Degree-layer theory of network topology
  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. 
Bin Zhou, Bing-Hong Wang, He Zhe
physics.soc-ph cs.SI