(Preface -- Part 3 - Structuring Complex Systems - Warfield 1974)
SUMMARY

The problem considered in this monograph
may he described as one of finding  useful
graphical descriptions of single contextual relations
germain to a system. The problem is addressed as 
one of developing a process that minimizes
the logistical burden on the developer
of the product. The product is called an 
interpretive  structural model. The  latter is 
structurally isomorphic to a digraph, but contains both
substantive and structural information supplied
by the developer. All of the mathematical operations 
required by the process would be carried
out by a computer. The critical, substantive system
knowledge is supplied by the developer,
The partnership of developer and computer gives
the process its strength.


By a system is understood a complex of
entities S with interactions R. Knowledge concerning
(S, R) can he structural or substantive
or both, and is the consequence of perceptions of
the system. Some knowledge of the system is
presumed to be available. Total knowledge of a
system is thought normally to be unattainable. A
description of a system is approximate.


Let <S> be represented by a finite set S = {s1, s2, ..... sn}
and let <R> be represented by a finite set <R> = {<R1>, <R2>, ... <Rm>}.
Call si an element of the system, and call <Rj> a contextual 
relation of the system. A contextual relation is 
a phrase in a colloquial language.

Let <Rj>, be represented by Rj, where Rj is a
binary relation on S x S. Let Aij be a binary
matrix indexed by (S, S) which portrays both Rj
and its complement, not Rj. Let Mij be the transitive
closure of Aij + I, where I is the identity matrix.


Then, there exists a permutation matrix P
such that the matrix Mj = (1/P)MijP induces a
unique partition on S into subsets called levels,
and each submatrix of Mj indexed by a level is
symmetric.

For any submatrix indexed by a level, there
exists a permutation such that its block diagonal
submatrices form a minimum set of universal
matrices. Each nontrivial index set of one or more of
these is called a maximal cycle set. If an index
set contains only one element, the set is trivial
and the element is an isolate within the level,
though it may connect. to other levels.

If appropriate modifications are made to Mj,
it can be converted to a matrix M’j’ such that
every maximal cycle set is replaced by a proxy
element, and reachability is preserved by following a
principle of condensation.

The digraph of the condensed matrix is a
hierarchy with the same number of levels as
Mij and Mj. There exists a binary matrix Aj for
which M’j’ is the transitive closure, called a
skeleton matrix, such that there is no other
matrix less than Aj that has M’j’ as its transitive
closure. The digraph D(Aj) is called the skeleton
digraph of Mij. It is the minimum-edge digraph
that preserves condensed reachability or Aij.


To every maximal cycle set, there corresponds 
a minimum-edge digraph that preserves
reachability within the cycle set. If every proxy
vertex on D(Aj) is replaced with such a minimum 
edge digraph, the resulting digraph is a
minimum-edge approximation of D(Aij).

Because the matrix Aij is often difficult to
develop, the matrix Mij is developed instead. In
developing Mij transitivity is enforced. While
it is impossible to determine Aij from Mij, it is
possible to determine Aj.


Since a minimum-edge representation of a
maximal cycle is often an inadequate approximation
to the structure of a maximal cycle, additional
information concerning the structure of
the cycle may be sought. Such structure is
based upon a weighting matrix W for the maximal
cycle, wherein the intensity of some contextual 
relation is supplied on a specified scale.
From such a weighting matrix, it is possible to
compute a binary matrix that represents a threshold
of intensity and preserves reachability
within the maximal cycle. This binary matrix
permits construction of a reachability-preserving
digraph for the maximal cycle which normally 
has fewer edges than a complete graph for
the cycle.

If those digraphs that result from application
of an intensity threshold to weighting matrices
of the various maximal cycles are substituted
for the proxy elements in the skeleton digraph
D(Aij) while preserving overall reachability, it
is believed a good approximation to the system
structure results insolar as the relation Rj is
concerned. The digraph so formed may be complicated
by the presence of many vertexes,
edges, and crossings of the edges. If so, additional
interpretive digraphs may be useful.

A maximal geodetic cycle specified on
(si, sj) is a maximal cycle consisting of a pair
of (geodetic) paths, one path originating at
vertex si and terminating at sj, the other originating
at sj and terminating at si. It is possible
to compute, for a given matrix Aij, a set of
maximal geodetic cycles. To each maximal
geodetic cycle there corresponds a cycle set,
which is a subset of S. The set of all such cycle