Establishing Correlation Between Structural and Spectral Property in K-Shell Structure

Degeneracy in graph theory, often known as k-shell decomposition in the graph is a well-studied problem. It is used to generate a core-periphery structure of a network that defines the levels of importance of the nodes of a network. The core nodes are the most influential nodes of a network. In this paper, we investigate the cascade capacity of a node inside a shell. We found that the intra-shell cascade capacity of a node in the core is more than the others because the Clustering Coefficient (CC) of the core is high and the Localization factor is less. There exist a negative correlation between CC and IPR. We have established this relationship between structural property CC and spectral property of Principal Eigen Vector (PEV) Localization using IPR and validated the intra-shell cascade capacity using Independent Cascade(IC) Model.