**5. Discussion**

In this paper we have reviewed some computational aspects of the Ollivier–Ricci curvature for networks, and shown a few simple computational bounds. As already mentioned in Section 1, there are other notions of network curvature that is also used by researchers and therefore this review should not be viewed as championing the Ollivier–Ricci curvature over other curvatures. We hope that this review will motivate further research on the exciting interplay between notions of curvatures from network and non-network domains. Some applications of network curvatures for real-world networks appear in references such as [11,13,15,16,18].

We conclude our article by mentioning an interesting application of the Ollivier–Ricci curvature for Markov chains for graph coloring and other problems (recise technical descriptions of these results are beyond the scope of this introductory review). The probability distributions on nodes used to compute EMD in the Ollivier–Ricci curvature can be naturally associated with a Markov process on the given graph (as a very simplified illustration, one can use a "normalized version" of EMD(*Vu*,*v*, P*<sup>u</sup>*, <sup>P</sup>*v*) as the probability of transition between the states corresponding to nodes *u* and *v*). Such associations have a long history in the Markov chain literature under various names such as path coupling [41] and the values of RIC(*<sup>u</sup>*, *v*)'s have been used (explicitly or implicitly) to prove useful properties of the Markov chain, such as fast convergence to its stationary distribution, in many settings such as graph colouring [41] and sampling of paths with constraints [42].

**Author Contributions:** The author contributions are as follows: Conceptualization, N.A., P.S. and B.D.; methodology, N.A., P.S. and B.D.; software, N.A., P.S. and B.D.; validation, N.A., P.S. and B.D.; formal analysis, N.A., P.S. and B.D.; investigation, N.A., P.S. and B.D.; resources, N.A., P.S. and B.D.; data curation, N.A., P.S. and B.D.; writing–original draft preparation, N.A., P.S. and B.D.; writing–review and editing, N.A., P.S. and B.D.; visualization, N.A., P.S. and B.D.; supervision, B.D.; project administration, B.D.; funding acquisition, B.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by NSF gran<sup>t</sup> number IIS-1814931.

**Conflicts of Interest:** The authors declare no conflict of interest.
