  A network is formed using the $N$ sites of an one-dimensional lattice in the
shape of a ring as nodes and each node with the initial degree $k_{in}=2$. $N$
links are then introduced to this network, each link starts from a distinct
node, the other end being connected to any other node with degree $k$ randomly
selected with an attachment probability proportional to $k^{\alpha}$. Tuning
the control parameter $\alpha$ we observe a transition where the average degree
of the largest node $<k_m(\alpha,N)>$ changes its variation from $N^0$ to $N$
at a specific transition point of $\alpha_c$. The network is scale-free i.e.,
the nodal degree distribution has a power law decay for $\alpha \ge \alpha_c$.
