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

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

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

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Networks can represent relationships between pairs, Or relations between any number of things … Hypernetworks

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

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The generalisation of an edge in a network is a simplex A p-dimensional simplex has p+1 vertices 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 v 0, v 1, … v p has p+1 vertices

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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

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set of vertices simplex clique cold, yellow, soft, sweet, vanilla From Networks to Hypernetworks

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

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

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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

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Polyhedra can be q-connected through shared faces

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

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

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

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Polyhedral Connectivity & q-transmission

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change is not transmitted across the low dimensional face

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

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s 0, s 1, …..s 95 R offset s 0, s 1, …..s 95 R aligned illusion: Squares narrow horizontally No illusion Richard Gregory’s café wall i llusion

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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

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Example: multiple choice questions … … … … … … … … … … … … … … … … … … … … …

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Most questions have a majority answer, e.g. of 45 students all the students give answers A 3 and A 5 40+ students give C 1, C 7, C 12, G 17

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Most questions have a majority answer, e.g. of 45 students all the students give answers A 3 and A 5 40+ students give C 1, C 7, C 12, G 17 30+ students give the same answers to 17 of 20 questions

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Most questions have a majority answer, e.g. of 45 students all the students give answers A 3 and A 5 40+ students give C 1, C 7, C 12, G 17 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 F 6 is the majority by one student – is it correct ?

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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

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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 M 1 M 2 M 3 M 4 M 1 & M 2 probably goodM 3 or M 4 is bad

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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 M 1 M 2 M 3 M 4 M 1 & M 2 & M 5 probably goodM 3 or M 4 M 6 is bad, … M5M5 M6M6

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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 M 1 M 2 M 3 M 4 M 1 & M 2 & M 5 probably goodM 3 or M 4 M 6 is bad, … M5M5 M6M6

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Example: Étoile Peer Marking Questions Answers +

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

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

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

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Example: Etoile students URLs 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.

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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

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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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Creating a simplicial complex Step 0.) Start by adding 0-dimensional vertices (0-simplices)

Creating a simplicial complex Step 0.) Start by adding 0-dimensional vertices (0-simplices)

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