4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 From Networks to Hypernetworks for a Science of Complex Systems Jeffrey Johnson.

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


Relational Structure

Binary relation 3-ary relation 4-ary relation



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.




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.




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.



Every network is a simplicial complex whose simplices have dimension q = 0 or q = 1.

Simplicial complexes are a multidimensional generalisation of networks.


Gestalt Psychologist Katz:

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

it is a Gestalt – experienced as a whole

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

Definition

A hypernetwork is a set of relational simplices



cold + yellow + soft + sweet + vanilla; RVanilla_Ice_Cream

Example: Road Accidents

upset

rain

speed

tired

The accident is a whole


Example: Road Accidents

upset

rain

speed

tired

The accident is a whole

the individual parts may not cause an accident



Interesting structures


polyhedron representation Euler Polygon Representation




q-near




q-near

eccentricity() = | | - | ’|

| |

high eccentricity low eccentricity




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



1- near polyhedra

0- near polyhedra





0- near polyhedra



1- near polyhedra

(and also 0-near)



1- near polyhedra

2- near polyhedra

0- near polyhedra


Polyhedra can be q-connected

through shared faces





1-connected components





1-connected components

Q-analysis: listing q-components


Polyhedral Connectivity & q-transmission


change on some

part of the

system







change is not transmitted

across the low dimensional face



Intersections of simplices and dynamics

Shared faces are sites of interaction for pairs of simplices

What about the intersection of more than two simplices?

























star

hub

star-hub relationship is a Galois connection


relational simplices have rich connectivity structures




(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




. . . . . . . . . . .

. . . 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


e.g. take a set of 3 blocks assembled by a 3-ary relation


{ }R


e.g. take a set of 3 blocks assembled by a 3-ary relation R

The structure has an emergent property


{ }R


Level N+1

Level N

n-ary relation

assembles elements

into named

structures at a higher

level


{ }R


Archn-ary relation

assembles elements

into named

structures at a higher

level

R


AND and OR aggregations in multilevel systems


Multilevel patterns of numbers on the structure


Emergentcapability

Capabilities


Multilevel patterns of numbers on the structure

System dynamics as traffic on a fixed multilevel backcloth

Dynamics on the hypernetwork backcloth


Dynamics


System time and System Events





Planning involves changing relations



System dynamics involves changing relations

… trajectories of multidimensional events


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



E.g. Cooperation – Making a Project


Multilevel Systems

Macro-level “The System”

Meso-levels

Micro-Level “The Atoms”


Multilevel Systems

Macro-level “The Project”

Meso-levels

Micro-Level “The Atoms”

Stuff ?

Lo

ts of In

termed

iate S

tuff


Multilevel Systems


Meso-levels

Micro-Level

Lo

ts of In

termed

iate S

tuff

team-A task-B


Multilevel Systems


Meso-levels

Micro-Level

team-A task-B

€ n m

€ 200 k

€ 200 k

XXX


Multilevel Systems


Meso-levels

Micro-Level

team-A task-B

€ n m

€ 200 k

€ 213 k

XXX WP1 WP2 …

€ 30 k




We agree to make a project





The proposal is ready – submit !






The proposal is accepted – hooray !






The proposal is accepted – hooray !

Big research breakthrough !

Conclusions

Relations can be multidimensional

need polyhedral hypernetworks

Multilevel systems


Dynamics involves structures & numbers


Hypernetworks are necessary if not sufficient to model complex

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

4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010 From Networks to Hypernetworks for a Science of Complex Systems Jeffrey Johnson.

Documents

complexity science

th chinaeurope summer

cream slide

relational structure

abstract psimplex

relational simplex

set of simplices

set of relational simplices