Permutation graphs and copermutations

A wiki demonstrates a graph of connected concepts (though not a 'knowledge graph' in the strict sense of ontology). ^TODO:verify

A permutation graph can be applied to a network of concepts, but what's subsequently viewed are not nodes from the original concept map, nor edges of the graph of alterations made by a reordering (the 2 intuitive features of regular graphs), but rather: the edges that cross on a matching diagram of the permutation against that structure which is permuted. Understanding this, I interpret them for a knowledge graph as 'attentional verticals'.

Figure i. Matching diagram for the permutation (4,3,5,1,2) and the corresponding permutation graph, via Wiki: Permutation graph

[This analogy is likely unclear as I'll clarify in revision, and through graphical depiction] ^TODO:clarify

The analogy goes like:

• in business terminology, 'vertical' is shorthand for 'vertical market' and has 2 implications:
  1. a population of interest (i.e. multiplicity, or a domain)
  2. specificity (i.e. direction of attention, or a restricted range)
• ...^TODO:clarify
• ...when applied to a knowledge graph its meaning suggests a 'bottom up' approach to search space, by which items (nodes i.e. concepts etc) which are targetted as 'of interest' (i.e. given attention) act similarly to keywords.
  • This concept of items 'expressing a search space' is intuitive from recommendation systems, explained well in this video: ^{TODO:get video (Omnity vimeo, tweeted)}
  • see: Attentional verticals
• the connections of a permutation graph [applied over a knowledge graph, a.k.a. ontology] constitute modifications of a patterning applied to concepts (crossings on the matching diagram).
  • the 'modification' is nuanced: ^TODO:clarify
  • this metaphor gets deep and tricky, but thankfully there is already an extensive literature using "neural network" as a biological metaphor at the intersection of information theory and neuroscience by which to make it more intuitive and thus more effective as a narrative device (for both personal use and exposition).
• Crossings on the matching diagram / edges of the permutation graph ∴ have a subtle meaning, which I'm hoping to elucidate, but my initial induction* is that there is an indication within this alteration of concept-structure of a shift in attention of some overarching process, such as one that may be being deliberately carried out within an inference process (e.g. Hebbian ANN procedures, reweighting etc.). ^TODO:clarify
  • see: Wiki - Hebbian theory / Hebb's rule
  • NB: pattern analysis is more often unsupervised cf. classification more oft supervised.

A copermutation has a distinct meaning in French ^†, a theological concept which the likes of Pascal would surely have been aware of as he intermingled his Pensées later into a work entitled An Apology for the Christian Religion - the century following the Reformation, during which benefices (rewards for the clergy) were removed

• Pascal's writings on combinations, permutations as projective geometry/combinatorics problems applied to probability are thus not entirely inseparable from the intriguingly similarly named concept of 'copermutations' (relating to the aforementioned benefices, a legal entity relating to land ownership - or more strictly usufruct).

Pascal's fragmentary approach to writing (liasses, in modern French translated as 'bundle', 'batch', 'wad', 'unit-set', or 'sheaf') permitted and encouraged right up to the point of publication the free association of concepts such that the attention of the editor (i.e., the publisher) would rearrange sections and in turn their internal structure.

This approach has a deep and beautiful significance for his work, and in an age of computational shortcuts truly emphasises the restrictions imposed by filing systems' physical considerations (as regards memory, linearity, edition/versioning, and by contrast the objectifying effect of rigid textual structures).

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• to develop: 'Attentional verticals'
  • by analogy, converse (?) : Differential privacy (or rather, continuing with the attentional analogy what is the meaning of differential privacy on a knowledge graph permutation graph - private attention? "Inner thought" of an ANN ? (!)

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Web project forthcoming.

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• * abduction? deduction? ^TODO:clarify
  • (hunch / conjecture / hypothesis / view)
• ^† see:

  • Wiki.fr: Bénéfice ecclésiastique

  • Wiktionary: copermuter

    "Dans le langage ecclésiastique, échanger un bénéfice contre un autre"
    "In ecclesiastical language, to exchange a benefice (for/into) another [form]"

  • NB: first, note Pascal rejected mind/body dualism ^TODO:source and regarded man as 'automaton' (one could also say 'machinic') in a way that here is to be interpreted w.r.t. information theory [cf. theology]. He took from [biblical reading] Ecclesiastes, 'man is wretched because of his awareness through reason'. To me, this gives the term 'ecclesiastical' an information theoretical sense, and as such allows interpretation of 'copermutations' through further analogy. ^TODO:TBC

    • Wiki.fr: Bénéfice ecclésiastique ^{TODO:translate}