# A Multilevel Algorithm for Force-Directed Graph-Drawing

**C. Walshaw**

### Abstract:

We describe a heuristic method for drawing graphs which uses a multilevel framework combined with a force-directed placement algorithm.
The multilevel technique matches and coalesces pairs of adjacent vertices to define a new graph and is repeated recursively to create a hierarchy of increasingly coarse graphs, .
The coarsest graph, , is then given an initial layout and the layout is refined and extended to all the graphs starting with the coarsest and ending with the original.
At each successive change of level, *l*, the initial layout for is taken from its coarser and smaller child graph, , and refined using force-directed placement.
In this way the multilevel framework both accelerates and appears to give a more global quality to the drawing.
The algorithm can compute both 2 & 3 dimensional layouts and we demonstrate it on examples ranging in size from 10 to 225,000 vertices.
It is also very fast and can compute a 2D layout of a sparse graph in around 12 seconds for a 10,000 vertex graph to around 5-7 minutes for the largest graphs.
This is an order of magnitude faster than recent implementations of force-directed placement algorithms.

**Keywords:**
Graph-drawing, multilevel optimisation, force-directed placement.

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Fri Aug 13 13:39:45 BST 2004