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The importance of studying properties of networks is manifest in diversefields ranging from biology, engineering, physics, chemistry, neuroscience, andmedicine. The functionality of networks with regard to performance, throughput,reliability and robustness is strongly linked to the underlying geometric andtopological properties of the network and this is the focus of this paper,especially as applied to certain biological networks. The fundamentalmathematical abstraction of a network as a weighted graph brings to bear thetools of graph theory--a highly developed subject of mathematical research. Butmore importantly, recently proposed geometric notions of curvature on verygeneral metric measure spaces allow us to utilize a whole new set of tools andideas that help quantify functionality and robustness of graphs. In particular,robustness is closely connected to network entropy which, in turn, is veryclosely related to curvature. We will see that there are a number ofalternative notions of discrete curvature that are compatible with theclassical Riemannian definition, each having its own advantages anddisadvantages, and are relevant to networks of interest. We will concentrate onthe role of curvature for certain key cancer networks in order toquantitatively indicate their apparent functional robustness relative to theirnormal counterparts.

Graph Curvature and the Robustness of Cancer Networks