The structure connectivity of Data Center Networks

TL;DR

This paper calculates the structure connectivity of data center networks, specifically DCell and BCDC networks, with respect to common subgraph structures, providing formulas for fault-tolerance analysis.

Contribution

It introduces exact formulas for the structure and substructure connectivity of DCell and BCDC networks concerning various subgraphs, advancing fault-tolerance understanding.

Findings

Derived formulas for D_{m,n} with star and complete subgraphs.

Computed structure connectivity for B_n with specific subgraphs.

Enhanced understanding of fault-tolerance in data center network topologies.

Abstract

Last decade, numerous giant data center networks are built to provide increasingly fashionable web applications. For two integers $m\geq 0$ and $n\geq 2$, the $m$-dimensional DCell network with $n$-port switches $D_{m,n}$ and $n$-dimensional BCDC network $B_{n}$ have been proposed. Connectivity is a basic parameter to measure fault-tolerance of networks. As generalizations of connectivity, structure (substructure) connectivity was recently proposed. Let $G$ and $H$ be two connected graphs. Let $\mathcal{F}$ be a set whose elements are subgraphs of $G$, and every member of $\mathcal{F}$ is isomorphic to $H$ (resp. a connected subgraph of $H$). Then $H$-structure connectivity $\kappa(G; H)$ (resp. $H$-substructure connectivity $\kappa^{s}(G; H)$) of $G$ is the size of a smallest set of $\mathcal{F}$ such that the rest of $G$ is disconnected or the singleton when removing $\mathcal{F}$. Then it is meaningful to calculate the structure connectivity of data center networks on some common structures, such as star $K_{1,t}$, path $P_k$, cycle $C_k$, complete graph $K_s$ and so on. In this paper, we obtain that $\kappa (D_{m,n}; K_{1,t})=\kappa^s (D_{m,n}; K_{1,t})=\lceil \frac{n-1}{1+t}\rceil+m$ for $1\leq t\leq m+n-2$ and $\kappa (D_{m,n}; K_s)= \lceil\frac{n-1}{s}\rceil+m$ for $3\leq s\leq n-1$ by analyzing the structural properties of $D_{m,n}$. We also compute $\kappa(B_n; H)$ and $\kappa^s(B_n; H)$ for $H\in \{K_{1,t}, P_{k}, C_{k}|1\leq t\leq 2n-3, 6\leq k\leq 2n-1 \}$ and $n\geq 5$ by using $g$-extra connectivity of $B_n$.