PMID: 22587160May 17, 2012Paper

Heat diffusion: thermodynamic depth complexity of networks

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
Francisco EscolanoMiguel A Lozano

Abstract

In this paper we use the Birkhoff-von Neumann decomposition of the diffusion kernel to compute a polytopal measure of graph complexity. We decompose the diffusion kernel into a series of weighted Birkhoff combinations and compute the entropy associated with the weighting proportions (polytopal complexity). The maximum entropy Birkhoff combination can be expressed in terms of matrix permanents. This allows us to introduce a phase-transition principle that links our definition of polytopal complexity to the heat flowing through the network at a given diffusion time. The result is an efficiently computed complexity measure, which we refer to as flow complexity. Moreover, the flow complexity measure allows us to analyze graphs and networks in terms of the thermodynamic depth. We compare our method with three alternative methods described in the literature (Estrada's heterogeneity index, the Laplacian energy, and the von Neumann entropy). Our study is based on 217 protein-protein interaction (PPI) networks including histidine kinases from several species of bacteria. We find a correlation between structural complexity and phylogeny (more evolved species have statistically more complex PPIs). Although our methods outperform the alter...Continue Reading

References

Mar 24, 2005·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Benjamin Machta, Jonathan Machta
Mar 4, 2006·Science·Francesca D CiccarelliPeer Bork
May 27, 2006·IEEE Transactions on Pattern Analysis and Machine Intelligence·Andrea Torsello, Edwin R Hancock
May 10, 2008·Genome Biology·Jia Qian Wu, Michael Snyder
Oct 23, 2008·Nucleic Acids Research·Lars J JensenChristian von Mering
Jan 15, 2011·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Ernesto Estrada

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Citations

Oct 30, 2014·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Cheng YeEdwin R Hancock
Mar 15, 2015·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Luca RossiEdwin R Hancock
Oct 16, 2015·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Cheng YeEdwin R Hancock

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