We introduce the Stability-Incorporated Neighborhood Graph (SING), a novel density-aware structure designed to capture the intrinsic geometric properties of a point set. We improve upon the spheres-of-influence graph by incorporating additional features to offer more flexibility and control in encoding proximity information and capturing local density variations. Through persistence analysis on our proximity graph, we propose a new clustering technique and explore additional variants incorporating extra features for the proximity criterion. Alongside the detailed analysis and comparison to evaluate its performance on various datasets, our experiments demonstrate that the proposed method can effectively extract meaningful clusters from diverse datasets with variations in density and correlation. Our application scenarios underscore the advantages of the proposed graph over classical neighborhood graphs, particularly in terms of parameter tuning.