## 4 Topology Control
### Pre-Deployment
#### 4.2.1 Random Deployment  
If the sensor nodes are randomly deployed, it is commonly assumed that the node position has a uniform distribution. The question is how many neighbors (or node degrees) a node needs so that the whole network will be \(k-connected\). In the uniform distribution network, [59] states that six to 10 neighbors are able to make sure that the network is connected with high probability. The work in [60] concludes that the \"lower bound\" to make a network disconnected and the \"upper bound\" to make a network connected are \(0.074\log n\) and \(5.1774\log n\), respectively, where \(n\) is the total number of nodes. The work in [61] observes that the upper bound is \(\alpha e\log n\), where \(\alpha>1\), \(e\) is the naturalbase. The work in [62] shows that if \(k\leq 0.3043\log n\), then the graph is not connected with a high probability, and if \(k\geq 0.5193\log n\), the network is connected with a high probability as \(n\rightarrow+\infty\). The \(k-connected\) network is asymptotically the same as that for \(1-connected\)[63], which indicates that once the network  
Figure 5: Topology control: adjust the transmission range of Node 3 from R1 to R2 in order to improve connectivity. Figure 6: Topology control: the new Node 6 is added to recover the connection. is \(1-connected\), it achieves \(k-connected\) immediately in the uniform distribution network. However, those asymptotical conclusions hold theoretically only when \(n\rightarrow+\infty\), which is not so practical. Our previous study [40] leverages a probabilistic approach to analyze the network connection probability and proves that if the probability of the network being connected is 0.36\(\varepsilon\), then the probability of the network being connected is at least \(1-\varepsilon\) by means of increasing the communication range by constant \(C(\varepsilon)\), where \(0<\varepsilon<e^{-1}\). Explicit function \(C(\varepsilon)\) is found. Furthermore, the localized control algorithms based on fuzzy logic are proposed in our paper [46,64] to achieve the desired node degree. The simulation results show that the average node degree can be achieved in dynamic WSNs due to the closed-loop feedback of the control system. The proposal in [46] improves the fault-tolerate performance in the presence of random attacks. By combining the theory in [40] with the control algorithms in [46,64], a network being \(k-connected\) with a high probability can be designed, which will tolerate at most \(k-1\) node failures. If the nodes are uniformly and randomly deployed, the degree distribution follows the Poisson distribution, but in real life, degree distributions are strongly non-Poisson distribution, often taking the power law, truncated power-law or exponential forms [65]. The work in [66] shows that the degree of nodes in technological networks, such as the World Wide Web, the Internet and airplane connection networks, is a power law distribution. Some biological systems, such as metabolic networks and protein networks, are different from the uniform distribution networks and have a power law degree distribution with an exponent that ranges between two and three. The power law degree distribution is more vulnerable to deliberate attack than the Poisson degree distribution, but the power law degree distribution is more resilient to random failure than the Poisson degree distribution [65,66,67]. The reason is straightforward. For the uniform distribution networks, each node contributes equally to the network, so randomly removing nodes causes the same amount of damage. By contrast, for the non-uniform distribution network, each node plays a different role and attacks the most connected nodes (namely higher degree) causing heavy damage. The work in [68] shows a real case of an earthquake and tsunami that happened in Japan. In this case, the Internet was impressively resilient to the disaster due to its heterogeneity.