  In this paper, a baseline model termed as random birth-and-death network
model (RBDN) is considered, in which at each time step, a new node is added
into the network with probability p (0<p <1) connect it with m old nodes
uniformly, or an existing node is deleted from the network with probability
q=1-p. This model allows for fluctuations in size, which may reach many
different disciplines in physics, ecology and economics. The purpose of this
study is to develop the RBDN model and explore its basic statistical
properties. For different p, we first discuss the network size of RBDN. And
then combining the stochastic process rules (SPR) based Markov chain method and
the probability generating function method, we provide the exact solutions of
the degree distributions. Finally, the characteristics of the tail of the
degree distributions are explored after simulation verification. Our results
show that the tail of the degree distribution for RBDN exhibits a Poisson tail
in the case of 0<p<=1/2 and an exponential tail as p approaches to 1.
