A New Perspective for Risk Management: A Study of the Design of Generic Technology with a Matroid Model in C-K Theory - Principia Designae: Pre-Design, Design, and Post-Design - page 199

Fig. 13.4 A graph G

f 4

f 1

e 45

e 46

e 13

e 12

f 5

f 6

f 2

e 23

f 3

1. In K: We consider that K contains a graph G. One interpretation of this graph can
be as follows: the vertices are some functions fi; an edge represents a technology
to address a pair of functions. A path of edges defines a technology (a combi-
nation of technologies) to address all the vertices in the path. The graph G below
can be interpreted as a synthesis of the technological know-how of the designer.
For instance in the Fig. 13.4 the designer knows how to address {f 4 ; f 5 } (with the
edge e 45 ); and he knows several solutions to address {f 1 ; f 2 ; f 3 } (e 12 -e 23 or
e 13 -e 23 ); he doesn't know any solution to address {f 5 ; f 6 } or {f 3 ; f 5 }. This
graph of designer's knowledge is a matroid.
Note that this matroid can also be characterized by its basis or its cycles. It has
a certain rank, which actually corresponds to the size of the largest independent
set. In a graph G, we have: rank(G)¼jV(G)j-ncc(G) where jV(G)j is the number
of vertices in G and ncc(G) is the number of connected components in G. (rank
(G) ¼4 in the example below). The rank illustrates the most “complex” tech-
nologies that can be built by the a user of the K-space: at most the graph G below
enables to separately address four independent edges.
The matroid also has so-called “flats”. A flat is a set of elements such that it is
impossible to add a new element to it without changing its ranks. One says that
a flat is a “closed” set in a matroid. In G below, {e 12 ,e 23 ,e 13 } is a flat and
{e 12 ,e 13 } is not a flat. The flat represent stable “substructures” in a matroid.
We also represent in K the context invariance, representing all the states of
nature that the design could be facing. We consider for instance that the designer
might need to address any combination of functions taken from the set of
functions F¼ {f 1 ; f 2 , ... f n }. And we consider (for sake of simplicity) that
any market is independent of the other (the market {f 1 ; f 2 } is independent of
{f 1 ; f 2 ; f 3 ;} even if the functions of the former are included in the functions of the
latter). This context invariance is also a matroid: it is one of the so-called “uniform
matroids” U p,n where the elements are the n-first integers and the independence
sets are all subsets of E of size equal or less than p. The contexts can hence
be represented by a matroid U n,n , in which all the subsets of E are independent.
With these two matroids we have the structures of the known, in K space.
2. In C: In the model, designing the generic concept consists in designing addi-
tional edges in G to address all the markets in U n,n . It actually means to design
the missing edges so that the graph built on F becomes complete—ie: each
vertex is linked to all the others by one single edge. This complete graph is called
K n-1 in matroid theory. It means that if F¼V(G) and G is complete, then the
designer knowledge is a generic technology for U n,n built on F.