## 5 Application
Figure 7:0 Exemplary mesh topologies for the radius of 0.1: (**a**) \(n=100\); (**c**) \(n=150\); (**e**) \(n\)=250; (**g**) \(n=1000\); and structural phase transition for these topologies: (**b**) \(n=100\); (**d**) \(n=150\); (**f**) \(n=250\); (**h**) \(n=1000\). Figure 7:1 Exemplary mesh topologies for the radius of 0.4: (**a**) \(n=100\); (**c**) \(n=150\); (**e**) \(n=250\); (**g**) \(n=1000\); and structural phase transition for these topologies: (**b**) \(n=100\); (**d**) \(n=150\); (**f**) \(n=250\); (**h**) \(n=1000\). Analyzing the results obtained from the point of view of fault tolerance, the situation is then the opposite; i.e., in the case of Figures 11 and 12, despite the narrow interfacial area, it is characterized by high resistance to the potential for failure of individual nodes. On the other hand, in the case of Figures 6 and 7, the wide interfacial area does not guarantee consistency when the number of active nodes is reduced. Additionally, one can imagine a situation in which the interfacial area could be very narrow and almost tangential to the vertical axis; i.e., the value near \(r=0\). Such a situation is not economical, as it requires the use of a huge number of nodes. However, comparing Figures 6b, 11b and 12b, it can be stated that by oscillating radius values in the range from 0.2 to 0.6, network coherence can be achieved by using a much smaller number of nodes. Therefore, by increasing the radius value, we are also broadening the interfacial area, and thus the critical and warning areas are much wider. For example: the diagrams related to the interfacial area from Figures 11a and 12a are very similar to each other, but in the case of the topology corresponding to Figure 11a there is a loss of network consistency which results directly from a small radius value \(r=0.1\) (in the case of topology from Figure 12a \(r=0.4\)). For the example under consideration, it is important to determine the appropriate probability value \(p_{t}\), which will define the demarcation lines between the critical and warning areas. Of course, as it was previously described in this paper, this value can be iteratively determined during the system's operation; however, it should be noted that the value of \(p_{t}\) determined for the network from Figure 12a will be much higher than the value of \(p_{t}\) for the network from Figure 12b. The network management system (NMS) plays a key role in responding to a failure in the proposed architecture. In short, it is a central IT system (most often an application) for IT system management including a computer network. Recently, the paradigm of NMS systems has changed and has evolved towards SDN networks [47] dedicated to automatic network management. In the research we conducted, two mechanisms of NMS system response to critical or warning messages reported by the arbiter were considered. In the first reactive approach, NMS system informs technical staff about the need to take action. In the second case, first NMS system automatically takes independent actions to repair the network. Figure 13 proposes the architecture of the MESH system with inactive transmitters marked in red. For a given type of messages, the NSM system can activate all or select transmitters to improve connection conditions of the entire system. The application of test results presented in this chapter is one of many possible scenarios. The research conducted by the team is closely related to industrial applications, Industry 4.0, and IoT systems, which is why this class of systems has been chosen to present the possibilities of using research related to phase transition analysis in wireless networks built in MESH topology. Figure 7:2 MESH network with inactive transmitters: (**a**) simulation for a sample network; (**b**) presentation of ideas. Existing work in the literature does not deal efficiently to resolve the issue of early detection of network coherence, especially for wireless MESH networks.