Modeling using Nested Graph An introduction BRIDE Hadrien - ROUSSET Alban Seminex

Outline ● ● ●

Graphs in Computer Science Polygraph : A generalization of directed graph Nested Graphs ○ ○ ○ ○

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Definition Example Pros/Cons Implementation

Usage ○ ○

Fractal PMAS : A Parallel Multi-Agent Simulator Hadara Ad-Simul : A workflow nets analysis framework

Graphs Timeline Apparition of Graph Th.

Shortest paths

Coloring problem

Leonard Euler 1736

Dijkstra 1956

Heinrich Heesch 1969

First book on Gr. Th.

Maximum flow

Finding SCC

Denes König 1936

L. R. Ford, Jr. and D.R. Fulkerson Tarjan 1956 1972

Graphs in Computer Science

● Definition ● Applications field

Graphs: Definition G = (V, E) where: ● ●

V is a set vertices and E is a set of edges.

Graphs are a structure describing relationships between objects. (The vertices denote the objects and the edges represent the relationship).

Graphs: Applications field Graphs are used in different fields such as : ●

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Biology, Sociology, Physics, Chemistry, … Computer science ○ Big data analysis ○ Artificial Intelligence ○ Compilation ○ ...

One if not the most used data structure in computer science. Famous example: World Wide Web

Polygraphs

● A generalization of directed graph

Polygraphs ● ●

Introduced by Albert Burroni Generalize directed graph to higher dimension

0_Cell

1_Cell (0_Cell => 0_Cell)

Polygraphs

1_Path (succession of 1_Cells)

2_Cell (1_Path => 1_Path)

Polygraphs

2_Path (succession of 2_Cells)

Polygraphs Polygraph can be used to model complex systems. ● ●

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0-Polygraph: A set of 0_Cell (i.e. a set of points) 1-Polygraph: A set of 1_Cell (i.e. a directed graph) ○ “State” of the modelled system 2-Polygraph: A set of 2_Cell (i.e. a set of graph transformations) ○ “Behaviours” of the modelled system 3-Polygraph: A set of 3_Cell (i.e. a set of graph transformations transformations) ○ “Reconfiguration” of the modelled system

Nested Graphs

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Definition Example Pros/Cons Implementation

Nested Graphs : Definition G’ ∈ G ⇔ G = (G, A, T, L, V) where: ● ● ● ● ●

G’ ⊆ G is a set of nodes A ⊆ G × G is a set of directed arcs T : G → Γ is a typing function assigning a type to each nodes L : G → Σ is a labelling function assigning a label to each nodes V : A → Λ is a weight function assigning a value to any arc

Generalizing Nested Graphs: defining 0_Cell as Nested Graphs

Poly-Nested Graphs : Example Prey-Predator Model: ● ●

Sheep/Wolfs move inside 2D environment Each Sh.W. have vital “Force” (mod. arcs)

Poly-Nested Graphs : Example Reproduce behaviour of a Sheep: ● ●

A sheep can be duplicated by itself It produce a new Sheep with an Init. Force

Nested Graphs : Pros/Cons Pros: ● ● ● ●

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Hierarchical structure Multi-level Abstraction Graphic representation Efficiency ○ storage ○ manipulation Very expressive

Cons: ●

Might not be intuitive for everyone

Nested Graphs : Implementation ● ●

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A C++ Framework Available with : ○ vector ○ set ○ hashmap Basic functions: ○ Add/Del node(s) ○ Add/Del arc(s) ○ Get/Set neighbour(s) ○ Import/Export from/to XML Basic algorithms: ○ BFS ○ RBFS

Nested Graphs Framework is only based on the Standard library

cSub2->AddNode(cSub2Sub1);

Nested Graphs : Example Code: //The lower CGraph CGraph CGraph

level (0) nodes * cSub2Sub1Sub1 * cSub2Sub1Sub2 * cSub2Sub1Sub3

of the fractal graph on the right = new CGraph("C1"); = new CGraph("C2"); = new CGraph("C3");

//The nodes of level (1) of the fractal graph on the right CGraph * cSub2Sub1 = new CGraph("C"); cSub2Sub1->AddNode(cSub2Sub1Sub1); cSub2Sub1->AddNode(cSub2Sub1Sub2); cSub2Sub1->AddNode(cSub2Sub1Sub3); CArc * a3 = new CArc("a3",cSub2Sub1Sub1, cSub2Sub1Sub2,6); CArc * a4 = new CArc("a4",cSub2Sub1Sub2, cSub2Sub1Sub3,4); CArc * a5 = new CArc("a5",cSub2Sub1Sub3, cSub2Sub1Sub1,9); cSub2Sub1->AddArc(a3); cSub2Sub1->AddArc(a4); cSub2Sub1->AddArc(a5); //The left fractal graph of level (2) CGraph * cSub2 = new CGraph("Sub2");

//The main graph level (3) CGraph * cGraph = new CGraph("MainGraph"); cGraph->AddNode(cSub1); cGraph->AddNode(cSub2); CArc * a6 = new CArc("a6",cSub1,cSub2,1); CArc * a7 = new CArc("a7",cSub2,cSub1,2); cGraph->AddArc(a6); cGraph->AddArc(a7); //Export as dot cGraph->SaveAsDot("graph.dot");

You can also contribute!

Information Implementation available on GitHub under GNU licence : ●

https://github.com/LoW12/FractalGraph

Usage

● Fractal PMAS ● Hadara Ad-Simul

Fractal PMAS ●

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Use of CGraph at 2 levels: ○ Modelisation of agents ○ Representation of Distribution and Load Balancing CGraph is coupled with a parallel graph partitioning framework (Zoltan) Making large scale simulations

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Fractal PMAS

Fractal PMAS ●

Simulation with: ○ more than 600 000 nodes

Work published in PDP’2016

Hadara Ad-Simul ●

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Implementation Based on CGraph ○ Workflow nets are Graphs ○ Support hierarchical workflows Generate workflow nets (large scale) ○ Benchmarking Reduce workflow nets (large scale) ○ Semi-decide Generalised Soundness

Thanks for your attention Any questions ?