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Based on two classical notions of curvature for curves in general metricspaces, namely the Menger and Haantjes curvatures, we introduce new definitionsof sectional, Ricci and scalar curvature for networks and their higherdimensional counterparts. These new types of curvature, that apply to weightedand unweighted, directed or undirected networks, are far more intuitive andeasier to compute, than other network curvatures. In particular, the proposedcurvatures based on the interpretation of Haantjes definition as geodesiccurvature, and derived via a fitting discrete Gauss-Bonnet Theorem, are quiteflexible. We also propose even simpler and more intuitive substitutes of theHaantjes curvature, that allow for even faster and easier computations inlarge-scale networks.

A Simple Differential Geometry for Networks and its Generalizations