  In the present work, we study the properties of biological networks by
applying analogous notions of fundamental concepts in Riemannian geometry and
optimal mass transport to discrete networks described by weighted graphs.
Specifically, we employ possible generalizations of the notion of Ricci
curvature on Riemannian manifold to discrete spaces in order to infer certain
robustness properties of the networks of interest. We compare three possible
discrete notions of Ricci curvature (Olivier Ricci curvature, Bakry-\'Emery
Ricci curvature, and Forman Ricci curvature) on some model and biological
networks. While the exact relationship of each of the three definitions of
curvature to one another is still not known, they do yield similar results on
our biological networks of interest. These notions are initially defined on
positively weighted graphs; however, Forman-Ricci curvature can also be defined
on directed positively weighted networks. We will generalize this notion of
directed Forman Ricci curvature on the network to a form that also considers
the signs of the directions (e. g., activator and repressor in transcription
networks). We call this notion the \emph{signed-control Ricci curvature}. Given
real biological networks are almost always directed and possess positive and
negative controls, the notion of signed-control curvature can elucidate the
network-based study of these complex networks. Finally, we compare the results
of these notions of Ricci curvature on networks to some known network measures,
namely, betweenness centrality and clustering coefficients on graphs.
