## 1 Introduction
The implementation of these solutions in a real network would, unfortunately, be difficult due to the need for ongoing monitoring of all inter-node communication, which in consequence introduces large communication overheads. In [32], the authors analyzed the possibility of using \"hello messages\" to quickly detect failure in a mesh network. The paper [33] proposed how to accelerate the failure detection in MESH network nodes using data-mining-based link failure detection and using the MAC layer to predict connection failures. The literature also includes the use of a number of other mechanisms for detecting MESH network damage, such as Cooperative Watchdog [34], Software Defined Networks (SDN) [35], and the use of redundant elements [28]. Frequent exchange of control messages between nodes may be required for routing protocols that support fault tolerance to identify current network topology. Mechanisms known from wired networks (e.g., Bidirectional Forwarding Detection) can also be used in conjunction with the alternative routes table for each topology node. In the case of using tolerance and fault detection mechanisms at the level of routing protocols after damage detection or during the diagnostic process, increased communication of the MESH network nodes with the network supervision node occurs, which may adversely affect the time of failure removal and network load [35; 36; 37]. Formal methods applied to diagnose failures in wired mesh networks in the case of supercomputers and on Networks on Chip (NoCs) networks can be adapted for early detection or estimation of the probability of failure [38; 39]. However, direct application of the results is difficult due to the inability to take into account the current, variable state of the network in the proposed model. The solutions regarding MESH connection network failure described in the literature are associated with a reactive approach based on the early detection of a failure to a node or a communication channel. In the cases analyzed, the prediction of network failure is usually based on the use of artificial intelligence methods or data mining techniques to analyze sets of a wide spectrum of network operational parameters, and the traffic that is transmitted through them. There are no solutions that,based on limited information, are able to predict network failures without having to analyze all traffic parameters. The methods presented above are based on mechanisms specific to layers 2 and 3 of the ISO/OSI model. This paper proposes a method of prediction of a failure in the physical layer (topology) of the MESH network using a phase transition model with an arbiter and a set of limited diagnostic information. We concentrated on the relationship between the change in the strength of wireless transmitters (which has a direct impact on their range) and the consistency of physical topology. Thanks to this analysis, the phase changes that accompany them may be presented. The conducted simulations were carried out for random topological structures using a variable power of transmitters. The results of these simulations give a completely new look at the issue of phase transitions in computer networks and provide the basis for further research.