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Hypernetworks in Scalable Open Education Jeffrey Johnson Cristian Jimenez-Romero Alistair Willis

Hypernetworks in Scalable Open Education Jeffrey Johnson Cristian Jimenez-Romero Alistair Willis European TOPDRIM (DYM-CS), Etoile, & GSDP Projects & Complexity and Design Research Group www.complexitanddesign.org The Open University, UK ECCS 2013 Barcelona 16 th September 2013.

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Hypernetworks in Scalable Open Education Jeffrey Johnson Cristian Jimenez-Romero Alistair Willis

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  1. Hypernetworks in Scalable Open Education Jeffrey Johnson Cristian Jimenez-Romero Alistair Willis European TOPDRIM (DYM-CS), Etoile, & GSDP Projects & Complexity and Design Research Group www.complexitanddesign.org The Open University, UK ECCS 2013 Barcelona 16th September 2013

  2. Hypernetworks Networks can represent relationships between pairs, < x, y > e.g. student x studies with student y

  3. Hypernetworks Networks can represent relationships between pairs, < x, y > e.g. student x studies with student y What about relationships between three students, < x, y, z > e.g. x, y and z all study together.

  4. Hypernetworks Networks can represent relationships between pairs, < x, y > e.g. student x studies with student y What about relationships between three students, < x, y, z > e.g. x, y and z all study together. Or a relation between 4 ?

  5. Hypernetworks Networks can represent relationships between pairs, < x, y > Or relations between any number of things …

  6. The generalisation of an edge in a network is a simplex Simplices can represent n-ary relation between n vertices

  7. A 1-simplex a, b has 2 vertices A 2-simplex a, b, c has 3 vertices A 3-simplex a, b, c, d has 4 vertices A p-simplex v0, v1, … vp has p+1 vertices The generalisation of an edge in a network is a simplex A p-dimensional simplex has p+1 vertices

  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 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. Multidimensional Connectivity

  14. Multidimensional Connectivity Polyhedra can be q-connected through shared faces

  15. Multidimensional Connectivity Polyhedra can be q-connected through shared faces 1-connected components

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

  22. Richard Gregory’s café wall illusion s0, s1, …..s95 Roffset  s0, s1, …..s95 Raligned  illusion: Squares narrow horizontally No illusion

  23. A hypernetwork is a set of relational simplices Hypernetworks augment and are consistent with all other network and hypergraph approaches to systems modelling: Hypernetworks and networks can & should work together

  24. Example: multiple choice questions … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … …

  25. Most questions have a majority answer, e.g. of 45 students all the students give answers A3 and A5 40+ students give C1, C7, C12, G17

  26. Most questions have a majority answer, e.g. of 45 students all the students give answers A3 and A5 40+ students give C1, C7, C12, G17 30+ students give the same answers to 17 of 20 questions

  27. Most questions have a majority answer, e.g. of 45 students all the students give answers A3 and A5 40+ students give C1, C7, C12, G17 30+ students give the same answers to 17 of 20 questions but majority answer for 3 questions is close to 45/2 = 23.5 answer F6 is the majority by one student – is it correct ?

  28. The most highly connected students all give the minority answer The majority of highly connected students give the minority answer The more disconnected connected students all give the majority answer

  29. Example: Peer marking Each student does an assignment Each student marks or grades 3 other students Bootstrap Problem: which students are good markers? As before the better markers will be more highly connected M1 & M2 probably good M3 or M4 is bad M1 M2 M3 M4

  30. Example: Peer marking Each student does an assignment Each student marks or grades 3 other students Bootstrap Problem: which students are good markers? As before the better markers will be more highly connected M1 & M2 & M5 probably good M3 or M4 M6 is bad, … M1 M2 M3 M4 M6 M5

  31. Example: Peer marking Each student does an assignment Each student marks or grades 3 other students Bootstrap Problem: which students are good markers? As before the better markers will be more highly connected M1 & M2 & M5 probably good M3 or M4 M6 is bad, … M1 M2 M3 M4 M6 M5

  32. Example: Étoile Questions Answers + Peer Marking

  33. Example: Etoile Similar students are highly connected Attractive URLS student student Attractive URLS Attractive URLS student

  34. Example: Etoile Students shared by URLs ULs shared by students URL-3 Student-3 URL-4 Student-1 Student-2 URL-1 URL-2 Galois pair:  S-1, S-2, S-3  U-1, U-2, U3, U-4 towards personalised education

  35. Example: Etoile URLs students 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Maximal rectangles determine Galois pairs

  36. Example: Etoile URLs students 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 Q-connected components more tolerant of missing 1s - may tame the combinatorial explosion of the Galois lattice.

  37. Example: Etoile Other Big Data bipartite relations include Students – Questions on which they perform well Students – Subjects in which they do well Questions – lecturers selecting questions for their tests etc

  38. Conclusions Hypernetworks Q-analysis gives syntactic structural clustering High q-connectivity more likely to indicate consistency Galois pairs give syntactic paired structural clusters Q-analysis more tolerance of noise that Galois lattice These structures can support personalised education Etoile provides crowd-sourced learning resources Uses crowd sourced learning resource + peer marking There are many hypernetwork structures in Étoile data Experiments planned to test these ideas with many students

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