{"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/assigning-stationary-distributions-to-sparse","title":"Assigning Stationary Distributions to Sparse Stochastic Matrices","arxiv_id":"2312.16011","date":"2023-12-26","proceeding":null,"authors":["Nicolas Gillis","Paul Van Dooren"],"abstract":"The target stationary distribution problem (TSDP) is the following: given an irreducible stochastic matrix $G$ and a target stationary distribution $\\hat \\mu$, construct a minimum norm perturbation, $\\Delta$, such that $\\hat G = G+\\Delta$ is also stochastic and has the prescribed target stationary distribution, $\\hat \\mu$. In this paper, we revisit the TSDP under a constraint on the support of $\\Delta$, that is, on the set of non-zero entries of $\\Delta$. This is particularly meaningful in practice since one cannot typically modify all entries of $G$. We first show how to construct a feasible solution $\\hat G$ that has essentially the same support as the matrix $G$. Then we show how to compute globally optimal and sparse solutions using the component-wise $\\ell_1$ norm and linear optimization. We propose an efficient implementation that relies on a column-generation approach which allows us to solve sparse problems of size up to $10^5 \\times 10^5$ in a few minutes. We illustrate the proposed algorithms with several numerical experiments.","url_abs":"https://arxiv.org/abs/2312.16011v3","url_pdf":"https://arxiv.org/pdf/2312.16011v3.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":"assigning-stationary-distributions-to-sparse","repo_url":"https://gitlab.com/ngillis/tsdp","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"assigning-stationary-distributions-to-sparse","repo_url":"https://github.com/Cirdans-Home/enforce-katz-and-pagerank","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}