Abstract. We develop a discrete version of the classical Finsler-Haantjes met-
ric curvature. This allows us to define discrete analogues of geodesic, sectional,
mean and Gauss curvatures for weighted graphs. Furthermore, we endow each
weighted graph with an intrinsic metric induced by the weights, yielding it as
a geodesic metric space.

These methods combined allow us to approach a variety of problems, mainly
to networks’ geometrization. In particular, we examine the use of our discrete
curvature in clustering problems, both for communication networks and in
DNA microarray analysis. Moreover, the relations between the weights given
by the internal logic of the network and the geometric structure presented
herein are investigated. This is done mainly via a discrete form of the Euler
characteristic and its comparison to the one of the physical layout of a given
network. In addition, computation of the Cheeger constant is also proposed.
Moreover, we apply this metric curvature in determining quasi-geodesics in
graphs, with applications both for P L-surfaces and for determining straight-
est paths and holes in networks.