## 4 Communication Network Interdiction
Then to find the minimum number of jamming devices that will allow covering \(\alpha\cdot 100\%\) of the network nodes with prescribed levels of jamming \(C_{i}\), we must solve the following integer program \[\min\sum_{j=1}^{n}c_{j}x_{j}\] (40) s.t. \[\sum_{i=1}^{m}y_{i}\geq\alpha m, \tag{41}\] \[\sum_{j=1}^{n}d_{ij}x_{j}\geq C_{i}y_{i},\quad i=1,2,\ldots,m,\] (42) \[x_{j}\in\{0,1\},\quad j=1,2,\ldots,n,\] (43) \[y_{i}\in\{0,1\},\quad i=1,2,\ldots,m. \tag{44}\]  
The \(\alpha\)-CVaR optimization model for network covering can be formulated as a mixed integer linear program using a standard linearization framework:  
\[y_{ij}\text{ is consistent with the network and }z_{i}\]0 (45) s.t. \[y_{ij}\text{ is consistent with the network and }z_{i}\]1 \[y_{ij}\text{ is consistent with the network and }z_{i}\]2 (47) \[y_{ij}\text{ is consistent with the network and }z_{i}\]3  
The VaR and CVaR models can also be written for connectivity suppression models in the similar fashion. We refer the reader to [33] for details. The deterministic formulations of the wireless network jamming problem are extended in [34] to tackle the stochastic jamming problem formulations using percentile type constraints. These formulations consider the case when the exact topology of the network to be jammed is not known, but we know the distribution of network parameters. Since the exact locations of the network nodes are unknown, it is assumed that a set of intelligence data has been collected and from that a set \(\mathcal{S}\) of the most likely scenarios have been compiled. Scenario \(s\in\mathcal{S}\) contains both the node locations \(\{(\xi_{1}^{s},\eta_{1}^{s}),(\xi_{2}^{s},\eta_{2}^{s}),\ldots,(\xi_{m}^{s}, \eta_{m}^{s})\}\) and the set of jamming thresholds \(\{C_{1}^{s},C_{2}^{s},\ldots,C_{m}^{s}\}\). For each scenario \(s\in\mathcal{S}\), the set \(\mathcal{M}_{s}=\{1,2,\ldots,m_{s}\}\) needs to be jammed. Taking into account all the scenarios we can write a mathematical program for node covering problem:  
\[y_{ij}\text{ is consistent with the network and }z_{i}\]4 (49) s.t. \[y_{ij}\text{ is consistent with the network and }z_{i}\]5 \[y_{ij}\text{ is consistent with the network and }z_{i}\]6  
It is unlikely to find the solution that can provide effective jamming strategy for all scenarios. Therefore, the notion of percentile-based risk measures can be utilized to develop formulations of the robust jamming problems incorporating these risk constraints. The robust node covering problem with Value-at-Risk constraints can be formulated as  
\[y_{ij}\text{ is consistent with the network and }z_{i}\]7 (52) s.t. \[y_{ij}\text{ is consistent with the network and }z_{i}\]8 \[y_{ij}\text{ is consistent with the network and }z_{i}\]9 (54) \[x_{j}\in\{0,1\},\;\forall\;j\in\mathcal{N},\]0 (55) \[x_{j}\in\{0,1\},\;\forall\;j\in\mathcal{N},\]1  
The loss function can be considered as the difference between the energy required to jam network node \(i\), namely \(C_{i}^{s}\), and the cumulative amount of energy received at node \(i\) due to \(x\) over each scenario. With this the robust node covering problem with CVaR constraints is formulated as follows. \[x_{j}\in\{0,1\},\;\forall\;j\in\mathcal{N},\]2 (57) s.t. \[x_{j}\in\{0,1\},\;\forall\;j\in\mathcal{N},\]3 \[x_{j}\in\{0,1\},\;\forall\;j\in\mathcal{N},\]4 (59) \[x_{j}\in\{0,1\},\;\forall\;j\in\mathcal{N},\]5 (60) \[x_{j}\in\{0,1\},\;\forall\;j\in\mathcal{N},\]6  
The CVaR constraint (59) implies that for the \((1-\alpha)\cdot 100\%\) of the worst (least) covered nodes, the average value of \(f(x)\) is less than or equal to \(0\). Numerical experiments demonstrate that problems of moderate size can be solved by ILOG CPLEX. Finding effective heuristics and approximations for sensors network jamming problems of large dimensions is an interesting and important research direction.