Clarification of a Mathematical Challenge for Systems Science
- / -
Paper prepared for the Annual Meeting of the International Society for Systems
Sciences (Amsterdam, 1995)
Data has been collected and partially ordered on some 12,000 'world problems'
as perceived by the universe of international organizations. The unique feature of this
data set is that some 120,000 relationships have been identified between such problems --
effectively constituting a complex network of systemic relationships. The data is
published as part of the Encyclopedia of
World Problems and Human Potential (1994).
The partial ordering of the data has taken the form of distinguishing hierarchical
relationships between problems where possible, namely establishing relationships to those
which are more general ('broader') and to those which are more specific
('narrower'). A given problem may however point to several more general ones.
Where such identification of levels seems questionable, problems may simply be identified
as 'related' to others. Clearly broader/narrower and related/related
relationships are reciprocal.
From a systemic perspective, of greater interest are other relationships whereby a
given problem is identified as 'aggravating' one or more other problems. These
too are reciprocated through 'aggravated by' relationships. There are some
30,000 aggravating relationships. Also of interest, but rarer, are cases of problems
'reducing' other problems, and in turn being 'reduced' by others.
These may be seen as different types of feedback loop.
The question of interest concerns useful ways of analysing the data, especially that on
aggravating relationships. There are several reasons:
- the data is prepared through what amounts to a hypertext editing process that can
generate redundant relationships which it is difficult to identify in a linear editing
environment; more powerful techniques are required to identify them
- even when redundancies are removed the resulting network is exceedingly complex and does
not lend itself to visual inspection, especially when local properties of the network (in
systemic terms) may be easily confused with global properties of greater significance
- since it represents a global system of relationships, it is useful to ask in what way
may the network be manipulated, preserving its topological properties, in order to render
its global properties more comprehensible.
Focusing only on aggravating relationships, a computer program has been run to identify
loops or cycles amongst the problems. These have been sought up to loops of seven links.
Using ordinary PCs, even running in parallel, this operation can take up to a week because
of the number of pathways that have to be searched to detect such loops.
Some 19,000 loops have been identified. Some of these will include loops resulting from
editing errors (as mentioned above). It is however interesting to shift the unit of
analysis from specific problems to problem loops.
Some idea of the resulting data set can be obtained from the tables.
Each problem has been tentatively allocated to a level of generality through use of a
code letter: B=general down to G=very specific, with F=unallocateable. Because of this it
is possible to get some idea of the probability of errors in the loops. For example:
- a loop of the type BBEBBB, raises questions as to whether the E problem should be
included rather than a more general problem. This raises questions both with regard to the
tentative level allocation, and to the positioning in the problem hierarchy which
partially determined it.
- a loop of the type DDD, raises questions as to whether such a small loop is not the
result of an editing error
With this framework, it is useful to think of the more specific problems as being more
'local' and the more general problems as being more 'global' -- in a
topological rather than a geographical sense.
Clarifying the larger challenge
It is interesting to hypothesize the existence of major 'pathways' formed by
elements of different loops. Such pathways can be envisaged as being like rivers from
which local loops break off (as whorls). A number of such pathways may intersect. It would
then be the interlocking of these pathways which ensured the integrity of the system of
problems as a whole.
This approach becomes especially interesting if it is hypothesized that such pathways
are themselves necessarily circular. The question can then be formulated in terms of the
nature of the surface onto which the pattern of loops can be usefully projected or mapped
so as best to bring out the systemic integrity.
A flat surface is not especially interesting for such an exercise (although this has
been done for metabolic pathways). More interesting in comparison is a sphere or a torus.
In the case of a sphere, any such major pathways could emerge as 'great circles'
in a geometrical sense. It would then be the interlocking of such great circles which
provided the integrity of the structure. Lesser circles could encompass portions of the
sphere. Local circles could then be positioned appropriately in relationship to them.
Graphically displayed on computer, they could be accessible by zooming when detail was
required, but the overall pattern would not be lost.
Clearly the projection of the loops onto such a surface constitutes a special
challenge. What mathematical operation is necessary to effectively 'massage'
the loops into the positions on the surface which most effectively bring out the integrity
of any great circle-type phenomena? How should this question be formulated in mathematical
Clearly the number of loops of the type detected can be considered very large, or even
infinite, when the number of elements in a loop is increased. What then is the
mathematical constraint which usefully excludes certain loops, so limiting the number that
can be detected? This question would seem to have something to do with maximizing the
number of tangential relationships between loops, notably by 'nesting' the
maximum number of loops within other loops -- minimizing the number of loops which
need to be represented non-tangentially as crossing other loops.
It might be argued that this approach to the analysis of the data is more complex than
other more conventional forms available from graph theory. The assumption made here is
that the constraint of representation on a surface comprehensible to the human mind is of
immediate relevance to the ability to make informed decisions on such matters at a policy
level. The existing ability to provide specialized analysis of what amounts to local loops
in isolation has been well demonstrated, as has the inability to act on the larger loops
to which these may contribute. A more comprehensive approach is required to 'thinking
globally and acting locally', whether in the geographical or the systemic sense (as
As in the case of studies currently supported by computer graphics (CAD, PCBD,
molecular chemistry, etc), and even dependent on them, it is suspected that this approach
might offer an entirely new grasp of global system properties of very large systems.
The data on 'world problems' discussed above, is part of a related set of
databases. These include databases on: some 30,000 international organizations (80,000
relationships), some 12,000 organization strategies (in excess of 20,000 relationships),
some 5,000 human values (23,000 relationships).
These databases are themselves linked between each other and to the problems database.
Thus international organizations have strategies which focus on problems. Human values are
fundamental to clear definition of both strategies (positive values) and problems
The analysis of problems suggested above could provide the basis for a similar analysis
of strategies. Again the data is ordered in terms of both hierarchical and functional
relationships (namely strategies 'undermining' or 'facilitating' each
There is a strong case for developing the ability to detect strategies capable of
reinforcing each other in order to contain problems, as well as of identifying those
strategies which tend to undermine this process. As with the problems, the major pathways
of this process may be effectively beyond the present human 'comprehension span'
when the data is presented in conventional forms. For, in effect, different international
organizations tend to have as their mandate particular (systemically 'local')
problems and strategies. They have no mandate or capacity to respond to problems in larger
(systemically 'global' loops) and have little understanding of how their
particular strategies could contribute to the larger pattern which might contain such
An approach of the kind recommended could explore the feasibility of constructing
interlocking strategies capable of responding in a sustainable manner to the system of
problems (which has certainly demonstrated its own sustainability).
In operations research terms, the problem seems to bear some relationship to that of
optimizing truck delivery routes. The problem also recalls some of the issues in
'rubber-banding' line representations in computer-aided design, although here
any relationship is not simplified to a straight line but to a circle.
The issue at the policy level is how to grasp the nature of networks of problems,
organizations or strategies and detect where relevant action is appropriate. Let us assume
that some strategies are less fit than others, whether some are outdated or simply
inefficient. Let us also assume that any global strategy to 'deliver' solutions
has to work through several branching and/or converging networks of strategies (as in PERT
charts). The issue is to discover what is the best group of pathways to use (eg fund or
subcontract through) at any one time or alternatively which ones in a critical pathway
need to be upgraded in some way -- possibly by being replaced by more effective
strategies, possibly by additional infrastructure funding, etc. These are important
questions for policy-makers.
There is a suspicion that the level of complexity implied by comprehensive systems of
the kind discussed above calls for higher orders of consensus in any organized response.
By this is meant that new ways of organizing disagreement are required to embody the
necessary diversity of perspectives to handle that complexity.
In this respect a number of clues deriving from tensegrity organization may prove
useful, especially in the light of the structural importance they give to great circle
properties. Such insights have recently been explored in relationship to group
organization in the work of Stafford Beer based on his own work on cybernetics in
Of special interest is the possibility of incorporating insights from such analysis
into the design of computer protocols to facilitate management of communication gateways
been sets of people with diverse but complementary interests. It is perhaps a fundamental
conceptual error to envisage the 'information highway' in linear terms. How it
patterns itself in channelling content is of fundamental significance to the ability to
provide any higher order response.
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