Evolving spanning trees using the heat equation

Authors:
Fan Zhang;Huaijun Qiu;Edwin R. Hancock
Affiliations:
Department of Computer Science, University of York, York, UK;Department of Computer Science, University of York, York, UK;Department of Computer Science, University of York, York, UK
Venue:
CAIP'05 Proceedings of the 11th international conference on Computer Analysis of Images and Patterns
Year:
2005

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Abstract

This paper explores how to use the heat kernel to evolve the minimum spanning tree of a graph with time. We use the heat kernel to weight the edges of the graph, and these weights can be computed by exponentiating the Laplacian eigensystem of the graph with time. The resulting spanning trees exhibit an interesting behaviour as time increases. Initially, they are bushy and rooted near the centre of graph, but as time evolves they become string-like and hug the boundary of the graph. We characterise this behaviour using the distribution of terminal nodes with time, and use this distribution for the purposes of graph clustering and image segmentation.