4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 From Networks to Hypernetworks for a Science of Complex Systems Jeffrey Johnson.
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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
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
Relational Structure
Binary relation 3-ary relation 4-ary relation
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Relational Structure
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
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
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.
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
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.
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
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
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
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
Example: Road Accidents
upset
rain
speed
tired
The accident is a whole
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Example: Road Accidents
upset
rain
speed
tired
The accident is a whole
the individual parts may not cause an accident
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
From networks to simplicial complexes
Interesting structures
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
polyhedron representation Euler Polygon Representation
From networks to simplicial complexes
Interesting structures
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
q-near
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
From networks to simplicial complexes
Interesting structures
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
q-near
q-neighbourhood of
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
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
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
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
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)
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Polyhedral Connectivity
1- near polyhedra
2- near polyhedra
0- near polyhedra
Polyhedral Connectivity
Polyhedra can be q-connected
through shared faces
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Polyhedral Connectivity
Polyhedra can be q-connected
through shared faces
1-connected components
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Polyhedral Connectivity
Polyhedra can be q-connected
through shared faces
1-connected components
Q-analysis: listing q-components
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Polyhedral Connectivity & q-transmission
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
change on some
part of the
system
Polyhedral Connectivity & q-transmission
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Polyhedral Connectivity & q-transmission
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
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
Polyhedral Connectivity & q-transmission
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Intersections of simplices and dynamics
Shared faces are sites of interaction for pairs of simplices
What about the intersection of more than two simplices?
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Intersections of simplices and dynamics
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Intersections of simplices and dynamics
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Intersections of simplices and dynamics
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Intersections of simplices and dynamics
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Intersections of simplices and dynamics
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Intersections of simplices and dynamics
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Intersections of simplices and dynamics
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Intersections of simplices and dynamics
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Intersections of simplices and dynamics
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Intersections of simplices and dynamics
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Intersections of simplices and dynamics
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
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
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
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 . . .
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
{ }R
R
Formation of simplices hierarchical structure
e.g. take a set of 3 blocks assembled by a 3-ary relation
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
{ }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
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
{ }R
Formation of simplices hierarchical structure
Level N+1
Level N
n-ary relation
assembles elements
into named
structures at a higher
level
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
{ }R
Formation of simplices hierarchical structure
Archn-ary relation
assembles elements
into named
structures at a higher
level
R
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
AND and OR aggregations in multilevel systems
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Multilevel patterns of numbers on the structure
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Emergentcapability
Capabilities
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Multilevel patterns of numbers on the structure
System dynamics as traffic on a fixed multilevel backcloth
Dynamics on the hypernetwork backcloth
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Dynamics
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
System time and System Events
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
System time and System Events
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
System time and System Events
Planning involves changing relations
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
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
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
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
E.g. Cooperation – Making a Project
4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
Multilevel Systems
Macro-level “The System”
Meso-levels
Micro-Level “The Atoms”
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
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
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
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
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
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 !
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 !
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 !
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
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