Hypernetworks in Scalable Open Education Jeffrey Johnson Cristian Jimenez-Romero Alistair Willis
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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
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Networks can represent relationships between pairs, < x, y >
e.g. student x studies with student y
Hypernetworks
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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
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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
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Networks can represent relationships between pairs, < x, y >
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 v0, v1, … vp has p+1 vertices
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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
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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 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
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Multidimensional Connectivity
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Multidimensional Connectivity
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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Polyhedral Connectivity & q-transmission
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Polyhedral Connectivity & q-transmission
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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s0, s1, …..s95 Roffset s0, s1, …..s95 Raligned
illusion: Squares narrow horizontally No illusion
Richard Gregory’s café wall illusion
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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 A3 and A5
40+ students give C1, C7, C12, G17
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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
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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 ?
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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 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
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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
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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
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Example: Étoile
Peer Marking
Questions
Answers +
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Example: Etoile
studentAttractive 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-2URL-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: 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
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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.
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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