  The network topology can be described by the number of nodes and the
interconnections among them. The degree of a node in a network is the number of
connections it has to other nodes and the degree distribution is the
probability distribution of these degrees over the whole network. Therefore,
the degree is very important structural parameter of network topology. However,
given the number of nodes and the degree of each node in a network, the
topology of the network cannot be determined. Therefore, we propose the
degree-layer theory of network topology to describe deeply the network
topology. First, we propose the concept of degree-tree with the breadth-first
search tree. The degrees of all nodes are layered and have a hierarchical
structure. Second,the degree-layer theory is described in detail. Two new
concepts are defined in the theory. An index is proposed to quantitatively
distinguish the two network topologies. It also can quantitatively measure the
stability of network topology built by a model mechanism. One theorem is given
and proved, furthermore, and one corollary is derived directly from the
theorem. Third, the applications of the degree-layer theory are discussed in
the ER random network, WS small world network and BA scale-free network, and
the influences of the degree distribution on the stability of network topology
are studied in the three networks. In conclusion, the degree-layer theory is
helpful for accurately describing the network topology, and provides a new
starting point for researching the similarity and isomorphism between two
network topologies.
