Network structures sustained by internal links and distributed lifetime of old nodes in stationary state of number of nodes

Abstract

In network models that take into account growth properties, deletion of old nodes has a serious impact on degree distributions, because old nodes tend to become hub nodes. In this study, we aim to provide a simple explanation for why hubs can exist even in conditions where the number of nodes is stationary due to the deletion of old nodes. We show that an exponential increase in the degree of nodes is a natural consequence of the balance between the deletion and addition of nodes as long as a preferential attachment mechanism holds. As a result, the largest degree is determined by the magnitude relationship between the time scale of the exponential growth of degrees and lifetime of old nodes. The degree distribution exhibits a power-law form ∼ k−γ with exponent γ = 1 when the lifetime of nodes is constant. However, various values of γ can be realized by introducing distributed lifetime of nodes.