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Graph Curvature and the Robustness of Cancer Networks

The importance of studying properties of networks is manifest in diverse fields ranging from biology, engineering, physics, chemistry, neuroscience, and medicine. The functionality of networks with regard to performance, throughput, reliability and robustness is strongly linked to the underlying geometric and topological properties of the network and this is the focus of this paper, especially as applied to certain biological networks. The fundamental mathematical abstraction of a network as a weighted graph brings to bear the tools of graph theory--a highly developed subject of mathematical research. But more importantly, recently proposed geometric notions of curvature on very general metric measure spaces allow us to utilize a whole new set of tools and ideas that help quantify functionality and robustness of graphs. In particular, robustness is closely connected to network entropy which, in turn, is very closely related to curvature. We will see that there are a number of alternative notions of discrete curvature that are compatible with the classical Riemannian definition, each having its own advantages and disadvantages, and are relevant to networks of interest. We will concentrate on the role of curvature for certain key cancer networks in order to quantitatively indicate their apparent functional robustness relative to their normal counterparts.

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