  Understanding the structure and evolution of online bipartite networks is a
significant task since they play a crucial role in various e-commerce services
nowadays. Recently, various attempts have been tried to propose different
models, resulting in either power-law or exponential degree
distributions.However, many empirical results show that the user degree
distribution actually follows a shifted power-law distribution, so-called
\emph{Mandelbrot law}, which cannot be fully described by previous models. In
this paper, we propose an evolving model, considering two different user
behaviors: random and preferential attachment. Extensive empirical results on
two real bipartite networks, \emph{Delicious} and \emph{CiteULike}, show that
the theoretical model can well characterize the structure of real networks for
both user and object degree distributions. In addition, we introduce a
structural parameter $p$, to demonstrate that the hybrid user behavior leads to
the shifted power-law degree distribution, and the region of power-law tail
will increase with the increment of $p$. The proposed model might shed some
lights in understanding the underlying laws governing the structure of real
online bipartite networks.
