{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/interesting-paths-in-the-mapper","title":"Interesting Paths in the Mapper","arxiv_id":"1712.10197","date":"2017-12-29","proceeding":null,"authors":["Ananth Kalyanaraman","Methun Kamruzzaman","Bala Krishnamoorthy"],"abstract":"The Mapper produces a compact summary of high dimensional data as a simplicial complex. We study the problem of quantifying the interestingness of subpopulations in a Mapper, which appear as long paths, flares, or loops. First, we create a weighted directed graph G using the 1-skeleton of the Mapper. We use the average values at the vertices of a target function to direct edges (from low to high). The difference between the average values at vertices (high-low) is set as the edge's weight. Covariation of the remaining h functions (independent variables) is captured by a h-bit binary signature assigned to the edge. An interesting path in G is a directed path whose edges all have the same signature. We define the interestingness score of such a path as a sum of its edge weights multiplied by a nonlinear function of their ranks in the path. Second, we study three optimization problems on this graph G. In the problem Max-IP, we seek an interesting path in G with the maximum interestingness score. We show that Max-IP is NP-complete. For the special case when G is a directed acyclic graph (DAG), we show that Max-IP can be solved in polynomial time - in O(mnd_i) where d_i is the maximum indegree of a vertex in G. In the more general problem IP, the goal is to find a collection of edge-disjoint interesting paths such that the overall sum of their interestingness scores is maximized. We also study a variant of IP termed k-IP, where the goal is to identify a collection of edge-disjoint interesting paths each with k edges, and their total interestingness score is maximized. While k-IP can be solved in polynomial time for k <= 2, we show k-IP is NP-complete for k >= 3 even when G is a DAG. We develop polynomial time heuristics for IP and k-IP on DAGs.","url_abs":"https://arxiv.org/abs/1712.10197v2","url_pdf":"https://arxiv.org/pdf/1712.10197v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"interesting-paths-in-the-mapper","repo_url":"https://github.com/xperthut/HYPPO-X","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}