1. 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 From Networks to Hypernetworks for a Science of Complex Systems Jeffrey Johnson Open University UK

2. 3 binary relations  one 3-ary relation Binary relations are not rich enough 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

3. Relational Structure Binary relation 3-ary relation 4-ary relation 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

4. Relational Structure

5. From networks to simplicial complexes An abstract p-simplex is an ordered set of vertices,  p =  v 0, v 1, v 2, …, v p . v1v1 v0v0 v2v2 v3v3  3 =  v 0, v 1, v 2, v 3 . 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

6. From networks to simplicial complexes An abstract p-simplex is an ordered set of vertices,  p =  v 0, v 1, v 2, …, v p . e.g. the tetrahedron A face is a sub-simplex. e.g. a triangle v1v1 v0v0 v2v2 v3v3  3 =  v 0, v 1, v 2, v 3 .  3 =  v 0, v 1, v 3 . 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

7. From networks to simplicial complexes An abstract p-simplex is an ordered set of vertices,  p =  v 0, v 1, v 2, …, v p . e.g. the tetrahedron A face is a sub-simplex. e.g. a triangle A simplicial complex is a set of simplices with all their faces v1v1 v0v0 v2v2 v3v3  3 =  v 0, v 1, v 2, v 3 . 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

8. From networks to simplicial complexes Every network is a simplicial complex whose simplices have dimension q = 0 or q = 1.  Simplicial complexes are a multidimensional generalisation of networks. 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

9. 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 From Networks to Hypernetworks 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

10. Gestalt Psychologist Katz: V anilla I ce C ream  c old + y ellow + soft + s weet + v anilla it is a Gestalt. It is a relational simplex From Networks to Hypernetworks 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010  cold + yellow + soft + sweet + vanilla; R Vanilla_Ice_Cream 

11. Definition A hypernetwork is a set of relational simplices From Networks to Hypernetworks 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010  cold + yellow + soft + sweet + vanilla; R Vanilla_Ice_Cream 

12. Example: Road Accidents upset rain speed tired The accident is a whole 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

13. Example: Road Accidents upset rain speed tired The accident is a whole the individual parts may not cause an accident 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

14. From networks to simplicial complexes Interesting structures 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 polyhedron representation Euler Polygon Representation

15. From networks to simplicial complexes Interesting structures 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 q-near

16. From networks to simplicial complexes Interesting structures 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 q-near eccentricity(  ) = |  | - |    ’| |  | high eccentricity low eccentricity

17. From networks to simplicial complexes Interesting structures 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 q-near q-neighbourhood of  

18. Polyhedral Connectivity 0- near polyhedra The intersection of two simplices is called their shared face. They are q-near if their shared face has dimension  q 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

19. Polyhedral Connectivity 1- near polyhedra 0- near polyhedra The intersection of two simplices is called their shared face. They are q-near if their shared face has dimension  q 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

20. Polyhedral Connectivity 0- near polyhedra The intersection of two simplices is called their shared face. They are q-near if their shared face has dimension  q 1- near polyhedra (and also 0-near) 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

21. Polyhedral Connectivity 1- near polyhedra 2- near polyhedra 0- near polyhedra

22. Polyhedral Connectivity Polyhedra can be q-connected through shared faces 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

23. Polyhedral Connectivity Polyhedra can be q-connected through shared faces 1-connected components 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

24. Polyhedral Connectivity Polyhedra can be q-connected through shared faces 1-connected components Q-analysis: listing q-components 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

25. Polyhedral Connectivity & q-transmission 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 change on some part of the system

26. Polyhedral Connectivity & q-transmission 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

27. Polyhedral Connectivity & q-transmission 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

28. Polyhedral Connectivity & q-transmission 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 change is not transmitted across the low dimensional face

29. Polyhedral Connectivity & q-transmission 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

30. Intersections of simplices and dynamics Shared faces are sites of interaction for pairs of simplices What about the intersection of more than two simplices? 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

31. Intersections of simplices and dynamics 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

32. Intersections of simplices and dynamics 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

33. Intersections of simplices and dynamics 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

34. Intersections of simplices and dynamics 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

35. Intersections of simplices and dynamics 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

36. Intersections of simplices and dynamics 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

37. Intersections of simplices and dynamics 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

38. Intersections of simplices and dynamics 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

39. Intersections of simplices and dynamics 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

40. Intersections of simplices and dynamics 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

41. Intersections of simplices and dynamics 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

42. Intersections of simplices and dynamics star hub star-hub relationship is a Galois connection 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 relational simplices have rich connectivity structures

43. Intersections of simplices and dynamics star-hub relationship is a Galois connection 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010  (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

44. Intersections of simplices and dynamics star-hub relationship is a Galois connection 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010.............. 1 1 1 1 1.............. …a1a2a3a4……a1a2a3a4…... b 1 b 2 b 3 b 4 b 5...

45. e.g. take a set of 3 blocks Formation of simplices  hierarchical structure {} 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

46. {} R R Formation of simplices  hierarchical structure e.g. take a set of 3 blocks assembled by a 3-ary relation 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

47. {} 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 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

48. {} R Formation of simplices  hierarchical structure Level N+1 Level N n-ary relation assembles elements into named structures at a higher level 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

49. {} R Formation of simplices  hierarchical structure Arch n-ary relation assembles elements into named structures at a higher level R 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

50. AND and OR aggregations in multilevel systems 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

51. Multilevel patterns of numbers on the structure 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 Emergent capability Capabilities

52. 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 Multilevel patterns of numbers on the structure

53. System dynamics as traffic on a fixed multilevel backcloth Dynamics on the hypernetwork backcloth 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

54. Dynamics 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

55. System time and System Events 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

56. System time and System Events 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

57. System time and System Events Planning involves changing relations 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

58. System time and System Events System dynamics involves changing relations … trajectories of multidimensional events 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

59. Representing Complex Systems & Dynamics Backcloth, Traffic & Type-1 Dynamics Traffic on the multi-level backcloth - coherence Cannot look at just one level Traffic must aggregate coherently over the structure 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

60. E.g. Cooperation – Making a Project

61. 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 Multilevel Systems Macro-level“The System” Meso-levels Micro-Level“The Atoms”

62. 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 Multilevel Systems Macro-level“The Project” Meso-levels Micro-Level“The Atoms” Stuff ? Lots of Intermediate Stuff

63. 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 Multilevel Systems Macro-level“The Project” Meso-levels Micro-Level Lots of Intermediate Stuff team-A task-B

64. 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 Multilevel Systems Macro-level“The Project” Meso-levels Micro-Level team-A task-B € n m € 200 k XXX

65. 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 Multilevel Systems Macro-level“The Project” Meso-levels Micro-Level team-A task-B € n m € 200 k € 213 k XXX WP1 WP2 … € 30 k

66. System time and System Events … trajectories of multidimensional events 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 We agree to make a project

67. System time and System Events … trajectories of multidimensional events 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 We agree to make a project The proposal is ready – submit !

68. System time and System Events … trajectories of multidimensional events 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 We agree to make a project The proposal is ready – submit ! The proposal is accepted – hooray !

69. System time and System Events … trajectories of multidimensional events 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 We agree to make a project The proposal is ready – submit ! The proposal is accepted – hooray ! Big research breakthrough !

70. Conclusions Relations can be multidimensional  need polyhedral hypernetworks Multilevel systems  need polyhedral hypernetworks Dynamics involves structures & numbers  need polyhedral hypernetworks Hypernetworks are necessary if not sufficient to model complex systems 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010