Exploring topological features of a complex network of prime factors and composite numbers

Abstract

Prime numbers are mathematical concepts of particular interest because of their relative rarity among integers, as well as the relative unpredictability of their occurrence and density along the integer sequence. They constitute the building blocks of all other composite numbers, giving rise to the interesting and important concept of prime factorization, a process guaranteed by the fundamental theorem of arithmetic. 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. In this paper, we derive some topological features of a bipartite complex network of prime factors and composite numbers. More specifically, we prove mathematically that the network's diameter is exactly five regardless of the number of nodes. 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. The last shows that this network is of small-world type. 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. Finally, we focus on the computation of the information dimensions on the basis of Shannon and Tsallis entropies of the studied complex network. The results show both decreasing as the number of nodes increases. 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.