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Hypernetworks in systems of systems of systems Jeffrey Johnson The Open University, UK TOPDRIM - Topology-driven methods in CS NESS - Non-Equilibrium Social Science GSDP – Global Systems Dynamics & Policy Étoile – Enhanced Technology for Open Intelligent Learning Environments

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We have no formalism to combine the micro-, meso- and macro-level dynamics of systems in any field. Level N+h Level N+h-1 Level N+h-2 … Level N+2 Level N+1 Level N Societies …. Individual animals … cells …. Proteins, fluids Biological systems have many identifiable levels

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We have no formalism to combine the micro-, meso- and macro-level dynamics of systems in any field. Nations, Regions, Cities, … neighbourhoods Houses, shops, roads Rooms, gardens,.. Built environment has many identifiable levels Level N+h Level N+h-1 Level N+h-2 … Level N+2 Level N+1 Level N

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As scientists we need to: (1) prove that such a formalism can’t exist (2) … or construct that formalism We have no formalism to combine the micro-, meso- and macro-level dynamics of systems in any field. Hypernetwork theory is an attempt to do this

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From networks to hypernetworks

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From hypergraphs & Galois pairs to hypernetworks

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Hypergraphs are set theoretic & not rich enough Same set of parts but arranged differently The vertices need to be ordered to discriminate them

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Gestalt Psychologist Katz: V anilla I ce C ream c old + y ellow + soft + s weet + v anilla it is a Gestalt – experienced as a whole cold, yellow, soft, sweet, vanilla From Networks to Hypernetworks

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Set of vertices simplex clique relational simplex cold, yellow, soft, sweet, vanilla

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Simplices represent wholes … remove a vertex and the whole ceases to exist.

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A set of simplices with all its faces is called a simplicial complex Simplices have multidimensional faces Multidimensional Connectivity

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Simplices have multidimensional connectivity through their faces Share a vertex 0 - near Share an edge 1 - near Share a triangle 2 - near A network is a 1-dimensional simplicial complex with some 1-dimensional simplices (edges) connected through their 0-dimensional simplices (vertices) Multidimensional Connectivity

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From networks to simplicial complexes Interesting structures q-near

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From networks to simplicial complexes Interesting structures q-near is q-connected to ’ ’

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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 Q-analysis: listing q-components

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Polyhedral Connectivity & q-transmission change on some part of the system (q-percolation)

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Polyhedral Connectivity & q-transmission

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change is not transmitted across the low dimensional face

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Intersections of simplices and dynamics star-hub relationship is a Galois connection (a 1 ) (a 2 ) (a 3 ) (a 4 ) (b 1 ) (b 2 ) (b 3 ) (b 4 ) (b 5 ) (a 1 ) (a 2 ) (a 3 ) (a 4 ) (b 1 ) (b 2 ) (b 3 ) (b 4 ) (b 5 ) a1a1 a2a2 a3a3 a4a4 a5a5 b1b1 b2b2 b3b3 b4b4

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star-hub relationship is a Galois connection …a1a2a3a4……a1a2a3a4…... b 1 b 2 b 3 b 4 b 5...

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star-hub relationship is a Galois connection …a1a2a3a4……a1a2a3a4…... b 1 b 2 b 3 b 4 b 5... q-connect components ! loser clusters of simplices

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From Complexes to Hypernetworks Simplices are not rich enough to discriminate things Same parts, different relation, different structure & emergence We must have relational simplices

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Definition A hypernetwork is a set of relational simplices cold + yellow + soft + sweet + vanilla; R Vanilla_Ice_Cream e.g.

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Relational Simplices and Multilevel Systems

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Multilevel Systems Can highly entangled multilevel systems separated into well-defined levels ?

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Multilevel Systems The Intermediate Word Problem Hierarchical Soup

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Multilevel Systems

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e.g. take a set of 3 blocks Formation of simplices hierarchical structure {}

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{} R R 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 Arch n-ary relation assembles elements into named structures at a higher level R

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AND and OR aggregations in multilevel systems Sets, classes Structures Sets of parts Conventional classification trees don’t have alpha aggregations

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Observing multilevel systems of systems of systems Hypothesis 1 When we look at systems we see the whole & the parts Hypothesis 2 Our brains create new multilevel structures

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Aggregation – deconstruction downward dynamics in representing systems Level N+1 Level N

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Level N+1 Level N Create a new object at Level N ! Aggregation – deconstruction downward dynamics in representing systems

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Level N+1 Level N Create a new level - Level N-1 ! And new objects at this level Aggregation – deconstruction downward dynamics in representing systems

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Level N+1 Level N Create a new object at Level N Aggregation – deconstruction downward dynamics in representing systems

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Level N+1 Level N Create new objects at Level N+1 Aggregation – deconstruction downward dynamics in representing systems

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Mereology Parts and wholes goes back millennia to Plato and Aristotle. mereology was coined in 1927 by Stanislaw Lesniewski A mereological system - Objects, X, and a binary relation parthood, ‘x is a part of y’.

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1.8 Mereology Paying is part of shopping

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Multilevel patterns of numbers on the structure costs Wages etc Traffic Backcloth

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Multilevel patterns of numbers on the structure Traffic Backcloth Costs Wages etc Profits Costs Income taxes

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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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Backcloth dynamics: System time and System Events

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System dynamics involves changing relations … trajectories of multidimensional events Backcloth dynamics: System time and System Events

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Policy predictions as fans of connected multilevel multidimensional trajectories

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Policy as designing the future is entangled with complexity science and design Policy Science Design Policy is designing the future

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Design is an Intermediate Word Problem What are the intermediate structures ? What shall we call them ? Hierarchical Soup Policy is designing the future

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Policy involves creating artificial systems Creating artificial systems involves Design Design a co-evolution between what you think you want & what you think you can have Policy is designing the future

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“The System” – an abstraction it doesn’t exist (yet) “The parts” in the soup concrete stuff - components Top-down – imagine structures Bottom-up – instantiate structures The soup The System Design: hypothesing intermediate structures, Top-down and bottom-up

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A fully instantiated design is a blueprint – the system can now be constructed Design: hypothesing intermediate structures, Top-down and bottom-up

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Hypernetworks, Global System Science & Policy Consider a policy issue P, e.g. land use and transportation planning P What is relevant to the dynamics of P ?

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Consider a policy issue P, e.g. international emergency P What Are The Intermediate Words? Hypernetworks, Global System Science & Policy

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P What Are The Intermediate Words? Hypernetworks, Global System Science & Policy Consider a policy issue P, e.g. international emergency

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P What Are The Intermediate Words? What are the relations? Hypernetworks, Global System Science & Policy Consider a policy issue P, e.g. international emergency

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P What Are The Intermediate Words? What are the relations? What are the (hyper) networks? How do they work together? Hypernetworks, Global System Science & Policy Consider a policy issue P, e.g. international emergency

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Conclusions (1) Hypernetworks generalise networks and hypergraphs (2) Hypernetworks have new kinds of connectivity structure (3) Hypernetworks give a way to represent multilevel systems (4) Policy is deigning the future (5) We build the science we need to answer (policy) questions (6) Hypernetworks - necessary if not sufficient for ML systems (pre-print of forthcoming book) Hypernetworks in Systems of Systems of Systems

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