  We propose a model that generates a new class of networks exhibiting
power-law degree distribution with a spectrum of exponents depending on the
number of links ($m$) with which incoming nodes join the existing network.
Unlike the Barab\'{a}si-Albert (BA) model, each new node first picks an
existing node at random, and connects not with this but with $m$ of its
neighbors also picked at random. Counterintuitively enough, such a
mediation-driven attachment rule results not only in preferential but
super-preferential attachment, albeit in disguise. We show that for small $m$,
the dynamics of our model is governed by winners take all phenomenon, and for
higher $m$ it is governed by winners take some. Besides, we show that the mean
of the inverse harmonic mean of degrees of the neighborhood of all existing
nodes is a measure that can well qualify how straight the degree distribution
is.
