New method to optimize Force-directed layouts of large graphs Meva DODO, Fenohery ANDRIAMANAMPISOA, Patrice TORGUET, Jean Pierre JESSEL IRIT University of Toulouse WSCG 2008

Introduction Category of graph drawing algorithms : Hierarchical algorithms that distribute nodes according to their hierarchy Algorithms based on physical models Main idea : Optimizing the layout by equitably distributing vertices in space Using a numerical optimization to compute the final position of vertices

Problem Different strategies to draw general undirected graph [Ead84a]: Planar embedding algorithm Using an Eulerian path and use a directed graph drawing algorithm Force-directed algorithms Spring Model: Most popular force-directed algorithm, originally developed by Eades[Ead84a], improved by Kamada [Kam89a] Vertices are initially assigned random position and a system of differential equations is solved to find the state with the minimum energy

Problem Total energy of the system : : the natural length between p i and p j : the strength of the spring between p i and p j Two methods to minimize E:  Using a simulation of repulsive and attractive forces  Resolving a system of differential equations

Solution Methods combined with Force-directed algorithms : Multilevel technique [Wal03a]: coalesces clusters of vertices to define a new graph and recursively iterates this procedure to create a hierarchy of increasingly coarse graphs: MIS (Maximal Independent Set) [Gaj04a]: to create a filtration of vertices of the given graph: Multi-grid technique of [Fru91a] that allows a portion of graph to be moved [Dav96a], [Har02a], …

Our approach 1 - Vertices distribution approach Aesthetic criteria : An equitable distribution of vertices No collision between vertices Minimization of unused space S0S0 S 01 S 02 S 03 Node2 Node1 Node3 Area Initial sphere sub-sphere

Our approach 2 – Attractive forces, repulsive forces Force applied to a vertex : Attractive force Repulsive force Final state of the system is found when :

Our approach 3 – Total energy According to Kamada & Kawai:

Our approach 3 – Total energy P i (x i, y i, z i ) the initial position of vertex i, We need to find P’ i (x’ i, y’ i, z’ i ) the final position of vertex i which minimizes E, Ti(tx i, ty i, tz i ) the translation of P i, x’ i =x i +tx i, y’ i =y i +ty i, z’ i =z i +tz

Our approach 4 – Minimizing total energy, conjugated gradient[Fle64a] Final state of the system: Where X i = x i +tx i, X j = x j +tx j, Y i = y i +ty i, Y j = y j +ty j, Z i = z i +tz i, Z j = z j +tz j

Algorithm Initialisation : Assign random placement of vertices (x i, y i, z i ), initialize the translations to zero (tx i, ty i, tz i ) Compute the gradients of E for all (tx i,ty i, tz i ) and compute Δ i Find the vertex m with Δ m ≥ Δ i Δ m > ε Compute the optimal translation for the vertex m through the conjugated gradient Compute the gradients of E for all (tx i, ty i, tz i ) and compute Δ i with the new translation value of vertex m Find the vertex m with Δ m ≥ Δ i, i=1...n i=1...n True En d False i=1...n

Application and Results 3D Simulation of large graphs: trees, undirected graph 3D Visualization of network computing system, data structures, databases 15 vertices110 vertices (0.30s) 660 vertices (2.0s) 1051 vertices (2.0s)

Result of the computation for the first hierarchy Result of the computation for the second hierarchy. Each vertex in the first hierarchy has 15 vertices children Result of the computation for the third hierarchy. Each vertex in the first hierarchy has 15 vertices children Application and Results

Conclusion and Future work  Minimization of edges crossing  No collision between vertices  Minimization of unused space  Efficient energy function  Introducing the notion of vertex weight to subdivide the space  Improving the numerical optimization for drawing hypergraphs

End Questions ?