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Specifically, we first propose an evolving deterministic network\n            <jats:inline-formula content-type=\"math\/tex\">\n              <jats:tex-math notation=\"LaTeX\" version=\"MathJax\">\\(N(t)\\)<\/jats:tex-math>\n            <\/jats:inline-formula>\n            using three types of growth ways. Then, we study some topological structural parameters including degree distribution, diameter, and clustering coefficient on network\n            <jats:inline-formula content-type=\"math\/tex\">\n              <jats:tex-math notation=\"LaTeX\" version=\"MathJax\">\\(N(t)\\)<\/jats:tex-math>\n            <\/jats:inline-formula>\n            . The results demonstrate that the proposed network has scale-free feature and small-world property. In the meantime, we obtain an interesting finding, i.e., the first handshake between Fibonacci series and the \u201cpure\u201d preferential attachment mechanism. Next, we enumerate spanning trees on network\n            <jats:inline-formula content-type=\"math\/tex\">\n              <jats:tex-math notation=\"LaTeX\" version=\"MathJax\">\\(N(t)\\)<\/jats:tex-math>\n            <\/jats:inline-formula>\n            and derive the closed-form solution of spanning trees number. Second, we introduce randomness into the growth process of network\n            <jats:inline-formula content-type=\"math\/tex\">\n              <jats:tex-math notation=\"LaTeX\" version=\"MathJax\">\\(N(t)\\)<\/jats:tex-math>\n            <\/jats:inline-formula>\n            to further establish evolving stochastic networks\n            <jats:inline-formula content-type=\"math\/tex\">\n              <jats:tex-math notation=\"LaTeX\" version=\"MathJax\">\\(\\mathfrak{N}(t)\\)<\/jats:tex-math>\n            <\/jats:inline-formula>\n            that follow the same degree distribution as network\n            <jats:inline-formula content-type=\"math\/tex\">\n              <jats:tex-math notation=\"LaTeX\" version=\"MathJax\">\\(N(t)\\)<\/jats:tex-math>\n            <\/jats:inline-formula>\n            and also determine some topological structural parameters so as to investigate effect of randomness on structural properties. We show analytically that such a randomization approach makes the resulting stochastic networks not only to greatly inherit some fundamental structural properties from deterministic network\n            <jats:inline-formula content-type=\"math\/tex\">\n              <jats:tex-math notation=\"LaTeX\" version=\"MathJax\">\\(N(t)\\)<\/jats:tex-math>\n            <\/jats:inline-formula>\n            but also to considerably improve the robustness of network when encountering deliberate removal of edge. Lastly, we list out some open problems.\n          <\/jats:p>","DOI":"10.1145\/3748512","type":"journal-article","created":{"date-parts":[[2025,7,14]],"date-time":"2025-07-14T19:54:33Z","timestamp":1752522873000},"page":"1-32","update-policy":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["Growth Scale-Free Networks by Various Generative Ways"],"prefix":"10.1145","volume":"19","author":[{"ORCID":"https:\/\/2.zoppoz.workers.dev:443\/https\/orcid.org\/0000-0002-4269-2565","authenticated-orcid":false,"given":"Fei","family":"Ma","sequence":"first","affiliation":[{"name":"School of Computer Science, Northwestern Polytechnical University, Xi\u2019an, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/2.zoppoz.workers.dev:443\/https\/orcid.org\/0000-0002-8854-2079","authenticated-orcid":false,"given":"Ping","family":"Wang","sequence":"additional","affiliation":[{"name":"National Engineering Research Center for Software Engineering, Peking University, Beijing, China, School of Software and Microelectronics, Peking University, Beijing, China, and Key Laboratory of High Confidence Software Technologies (PKU), Ministry of Education, Peking University, Beijing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2025,9,8]]},"reference":[{"key":"e_1_3_3_2_2","doi-asserted-by":"publisher","DOI":"10.1038\/nphys1130"},{"key":"e_1_3_3_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.physrep.2018.10.005"},{"key":"e_1_3_3_4_2","doi-asserted-by":"publisher","DOI":"10.1038\/s42254-023-00676-y"},{"key":"e_1_3_3_5_2","doi-asserted-by":"publisher","DOI":"10.1109\/TKDE.2023.3274547"},{"key":"e_1_3_3_6_2","doi-asserted-by":"publisher","DOI":"10.1109\/TKDE.2020.3036013"},{"key":"e_1_3_3_7_2","doi-asserted-by":"publisher","DOI":"10.1145\/3674146"},{"issue":"1","key":"e_1_3_3_8_2","article-title":"Networks with growth and preferential attachment: Modelling and applications","volume":"9","author":"Piva G. 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