  In this paper, we first discuss the origin of preferential attachment. Then
we establish the generalized preferential attachment which has two new
properties; first, it encapsulates both the topological and weight aspects of a
network, which makes it is neither entirely degree preferential nor entirely
weight preferential. Second, it can tell us not only the chance that each
already-existing vertex being connected but also how much weight each new edge
has. The generalized preferential attachment can generate four power-law
distributions, besides the three for vertex degrees, vertex strengths, and edge
weights, it yield a new power-law distribution for the subgraph degrees.
