Growth network models with random number of attached links

Abstract

In the process of many real networks’ growth, each new node joins a few existing nodes, the number of which is unknown in advance. However, classical network growth models assume that the number of nodes to which each newborn node links is constant. In this regard, the main research question of this paper is as follows: how do structural properties of networks generated in accordance with growth models with random number of attached links, change (in comparison with networks with a constant number of attached links)? In this paper, we restrict our analysis to the two growth network models: the Barabási–Albert (BA) model and the triadic closure (TC) model. However, both in the BA and in the TC models the number of links attached to each new node of the network is constant on every iteration, which is not quite realistic. In this paper we extend both the BA and the TC models by allowing the number of newly added links to be random, under some mild assumptions on its distribution law. We examine the geometric properties of networks generated by the proposed models analytically and empirically to show that their properties differ from those of the classical BA and TC models. In particular, while the degree distributions of the generated networks follow the power law, their exponents vary depending on the thickness of the tail of the distribution that generates the number of attached links, and may differ significantly from the value of the corresponding exponent γ=−3 for the networks generated by the classical BA and TC model. The ‘random’ versions of models have a useful feature: they are capable of creating new nodes with a high degree at any iteration. This greatly distinguishes them from classical models, in which the nodes that appeared in the early iterations gain a significant advantage. The property of the proposed model, associated with the possibility of late arrivals to receive a random number of neighbors (which can be arbitrarily large), seems to be capable to simulate the temporal behavior of real networks in a more realistic way.