Suitable notions of shapes play a critical role in investigating objects in mathematics, physics and other research areas. Various kinds of curvatures are very natural measures of shapes of higher dimensional objects in mainstream physics and mathematics. However, any attempt to extend notions of these kinds of measures to networks needs to overcome several key challenges. In this article we review several curvature measures for networks such as (i) Gromov-hyperbolic curvature, (ii) extension of discretization of Ricci curvature for polyhedral complexes, and (iii) extension of discretization of Ricci curvature via mass transportation distances, and the corresponding flow technique. We finally review the bioinformatics applications of these measures for several biological networks such as E. coli transcriptional network, metabolic network of M. tuberculosis, protein-protein interaction networks in humans and network of functional correlations between brain regions for attention deficit hyperactivity disorder.
