  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 $0< r <1$.
