Curvature of a smooth manifold is quite intuitive, and has been studied in differential geometry for a long time. However, the notion of curvature for metric spaces in general, and for graphs in particular, is a relatively recent idea. In 2015, graph Ricci curvature was introduced as a framework to consider neighborhood to neighborhood interactions within a weighted undirected graph. In this thesis, we generalize graph Ricci curvature for weighted directed graphs, and apply this notion to analyze the spread of the Coronavirus disease 2019 (COVID-19) across the counties in the state of California.

We use real data for the daily traffic across different counties in California, and the daily COVID-19 case counts from March 2020 to March 2021. We demonstrate that graph Ricci curvature, and curvatures derived from it--such as graph scalar curvature--are particularly suited to dynamically predict and locate the onset and intensity of virus spread. The outcome of this thesis is a novel geometric data-driven risk analytics methodology to identify time-varying network-level risks for a virus spread. We envisage that our ideas will be useful for designing dynamic nonpharmaceutical intervention (NPI) strategies across the network to optimally mitigate the spread of the virus.