Graph embedding approaches attempt to project graphs into geometric entities, {\em i.e.}, manifolds. At its core, the idea is that the geometric properties of the projected manifolds are helpful in the inference of graph properties. However, the choice of the embedding manifold is critical and, if incorrectly performed, can lead to misleading interpretations due to incorrect geometric inference. We argue that the classical embedding techniques cannot lead to correct geometric interpretation as the microscopic details, {\em e.g.}, curvature at each point, of manifold, that are needed to derive geometric properties in Riemannian geometry methods are not available, and we cannot evaluate the impact of the underlying space on geometric properties of shapes that lie on them. We advocate that for doing correct geometric interpretation the embedding of a graph should be done over regular constant curvature manifolds. To this end, we present an embedding approach, the discrete Ricci flow graph embedding (dRfge) based on the discrete Ricci flow that adapts the distance between nodes in a graph so that the graph can be embedded onto a constant curvature manifold that is homogeneous and isotropic, {\em i.e.}, all directions are equivalent and distances comparable, resulting in correct geometric interpretations. A major contribution of this paper is that for the first time, we prove the convergence of discrete Ricci flow to a constant curvature and stable distance metric over the edges. A drawback of using the discrete Ricci flow is the high computational complexity that prevented its usage in large-scale graph analysis. Another contribution of our work is a new algorithmic solution that makes it feasible to calculate the Ricci flow for graphs of up to 50k nodes, and beyond. The intuitions behind the discrete Ricci flow make it possible to obtain new insights into the structure of large-scale graphs.