  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.
