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From Networks to Hypernetworks for a Science of Complex Systems

From Networks to Hypernetworks for a Science of Complex Systems. Jeffrey Johnson Open University UK. 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010. Binary relations are not rich enough. 3 binary relations  one 3-ary relation.

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From Networks to Hypernetworks for a Science of Complex Systems

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

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

  5. From networks to simplicial complexes An abstract p-simplex is an ordered set of vertices, p =  v0, v1,v2, … , vp. v0 v2 v1 v3 3=  v0, v1,v2, v3. 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 =  v0, v1,v2, … , vp. e.g. the tetrahedron A face is a sub-simplex. e.g. a triangle v0 v2 v1 v3 3=  v0, v1,v3. 3=  v0, v1,v2, v3. 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 =  v0, v1,v2, … , vp. 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 v0 v2 v1 v3 3=  v0, v1,v2, v3. 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. From Networks to Hypernetworks Gestalt Psychologist Katz: Vanilla Ice Cream cold + yellow + soft + sweet + vanilla it is a Gestalt – experienced as a whole 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

  10. From Networks to Hypernetworks Gestalt Psychologist Katz: Vanilla Ice Cream cold + yellow + soft + sweet + vanilla it is a Gestalt.It is a relational simplex  cold + yellow + soft + sweet + vanilla; RVanilla_Ice_Cream 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

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

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

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

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

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

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

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

  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 1- near polyhedra (and also 0-near) 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

  21. Polyhedral Connectivity 1- near polyhedra 0- near polyhedra 2- 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 change on some part of the system 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

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

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

  43. 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) 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010

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

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

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

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

  48. Formation of simplices  hierarchical structure Level N+1 Level N 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

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

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