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Hypernetworks in systems of systems of systems

Hypernetworks in systems of systems of systems. Jeffrey Johnson The Open University, UK TOPDRIM - Topology-driven methods in CS NESS - Non-Equilibrium Social Science GSDP – Global Systems Dynamics & Policy Étoile – Enhanced Technology for Open

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Hypernetworks in systems of systems of systems

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  1. Hypernetworks in systems of systems of systems Jeffrey Johnson The Open University, UK TOPDRIM - Topology-driven methods in CS NESS - Non-Equilibrium Social Science GSDP – Global Systems Dynamics & Policy Étoile – Enhanced Technology for Open Intelligent Learning Environments

  2. We have no formalism to combine the micro-, meso- and macro-level dynamics of systems in any field. Biological systems have many identifiable levels Level N+h Level N+h-1 Level N+h-2 … … Level N+2 Level N+1 Level N Societies …. Individual animals … … cells …. Proteins, fluids

  3. We have no formalism to combine the micro-, meso- and macro-level dynamics of systems in any field. Built environment has many identifiable levels Level N+h Level N+h-1 Level N+h-2 … … Level N+2 Level N+1 Level N Nations, Regions, Cities, … … neighbourhoods Houses, shops, roads Rooms, gardens, ..

  4. We have no formalism to combine the micro-, meso- and macro-level dynamics of systems in any field. • As scientists we need to: • prove that such a formalism can’t exist • (2) … or construct that formalism Hypernetwork theory is an attempt to do this

  5. From networks to hypernetworks

  6. From hypergraphs & Galois pairs to hypernetworks

  7. Hypergraphs are set theoretic & not rich enough Same set of parts but arranged differently The vertices need to be ordered to discriminate them

  8. From Networks to Hypernetworks Gestalt Psychologist Katz: Vanilla Ice Cream cold + yellow + soft + sweet + vanilla it is a Gestalt – experienced as a whole  cold, yellow, soft, sweet, vanilla 

  9. From Networks to Hypernetworks relational simplex Set of vertices  simplex  clique  cold, yellow, soft, sweet, vanilla 

  10. Simplices represent wholes … remove a vertex and the whole ceases to exist.

  11. Multidimensional Connectivity Simplices have multidimensional faces A set of simplices with all its faces is called a simplicial complex

  12. Multidimensional Connectivity Simplices have multidimensional connectivity through their faces Share a vertex 0 - near Share an edge 1 - near Share a triangle 2 - near A network is a 1-dimensional simplicial complex with some 1-dimensional simplices (edges) connected through their 0-dimensional simplices (vertices)

  13. From networks to simplicial complexes Interesting structures q-near

  14. From networks to simplicial complexes Interesting structures  ’  q-near  is q-connected to ’

  15. Polyhedral Connectivity Polyhedra can be q-connected through shared faces

  16. Polyhedral Connectivity Polyhedra can be q-connected through shared faces 1-connected components Q-analysis: listing q-components

  17. Polyhedral Connectivity & q-transmission (q-percolation) change on some part of the system

  18. Polyhedral Connectivity & q-transmission

  19. Polyhedral Connectivity & q-transmission

  20. Polyhedral Connectivity & q-transmission change is not transmitted across the low dimensional face

  21. Intersections of simplices and dynamics star-hub relationship is a Galois connection (b4) (b5) (a3) a5 (a4) b2 a4 (b3) b3 b1 a1 a3 a2 (a1) b4 (b2) (b1) (a2) (a1) (a2) (a3) (a4) (b1) (b2) (b3) (b4) (b5)

  22. star-hub relationship is a Galois connection . . . b1 b2 b3 b4 b5 . . . … a1 a2 a3 a4 … . . . . . . . . . . . . . . 1 1 1 1 1 . . . . . . 1 1 1 1 1 . . . . . . 1 1 1 1 1 . . . . . . 1 1 1 1 1 . . . . . . . . . . . . . .

  23. star-hub relationship is a Galois connection . . . b1 b2 b3 b4 b5 . . . … a1 a2 a3 a4 … . . . . . . . . . . . . . . 1 1 1 1 1 . . . . . . 1 0 1 1 1 . . . . . . 1 1 1 0 1 . . . . . . 1 1 1 1 1 . . . . . . . . . . . . . . q-connect components ! loser clusters of simplices

  24. From Complexes to Hypernetworks Simplices are not rich enough to discriminate things Same parts, different relation, different structure & emergence We must have relational simplices

  25. Definition A hypernetwork is a set of relational simplices e.g. cold + yellow + soft + sweet + vanilla; RVanilla_Ice_Cream 

  26. Relational Simplices and Multilevel Systems

  27. Multilevel Systems Can highly entangled multilevel systems separated into well-defined levels ?

  28. Multilevel Systems Hierarchical Soup The Intermediate Word Problem

  29. Multilevel Systems

  30. Formation of simplices  hierarchical structure e.g. take a set of 3 blocks { }

  31. Formation of simplices  hierarchical structure e.g. take a set of 3 blocks assembled by a 3-ary relation R R { }

  32. Formation of simplices  hierarchical structure e.g. take a set of 3 blocks assembled by a 3-ary relation R The structure has an emergent property R { }

  33. Formation of simplices  hierarchical structure Level N+1 Level N n-ary relation assembles elements into named structures at a higher level R { }

  34. Formation of simplices  hierarchical structure Arch n-ary relation assembles elements into named structures at a higher level R R { }

  35. AND and OR aggregations in multilevel systems Sets, classes Structures Sets of parts Conventional classification trees don’t have alpha aggregations

  36. Observing multilevel systems of systems of systems Hypothesis 1 When we look at systems we see the whole & the parts Hypothesis 2 Our brains create new multilevel structures

  37. Aggregation – deconstruction downward dynamics in representing systems Level N+1 Level N

  38. Aggregation – deconstruction downward dynamics in representing systems Level N+1 Level N Create a new object at Level N !

  39. Aggregation – deconstruction downward dynamics in representing systems Level N+1 Level N Create a new level - Level N-1 ! And new objects at this level

  40. Aggregation – deconstruction downward dynamics in representing systems Level N+1 Level N Create a new object at Level N

  41. Aggregation – deconstruction downward dynamics in representing systems Create new objects at Level N+1 Level N+1 Level N

  42. Mereology Parts and wholes goes back millennia to Plato and Aristotle. mereology was coined in 1927 by Stanislaw Lesniewski A mereological system - Objects, X, and a binary relation parthood, ‘x is a part of y’.

  43. 1.8 Mereology Paying is part of shopping

  44. Multilevel patterns of numbers on the structure Traffic Backcloth costs Wages etc

  45. Multilevel patterns of numbers on the structure Traffic Backcloth Profits Costs Income taxes Costs Wages etc

  46. Dynamics on the hypernetwork backcloth System dynamics as traffic on a fixed multilevel backcloth

  47. Dynamics

  48. Backcloth dynamics: System time and System Events

  49. Backcloth dynamics: System time and System Events

  50. Backcloth dynamics: System time and System Events System dynamics involves changing relations … trajectories of multidimensional events

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