  It is commonly believed that real networks are scale-free and fraction of
nodes $P(k)$ with degree $k$ satisfies the power law $P(k) \propto k^{-\gamma}
\text{ for } k > k_{min} > 0$. Preferential attachment is the mechanism that
has been considered responsible for such organization of these networks. In
many real networks, degree distribution before the $k_{min}$ varies very slowly
to the extent of being uniform as compared with the degree distribution for $k
> k_{min}$ . In this paper, we proposed a model that describe this particular
degree distribution for the whole range of $k>0$. We adopt a two step approach.
In the first step, at every time stamp we add a new node to the network and
attach it with an existing node using preferential attachment method. In the
second step, we add edges between existing pairs of nodes with the node
selection based on the uniform probability distribution. Our approach generates
weakly scale-free networks that closely follow the degree distribution of
real-world networks. We perform comprehensive mathematical analysis of the
model in the discrete domain and compare the degree distribution generated by
these models with that of real-world networks.
