Viewing the network parameters and $\mathcal{H}$-factors from the perspective of geometry

Recent studies have shown that there is a profound connection between the existence of $\mathcal{H}$-factors under attack circumstances and the parameters to measure the vulnerability of the network. The disadvantage of the previous theoretical conclusions is that the connectivity is often regarded as fixed, and the relationship between other network parameters and the $\mathcal{H}$-factor is explored. However, in a real situation, as the network structure changes, connectivity becomes a dynamically variational parameter, and its changing will affect other parameters to make corresponding changes. In this study, we treat all parameters as a complex system that restricts each other, consider their mutual constraints from a geometric point of view, and apply high-dimensional surfaces to characterize their relationships. Seven network parameters including toughness and binding number are considered, and the concrete expression form of the surfaces in several specific settings are obtained. Furthermore, the local version for these seven network parameters are introduced.