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From Networks to Hypernetworks for a Science of Complex Systems. Jeffrey Johnson Open University UK. 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010. Binary relations are not rich enough. 3 binary relations  one 3-ary relation.

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From Networks to Hypernetworks for a Science of Complex Systems

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From networks to hypernetworks for a science of complex systems

From Networks to Hypernetworks for a

Science of Complex Systems

Jeffrey Johnson

Open University

UK

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Binary relations are not rich enough

3 binary relations  one 3-ary relation

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Relational Structure

Binary relation 3-ary relation 4-ary relation

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Relational Structure

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

From networks to simplicial complexes

An abstract p-simplex is an ordered set of vertices,

p =  v0, v1,v2, … , vp.

v0

v2

v1

v3

3=  v0, v1,v2, v3.

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

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

v0

v2

v1

v3

3=  v0, v1,v3.

3=  v0, v1,v2, v3.

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

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

v0

v2

v1

v3

3=  v0, v1,v2, v3.

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

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


From networks to hypernetworks for a science of complex systems

From Networks to Hypernetworks

Gestalt Psychologist Katz:

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

it is a Gestalt – experienced as a whole

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

From Networks to Hypernetworks

Gestalt Psychologist Katz:

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

it is a Gestalt.It is a relational simplex

 cold + yellow + soft + sweet + vanilla; RVanilla_Ice_Cream

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

From Networks to Hypernetworks

Definition

A hypernetwork is a set of relational simplices

 cold + yellow + soft + sweet + vanilla; RVanilla_Ice_Cream

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Example: Road Accidents

The accident is a whole

speed

tired

rain

upset

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Example: Road Accidents

The accident is a whole

the individual parts may not cause an accident

speed

tired

rain

upset

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

From networks to simplicial complexes

Interesting structures

polyhedron representation Euler Polygon Representation

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

From networks to simplicial complexes

Interesting structures

q-near

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

eccentricity() = |  | - |   ’|

|  |

From networks to simplicial complexes

Interesting structures

q-near

high eccentricity low eccentricity

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

From networks to simplicial complexes

Interesting structures

q-near

q-neighbourhood of 

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

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


From networks to hypernetworks for a science of complex systems

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


From networks to hypernetworks for a science of complex systems

Polyhedral Connectivity

1- near polyhedra

(and also 0-near)

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


From networks to hypernetworks for a science of complex systems

Polyhedral Connectivity

1- near polyhedra

0- near polyhedra

2- near polyhedra


From networks to hypernetworks for a science of complex systems

Polyhedral Connectivity

Polyhedra can be q-connected

through shared faces

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

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


From networks to hypernetworks for a science of complex systems

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


From networks to hypernetworks for a science of complex systems

Polyhedral Connectivity & q-transmission

change on some

part of the

system

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Polyhedral Connectivity & q-transmission

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Polyhedral Connectivity & q-transmission

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Polyhedral Connectivity & q-transmission

change is not transmitted

across the low dimensional face

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Polyhedral Connectivity & q-transmission

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

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


From networks to hypernetworks for a science of complex systems

Intersections of simplices and dynamics

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Intersections of simplices and dynamics

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Intersections of simplices and dynamics

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Intersections of simplices and dynamics

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Intersections of simplices and dynamics

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Intersections of simplices and dynamics

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Intersections of simplices and dynamics

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Intersections of simplices and dynamics

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Intersections of simplices and dynamics

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Intersections of simplices and dynamics

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Intersections of simplices and dynamics

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Intersections of simplices and dynamics

star-hub relationship is a Galois connection

hub

star

relational simplices have rich connectivity structures

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Intersections of simplices and dynamics

star-hub relationship is a Galois connection

(b4)

(b5)

(a3)

a5

(a4)

b2

a4

(b3)

b3

b1

a1

a3

a2

(a1)

b4

(b2)

(b1)

(a2)

(a1) (a2) (a3) (a4) (b1) (b2) (b3) (b4) (b5)

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Intersections of simplices and dynamics

star-hub relationship is a Galois connection

. . . b1 b2 b3 b4 b5 . . .

a1

a2

a3

a4

. . . . . . . . . . .

. . . 1 1 1 1 1 . . .

. . . 1 1 1 1 1 . . .

. . . 1 1 1 1 1 . . .

. . . 1 1 1 1 1 . . .

. . . . . . . . . . .

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Formation of simplices  hierarchical structure

e.g. take a set of 3 blocks

{

}

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Formation of simplices  hierarchical structure

e.g. take a set of 3 blocks

assembled by a 3-ary

relation

R

R

{

}

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

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

R

{

}

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Formation of simplices  hierarchical structure

Level N+1

Level N

n-ary relation assembles elements into named structures at a higher level

R

{

}

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Formation of simplices  hierarchical structure

Arch

n-ary relation assembles elements into named structures at a higher level

R

R

{

}

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

AND and OR aggregations in multilevel systems

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Multilevel patterns of numbers on the structure

Emergent

capability

Capabilities

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Multilevel patterns of numbers on the structure

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Dynamics on the hypernetwork backcloth

System dynamics as traffic on a fixed multilevel backcloth

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Dynamics

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

System time and System Events

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

System time and System Events

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

System time and System Events

Planning involves changing relations

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

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


From networks to hypernetworks for a science of complex systems

Representing Complex Systems & Dynamics

Backcloth, Traffic & Type-1 Dynamics

Cannot look at just one level

Traffic must aggregate coherently over the structure

Traffic on the multi-level backcloth - coherence

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

E.g. Cooperation – Making a Project

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Multilevel Systems

Macro-level“The System”

Meso-levels

Micro-Level“The Atoms”

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Multilevel Systems

Macro-level“The Project”

Meso-levels

Micro-Level“The Atoms”

Lots of Intermediate Stuff

Stuff ?

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Multilevel Systems

Macro-level“The Project”

Meso-levels

Micro-Level

Lots of Intermediate Stuff

team-A task-B

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Multilevel Systems

Macro-level“The Project”

Meso-levels

Micro-Level

€ n m

XXX

team-A task-B

€ 200 k

€ 200 k

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

Multilevel Systems

Macro-level“The Project”

Meso-levels

Micro-Level

€ n m

WP1 WP2 …

XXX

€ 200 k

€ 213 k

team-A task-B

€ 30 k

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

System time and System Events

We agree to make a project

… trajectories of multidimensional events

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

System time and System Events

We agree to make a project

The proposal is ready – submit !

… trajectories of multidimensional events

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

System time and System Events

We agree to make a project

The proposal is accepted – hooray !

The proposal is ready – submit !

… trajectories of multidimensional events

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

System time and System Events

We agree to make a project

The proposal is accepted – hooray !

The proposal is ready – submit !

Big research breakthrough !

… trajectories of multidimensional events

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010


From networks to hypernetworks for a science of complex systems

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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