Polytopal Graph Complexity, Matrix Permanents, and Embedding

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.