Topological Density Landscapes: A Graph-Theoretic Foundation for Histograms

This paper introduces the concept of Topological Density Landscapes (TDLs) as a novel, robust alternative to traditional histograms for data density visualization and analysis. Standard histograms are highly sensitive to the choice of binning parameters, such as bin width and origin, which can obscure or misrepresent the underlying structure of the data. TDLs address this limitation by leveraging principles from graph theory and topological data analysis (TDA). We construct a graph from the data points, where vertices represent the data and edges encode proximity. A density function defined on the vertices of this graph creates a landscape whose topological features, such as connected components corresponding to data modes, are analyzed using persistent homology. This approach yields a multi-scale representation of the data's density structure that is invariant to arbitrary partitioning. We demonstrate through synthetic and real-world examples that TDLs provide a more stable and informative visualization, consistently identifying key distributional features like modality and clusters without the need for manual parameter tuning. The resulting framework offers a rigorous, graph-theoretic foundation for density estimation that overcomes the fundamental frailties of classical histogram methods.