Chordless cycle filtrations for dimensionality detection in complex networks via topological data analysis

The rapid growth of data has created significant challenges in science and technology. Large datasets from fields like biology (e.g., gene expression and protein interactions), social networks (e.g., agent behavior), and physics (e.g., cosmological simulations) often contain rich structural information hidden in their complex nature. Capturing this information requires tools capable of detecting patterns in data. Topological data analysis (TDA) [1-4] is one of such approaches, using ideas from topology to identify features that persist across scales. In TDA, scales are usually defined through filtrations, a systematic way to build a sequence of simplicial complexes that encode the geometric and topological structure of a dataset across multiple levels of resolution. Central to the analysis of such simplicial complexes is persistent homology [2, 3], the main TDA computational method that tracks the emergence, persistence, and disappearance of topological features --such as connected components or loops-- in a filtered simplicial complex as the filtration parameter evolves. In this context, graphs can be treated as simplicial complexes and can be filtered by assigning weights to their nodes and/or to their edges. The success of persistent homology depends critically on the choice of a filtration and topological features of interest, and the selection of these depends on the problem being addressed.