Polytopal Graph Complexity, Matrix Permanents, and Embedding

Authors:
Francisco Escolano;Edwin R. Hancock;Miguel A. Lozano
Affiliations:
Departamento de Ciencia de la Computación e Inteligencia Artificial, University of Alicante, Spain;Department of Computer Science, University of York, UK;Departamento de Ciencia de la Computación e Inteligencia Artificial, University of Alicante, Spain
Venue:
SSPR & SPR '08 Proceedings of the 2008 Joint IAPR International Workshop on Structural, Syntactic, and Statistical Pattern Recognition
Year:
2008

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Abstract

In this paper, we show how to quantify graph complexity in terms of the normalized entropies of convex Birkhoff combinations. We commence by demonstrating how the heat kernel of a graph can be decomposed in terms of Birkhoff polytopes. Drawing on the work of Birkhoff and von Neuman, we next show how to characterise the complexity of the heat kernel. Finally, we provide connections with the permanent of a matrix, and in particular those that are doubly stochastic. We also include graph embedding experiments based on polytopal complexity, mainly in the context of Bioinformatics (like the clustering of protein-protein interaction networks) and image-based planar graphs.