Supremacy distribution in evolving networks

TL;DR

This paper investigates the distribution of a new measure called supremacy in evolving Barabasi-Albert networks, revealing how it scales with network parameters and confirming findings through analytical and numerical methods.

Contribution

It introduces the concept of supremacy in evolving networks and derives its distribution and scaling laws, supported by analytical and numerical validation.

Findings

Supremacy of nodes increases as t^{(1+m)/2} with network age.

Supremacy distribution follows a power law with exponent -1-2/(1+m).

Relation s(k) ~ k^{m+1} links supremacy to node degree.

Abstract

We study a supremacy distribution in evolving Barabasi-Albert networks. The supremacy $s_i$ of a node $i$ is defined as a total number of all nodes that are younger than $i$ and can be connected to it by a directed path. 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 $s(k) \sim k^{m+1}$ 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.