A Novel Measurement for Network Reliability

Abstract

The attackers in a network may have a tendency of targeting on a group of clustered nodes, and they hope to avoid the existence of significant large communication groups in the remaining network, such as botnet attack, DDoS attack, and Local Area Network Denial attack. Current various kinds of connectivity do not well reflect the fault tolerance of a network under these attacks. This observation inspires a new measure for network reliability to resist the block attack by taking into account of the dispersity of the remaining nodes. Let $G$ G be a network, $C\subset V(G)$ C ⊂ V ( G ) and $G[C]$ G [ C ] be a connected subgraph . Then $C$ C is called an $h$ h -faulty-block of $G$ G if $G-C$ G - C is disconnected, and every component of $G-C$ G - C has at least $h+1$ h + 1 nodes. The minimum cardinality over all $h$ h -faulty-blocks of $G$ G is called $h$ h -faulty-block connectivity of $G$ G , denoted by ${FB}\kappa _h(G)$ F B κ h ( G ) . In this article, we determine ${FB}\kappa _h(Q_n)$ F B κ h ( Q n ) for $n$ n -dimensional hypercube $Q_n$ Q n ( $n\geq 4$ n ≥ 4 ), a classic interconnection network. We establish that ${FB}\kappa _h(Q_n)=(h+2)n-3h-1$ F B κ h ( Q n ) = ( h + 2 ) n - 3 h - 1 for $0\leq h\leq 1$ 0 ≤ h ≤ 1 , and ${FB}\kappa _h(Q_n)=(h+2)n-4h+1$ F B κ h ( Q n ) = ( h + 2 ) n - 4 h + 1 for $2\leq h\leq n-2$ 2 ≤ h ≤ n - 2 , respectively. Larger $h$ h -faulty-block connectivity implies that an attacker will have to stage an attack to a bigger block of connected nodes, so that each remaining components will not be too small, which will in turn limit the size of large components. In other words, there will not be great disparity in sizes between any two remaining components, and hence there will less likely be a significantly large remaining communication group. The larger the $h$ h -faulty-block, the more difficult for an attacker to achieve that goal. As a consequence, the resistance of the network against the attacker will increase. Our experiments also show that as $h$ h increases, the $h$ h -faulty-block gets larger, and the size disparity between any two remaining components decreases. In turn, as expected, the size of the largest remaining communication group becomes smaller.