This presentation, delivered by Jeffrey Johnson from the Open University at the 4th China-Europe Summer School on Complexity Science in Shanghai (August 2010), explores the evolution from traditional networks to hypernetworks. It emphasizes the limitations of binary relations and introduces the concept of relational simplices. By employing examples such as road accidents and reflecting on polyhedral connectivity, the talk illustrates how hypernetworks can represent complex interactions within systems. This approach provides a richer framework for understanding the intricate structures of complex systems.
01 / 70From Networks to Hypernetworks for a Science of Complex Systems
1 / 17
Télécharger la présentation
From Networks to Hypernetworks for a Science of Complex Systems
Une image / un lien ci-dessous est fourni (tel quel) pour télécharger la présentation
Politique de téléchargement : le contenu du site vous est fourni TEL QUEL pour votre information et votre usage personnel ; il ne peut être vendu, concédé sous licence ni partagé sur d’autres sites sans l’accord de son auteur.Le contenu vous est fourni TEL QUEL, pour votre information et votre usage personnel uniquement.Téléchargez la présentation en cliquant sur ce lien.Lors du téléchargement, si vous ne parvenez pas à télécharger une présentation, l’éditeur a peut-être supprimé le fichier de son serveur.Pendant le téléchargement, si vous ne parvenez pas à obtenir une présentation, le fichier a peut-être été supprimé par l’éditeur.
F I N
Plus de contenu associé
Transcription de la présentation
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 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
Relational Structure 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. v0 v2 v1 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 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 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 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 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 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 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
Example: Road Accidents The accident is a whole speed tired rain upset 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
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 simplicial complexes Interesting structures polyhedron representation Euler Polygon Representation 4th China-Europe Summer School on Complexity Science. Shanghai 12-Aug-2010
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 simplicial complexes Interesting structures q-near q-neighbourhood of 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 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 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
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 change is not transmitted across the low dimensional face 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 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
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
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
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
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
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
