Range changeable local structural information of nodes in complex networks

In the research of complex networks, structural analysis can be explained as finding the information hidden in the network's topological structure. Thus, the way and the range of the structural information collection decide what kinds of information can be found in the structural analysis. In this work, based on the definition of Shannon entropy and the changeable range of structural information collecting (changeable local network for each node), the local structural information (LSI) of nodes in complex networks is proposed. According to the definition, when the range of the local network converges to the node itself, the LSI is their original structural properties, e.g. node's degree, betweenness and clustering coefficient, but when the range of the local network extends to the whole network (order of the local network equal to the diameter of networks), the LSI is equivalent to the structural entropy of the entire static network, e.g. degree structural entropy, betweenness structural entropy. We also find that the local degree structural information can be used to classify the nodes in the network, and the proportion of the "bridge" nodes in the network is a new indicator of the network's robustness, the bigger this proportion of bridge nodes in the network, the more robust the network. This finding also explains why the regular networks or the lattice is so stable, as almost all the nodes in those systems are the "bridge" nodes that are identified by the local degree structural information.