{"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/vulnerability-and-hierarchy-of-complex","title":"Vulnerability and Hierarchy of Complex Networks","arxiv_id":"cond-mat/0409298","date":"2004-09-13","proceeding":null,"authors":["V. Gol'dshtein","G. A. Koganov","G. I. Surdutovich"],"abstract":"We suggest an approach to study hierarchy, especially hidden one, of complex networks based on the analysis of their vulnerability. Two quantities are proposed as a measure of network hierarchy. The first one is the system vulnerability V. We show that being quite suitable for regular networks this characteristic does not allow one to estimate the hierarchy of large random networks. The second quantity is a relative variance h of the system vulnerability that allows us to characterize a \"natural\" hierarchy level of random networks. We find that hierarchical properties of random networks depend crucially on a ratio between the number of nodes and the number of edges. We note that any graph with a transitive isometry group action (i.e. an absolutely symmetric graph) is not hierarchical. Breaking such a symmetry leads to appearance of hierarchy.","url_abs":"https://arxiv.org/abs/cond-mat/0409298v1","url_pdf":"https://arxiv.org/pdf/cond-mat/0409298v1.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":"vulnerability-and-hierarchy-of-complex","repo_url":"https://github.com/gioguarnieri/tamanduatei_vulnerability","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"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}