Time-changing Data Structures Antoine Spicher - PowerPoint PPT Presentation

Modeling Complex Systems in Time-changing Data Structures Antoine Spicher www.spatial-computing.org/mgs AgreenSkills Reasearch School, Météopole, Toulouse October 2014

Outline  Modeling Morphogenesis  Our Approach  Current & Future Work AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

(Dynamical) System  “ Object under study taken from the rest of the world ”  Frontier between inside and outside  Dynamical => specific interest in its evolution Zebra fish embryogenesis Bouncing ball AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Model  “ Abstraction of a system ” state evolution  Description relying on a formalism time (generally mathematical) language  Requires 3 elements  Representation of the state (observables) Position: 𝐪 = (𝑞 𝑦 , 𝑞 𝑧 ) ∈ ℝ 2  Velocity: 𝐰 = (𝑤 𝑦 , 𝑤 𝑧 ) ∈ ℝ 2   Representation of time 𝑢 ∈ ℝ + ⇒ 𝑞 𝑦 𝑢 , 𝑞 𝑧 𝑢 , 𝑤 𝑦 𝑢 , 𝑤 𝑧 𝑢  Evolution function specification 𝜖 2 𝐪 𝜖𝑢 2 = 𝐺ext 𝑛  Newton’s motion law 𝑢 + 𝐺imp. 𝑒𝑢 = 𝑛(𝐰 + − 𝐰 − )  Newton’s impact equations 𝑢 − AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Simulation  “ Production of trajectories w.r.t. a model ” state evolution time H H H* ℕ state t-1 état t state t state t+1 ℝ state t-dt state t state t+dt  H ( t )d t AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Complex Systems  “ ( Dyn.) Systems composed of a population of entities ”  Some properties  Local interactions  Emergent phenomena  Non-linearity, feedback loops, etc. AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Modeling of Complex Systems state evolution time  Spatial Representation  Without space AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Modeling of Complex Systems state evolution time  Spatial Representation  Without space  With space Discrete vs. continuous  Absolute vs. relative  (Newton vs. Leibniz) AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Modeling of Complex Systems state evolution time  Spatial Representation AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Modeling of Complex Systems x => . AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Modeling of Complex Systems x => . AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Modeling of Complex Systems state evolution time  Time Representation  Without time AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Modeling of Complex Systems state evolution time  Time Representation  Without time  With time Discrete vs. continuous  Absolute vs. relative  (Newton vs. Leibniz) AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Modeling of Complex Systems state evolution time  Evolution Function  Several events at different places  About synchrony Synchronous  All events in parallel  Sequential One event at a time Asynchronous  At least one event AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Dynamical Structure  What about morphogenesis?  Bouncing ball At any time, the state is defined by exactly two vectors (position & speed)  Developing embryo At a given time, the state is defined  A variable number of cells (geometry, concentration, …)  A variable organization (division, migration, apoptosis, …) VS . Static structure Dynamic structure AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Dynamical Structure  Examples of (DS)²  In biology Molecular bio., developmental bio.   In physics P. Prusinkiewicz  Soft matter mechanics, D. Goodsell multi-scale systems General relativity   In SHS Urbanism,  traffic control  Economics  In computer science Internet, social network  Reconfigurable robots  M. Satoshi AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Dynamical Structure  The bootstrap issue Dynamics OF the shape Dynamics ON the shape The state as well as the structure of the state is changing in time Formally: The structure of the state is needed to specify the evol. fun. but, the evol. fun. computes the structure of the state AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Computer Sc. and Morphogenesis A. M. Turing (1912-1954) AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Computer Sc. and Morphogenesis A. M. Turing (1912-1954) AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Computer Sc. and Morphogenesis A. M. Turing (1912-1954) AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Computer Sc. and Morphogenesis  What’s up nowadays? P. Prusinkiewicz , 2003. Diffusion and reaction in a deformable surface. Based on E. Coen’s expanding canvas metaphor. Spring-mass system. No topological change. AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Computer Sc. and Morphogenesis  Patterning vs . development Modeling and simulation tools Usual tools (patterning) C : continuous, Coupled Iteration of Cellular MGS PDE D : discrete ODE functions automata C C C D D State C C D D D Time C D D D D Space Generative formalisms n D combinatorial Topology : Multiset Sequence Uniform Arbitrary graph structures Map L systems Membrane MGS Formalism: L systems GBF systems Graph grammars AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Outline  Modeling Morphogenesis  Our Approach  Current & Future Work AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Topology of Interactions  Let’s observe the system  State of the system given by observation  Structure is dynamic ⇒ structure is an observable System described by a state Environment (determined by observation) characterized by its effects on the system AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Topology of Interactions A system in some state Part of a system that evolves Can be identified by comparison with the previous global state AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Topology of Interactions t = 1 AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Topology of Interactions t = 2 AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Topology of Interactions t = 3 AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Topology of Interactions  Decompose a system in parts following the interactions AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Topology of Interactions  Decompose a system in parts following the interactions The interactions decomposes the systems into elementary parts An interaction implies one or several elementary parts AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Topology of Interactions The inclusion structure between the elementary and interacting parts is a lattice A (simplicial) complex is a (topological) equivalent representation AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Topology of Interactions AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures

Topology of Interactions  Interaction based modeling/simulation  Interactions in dynamical system 𝑡(𝑢) : state of the system at time 𝑢  𝑢 : i th sub-system where an interaction occurs at time 𝑢 𝑇 𝑗   The successive partitions give rise to a topology on 𝑇  Basic elements in interaction: points  Spatial organization of the interactions: topology of interactions  Different kinds of interaction: local evolution laws AgreenSkills Research School 2014 - A. Spicher - Modeling Complex Systems in Time-changing Data Structures