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Clarification of a Mathematical Challenge for Systems Science

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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:

Preliminary analysis

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:

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 terms?

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 suggested here).

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.

Related issues

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 (negative values).

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 other).

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 problem loops.

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).

Further clues

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 organizations.

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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