Supremacy distribution in evolving networks

Analysis of a supremacy distribution in evolving Barabasi-Albert networks finds a relation s(k) approximately k(m+1) between a node degree k and its supremacy s, and the supremacy distribution P(s) scales as s(-1-2/(1+m)) .

Abstract

We study a supremacy distribution in evolving Barabasi-Albert networks. The supremacy s(i) of a node i is defined as the total number of all nodes that are not older than i and can be linked to it by a directed path (including the node i ). The nodes form a basin connected to the node i as its in-component. For a network with a characteristic parameter m=1,2,3,... , the supremacy of an individual node increases with the network age as t((1+m)/2) in an appropriate scaling region. It follows that there is a relation s(k) approximately k(m+1) between a node degree k and its supremacy s , and the supremacy distribution P(s) scales as s(-1-2/(1+m)) . Analytic calculations basing on a continuum theory of supremacy evolution and on a corresponding rate equation have been confirmed by numerical simulations.