In the present work, we apply a geometric network approach to study common biological features of anticancer drug response. We use for this purpose the panel of 60 human cell lines (NCI-60) provided by the National Cancer Institute. Our study suggests that mathematical tools for network-based analysis can provide novel insights into drug response and cancer biology. We adopted a discrete notion of Ricci curvature to measure, via a link between Ricci curvature and network robustness established by the theory of optimal mass transport, the robustness of biological networks constructed with a pre-treatment gene expression dataset and coupled the results with the GI50 response of the cell lines to the drugs. Based on the resulting drug response ranking, we assessed the impact of genes that are likely associated with individual drug response. For genes identified as important, we performed a gene ontology enrichment analysis using a curated bioinformatics database which resulted in biological processes associated with drug response across cell lines and tissue types which are plausible from the point of view of the biological literature. These results demonstrate the potential of using the mathematical network analysis in assessing drug response and in identifying relevant genomic biomarkers and biological processes for precision medicine.