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![Page 1: 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 From Networks to Hypernetworks for a Science of Complex Systems Jeffrey Johnson.](https://reader035.fdocuments.us/reader035/viewer/2022062516/56649e215503460f94b0d4a5/html5/thumbnails/1.jpg)
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
From Networks to Hypernetworks for a
Science of Complex Systems
Jeffrey Johnson
Open University
UK
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3 binary relations one 3-ary relation
Binary relations are not rich enough
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
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Relational Structure
Binary relation 3-ary relation 4-ary relation
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4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Relational Structure
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From networks to simplicial complexes
An abstract p-simplex is an ordered set of vertices,
p = v0, v1, v2, … , vp.
v1
v0
v2
v3
3 = v0, v1, v2, v3.
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
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From networks to simplicial complexes
An abstract p-simplex is an ordered set of vertices,
p = v0, v1, v2, … , vp.
e.g. the tetrahedron
A face is a sub-simplex.
e.g. a trianglev1
v0
v2
v3
3 = v0, v1, v2, v3.3 = v0, v1, v3.
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
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From networks to simplicial complexes
An abstract p-simplex is an ordered set of vertices,
p = v0, v1, v2, … , vp.
e.g. the tetrahedron
A face is a sub-simplex.
e.g. a triangle
A simplicial complex is a set of
simplices with all their faces
v1
v0
v2
v3
3 = v0, v1, v2, v3.
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From networks to simplicial complexes
Every network is a simplicial complex whose simplices have dimension q = 0 or q = 1.
Simplicial complexes are a multidimensional generalisation of networks.
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Gestalt Psychologist Katz:
Vanilla Ice Cream cold + yellow + soft + sweet + vanilla
it is a Gestalt – experienced as a whole
From Networks to Hypernetworks
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
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Gestalt Psychologist Katz:
Vanilla Ice Cream cold + yellow + soft + sweet + vanilla
it is a Gestalt. It is a relational simplex
From Networks to Hypernetworks
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
cold + yellow + soft + sweet + vanilla; RVanilla_Ice_Cream
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Definition
A hypernetwork is a set of relational simplices
From Networks to Hypernetworks
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
cold + yellow + soft + sweet + vanilla; RVanilla_Ice_Cream
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Example: Road Accidents
upset
rain
speed
tired
The accident is a whole
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Example: Road Accidents
upset
rain
speed
tired
The accident is a whole
the individual parts may not cause an accident
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From networks to simplicial complexes
Interesting structures
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
polyhedron representation Euler Polygon Representation
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From networks to simplicial complexes
Interesting structures
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
q-near
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From networks to simplicial complexes
Interesting structures
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
q-near
eccentricity() = | | - | ’|
| |
high eccentricity low eccentricity
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From networks to simplicial complexes
Interesting structures
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
q-near
q-neighbourhood of
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Polyhedral Connectivity
0- near polyhedra
The intersection of two simplices is called their shared face.
They are q-near if their shared face has dimension q
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Polyhedral Connectivity
1- near polyhedra
0- near polyhedra
The intersection of two simplices is called their shared face.
They are q-near if their shared face has dimension q
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Polyhedral Connectivity
0- near polyhedra
The intersection of two simplices is called their shared face.
They are q-near if their shared face has dimension q
1- near polyhedra
(and also 0-near)
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Polyhedral Connectivity
1- near polyhedra
2- near polyhedra
0- near polyhedra
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Polyhedral Connectivity
Polyhedra can be q-connected
through shared faces
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Polyhedral Connectivity
Polyhedra can be q-connected
through shared faces
1-connected components
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Polyhedral Connectivity
Polyhedra can be q-connected
through shared faces
1-connected components
Q-analysis: listing q-components
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Polyhedral Connectivity & q-transmission
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
change on some
part of the
system
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Polyhedral Connectivity & q-transmission
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Polyhedral Connectivity & q-transmission
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Polyhedral Connectivity & q-transmission
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
change is not transmitted
across the low dimensional face
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Polyhedral Connectivity & q-transmission
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Intersections of simplices and dynamics
Shared faces are sites of interaction for pairs of simplices
What about the intersection of more than two simplices?
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Intersections of simplices and dynamics
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Intersections of simplices and dynamics
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Intersections of simplices and dynamics
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Intersections of simplices and dynamics
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Intersections of simplices and dynamics
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Intersections of simplices and dynamics
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Intersections of simplices and dynamics
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Intersections of simplices and dynamics
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Intersections of simplices and dynamics
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Intersections of simplices and dynamics
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Intersections of simplices and dynamics
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Intersections of simplices and dynamics
star
hub
star-hub relationship is a Galois connection
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
relational simplices have rich connectivity structures
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Intersections of simplices and dynamics
star-hub relationship is a Galois connection
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
(a1) (a2) (a3) (a4) (b1) (b2) (b3) (b4) (b5)
(a1)
(a2)
(a3)
(a4)
(b1)(b2)
(b3)
(b4)(b5)
a1
a2
a3
a4
a5
b1
b2
b3
b4
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Intersections of simplices and dynamics
star-hub relationship is a Galois connection
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
. . . . . . . . . . .
. . . 1 1 1 1 1 . . .
. . . 1 1 1 1 1 . . .
. . . 1 1 1 1 1 . . .
. . . 1 1 1 1 1 . . .
. . . . . . . . . . .
…
a1
a2
a3
a4
…
. . . b1 b2 b3 b4 b5 . . .
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e.g. take a set of 3 blocks
Formation of simplices hierarchical structure
{ } 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
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{ }R
R
Formation of simplices hierarchical structure
e.g. take a set of 3 blocks assembled by a 3-ary relation
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{ }R
Formation of simplices hierarchical structure
e.g. take a set of 3 blocks assembled by a 3-ary relation R
The structure has an emergent property
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{ }R
Formation of simplices hierarchical structure
Level N+1
Level N
n-ary relation
assembles elements
into named
structures at a higher
level
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{ }R
Formation of simplices hierarchical structure
Archn-ary relation
assembles elements
into named
structures at a higher
level
R
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AND and OR aggregations in multilevel systems
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Multilevel patterns of numbers on the structure
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Emergentcapability
Capabilities
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4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Multilevel patterns of numbers on the structure
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System dynamics as traffic on a fixed multilevel backcloth
Dynamics on the hypernetwork backcloth
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Dynamics
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System time and System Events
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System time and System Events
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System time and System Events
Planning involves changing relations
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System time and System Events
System dynamics involves changing relations
… trajectories of multidimensional events
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
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Representing Complex Systems & Dynamics
Backcloth, Traffic & Type-1 Dynamics
Traffic on the multi-level backcloth - coherence
Cannot look at just one level
Traffic must aggregate coherently over the structure
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
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4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
E.g. Cooperation – Making a Project
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4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Multilevel Systems
Macro-level “The System”
Meso-levels
Micro-Level “The Atoms”
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4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Multilevel Systems
Macro-level “The Project”
Meso-levels
Micro-Level “The Atoms”
Stuff ?
Lo
ts of In
termed
iate S
tuff
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4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Multilevel Systems
Macro-level “The Project”
Meso-levels
Micro-Level
Lo
ts of In
termed
iate S
tuff
team-A task-B
![Page 64: 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 From Networks to Hypernetworks for a Science of Complex Systems Jeffrey Johnson.](https://reader035.fdocuments.us/reader035/viewer/2022062516/56649e215503460f94b0d4a5/html5/thumbnails/64.jpg)
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Multilevel Systems
Macro-level “The Project”
Meso-levels
Micro-Level
team-A task-B
€ n m
€ 200 k
€ 200 k
XXX
![Page 65: 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 From Networks to Hypernetworks for a Science of Complex Systems Jeffrey Johnson.](https://reader035.fdocuments.us/reader035/viewer/2022062516/56649e215503460f94b0d4a5/html5/thumbnails/65.jpg)
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Multilevel Systems
Macro-level “The Project”
Meso-levels
Micro-Level
team-A task-B
€ n m
€ 200 k
€ 213 k
XXX WP1 WP2 …
€ 30 k
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System time and System Events
… trajectories of multidimensional events
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
We agree to make a project
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System time and System Events
… trajectories of multidimensional events
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
We agree to make a project
The proposal is ready – submit !
![Page 68: 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 From Networks to Hypernetworks for a Science of Complex Systems Jeffrey Johnson.](https://reader035.fdocuments.us/reader035/viewer/2022062516/56649e215503460f94b0d4a5/html5/thumbnails/68.jpg)
System time and System Events
… trajectories of multidimensional events
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
We agree to make a project
The proposal is ready – submit !
The proposal is accepted – hooray !
![Page 69: 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 From Networks to Hypernetworks for a Science of Complex Systems Jeffrey Johnson.](https://reader035.fdocuments.us/reader035/viewer/2022062516/56649e215503460f94b0d4a5/html5/thumbnails/69.jpg)
System time and System Events
… trajectories of multidimensional events
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
We agree to make a project
The proposal is ready – submit !
The proposal is accepted – hooray !
Big research breakthrough !
![Page 70: 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 From Networks to Hypernetworks for a Science of Complex Systems Jeffrey Johnson.](https://reader035.fdocuments.us/reader035/viewer/2022062516/56649e215503460f94b0d4a5/html5/thumbnails/70.jpg)
Conclusions
Relations can be multidimensional
need polyhedral hypernetworks
Multilevel systems
need polyhedral hypernetworks
Dynamics involves structures & numbers
need polyhedral hypernetworks
Hypernetworks are necessary if not sufficient to model complex
systems 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010