  We study the Krapivsky-Redner (KR) network growth model but where new nodes
can connect to any number of existing nodes, $m$, picked from a power-law
distribution $p(m)\sim m^{-\alpha}$. Each of the $m$ new connections is still
carried out as in the KR model with probability redirection $r$ (corresponding
to degree exponent $\gamma_{\rm KR}=1+1/r$, in the original KR model). The
possibility to connect to any number of nodes resembles a more realistic type
of growth in several settings, such as social networks, routers networks, and
networks of citations. Here we focus on the in-, out-, and total-degree
distributions and on the potential tension between the degree exponent
$\alpha$, characterizing new connections (outgoing links), and the degree
exponent $\gamma_{\rm KR}(r)$ dictated by the redirection mechanism.
