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 200

C - Concept K - Knowledge

f 4

f 123 *

f 1

e 45

e 46

e C1

F = { f 1 ; f 2 , … f 6 }

e 13

e 12

f 456 *

f 5

f 6

e C2

f 2

e 23 f 3

Fig. 13.5 C-K structures with matroids in K

In C, it is possible to represent the graph of all the missing and required edges:
from the graph G, one represents all the missing and required edges; one
contracts all the edges that exists and are necessary; and one skip all the edges
that are known. If F¼ {f 1 ; f 2 , ... f 6 }, this gives the C-graph in Fig. 13.5 . Note
that this C-graph is also a matroid. It is a connected matroid, and, deleting the
loops, it is even a complete graph. Its rank equals the number of connected
components in K.
Let's see how the design of the missing edges occurs. There are two different
processes to add one single edge, and the matroid model will help us to
understand their critical features:

1. Constant-rank extension. This concerns the loop-edges in C. the graph G can be
“completed” with a new edge following a (single) extension process that does not
require changing the rank of the matroid. It means that the new edge does not add
a level of sophistication, a new dimension to the graph. It can be proven
(see (Oxley 2011 )) that any single extension of this kind corresponds to a flat
whose rank will be unchanged by the new edge. Such a flat is called a modular
cut. In graph G above, the edge e 56 linking f 5 and f 6 can be added without
changing the rank of the matroid; and it corresponds to the modular cut {e 45 ; e 46 }.
It can be shown that this process can be repeated until it completes all the
connected components.
The constant rank extension is a process to create new dependent sets.
For instance the new edge e 56 creates also {e 56 ;e 45 ; e 46 }. More generally, the
structure of the dependent sets can be measured by the rank of the dual of
the graph, also called the corank. If the graph is a spanning tree on n vertices,
the rank is n-1 and the corank is zero (no dependent set in the graph). If
the spanning tree is completed following the constant-rank extension, its
becomes a complete graph K n-1 , the rank is kept to n-1 but the corank will
reach (n-1)(n-2)/2.
This description corresponds to the fact that for a set of functions {f 4 ; f 5 ; f 6 ;},
all the associated markets will be addressed and for the most sophisticated
markets involving more than two functions, there will be several technological
combinations to reach it. The constant-rank extension hence helps to address all
the markets and, simultaneously, to offer multiple technological alternatives