Learning through Hamiltonian cycles and pathways

Monkeying with Global Governance (Part #13)

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In the light of the above argument, it is appropriate to note various ways in which the Hamiltonian cycle is explored in relation to learning (for exasmple: Ali Eshragh, et al., Hamiltonian Cycle Problem and Cross Entropy: passing through the nodes by learning, 2008; Dana Angluina and Jiang Chen, Learning a hidden graph using O(logn) queries per edge, Journal of Computer and System Sciences, 2008; Wieslaw Sienko and Wieslaw Citko, Hamiltonian Neural Networks Based Networks for Learning, InTech, January 2009).

Separately the potential value of polyhedra in relation to governance and thinking has been discussed (Polyhedral Empowerment of Networks through Symmetry: psycho-social implications for organization and global governance, 2008; Geometry of Thinking for Sustainable Global Governance, 2009). Of relevance to this argument is that the theory of Hamiltonian paths was first developed with respect to platonic polyhedra when represented as graphs.

Fig. 12: Dodecahedron (created with Stella Polyhedron Navigator)	Fig. 13: Hamiltonian cycle in a dodecahedron (reproduced from the icosian game in Wikipedia)

With respect to "learning", much is made of finding a Hamiltonian path, notably "learning a hidden subgraph". It is less evident that learning pathways are identified -- for purposes of education -- using insights from Hamiltonian pathways and cycles. The issue might be framed in terms of a matrix of topics to be learned -- as with some presentations of a curriculum -- with the challenge being to determine a fruitful pathway through all of them, as in the travelling salesman problem. The topics are then effectively "boxes to be ticked" -- with learning as a matter of "hitting all the spots". The arguments above for framing learning in terms of a "periodic table" can be understood in such terms.

Where the topics are such that they can only be deeply learned by being repeatedly "revisited", the concern then switches to the relevance of Hamiltonian cycles. With respect to governance and collective learning this revisiting can be recognized in the adage of Georges Sanatayana: "Those who cannot remember the past are condemned to repeat it". A potentially deeper form of learning is articulated in the much-cited verse of T. S. Eliot: "And the end of all our exploring. Will be to arrive where we started. And know it for the first time."

With regard to vulnerability to creative "monkeying", exemplified by the non-linear "trickiness" of the knight's move, it is the resulting cyclic patterns presented schematically in Fig. 4 which can then be understood as the central learning challenge. Failure to develop "streetwise" skills in response to such "monkeying" ensures vulnerability to those skilled in such trickiness -- as in chess and go. Of particular interest is the manner in which the skills of fencing and quarterstaff offer metaphors for such psychosocial skills as separately discussed (Modelling: authentic dialogue and quarterstaff combat?, 2003). The "philosophy" integral to fencing has long explored patterns of defence and attack, widely valued as indications of strategic postures (The Book of Five Rings).

The challenge becomes especially evident in relation to vicious cycles at the interpersonal or the collective level (Dysfunctional Cycles and Spirals: web resources on "breaking the cycle", 2002). Acquiring cognitive streetwise skills then merits exploration in terms of detecting Hamiltonian cycles and paths in behavioural dynamics.

This could well include the preoccupations of transactional analysis and the challenges addressed  by argument mapping. Ironically, rather than the travelling salesman problem, it is a matter of "seeing through" the salesman's manipulative lines of argument and focus switching -- a challenge similar to that of decoding the discourse of politicians or of charismatic leaders of manipulative sects (possibly then described as deprogramming). Ironically, whilst advertising and marketing campaigns of every kind are readily characterized as "monkey business", advertisers are more attentive to feedback on their questionable use of primates in the process (Less Monkey Business in Advertising, But Nonetheless Entertaining; Memo to Adland: Enough With the Monkey Business)

Hamiltonian cycles offer a formalization of the learning challenge -- then to be understood as finding a Hamilton cycle in a network of interrelated topics. An integrative comprehension of a subject area and its set of topics is, from this perspective, intimately related to internalization of knowledge of a Hamiltonian pathway or cycle -- in effect "knowing one's way around". The circularity resulting from the pattern of non sequiturs offers a contrast between "meta-education" and the linear progression (of "sequiturs") implied by "higher education", as separately discussed (¿ Higher Education ∞ Meta-education ? 2011).

Ironically, just as confidence can be built up in the ability to solve a formal Hamiltonian problem in mathematics, the more fundamental form of existential confidence is developed through the ability to transcend the behavioural cycles characterized by combinations of "monkeying" dynamics. Such transcendence -- "seeing" the path or the cycle -- then relates to that of recognizing a behavioural pattern. It is also a central theme of Atkin's q-analysis. It is in this sense that identity is emergent through a form of maturation of cognition -- through "embodying together" the dynamics of the "monkeys" (Emergence of Cyclical Psycho-social Identity: sustainability as "psyclically" defined, 2007). The wisdom of the "three wise monkeys" is then implicit in the integration of those dynamics -- thereby transcending their problematic individual functions and the vulnerabilities they enable.

The cyclic framing of emergent confidence and identity in this way is appropriate to their subtlety as experienced. In the light of the above argument, this then justifies further reflection on the nature of "circulation" within such learning cycles, as variously discussed (Circulation of the Light: essential metaphor of global sustainability? 2010; Enabling Moral Currency Circulation: reframing a stimulus package to avert moral bankruptcy, 2010; Potential Misuse of the Conveyor Metaphor: recognition of the circular dynamic essential to its appropriate operation, 2007).

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