# Secure Localization in the Presence of Colluders in WSNs
## 6 Experimental Results
### First Set of Simulations: Uniform Deployment of Nodes  
In this first set of simulations, we assume that a set of sensor nodes is uniformly deployed. Figure 4 depicts this scenario, where solid black dots are representing the anchor (i.e., sensor nodes equipped with GPSs) nodes (e.g., \(A\), \(C\), \(D\), \(E\)), and grey dots represent regular location-known sensors (e.g., \(B\)). The communication range of an anchor node is assumed to be \(2d\), where \(d\) equal to the distance between an anchor and any of its surrounding location-known sensor nodes. The communication range of a regular location-known sensor is \(\sqrt{2}d\).  
Any anchor node or regular sensor node may be a colluder. White-patterned-black dots represent location-unknown sensor nodes, e.g., sensor \(U\). The communication range of a location-unknown sensor node is also \(\sqrt{2}d\). We consider the worst case, in which the third category of colluders defined in Section 3 conduct the attacks. Under this scenario, node \(U\) (assumed to be a node that initially does not know its own position in the WSN) succeeds at detecting the attack of \(B\) if there are at least two more truth-tellers placed within the intersection of discs \(d_{B}\) and \(d_{U}\) (respectively centred at \(B\) and \(U\)) than the number of colluders.  
Simulation results are presented in Figures 5a and 6. Each simulation represents the deployment, on a 12-cell grid, of 99 location-known nodes (20 sensor nodes are anchors and the remainder 79 are regular location-known nodes) and 12 location-unknown sensor nodes. Any location-known sensor can be a colluder. At every run, attackers are distributed randomly and uniformly within every cell of the grid (e.g., cells depicted as \(ACDE\) in Figure 4). Figure 5a shows the detection rate of location-unknown nodes that successfully detect colluders after executing algorithm CC or SCC, under the uniform sensor deployment scenario with incremental percentage of location-known sensors acting as colluders ranging from 10% to 90%. Results are measured using 100 runs per each percentage-of-attackers case and 95% confidence intervals. Success means that all colluders in the neighbourhood of a location-unknown node (executing either the CC or SCC algorithm) are successfully removed from the final list of truth-telling nodes. Figure 6 shows different types of failures when executing each algorithm under the uniform sensor deployment scenarios. Counted failures are cases in which either colluders are not properly removed from the final list of truth-telling nodes or when the final list ends being wrongly empty. Failures are due to the fact that the detection rate of both algorithms is not always 100% in our simulations. This is because in our simulations, we are not filtering out topologies in which the hypotheses of the algorithms are not met. In order to maximize the detection rate for SCC, the rate of category three colluders among all other colluder types shall be inversely proportional to the rate of colluders among all anchor nodes. Since attackers are distributed randomly and uniformly in every simulation run, a percentage increase of colluders also represents an increase of detection failure due to hypothesis violations (e.g., percentage of truth-telling nodes less than twice the percentage of colluders in the intersection of discs \(d_{A}\) and \(d_{B}\) centred at \(A\) and \(B\) in Figure 4). The _advantage_ curve in Figure 5a represents the difference between the success rates of SCC and CC. It confirms that the detection success rate of SCC always outperforms that of CC. Looking at Figure 6, we can observe that the performance increase responds to the coverage of the colluding factor, so that more combinations of attackers are identified and removed from the final list of truth-telling nodes.  
Figure 4: Uniform deployment.