Results

To remove noise in our experiments, each data point was averaged over 100 trials.

Figure 1(a) shows a contour plot of log of the number of nodes in the network vs the fraction of nodes saved for the first experiment. The values of the other parameters were as follows; $\beta=2$, $\gamma =0.02$, $\tau=5$ and $\alpha$ takes on all even values between 2 and 10. Since it is easier to carry out simulations for much larger values of $n$ on the pseudo-random network, we vary the number of nodes for this network from $2^{12.58}$ to $2^{25.58}$. The size of the regular network varies from $2^{12}$ to $2^{22}$. We observe that there is a very small increase in the fraction of nodes saved in the pseudo-random network as $n$ crosses $2^{18.58}$, so in all our other experiments we have limited the network size of the pseudo-random network to the size of the random network. These results suggest that our algorithms for spreading alerts might be most effective in conjunction with other techniques (like throttling) that would enable the alerts to spread more quickly than the worm. There is a very small increase in the fraction of nodes saved as $n$ grows larger, implying that the results may be better for very large values of $n$.

Figure 1: (a) contour plot of log of $\char93$ of nodes vs fraction of nodes saved for $\alpha =2,4,6,8,10$ (b)number of nodes saved/infected in each round

For the second experiment, we plotted the fraction of nodes saved/infected in each round for both kind of networks. The network sizes considered were of the order of $10^7$ nodes. Here $\gamma =0.02$, $\beta=2$, $\alpha=10$ and $\tau=5$. For runs of the simulation where the number of rounds is less than the other runs, we repeat the final values to compensate for the missing rounds, while calculating the average of 100 trials. In Figure 1(b) we see results for the second experiment. We are able to save about $93\%$ and $91\%$ of the nodes for the random and pseudo-random networks respectively. Notice that the value of $\alpha$ is five times that of $\beta$. In practice, this condition may hold since the alerts are traveling through a predetermined overlay network and a technique such as throttling can ensure that alert messages received through the overlay are given priority over other types of messages. The number of nodes saved in the pseudo-random network is less than the number of nodes saved for the random network in all of our experiments. That is expected as the expansion of the pseudo-random network may be worse than the expansion of the random network.

Figure 2: alpha vs fraction of nodes saved

For our third experiment the number of nodes are fixed at $2^{23}$($2^{23.58}$ for the pseudo-random network). Here $\beta=2$ and $\gamma =0.02$. The value of $\tau$ is always adjusted so that the number of nodes which can be alerted due to a false alert is always $10^4$. So $\tau=\lfloor\log_{\alpha}{10^4}\rfloor$. In this experiment we find an improvement of around $20\%$ in the fraction of nodes saved for values of $\alpha$ between three and seven for both the network models. The results of this experiment are given in Figure 2. We note that there is a decrease in the number of nodes saved when we go from $\alpha=6$ to $\alpha=7$. We believe this happens because the value of $\tau$ at $\alpha=6$ is five and at $\alpha=7$ it decreases to four.