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to address each markets. This actually corresponds to a very powerful form of
robustness.
2. Rank-increase extension. The constant rank extension is not enough to address
all situations of generic technology design: the issue rises when the set
F contains vertices from different connected components in G. It is then required
to design a new edge. This new edge e has very different properties as the edge
designed by constant-rank extension: every flat of G will change rank when e is
added and no new dependent set is created. This extension process hence adds
one dimension to the graph (Oxley 2011 ) . For this reason, this can be assimilated
to an expansive partition.
Such an edge corresponds to the fact that two distinct sets of functions that
were not connected at all will now be connected by one single new edge. This
opens the possibility for more complex technologies. Note that adding such an
edge always means that the new “technology” is compatible with all already
existing technologies (since it enables all new paths that use the new edge and
other existing edges). A good example of such an “edge” is Watt and Boulton
“reciprocating movement” that enabled to link the steam engines technologies
(one connected component)
to the machine tools technologies (another
connected component)
The complete design process will (at least) then proceed as follows:
1. Constant-rank extension until all connected components are complete.
2. Connect connected components. There are several possibilities here, since the
C-graph is complete and only a spanning tree is required. Suppose that all
the C-edge have a certain cost: then the problem consists in finding the spanning
tree of minimal weight over a matroid structure. This can be done with the
Greedy algorithm (Kruskal algorithm in the case of graphs).
3. The resulting graph in K is now connected but is not complete. It is necessary to
identify in C the missing edges and to proceed with anew with a constant rank
extension. Some properties of this third step should be underlined:
a. The new edges that will be created are necessarily associated to a flat that
contains at least one edge resulting from an expansion process. In this sense
all the new edges are now the consequence of the expansive edges.
b. But on the other hand, each new edge of the third step creates a new circuit
that includes an expansive edge. This means that it also creates substitute for
the expansive edge.
Note that this process is not deterministic and keep the “generative” aspect of
design: for instance there are many possibilities for the spanning tree, depending on
the weight that the designer will put on all the C-edges. For instance the weight can
be based on the estimated cost of the technology development; but it can also be
linked to the expected difficulty to make this new edge compatible with already
existing edges in the two distinct connected components, ... Many other weight
systems are possible—note that whatever the weights chosen, the matroid structure
of the C-graphs warranties convergence of Greedy algorithm.
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