## 5 Application  
One of the basic areas of active management of a distributed computer system is the ongoing analysis of the consistency of communication network. It is particularly important from the perspective of dynamic structures based on radio communication for Industry 4.0 and IoT systems. One of the main pillars for the development of production lines designed in the spirit of industry 4.0 is the use of in-depth sensory diagnostics. For economic reasons, it is not possible to replace existing manufacturing components (machines) with new ones equipped with the desired sensors. One of the proposed solutions is the use of additional sensors placed on crucial elements of the machines that transfer data to the control system, anomaly detection, etc. Such a solution was proposed, e.g., in the paper [45], where the authors present the use of high-sensitivity LiNbO\({}_{3}\) vibration sensors for robotic arm monitoring. The MESH network was built to create a communication layer for the sensors. The possibilities of using MESH networks to connect IoT elements are obviously much wider and their use is practically unlimited, especially for data collection networks. Some of the industrially-used solutions use battery-powered MESH infrastructure elements. Their lifetime is limited, but a failure of one of the network elements does not cause its inconsistency. The node coherence of such networks is usually many times higher than 2. In order to manage such networks properly, it is necessary to develop a method for estimating the current network integrity. Of course, measures such as graph coherence or connectivity (understood as the ratio of the edges available in the topology being analyzed to the number of edges in the corresponding complete graph) can be used. However, their use does not determine the probability of loss of consistency in a holistic approach, but is rather used to determine whether the network is working or not. To assess the current ability to provide network communication, the authors proposed architecture and an algorithm based on the phase transition analysis discussed earlier. The system architecture is presented in Figure 8.  
Figure 7: Structural phase transition for mesh topology at \(n=25\): (**a**) \(r=1\); (**b**) \(r=0.65\); (**c**) \(r=0.28\); (**d**) \(r=0.2\); (**e**) \(r=0.1\); (**f**) probability of combination depending on \(r\). The key element of the system is the arbiter; i.e., the application that collects and controls the state of the MESH \(N_{1}\) network on an ongoing basis using three types of messages:  
* **Actual node radius**--in this message the nodes in the network transmit the current value of the transmitter power which is then transformed to the form of \(r\in\langle 0;1\rangle\). * **Keep alive**--is information periodically sent to the arbiter by each network node, informing it about its proper functioning, similar to the mechanisms used in classic routing protocols. The node may initiate sending information periodically or create a response to a message sent by the arbiter. It depends on the specificity of the network and the protocols used. In case this message is not received, the arbiter acknowledges that the node is unreachable and a response from technical staff is required. It is possible to define a threshold value for the number of unreceived messages, which allows us to conclude that a given node is permanently unavailable. * **Position**--this message provides information about the current position of the node (its location). The message is important for networks with mobile elements and should be treated as options. For static networks, the location of static nodes can be implemented at the network construction stage and remembered by the arbiter. Based on this data, the arbiter creates a virtual network model referred to as \(VN_{1}\) in the Figure 8. The model created in this way is passed for further analysis to the algorithm (blue blocks in Figure 8).