</a>Symmetry: competition vs complementarity?

Polyhedral Empowerment of Networks through Symmetry: psycho-social implications for organization and global governance (Part #17)

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The argument above points to the value of symmetry both in ensuring network robutness and comprehensibility of complexity. However, from a complexity science perspective as argued by Chris Lucas (personal communication):

Symmetry when applied to social networks seems to me to be a problematical basic concept. Whilst in ideal worlds a democracy implies a power symmetry, in real worlds such a state is unstable and rapidly becomes asymmetrical in as many ways as can be humanly found ! Given human nature, any 'fair' distribution soons suffers from the 'Tragedy of the Commons' and becomes unfair to some or most.

One of the criteria for self-organization is that if we start symmetrical then the self-organizational process will automatically generate structure - and what is structure if not dissymmetry? This relates again to the edge-of-chaos, in that an initially disconnected network will gain links and become more connected, whilst a fully connected one will lost links and become more disconnected. Prevention of this dynamic can only be achieved by force, i.e. hierarchy.

This framing highlights the question of how nodes are distinguished in a network and how vertices are distinguished in a polyhedron. In the case of a network, the strategic arguments for robustness (notably from a military perspective) stress a degree of uniformity and replaceability of nodes -- no node being significantly distinct from another.

However, from another perspective, it might be argued that if all the nodes were indistinguishable there would be no significance to relationships between them. Missing from the complexity argument is the sense in which a variety of nodes -- if only of different colour -- may be configured in a symmetrical array. Good examples of this are teams. The above argument clearly does not apply to a football team or to two opposing teams. There is no question of arguing for a different array of players -- 10 or 15 -- for that game. The same is true of a card game like bridge.

Of great management significance is the desirable range of roles in a management team as identified by Meredith Belbin (Management Teams, 1981). He distinguished nine key roles: plant, resource investigator, coordinator, shaper, monitor/evaluator, team worker, implementer, completer finisher, and specialist. Other such arrays have been suggested. How many distinct roles and functions are desirable for its sustainable development in any system -- whether an ecosystem or a psycho-social system?

This points to the complementarity implied by nodes in a symmetrical array. Significantly this is seldom explicit in the design of social networks, or the enthusiasm for that mode. People may have complementary roles but this tends to be recognized, if at all, through the dynamics of the network. Indeed, in practice, the roles identified by Belbin may be emergent rather than designed in. The problem for such a management team may arise if particular roles are missing or over-represented. This would suggest that in this sense the nodes in a "polyhedrally empowered network" are empowered precisely because they constitute an ordered diversity of skills or perspectives -- a viable system in the sense advocated by R Buckminster Fuller (Synergetics: explorations in the geometry of thinking, 1975/1979) and by Stafford Beer (Beyond Dispute: the invention of team syntegrity, 1994).

Intriguing in the case of the spherical symmetry explored above is the sense that some nodes, often half, are not visible from any perspective when the polyhedron is viewed. They may be understood to be in a "shadow" zone. The sphere has to be rotated to bring them successively into view. This suggests a way of thinking about "otherness", namely "them" rather than "us" -- typically a challenge to be met, possibly through the dynamics of competition.

The interplay between complementarity and competition is of course most evident in the archetypal symmetry of the relationship between man and woman. The challenge, exemplified in that context (and by the challenges of comprehending spherical symmetry), is how to internalize the "other". Another metaphor, using the dual form of polyhedra, is the implication of the possible need to alternate into the dual form, or to allow for its expression -- a transformation well-illustrated by morphing (Carl Erikson, Morphing Three Dimensional Polyhedral Objects, 1994; Wayne Carlson, et. al. Shape Transformation for Polyhedral Objects, 1992).

With respect to dynamics, Chris Lucas again argues:

I have no problem with the intuitive appeal of symmetry or the appeal of mapping social networks onto polyhedra, one advantage is that making all links explicit will show up just how little of the network is taken into account in normal decision processes (weak links or links from weak nodes are usually ignored). Null links are links too of course, so symmetry in this sense is the superset of asymmetry.

But the dynamic issue is perhaps a killer. It seems to me that any mappings (even partial or asymmetrical ones) do seem to lock in the network to a static form. In real social networks the dynamics are ever changing, so the nodes, links, values and strengths cannot be regarded as static "variables" and if allowed to vary the "network" rapidly diverges from the model. Taking a model that no longer resembles the reality and "solving" it is a very unscientific practice (even if so very common in number-crunching circles).

This argument is obviously valid in those terms. However it does not take account of classic examples such as the resonance hybrid dynamics of the benzene molecule so fundamental to the organic world. Again it does not take account of how symmetrical arrays of team players with specific roles engage with one another in football. Nor does it take account of the archetypal dance between man and women. Of great interest in relation to the above references to Galois lattices, is the lattice focus of the work of Patrick Heelan (Logic of Changing Classificatory Framework, 1974).

Of particular interest are the dynamics of possible relationships between polyhedral forms. As with a strategic play in footbll, a polyhedrally empowered network might may morph into another array, or between several arrays, as indicated in the images in the related paper (Configuring Global Governance Groups: experimental visualization of possible integrative relationships, 2008). Elsewhere it was suggested that sustainable development was perhaps to be understood as based on alternation (Policy Alternation for Development, 1984).

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