Enabling high reliability via matroidal connectivity and conditional matroidal connectivity on arrangement graph networks

Connectivity indicators are commonly used to evaluate system fault tolerance and reliability. However, with the high demand for multi-processor systems in high-performance computing and data center networks, the number of processors is getting larger, and the network is getting more complex. Thus, traditional connectivity and other indicators are hardly competent in assessing the reliability of complex networks. The matroidal connectivity and conditional matroidal connectivity are novel connectivity metrics that measure the actual fault-tolerant capability based on the constraints of each network dimension. In this paper, we study matroidal connectivity and conditional matroidal connectivity of the $(n,k)$-arrangement graph network $A_{n,k}$ from the natural perspective of the partition of edge dimension and obtain their theoretically accurate values. Moreover, we conduct numerical analysis to compare matroidal connectivity with other conditional edge connectivities in $A_{n,k}$. Additionally, we explore the distribution pattern of edge failure through simulation experiments in $A_{n,k}$ and attain the relation of conditional matroidal connectivity related to network scales. Our investigations include two famous network classes: alternating group graphs and star graphs.