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

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

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

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

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

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

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

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

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

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

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

Polyhedra can be q-connected through shared faces

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

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

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

Polyhedral Connectivity & q-transmission

change is not transmitted across the low dimensional face

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

 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

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

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

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

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

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

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

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

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

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

Example: Étoile Peer Marking Questions Answers +

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

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 

Example: Etoile students URLs Maximal rectangles determine Galois pairs

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

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

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