Divisibility patterns of natural numbers on a complex network

Abstract

Investigation of divisibility properties of natural numbers is one of the most important themes in the theory of numbers. Various tools have been developed over the centuries to discover and study the various patterns in the sequence of natural numbers in the context of divisibility. In the present paper, we study the divisibility of natural numbers using the framework of a growing complex network. In particular, using tools from the field of statistical inference, we show that the network is scale-free but has a non-stationary degree distribution. Along with this, we report a new kind of similarity pattern for the local clustering, which we call “stretching similarity”, in this network. We also show that the various characteristics like average degree, global clustering coefficient and assortativity coefficient of the network vary smoothly with the size of the network. Using analytical arguments we estimate the asymptotic behavior of global clustering and average degree which is validated using numerical analysis.

Divisibility Patterns within Pascal Divisibility Networks

“…The results concerning the local clustering coefficient are similar to the ones given by [2]. Besides, we have seen that when the size of the network grows, the global clustering coefficient tends to 0, and the divisibility networks approach more and more to being disassortative.…”

“…Applying the Maximum Likelihood method [2,10], we confirm that the divisibility network of Pascal matrices satisfies a scale-free law. This means that the network degree distribution follows, at least asymptotically, a power-law of the form p(k) = C · k −γ for all k ≥ k min .…”

“…Either in [2] or here, both growing networks present similar structures and characteristics to real based networks. This can be noticed when looking at the degree distribution, the global and local clustering coefficients, the assortativity, and the average path length.…”

“…We have considered Pascal matrices as a ground for constructing a growing divisibility network. Its structure has been studied, following the approach of [2]. The interest of this choice lies in the fact that PT contains all the natural numbers on each of their sides.…”

“…On the other hand, the numbers in these matrices are not consecutive, and when sequentially ordered, we can find big gaps between some of them. This suggests us to study whether the scale-free degree distribution holds and other network properties are still satisfied as to the ones shown in [2].…”