Entity Modelling

www.entitymodelling.org - entity modelling introduced from first principles - relational database design theory and practice - dependent type theory

## Information Structure Choices

In this section we discuss various ways of representing directed graphs, the different shapes that their entity models can take and the different organisations of information implied by the different shapes.

There is a reliance here, in this presentation, on the use of the work ‘abstract’ and on an understanding of what the mathematician means by this. The best that I can offer as explanation is that the mathematical idea of an abstract structure is that of a set of individuals and relations independent of any particular representation of the individuals and the mathematical concepts of graph and directed graph are cases in point. In the case of a graph the individuals are notionally vertices and edges and the relations are the incidence relations between them. It is certainly ironic and perhaps at first sight paradoxical, but with regard to any particular abstract structure, to describe it we need some kind of representation even though what we seek to describe is something without any particular representation. Mathematican Robin Gandy describes this to a non-mathematical audience in a lecture ‘Structure in Mathematics’ 1; he draws the love relationships to be found in Iris Murdoch's novel Severed Head as a directed graph and I have drawn an equivalent graph in figure 10 . If we abstract a directed graph from this then there are no labels, no lines, no points on paper just the facts of a set of notional individual entities and ther relationships between them 2.

 pLh hLp aLp pLa pLg mLp mLh aLx xLa gLx gLm mLg
Table 1
Gandy's relational depiction of the same love relationships.
Figure 10
Example from ‘Structure in Mathematics’ of a directed graph showing the love relationships to be found in Iris Murdoch's novel ‘Severed Head’ . The labels on the graph are simply to show the correspondence with the novel.

As an illustration of different representations of the same abstract structure, Gandy gives the relational representation shown in table 1 . In the relational representation, p occupies the same structural position as PALMER, h as HONOR, a as ANTONIA, g as GEORGIE, m as MARTIN, x as ALEXANDER. This relational description can be described by the message structure:

 table => loves*
 loves => lover, loved

Gandy's depicted love relationship is a many-many relationship and we can represent in an entity model like this:

There is of course nothing special about this example loves is a recursive many-many relationship and any such could be used to illustrate that a recursive many-many relationship is structurally a directed graph.

Gandy's relational representation shown in figure 10 could be rearranged in two ways from the point of view of the lover or the loved; these are shown in figures 11 and 12 .

aLp,x
gLx,m,g
hLp
mLp,h,g
pLh,a,g
xLa
Figure 11
This representation can be described by the message structure:
 table => loves*
 loves => lover, loved*
From this representation you may quickly find who is loved by a, who is loved by g and so on.
x,pLa
m,pLg
m,pLh
gLm
h,mLp
a,gLx
Figure 12
This representation is described by the message structure:
 table => loves*
 loves => lover*, loved
From this representation you may quickly determine who loves a, who loves g and so on.

But what communication structure do we associate with this? Well, this way of modelling directed graphs is akin to the matrix structure modelled in chapter one. Applied to the representation of the Severed Head love relationships it implies a communication in which it is apparent both (i) who loves each subject a, g, h and so on and (ii) who each subject a, g, h, etc. is loved by. This is made apparent by a matrix representation:

 a g h m p x a x x g x x h x x m p x x x x x x x

1 In Structuralism, edited by D. Robey, Wolfson College Lectures, 1972 .
2 According to Gandy: This indifference to form of representation is what Bertrand Russell referred to when he said that in mathematics we are not interested in what we are talking about. We return to the subject abstraction versus representation in a later section.