Hypernetworks in Scalable Open Education Jeffrey Johnson Cristian Jimenez-Romero Alistair Willis European TOPDRIM (DYM-CS), Etoile, & GSDP Projects & Complexity.

Hypernetworks in Scalable Open Education

Jeffrey JohnsonCristian Jimenez-Romero

Alistair Willis

European TOPDRIM (DYM-CS), Etoile, & GSDP Projects&

Complexity and Design Research Groupwww.complexitanddesign.org

The Open University, UK

ECCS 2013 Barcelona

16th September 2013

Networks can represent relationships between pairs, < x, y >

e.g. student x studies with student y

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.

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 ?

Hypernetworks

Networks can represent relationships between pairs, < x, y >

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 v0, v1, … vp has p+1 vertices

Gestalt Psychologist Katz:

Vanilla Ice Cream cold + yellow + soft + sweet + vanilla

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 facesShare 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

Multidimensional Connectivity

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

Polyhedral Connectivity & q-transmission

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

s0, s1, …..s95 Roffset s0, s1, …..s95 Raligned

illusion: Squares narrow horizontally No illusion

Richard Gregory’s café wall illusion

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 A3 and A5

40+ students give C1, C7, C12, G17

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

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 ?

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 assignmentEach 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

M3 M4

M1 & M2 probably good M3 or M4 is bad

Example: Peer marking

Each student does an assignmentEach 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

M3 M4

M1 & M2 & M5 probably good M3 or M4 M6 is bad, …

M5

M6

Example: Peer marking

Each student does an assignmentEach 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

M3 M4

M1 & M2 & M5 probably good M3 or M4 M6 is bad, …

M5

M6

Example: Étoile

Peer Marking

Questions

Answers +

Example: Etoile

studentAttractive 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-2URL-1

URL-3

URL-4

Galois pair: S-1, S-2, S-3 U-1, U-2, U3, U-4

Example: Etoilest

uden

tsURLs

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

Example: Etoilest

uden

ts

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.

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