## 4 Analysis of Phase Transitions in the MESH Network  
The simulations we conducted were based on randomly generated topologies of wireless networks. Thus, the random distribution of nodes determined the necessary power of the transmitters in order to ensure a proper set of connections. Taking into account the development of technology, antennas, protocols, and communication standards during the simulation, a general assumption was made in which the radius of a given node means \(r\in\langle 0;1\rangle\), where the value 1 means the guarantee of connection of each node with each in the infrastructure. Thus, this case corresponds to the topology, which is characterized by a high rate of connection redundancy (full mesh). The sample mesh topologies for different numbers of nodes and \(r=0.5\) are presented in Figure 5. It should be noted that the number of  
Figure 4: Tolerance to maintain connectivity in a wireless network.  
Figure 3: The power range of a transmitter in the context of providing connectivity.  
nodes randomly distributed in a given area affects the distances between them; therefore, it will affect the process of the phase transition.  
The simulations carried out indicate that as the radius value decreases, the number of available connections decreases. In addition, with a smaller number of nodes, there is a higher probability that the number of available connections will be significantly reduced with a relatively small change in the radius value. This dependence is shown in Figure 6 (for the purposes of the paper, this chart will be referred to as PTC; phase transition chart), which shows the relationship between the probability of connection and the radius value. These charts confirm previous observations. Thus, a smaller number of nodes causes a faster phase change (see Figure 6a), while the interface is much wider. However, in the case of a large number of nodes, e.g., 60 (see Figure 6d), the probability of providing connections (\(p=1\)) is much larger and the interphase is relatively narrow. Thus, the dynamics of the phase change process in the interfacial area is much larger, and a small change in the transmitter's power allows the meaning to reach the stability/consistency of the topology faster.  
Figure 5: Exemplary mesh topologies for the radius of 0.5 for: (**a**) \(n=10\); (**b**) \(n=25\); (**c**) \(n=60\).  
Figure 6: _Cont._  
Figure 7 presents selected simulation stages in which for a selected number of nodes, \(n=25\), the radius value was gradually reduced from 1 to 0.1. For a radius value of approximately 0.4, the interfacial area begins, which lasts up to approximately 0.2. At the same time, we obtain an incoherent structure at 0.2, so we are in the unstable phase from the perspective of the network structure.  
Figure 6: Phase transition in terms of providing connectivity in the MESH topology for values of: (a) \(n=3\); (b) \(n=10\); (c) \(n=25\); (d) \(n=60\).  
Figure 7: _Cont._  
As a result of the tests, it can be clearly stated that in the case of phase transition in the structure of the MESH wireless network, the interfacial area indicated in Figure 3 has no fixed size, and thus, for different distribution of nodes, may be larger or smaller. This is important for maintaining the stability of the network structure and for providing higher layer services. In the interfacial area, optimal parameters can be found to achieve, e.g., high energy savings, but it can also include parameters that will allow high redundancy of connections and devices.