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. 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 Biological systems have many identifiable levels

3. We have no formalism to combine the micro-, meso- and macro-level dynamics of systems in any field. Nations, Regions, Cities, … neighbourhoods Houses, shops, roads Rooms, gardens,.. 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

4. As scientists we need to: (1) prove that such a formalism can’t exist (2) … or construct that formalism We have no formalism to combine the micro-, meso- and macro-level dynamics of systems in any field. 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. Gestalt Psychologist Katz: V anilla I ce C ream  c old + y ellow + soft + s weet + v anilla it is a Gestalt – experienced as a whole  cold, yellow, soft, sweet, vanilla  From Networks to Hypernetworks

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

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

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

12. 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) Multidimensional Connectivity

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 change on some part of the system (q-percolation)

18. Polyhedral Connectivity & q-transmission

20. change is not transmitted across the low dimensional face

21. Intersections of simplices and dynamics star-hub relationship is a Galois connection  (a 1 )  (a 2 )  (a 3 )  (a 4 )  (b 1 )  (b 2 )  (b 3 )  (b 4 )  (b 5 )  (a 1 )  (a 2 )  (a 3 )  (a 4 )  (b 1 )  (b 2 )  (b 3 )  (b 4 )  (b 5 ) a1a1 a2a2 a3a3 a4a4 a5a5 b1b1 b2b2 b3b3 b4b4

22. star-hub relationship is a Galois connection.............. 1 1 1 1 1.............. …a1a2a3a4……a1a2a3a4…... b 1 b 2 b 3 b 4 b 5...

23. star-hub relationship is a Galois connection.............. 1 1 1 1 1...... 1 0 1 1 1...... 1 1 1 0 1...... 1 1 1 1 1.............. …a1a2a3a4……a1a2a3a4…... b 1 b 2 b 3 b 4 b 5... 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  cold + yellow + soft + sweet + vanilla; R Vanilla_Ice_Cream  e.g.

26. Relational Simplices and Multilevel Systems

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

28. Multilevel Systems The Intermediate Word Problem Hierarchical Soup

29. Multilevel Systems

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

31. {} R R e.g. take a set of 3 blocks assembled by a 3-ary relation

32. {} R 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

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

34. {} R Formation of simplices  hierarchical structure Arch n-ary relation assembles elements into named structures at a higher level 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. Level N+1 Level N Create a new object at Level N ! Aggregation – deconstruction downward dynamics in representing systems

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

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

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

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 costs Wages etc Traffic Backcloth

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

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

47. Dynamics

48. Backcloth dynamics: System time and System Events

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

51. Policy predictions as fans of connected multilevel multidimensional trajectories

52. Policy as designing the future is entangled with complexity science and design Policy Science Design Policy is designing the future

53. Design is an Intermediate Word Problem What are the intermediate structures ? What shall we call them ? Hierarchical Soup Policy is designing the future

54. Policy involves creating artificial systems Creating artificial systems involves Design Design a co-evolution between what you think you want & what you think you can have Policy is designing the future

56. “The System” – an abstraction it doesn’t exist (yet) “The parts” in the soup concrete stuff - components Top-down – imagine structures Bottom-up – instantiate structures The soup The System Design: hypothesing intermediate structures, Top-down and bottom-up

57. A fully instantiated design is a blueprint – the system can now be constructed Design: hypothesing intermediate structures, Top-down and bottom-up

58. Hypernetworks, Global System Science & Policy Consider a policy issue P, e.g. land use and transportation planning P What is relevant to the dynamics of P ?

59. Consider a policy issue P, e.g. international emergency P What Are The Intermediate Words? Hypernetworks, Global System Science & Policy

60. P What Are The Intermediate Words? Hypernetworks, Global System Science & Policy Consider a policy issue P, e.g. international emergency

61. P What Are The Intermediate Words? What are the relations? Hypernetworks, Global System Science & Policy Consider a policy issue P, e.g. international emergency

62. P What Are The Intermediate Words? What are the relations? What are the (hyper) networks? How do they work together? Hypernetworks, Global System Science & Policy Consider a policy issue P, e.g. international emergency

63. Conclusions (1) Hypernetworks generalise networks and hypergraphs (2) Hypernetworks have new kinds of connectivity structure (3) Hypernetworks give a way to represent multilevel systems (4) Policy is deigning the future (5) We build the science we need to answer (policy) questions (6) Hypernetworks - necessary if not sufficient for ML systems www.complexityanddesign.org jeff.johnson@open.ac.uk (pre-print of forthcoming book) Hypernetworks in Systems of Systems of Systems