  We introduce new definitions of sectional, Ricci and scalar curvature for
networks and their higher dimensional counterparts, derived from two classical
notions of curvature for curves in general metric spaces, namely, the Menger
curvature and the Haantjes curvature. These curvatures are applicable to
unweighted or weighted and undirected or directed networks, and are more
intuitive and easier to compute than other network curvatures. In particular,
the proposed curvatures based on the interpretation of Haantjes definition as
geodesic curvature allow us to give a network analogue of the classical local
Gauss-Bonnet theorem. Furthermore, we propose even simpler and more intuitive
proxies for the Haantjes curvature that allow for even faster and easier
computations in large-scale networks. In addition, we also investigate the
embedding properties of the proposed Ricci curvatures. Lastly, we also
investigate the behaviour, both on model and real-world networks, of the
curvatures introduced herein with more established notions of Ricci curvature
and other widely-used network measures.
