{"paper":{"title":"Exploring topological features of a complex network of prime factors and composite numbers","arxiv_id":"2610.33873","primary_cat":"math.NT","submitted_at":"2026-10-24T19:28:23+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. 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arithmetic.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.04,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":1.0,"id":"s2","own":false,"role":"background","text":"Among the several works that have been reported about prime numbers and their interrelationship with composite numbers, some have considered derived graphs and/or networks.","hedge":{"p":0.89,"label":"asserted"},"own_p":0.03,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":3,"p":1.0,"id":"s3","own":true,"role":"contribution","text":"In this paper, we derive some topological features of a bipartite complex network of prime factors and composite numbers.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.96,"main_claim_p":0.14,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":1.0,"interpretation":0.0}},{"i":4,"p":0.99,"id":"s4","own":true,"role":"result","text":"More specifically, we prove mathematically that the network's diameter is exactly five regardless of the number of nodes.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.49,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.99,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":5,"p":1.0,"id":"s5","own":true,"role":"result","text":"In particular, we show that certain nontrivial sub-networks have minimal diameter equal to four, which is, again, independent of the number of nodes as long as the network is large enough.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.97,"main_claim_p":0.14,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":6,"p":0.97,"id":"s6","own":true,"role":"interpretation","text":"The last shows that this network is of small-world type.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.9,"main_claim_p":0.24,"quantitative":false,"quantitative_p":0.1,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.03,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.97}},{"i":7,"p":0.9,"id":"s7","own":true,"role":"contribution","text":"We also characterize exactly the number of boundary nodes in the network and present a couple of algorithms to identify the most important, relevant, influential, or central nodes of the network.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.07,"quantitative":false,"quantitative_p":0.22,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.09,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.9,"interpretation":0.0}},{"i":8,"p":0.75,"id":"s8","own":true,"role":"contribution","text":"Finally, we focus on the computation of the information dimensions on the basis of Shannon and Tsallis entropies of the studied complex network.","hedge":{"p":0.82,"label":"asserted"},"own_p":0.94,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.25,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.75,"interpretation":0.0}},{"i":9,"p":1.0,"id":"s9","own":true,"role":"result","text":"The results show both decreasing as the number of nodes increases.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.1,"quantitative":true,"quantitative_p":0.85,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":10,"p":0.98,"id":"s10","own":true,"role":"result","text":"Furthermore, both entropies indicate that box coverings at the diameters \\(\\varepsilon=[3,4]\\), group nodes into roughly the same number of boxes, independent of the number of nodes in the network.","hedge":{"p":0.56,"label":"asserted"},"own_p":0.94,"main_claim_p":0.11,"quantitative":true,"quantitative_p":0.87,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.98,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.02}}]},"compiled":{"main_claim":{"p":0.86,"text":"The last shows that this network is of small-world type.","location":"abstract 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