F. (2008).

FORMALIZING A MODEL TO REPRESENT AND VISUALIZE CONCEPT SPACES IN E-LEARNING ENVIRONMENTS.

In Proceedings of the Fourth International Conference on Web Information Systems and Technologies, pages 339-346

DOI: 10.5220/0001525803390346

 SciTePress

of recreating it from scratch (Celik et al., 2006).

• They simplify the authoring process, in which the

user/learner may assume the role of an author

(see, e.g., Wikis and Wiki farms).

Considering the limitations highlighted by the study

of the current literature, we will focus our attention on

an innovative structure, proposed in (Nelson, 2004),

the zz-structure, that constitutes the main part of a

ZigZag system (Nelson, 1999).

Previous work in this direction has shown how ﬂex-

ible this structure is, and how it can be specialized

in different ﬁelds, such as, e.g., the modeling of an

information manager for mobile phones (zz-phones)

(Moore and Brailsford, 2004), of the London under-

ground train lines and stations (Nelson, 1999), of

bioinformatics workspaces (Moore et al., 2004), of

data grid systems (Dattolo and Luccio, 2007), of an

authoring system for electronic music (Archimedes)

(Canazza and Dattolo, 2007), or of web-based courses

(Andric et al., 2007). Although the work (Nel-

son, 2004) provides a reference description of zz-

structures, and the other previously mentioned works

use different aspects and features of the model, Nel-

son itself writes: “The ZigZag system is very hard to

explain, especially since it resembles nothing else in

the computer ﬁeld that we know of, except perhaps a

spreadsheet cut into strips and glued into loops ”.

Thus, in our opinion, a formal description of the struc-

ture may be very useful in simplifying the comprehen-

sion of the model.

Case Study. Our application ﬁeld is Web-based ed-

ucation; it has become a very important area of edu-

cational technology and a challenge for semantic Web

techniques. Web-based education enables learners

and authors (teachers) to access a wide quantity of

continuously updated educational sources. In order

to simplify the learning process of learners, and the

course creation/modiﬁcation/organization process of

authors, it is important to offer them tools to:

1. identify the collection of “interesting” documents,

for example applying semantic ﬁltering algo-

rithms (Brodnik et al., 2006), or proximity metrics

on the search engine results (Andric et al., 2007);

2. store the found collection of documents in ade-

quate structures, that are able to organize and vi-

sualize concept spaces;

3. create personalized adaptive paths and views for

learners.

These three topics are the guidelines of our current

research. In this paper, we focus our attention only on

point 2. We assume that an author has a collection

of available documents on a given topic that have to

be organized in concept maps, suitable for different

learners. E.g., some users could be preparing a degree

thesis, others could be studying for an examination on

a particular topic, others could be doing research on

a speciﬁc research area, and so on. Thus, the author

needs adequate tools to organize documents in a con-

cept space, and to create semantic interconnections

and personalized maps.

Contributions of this Work. The general goal of

this work is to propose a formal structure for repre-

senting and visualizing a concept space. This model

is based both on zz-structures and on graph theory.

We will show how identifying and deﬁning in an

analytic way the graph theoretical structure of zz-

structures can both provide interesting insights to ed-

ucational hypermedia designers (facilitating a deeper

understanding of which model might best support the

representation and interaction aims of their systems),

and to learners (offering them support for Web orien-

tation and navigation).

Our novel contributions are:

• a formal analytic graph-based description of zz-

structures. Particular attention has been devoted

to the formalization of two views (H and I views),

present into all ZigZag implementations;

• an extension of the concept of H and I views from

a number 2 towards a number n > 2 of dimen-

sions;

• a new concept map model for e-learning environ-

ments, based on our model.

The paper is organized as follows: in Section 2, we

introduce the reader to zz-structures and we present

some basic graph theory deﬁnitions; in Section 3, we

propose our formal deﬁnition of zz-structures, and we

use these structures as a reference model for repre-

senting concept maps. Finally, in Section 4 we ﬁrst

introduce the deﬁnition of the standard H and I view,

and we then extend this deﬁnition to the non-standard

n-dimensions view (with n > 2). Conclusion and fu-

ture works conclude the paper.

2 Zz-STRUCTURES AND GRAPH

THEORY

This section is introduced for consistency. If the

reader has a background on the ZigZag model and on

basic graph theory, can skip this section.

WEBIST 2008 - International Conference on Web Information Systems and Technologies

340

2.1 An Introduction to Zz-Structures

Zz-structures (Nelson, 2004) introduce a new, graph-

centric system of conventions for data and computing.

A zz-structure can be thought of as a space ﬁlled with

cells. Each cell may have a content (such as integers,

text, images, audio, etc.), and it is called atomic if

it contains only one unit of data of one type (Moore

et al., 2004), or it is called referential if it represents

a package of different cells. There are also special

cells, called positional, that do not have content and

thus have a positional or topographical function.

Cells are connected together with links of the

same color into linear sequences called dimensions.

A single series of cells connected in the same dimen-

sion is called rank, i.e., a rank is in a particular di-

mension. Moreover, a dimension may contain many

different ranks. The starting and an ending cell of

a rank are called, headcell and tailcell, respectively,

and the direction from the starting (ending) to the

ending (starting) cell is called posward (respectively,

negward). For any dimension, a cell can only have

one connection in the posward direction, and one in

the negward direction. This ensures that all paths are

non-branching, and thus embodies the simplest pos-

sible mechanism for traversing links. Dimensions are

used to project different structures: ordinary lists are

viewed in one dimension; spreadsheets and hierarchi-

cal directories in many dimensions.

The interesting part is how to view these struc-

tures, i.e., there are many different ways to arrange

them, choosing different dimensions and different

structures in a dimension. A raster is a way of se-

lecting the cells from a structure; a view is a way of

placing the cells on a screen. Generic views are de-

signed to be used in a big variety of cases and usually

show only few dimensions or few steps in each di-

mension. Among them the most common are the two-

dimensions rectangular views: the cells are placed,

using different rasters, on a Cartesian plane where the

dimensions increase going down and to the right. Ob-

viously some cells will not ﬁt in these two dimensions

and will have to be omitted. The simplest raster is the

row and column raster, i.e., two rasters which are the

same but rotated of 90 degrees from each other. A cell

is chosen and placed at the center of the plane (cursor

centric view). The chosen cell, called focus, may be

changed by moving the cursor horizontally and ver-

tically. In a row view I, a rank is chosen and placed

vertically. Then the ranks related to the cells in the

vertical rank are placed horizontally. Vice versa, in

the column view H, a rank is chosen and placed hor-

izontally and the related ranks are placed vertically.

All the cells are denoted by different numbers. Note

that in a view the same cell may appear in different

positions as it may represent the intersection of dif-

ferent dimensions.

2.2 Basic Graph Theory Deﬁnitions

In the following we introduce some standard graph

theory notation, for more details refer to (Harary,

1994).

A graph G is a pair G = (V, E), where V is a ﬁnite

non-empty set of elements called vertices and E is a

ﬁnite set of distinct unordered pairs {u, v} of distinct

elements of V called edges.

A multigraph is a triple MG = (V, E, f) where V is a

ﬁnite non-empty set of vertices, E is the set of edges,

and f : E → {{u, v} | u, v ∈ V, u 6= v} is a surjective

function.

An edge-colored multigraph is a triple ECMG =

(MG,C, c) where: MG = (V, E, f) is a multigraph, C

is a set of colors, c : E → C is an assignment of colors

to edges of the multigraph.

In a multigraph MG = (V, E, f ), edges e

, e

∈ E are

called multiple or parallel iff f(e

) = f(e

). Thus, a

graph as a particular multigraph G = (V, E, f) without

parallel edges.

Given an edge e = {u, v} ∈ E, we say that e is incident

to u and v; moreover u and v are neighboring vertices.

Given a vertex x ∈ V, we denote with deg(x) its de-

gree, i.e., the number of edges incident to x, and with

max

the maximum degree of the graph, i.e., d

max

z∈V

{deg(z)}. In an edge-colored (multi)graph

ECMG, where c

∈ C, we deﬁne deg

(x) the num-

ber of edges of color c

incident to vertex x. A vertex

of degree 0 is called isolated, a vertex of degree 1 is

called pendant.

A path P = {v

, v

, . . . , v

} is a sequence of neighbor-

ing vertices of G, i.e., {v

, v

i+1

} ∈ E, 1 ≤ i ≤ s − 1.

A graph G = (V, E) is connected if: ∀x, y ∈ V, ∃ a

path P = {x = v

, v

, . . . , v

= y}, with {v

, v

k+1

} ∈ E,

1 ≤ k ≤ s − 1. Two vertices x and y in a connected

graph are at distance dist if the shortest path connect-

ing them is composed of exactly dist edges.

Finally, a m × n mesh is a graph M

m,n

= (V, E) with

i, j

∈ V, 0 ≤ i ≤ m − 1, 0 ≤ j ≤ n − 1, and E con-

tains exactly the edges (v

i, j

, v

i, j+1

), j 6= n − 1, and

i, j

, v

i+1, j

), i 6= m− 1.

3 THE FORMAL MODEL

In this section, we formalize the model presented in

(Nelson, 2004) in terms of graph theory. In the rest

of this paper we describe formal deﬁnitions through a

simple example in the e-learning ﬁeld: an author has

a collection of available papers that ﬁrst wants to link

FORMALIZING A MODEL TO REPRESENT AND VISUALIZE CONCEPT SPACES IN E-LEARNING

ENVIRONMENTS

341

through different semantic paths and then wants to

merge into a unique concept space. Papers that have

been published in the proceedings of the same con-

ference, or papers that investigate a common topic, or

papers that share one author, are examples of seman-

tic paths, which automatically generate concept maps.

3.1 Zz-Structures

A zz-structure can be viewed as a multigraph where

edges are colored, with the restriction that every ver-

tex has at most two incident edges of the same color.

Differently from (McGufﬁn, 2004), but as mentioned

in (McGufﬁn and Schraefel, 2004; Dattolo and Luc-

cio, 2007), we consider undirected graphs, i.e., edges

may be traversed in both directions. A zz-structure is

formally deﬁned as follows.

Deﬁnition 1 (Zz-structure). A zz-structure is

an edge-colored multigraph S = (MG,C, c),

where MG = (V, E, f), and ∀x ∈ V, ∀k = 1, 2,

..., |C|, deg

(x) = 0, 1, 2. Each vertex of a zz-structure

is called zz-cell and each edge zz-link. The set of

isolated vertices is V

= {x ∈ V : deg(x) = 0}.

An example of a zz-structure is given in Figure 1. The

structure is a graph, where vertices v

, . . . , v

repre-

sent different papers, and edges of the same kind rep-

resent the same semantic connection.

In particular, in this example, thick edges connect

a sequence of papers published at the same confer-

ence (e.g., WEBIST2007), normal edges group pa-

pers that have at least an author in common, ﬁnally,

dotted lines link papers that have a keyword in com-

mon (e.g., wbe, that stands for web-based education).

3.2 Dimensions

An alternative way of viewing a zz-structure is a

union of subgraphs, each of which contains edges of

a unique color.

Proposition 1 Consider a set of colors

C = {c

, c

, ..., c

|C|

} and a family of indirect

Figure 1: A zz-structure where thick, normal and dotted

lines represent three different colors.

edge-colored graphs {D

, D

, ..., D

|C|

}, where

= (V, E

, f, {c

}, c), with k = 1, ..., |C|, is a graph

such that: 1) E

6= Ø; 2) ∀x ∈ V, deg

(x) = 0, 1, 2.

Then, S =

|C|

k=1

is a zz-structure.

Deﬁnition 2 (Dimension). Given a zz-structure S =

|C|

k=1

, then each graph D

, k = 1, . . . , |C|, is a dis-

tinct dimension of S.

From Figure 1 we can extrapolate three dimensions,

one for each different color (i.e., one for each differ-

ent semantic connection). As shown in Figure 2, we

associate thick lines to dimension D

conference

, normal

lines to dimension D

author

, and dotted lines to dimen-

sion D

wbe topic

Each dimension can be composed of isolated ver-

tices (e.g., vertices v

, v

in dimension D

author

), of

distinct paths (e.g., the three paths {v

, v

} and {v

, v

} in dimension D

author

and of distinct cycles (e.g., the unique cycle

, v

} in dimension D

wbe topic

3.3 Ranks

Deﬁnition 3 (Rank). Consider a dimension D

= (V,

, f, {c

}, c), k = 1, . . . , |C| of a zz-structure S =

∪

|C|

k=1

. Then, each of the l

connected components

of D

is called a rank.

Thus, each rank R

= (V

, E

, f, {c

}, c), i = 1, . . . , l

is an indirect, connected, edge-colored graph such

that: 1) V

⊆ V; 2) E

⊆ E

; 3) ∀x ∈ V

, 1 ≤

deg

(x) ≤ 2. A ringrank is a rank R

, where ∀x ∈

, deg

(x) = 2.

Note that the number l

of ranks differs in each

dimension D

, e.g. in Figure 2, dimension D

author

has three ranks ({ v

, v

}, {v

, v

} and

, v

}), and dimension D

conference

has a unique

rank ({v

, v

}). A ringrank is, e.g.,

conference

author

wbetopic

Figure 2: The three dimensions.

WEBIST 2008 - International Conference on Web Information Systems and Technologies

342

the cycle {v

, v

} of dimension

wbe topic

Deﬁnition 4 (Parallel Ranks). Given a zz-structure

S = ∪

|C|

k=1

, m ranks R

= (V

, E

, f, {c

}, c), (j =

1, 2, . . . , m, 2 ≤ m ≤ l

) are parallel ranks on the same

dimension D

, k ∈ {1, . . . , |C|} iff V

⊆ V, E

⊆

, ∀ j = 1, 2, . . . , m, and ∩

j=1

In Figure 2 the three ranks of dimension D

author

are parallel.

3.4 Cells and their Orientation

A vertex has local orientation on a rank if each of its

(1 or 2) incident edges has assigned a distinct label

(1 or -1). More formally (see also (Flocchini et al.,

1998)):

Deﬁnition 5 (Local Orientation). Consider a rank

= (V

, E

, f, {c

}, c) of a zz-structure S = ∪

|C|

k=1

Then, ∃ a function g

: E

→ {−1, 1}, such that, ∀x ∈

, if ∃y, z ∈ V

: {x, y}, {x, z} ∈ E

, then g

({x, y}) 6=

({x, z}). Thus, we say that each vertex x ∈ V

has a

local orientation in R

Deﬁnition 6 (Posward and Negward Directions).

Given an edge {a, b} ∈ E

, we say that {a, b} is in

a posward direction from a in R

, and that b is its

posward cell iff g

({a, b}) = 1, else {a, b} is in a neg-

ward direction and a is its negward cell. Moreover, a

path in rank R

follows a posward (negward) direc-

tion if it is composed of a sequence of edges of value

1 (respectively, -1).

For simplicity, given a rank R

, a way to represent

a path composed of a vertex x and a sequence of its

negward and posward cells, is by using the notation

. . . x

−2

−1

. . . , where, x

−1

represents the neg-

ward cell of x and x

the posward cell. In gen-

eral, x

−i

) is a cell at distance i in the negward

(posward) direction. We also assume that x

= x.

Deﬁnition 7 (Headcell and Tailcell). Given a rank

= (V

, E

, f, {c

}, c), a cell x is the headcell of R

iff ∃ its posward cell x

and 6 ∃ its negward cell x

−1

Analogously, a cell x is the tailcell of R

iff ∃ its neg-

ward cell x

−1

and 6 ∃ its posward cell x

4 VIEWS

We now formalize the standard notion of H and I

views in two dimensions, and we then propose a new

deﬁnition of H and I-views in n dimensions. We

also show some interesting applications of these new

higher dimensional views.

In the following, that we denote with x ∈ R

(x)

the

rank R

(x)

related to vertex x of color c

Deﬁnition 8 (H-view). Given a zz-structure S =

∪

|C|

k=1

, where D

= ∪

i=1

∪V

), and where R

, E

, f, {c

}, c), the H-view of size l = 2m + 1

and of focus x ∈ V = ∪

i=0

, on main vertical di-

mension D

and secondary horizontal dimension D

(a, b ∈ {1, ..., l

}), is deﬁned as a tree whose embed-

ding in the plane is a partially connected colored l ×l

mesh in which:

• the central node, in position ((m+ 1), (m+ 1)), is

the focus x;

• the horizontal central path (the m+1-th row) from

left to right, focused in vertex x ∈ R

(x)

is:

−g

. . . x

−1

. . . x

where x

∈ R

(x)

, for s =

−g, . . . , +p (g, p ≤ m).

• for each cell x

, s = −g, . . . , +p, the related

vertical path, from top to bottom, is:

)

−g

. . . (x

)

−1

)

. . . (x

)

, where

)

∈ R

)

, for t = −g

, . . . , +p

, p

≤ m).

Intuitively, the H-view extracts ranks along the two

chosen dimensions. Note that, the name H-view

comes from the fact that the columns remind the

vertical bars in a capital letter H. Observe also that

the cell x

−g

(in the m + 1-th row) is the headcell of

(x)

if g < m and the cell x

(in the same row) is the

tailcell of R

(x)

if p < m. Analogously, the cell x

−g

is the headcell of R

)

if g

< m and the cell x

the tailcell of R

)

if p

< m. Intuitively, the view is

composed of l × l cells unless some of the displayed

ranks have their headcell or tailcell very close (less

than m steps) to the chosen focus.

As an example consider Figure 3 left that refers

to the zz-structure of Figure 1. The main vertical

dimension is D

author

and the secondary horizon-

tal dimension is D

conference

. The view has size

l = 2m + 1 = 5, the focus is v

, the horizontal

central path is v

−2

−1

= {v

, v

}

(g, p = 2). The vertical path related to v

−1

= v

−1

)

−1

)(v

−1

)

−1

)

= {v

, v

} (g

= 1

and p

= 2), that is (v

−1

)

−1

= v

is the headcell of

the rank as g

= 1 < m = 2.

Analogously to the H-view we can deﬁne the I-view.

Deﬁnition 9 (I-view). Given a zz-structure S =

∪

|C|

k=1

, where D

= ∪

i=1

∪V

), and where R

FORMALIZING A MODEL TO REPRESENT AND VISUALIZE CONCEPT SPACES IN E-LEARNING

ENVIRONMENTS

343

conference

H-view I-view

author

conference

author

Figure 3: H-view and I-view, related to Figure 1.

, E

, f, {c

}, c), the I-view of size l = 2m + 1

and of focus x ∈ V = ∪

i=0

on main horizontal

dimension D

and secondary vertical dimension D

(a, b ∈ {1, ..., l

}), is deﬁned as a partially connected

colored l × l mesh in which:

• the central node, in position ((m+ 1), (m+ 1)) is

the focus x;

• the vertical central path (the m + 1-th column)

from top to bottom, focused in vertex x ∈ R

(x)

is:

−u

. . . x

−1

. . . x

where x

∈ R

(x)

, for s = −u,

. . . , +r (u, r ≤ m).

• for each cell x

, s = −u, . . . , +r, the related

horizontal path, from left to right, is:

)

−u

. . . (x

)

−1

)

. . . (x

)

, where

)

∈ R

)

, for t = −u

, . . . , +r

, r

≤ m).

Note that, the name I-view comes from the fact that

the rows remind the horizontal serif in a capital letter

I. Observe also that the cell x

−u

(in the m + 1-th

column) is the headcell of R

(x)

if u < m and the x

(in the same column) is the tailcell of R

(x)

if r < m.

Analogously, the cell x

−u

is the headcell of R

)

< m and the x

is the tailcell of R

)

if r

< m.

As example consider Figure 3 right. The main hori-

zontal dimension is D

conference

and the secondary ver-

tical dimension is D

author

. The view has size l =

2m+ 1 = 5, the focus is v

, the vertical central path is

−2

−1

= {v

, v

} (u, r = 2). The

horizontal path related to v

−1

= v

is (v

−1

)

−1

. . .

−1

)

= {v

, v

} (i.e., r = 2). Vice versa the

horizontal path related to v

= v

is {v

, v

} and

is the headcell. Finally, the horizontal path related

to v

= v

is {v

, v

We can now extend the known deﬁnition of H and I

views to a number n > 2 of dimensions. Intuitively,

we will build n− 1 different H-views (respectively, I-

views), centered in the same focus, with a ﬁxed main

dimension and a secondary dimension chosen among

the other n− 1 dimensions. Formally:

Deﬁnition 10 (n-Dimensions H-view). Given a zz-

structure S = ∪

|C|

k=1

, where D

= ∪

i=1

∪V

), and

where R

= (V

, E

, f, {c

}, c), the n-dimensions H-

view of size l = 2m+1 and of focus s x ∈ V = ∪

i=0

on dimensions D

, D

, . . . , D

is composed of n − 1

rectangular H-views, of main dimension D

and sec-

ondary dimensions D

, i = 2, . . . , n, all centered in the

same focus x.

Analogously, we have the following:

Deﬁnition 11 (n-Dimensions I-view). Given a zz-

structure S = ∪

|C|

k=1

, where D

= ∪

i=1

∪V

), and

where R

= (V

, E

, f, {c

}, c), the n-dimensions I-

view of size l = 2m+ 1 and of focus x ∈ V = ∪

i=0

on dimensions D

, D

, . . . , D

is composed of n − 1

rectangular I-views of main dimension D

, and sec-

ondary dimensions D

, i = 2, . . . , n, all centered in the

same focus x.

In Figure 3, we can distinguish only two dimensions

conference

and D

author

To display a 3-dimensions H-view we can add a

new dimension (let it be D

wbe topic

). This new H-view

has main dimension D

wbe topic

, and secondary dimen-

sions D

conference

and D

author

. To construct this view

we start from Figure 1 using v

as focus, and we con-

sider the two central paths (Figure 4 left), related to

the two secondary dimensions D

conference

and D

author

conference

author

conference

author

Figure 4: Two secondary dimensions cross the focus v

The same visualization is shown in Figure 4 right

under a different perspective.

Finally, in Figure 5 we obtain the 3-dimensions H-

view where the vertical paths on main dimension

wbe topic

are added.

conference

author

wbetopic

Figure 5: An example of a 3-dimensions H-view.

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344

We can now extend this example to the n-dimensions

case. In Figure 6, we show a 5-dimensions view, con-

sidering four secondary dimensions. In our example,

we have added other two dimensions (D

publication year

and D

publishing house

), representing the year of publi-

cation of the article and the publishing house. This

new view has focus v

, size l = 2m+ 1 = 5 and main

dimension D

publication year

conference

publishinghouse

publicationyear

author

wbetopic

Figure 6: A 5-dimensions H-view.

In the 3-dimensions case, we can extend the pre-

vious deﬁnition of a 3-dimensions H (or I) view. In-

tuitively, we build a standard 2-dimensions H (or I)

view and, starting from each of the related cells as fo-

cus, we display also the ranks in the third dimension.

Formally:

Deﬁnition 12 (3-Dimensions extended H-view).

Consider a zz-structure S = ∪

|C|

k=1

, where D

∪

i=1

∪ V

), and where R

= (V

, E

, f, {c

}, c).

The 3-dimensions extended H-view of size l = 2m+

1 and of focus x ∈ V = ∪

i=0

, on dimensions

, D

, is composed as follows:

• the central path (the m + 1-th row) from left to

right, focused in vertex x ∈ R

(x)

: x

−g

. . . x. . . x

where x

∈ R

(x)

, for s = −g, . . . , +p, g, p ≤ m and

g+ p+ 1 = l

′

;

• l

′

rectangular H-views of same size l and of fo-

cuses respectively x

−g

, . . . , x, . . . , x

, on main di-

mension D

and secondary dimension D

Analogously we can deﬁne a 3-dimensions extended

I-view.

Deﬁnition 13 (3-Dimensions extended I-

view). Consider a zz-structure S = ∪

|C|

k=1

where D

= ∪

i=1

∪ V

), and where

= (V

, E

, f, {c

}, c). The 3-dimensions ex-

tended I-view of size l = 2m + 1 and of focus

x ∈ V = ∪

i=0

, on dimensions D

, D

, is

composed as follows:

• the central path (the m+1-th column) from top to

bottom, focused in vertex x∈ R

(x)

: x

−u

. . . x. . . x

where x

∈ R

(x)

, for s = −u, . . . , +r, u, r ≤ m and

u+ r+ 1 = l

′′

;

• l

′′

rectangular I-views of same size l and of fo-

cuses respectively x

−u

, . . . , x, . . . , x

, on main di-

mension D

and secondary dimension D

As example, we start from Figure 4 and we consider

the related 2-dimensions H-view of size 5 and of fo-

cus v

, on main dimension D

conference

and secondary

dimension D

author

. We obtain the H-view shown in

Figure 7.

conference

author

Figure 7: Standard 2-dimensions H-view.

Now, for each cell of this view, we visualize the

related ranks in dimension D

wbe topic

. The result is

shown in Figure 8.

conference

author

wbetopic

Figure 8: A 3-dimensions extended H-view.

5 CONCLUSIONS

In this paper we have provided a description of zz-

structures, of H-view and I-view, and we have ex-

tended these deﬁnition to n-dimensions views. Our

aim is to use this formal model to represent concept

maps and to study their behavior in the Adaptive Ed-

ucational Hypermedia ﬁeld.

This paper represents a ﬁrst step in this direction and

it is part of larger project. Starting from the present

model, future works will focus on:

• automatic semantic ﬁltering methodologies;

FORMALIZING A MODEL TO REPRESENT AND VISUALIZE CONCEPT SPACES IN E-LEARNING

ENVIRONMENTS

345

• an extension of this model towards an open, dis-

tributed and concurrent agent based architecture;

• adaptive navigation and presentation for learners;

• authoring facilities for web-based courses.

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