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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