Supremacy distribution in evolving networks

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 $m=1,2,3,\mathrm{.}\mathrm{.}\mathrm{.}$ the supremacy of an individual node increases with the network age as $t^{(1+m)\mathrm{/}2}$ in an appropriate scaling region. It follows that there is a relation $s(k) \sim k^{m+1}$betweenanodedegree$k$anditssupremacy$s$ and the supremacy distribution $P(s)$ scales as $s^{-1-2\mathrm{/}(1+m)}$. Analytic calculations basing on a continuum theory of supremacy evolution and on a corresponding rate equation have been confirmed by numerical simulations.