Cellular interactions can be modeled as complex dynamical systems represented by weighted graphs. The functionality of such networks, including measures of robustness, reliability, performance, and efficiency, are intrinsically tied to the topology and geometry of the underlying graph. Utilizing recently proposed geometric notions of curvature on weighted graphs, we investigate the features of gene co-expression networks derived from large-scale genomic studies of cancer. We find that the curvature of these networks reliably distinguishes between cancer and normal samples, with cancer networks exhibiting higher curvature than their normal counterparts. We establish a quantitative relationship between our findings and prior investigations of network entropy. Furthermore, we demonstrate how our approach yields additional, non-trivial pair-wise (i.e. gene-gene) interactions which may be disrupted in cancer samples. The mathematical formulation of our approach yields an exact solution to calculating pair-wise changes in curvature which was computationally infeasible using prior methods. As such, our findings lay the foundation for an analytical approach to studying complex biological networks.
Bibliographical noteFunding Information:
This project was supported by in part by grants from the National Center for Research Resources (P41-RR-013218) and the National Institute of Biomedical Imaging and Bioengineering (P41-EB-015902) of the National Institutes of Health. This work was also supported by NIH grants R01 MH82918 and 1U24CA18092401A1 as well as AFOSR grants FA9550-12-1-0319 and FA9550-15-1-0045. We would also like to thank Chris Sander for very helpful and useful discussions regarding this work.