With the evolution of social networks, the network structure shows dynamic
nature in which nodes and edges appear as well as disappear for various
reasons. The role of a node in the network is presented as the number of
interactions it has with the other nodes. For this purpose a network is modeled
as a graph where nodes represent network members and edges represent a
relationship among them. Several models for evolution of social networks has
been proposed till date, most widely accepted being the Barab\'asi-Albert
\cite{Network science} model that is based on \emph{preferential attachment} of
nodes according to the degree distribution. This model leads to generation of
graphs that are called \emph{Scale Free} and the degree distribution of such
graphs follow the \emph{power law}. Several generalizations of this model has
also been proposed. In this paper we present a new generalization of the model
and attempt to bring out its implications in real life
