{"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/random-processes-for-generating-task","title":"Random processes for generating task-dependency graphs","arxiv_id":"2305.05205","date":"2023-05-09","proceeding":null,"authors":["Jesse Geneson","Shen-Fu Tsai"],"abstract":"We investigate random processes for generating task-dependency graphs of order $n$ with $m$ edges and a specified number of initial vertices and terminal vertices. In order to do so, we consider two random processes for generating task-dependency graphs that can be combined to accomplish this task. In the $(x, y)$ edge-removal process, we start with a maximally connected task-dependency graph and remove edges uniformly at random as long as they do not cause the number of initial vertices to exceed $x$ or the number of terminal vertices to exceed $y$. In the $(x, y)$ edge-addition process, we start with an empty task-dependency graph and add edges uniformly at random as long as they do not cause the number of initial vertices to be less than $x$ or the number of terminal vertices to be less than $y$. In the $(x, y)$ edge-addition process, we halt if there are exactly $x$ initial vertices and $y$ terminal vertices. For both processes, we determine the values of $x$ and $y$ for which the resulting task-dependency graph is guaranteed to have exactly $x$ initial vertices and $y$ terminal vertices, and we also find the extremal values for the number of edges in the resulting task-dependency graphs as a function of $x$, $y$, and the number of vertices. Furthermore, we asymptotically bound the expected number of edges in the resulting task-dependency graphs. Finally, we define a random process using only edge-addition and edge-removal, and we show that with high probability this random process generates an $(x, y)$ task-dependency graph of order $n$ with $m$ edges.","url_abs":"https://arxiv.org/abs/2305.05205v1","url_pdf":"https://arxiv.org/pdf/2305.05205v1.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":"random-processes-for-generating-task","repo_url":"https://github.com/jgeneson/tdgraphs","is_official":1,"mentioned_in_paper":1,"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}