Fault tolerance of hierarchical cubic networks based on cluster fault pattern

Connectivity is a meaningful metric parameter and indicator for estimating network reliability and evaluating network fault tolerance. However, the traditional connectivity and current conditional connectivity do not take into account the association between a certain node and its neighboring nodes. In fact, adjacent nodes are easily influenced by each other so that the failing probability of adjacent nodes around a faulty node is high. Therefore, cluster and super cluster connectivities are proposed to more intuitively measure the fault tolerance of the network. In this paper, we mainly explore the cluster connectivity and super cluster connectivity of the hierarchical cubic network $HCN_{n}$. In detail, we show that $\kappa (HCN_{n}\mid K_{1, 0}(K_{1, 0}<^>{*}))=n+1$, $\kappa (HCN_{n}\mid K_{1, 1}(K_{1, 1}<^>{*}))=\kappa <^>{\prime}(HCN_{n}\mid K_{1, 1}(K_{1, 1}<^>{*}))=n+1$, $\kappa (HCN_{n}\mid K_{1, m}(K_{1, m}<^>{*}))=\lceil n/2\rceil +1$ ($2\leq m\leq 4$), $\kappa <^>{\prime}(HCN_{n}\mid K_{1, 0}(K_{1, 0}<^>{*}))=2n$, and $\kappa <^>{\prime}(HCN_{n}\mid K_{1, m}(K_{1, m}<^>{*}))=n+1$ ($2\leq m\leq 3$) if $n$ is odd and $\kappa <^>{\prime}(HCN_{n}\mid K_{1, m}(K_{1, m}<^>{*}))=n$ ($2\leq m\leq 3$) if $n$ is even, where $n\geq 4$.