On proportional network connectivity

The reliability of a network is an important parameter to consider when building a network. Different characteristics of the network can become unreliable over time or from other outside forces. In a simple setting, we model a network as a graph where the vertices represent our objects and a connection between these objects are represented by an edge. Generally there are two things to consider when discussing the reliability of a network. The first is the conditions that need to be satisfied in order for the network to be operational. We can also consider what properties need to be satisfied in order to render our network inoperable. The second thing we need to consider is what properties of our network tend to fail. In certain applications edges are prone to failure and in others vertices are prone to failure. One of the first examples of a network reliability measure is the connectivity of a graph, which measures the minimum number of vertices (or edges) which can be removed in order to disconnect the graph. In this paper we extend this idea, however, instead of considering a network to be operational if there is a component of order larger than a fixed size, k, we define the network to be operational if there is a component of order at least some proportion of the original order. So as our networks become larger, we will need a component of proportionally larger order to remain if the network is to be in an operating state. This connectivity measure has been studied for $r=1/2$ and for the purposes of VLSI circuit design, but we explore the measure for all $0< r <1$.