Modeling Diversity

To better understand the impact of diversity skew, we simulate a system of hosts with various attributes . On the Internet, most hosts run some version of Windows with Internet Explorer as the web browser [8], so we biased the attribute distribution towards having some fixed subset of attributes. The size of this subset depends on the value $f$ chosen for the diversity of the system. To see this, consider a subset of size $\alpha$. Assuming that each attribute has $y$ possible values, for such a subset the total number of distinct configurations is $y^{\vert A\vert-\alpha}$. Thus, there is some integer $\alpha$, $\alpha \leq \vert A\vert$, that satisfies the following equation:

$$\frac{y^{\vert A\vert-\alpha}}{y^{\vert A\vert}} \geq (1-f) > \frac{y^{\vert A\vert-(\alpha + 1)}}{y^{\vert A\vert}}$$

$$\rightarrow \frac{1}{y^\alpha} \geq (1-f) > \frac{1}{y^{\alpha + 1}}.$$ (1)

In our simulations, we compute the value of $\alpha$ using Equation 1, and then pick a subset $A' \subseteq A$ of attributes such that $\vert A'\vert = \alpha$. For every attribute $a$ in $A'$, we fix the value of $a$ for a fraction $f$ of the hosts. We then randomly choose values for the remaining attributes for this fraction of hosts. For the remaining hosts, we pick attribute configurations at random, but we make sure that each configuration is not a configuration of any host in the first fraction.