  The network properties of a graph ensemble subject to the constraints imposed
by the expected degree sequence are studied. It is found that the linear
preferential attachment is a fundamental rule, as it keeps the maximal entropy
in sparse growing networks. This provides theoretical evidence in support of
the linear preferential attachment widely exists in real networks and adopted
as a crucial assumption in growing network models. Besides, in the sparse
limit, we develop a method to calculate the degree correlation and clustering
coefficient in our ensemble model, which is suitable for all kinds of sparse
networks including the BA model, proposed by Barabasi and Albert.
