{"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/optimization-on-the-smallest-eigenvalue-of","title":"Optimization on the smallest eigenvalue of grounded Laplacian matrix via edge addition","arxiv_id":"2309.17019","date":"2023-09-29","proceeding":null,"authors":["Xiaotian Zhou","Haoxin Sun","Wei Li","Zhongzhi Zhang"],"abstract":"The grounded Laplacian matrix $\\LL_{-S}$ of a graph $\\calG=(V,E)$ with $n=|V|$ nodes and $m=|E|$ edges is a $(n-s)\\times (n-s)$ submatrix of its Laplacian matrix $\\LL$, obtained from $\\LL$ by deleting rows and columns corresponding to $s=|S| \\ll n $ ground nodes forming set $S\\subset V$. The smallest eigenvalue of $\\LL_{-S}$ plays an important role in various practical scenarios, such as characterizing the convergence rate of leader-follower opinion dynamics, with a larger eigenvalue indicating faster convergence of opinion. In this paper, we study the problem of adding $k \\ll n$ edges among all the nonexistent edges forming the candidate edge set $Q = (V\\times V)\\backslash E$, in order to maximize the smallest eigenvalue of the grounded Laplacian matrix. We show that the objective function of the combinatorial optimization problem is monotone but non-submodular. To solve the problem, we first simplify the problem by restricting the candidate edge set $Q$ to be $(S\\times (V\\backslash S))\\backslash E$, and prove that it has the same optimal solution as the original problem, although the size of set $Q$ is reduced from $O(n^2)$ to $O(n)$. Then, we propose two greedy approximation algorithms. One is a simple greedy algorithm with an approximation ratio $(1-e^{-\\alpha\\gamma})/\\alpha$ and time complexity $O(kn^4)$, where $\\gamma$ and $\\alpha$ are, respectively, submodularity ratio and curvature, whose bounds are provided for some particular cases. The other is a fast greedy algorithm without approximation guarantee, which has a running time $\\tilde{O}(km)$, where $\\tilde{O}(\\cdot)$ suppresses the ${\\rm poly} (\\log n)$ factors. Numerous experiments on various real networks are performed to validate the superiority of our algorithms, in terms of effectiveness and efficiency.","url_abs":"https://arxiv.org/abs/2309.17019v1","url_pdf":"https://arxiv.org/pdf/2309.17019v1.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":"optimization-on-the-smallest-eigenvalue-of","repo_url":"https://github.com/kedges/kedges","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}