# Brain-Inspired Data Transmission in Dense Wireless Network
## 4 Network Reliability Assessment through the Percolation Theory
### Application of the Percolation Theory  
Stochastic geometry allows modeling and interpreting the impact of various processes and the random variables characterized by various spatial distributions. In other words, the impact of randomness in the spatial domain can be evaluated. Among various tools known from stochastic geometry, percolation theory allows specifying the value of the probability when the network will become blocked (no path may be identified between the source and the destination) for a given network structure (i.e., spatial distribution of nodes) [40; 41; 42]. As a physical phenomenon, percolation describes the mechanism by which porous material is permeated. A very common example here may be the coffee brewing process, where the water flows through coffee beans to extract the appropriate ingredients. Depending on the type of coffee and its amount (densely or loosely grounded coffee, amount of coffee, etc. ), the percolation process will run in a different way. One may apply percolation theory to describe this process, where the liquid flow is defined by the path in the graph, and the graph represents the material (coffee in this example). In the case of wireless networking, the theory of percolation allows analyzing the message passage through networks with various densities and spatial distributions, which we can call permeability. In that context, one may observe that for a given material (network structure) there exists a critical parameter called percolation threshold, which specifies the limit, and exceeding this limit informs about the lack of permeability. In particular, the percolation threshold defines the percentage of nodes that must be active to guarantee permeability. Thus, for a given set of nodes \(N\) and the achieved percolation threshold \(p_{T}\), the minimum required a number of nodes that guarantee the proper functioning of the network is equal to \(\lfloor p_{T}\cdot N\rfloor\), where \(\lfloor\cdot\rfloor\) is the floor function. Let us assume that the wireless network is represented in form of a graph, where the wireless transceiver corresponds to the node and the valid wireless link to the graph edge. Assuming some fixed maximum transmit power, the corresponding transmission range will be achieved. One may observe that, for the fixed range, when the density of nodes decreases, the probability of lack of permeability decreases as well. Analogous observations can be made when reducing the effective transmission range (due to, e.g., reduced transmit power or increased observed interference at the reception node). If the permeability is not achieved, the network graph contains some clusters (subgraphs). Following the authors of [40; 41; 42], the percolation threshold is reached at the moment when the size of the second-largest cluster (denoted hereafter as SG) in the graph reaches its maximum. Following [18], we derive the percolation threshold for the dense wireless node to show the reliability of the bio-inspired communications schemes, however, in this paper we consider a significant aspect, mainly the presence of self-interference, as illustrated in Figure 1. We assume that the network nodes are distributed according to the spatial Poisson process and are operating asynchronously. In our simplified case, we state that there exists a reliable wireless link between two wireless nodes, when the observed signal-to-interference-plus-noise ratio SINR is above some threshold \(\theta^{*}\), i.e., \(SINR\geq\theta^{*}\). As in [42], the desired signal may be expressed as \(S=\frac{P_{0}\prod_{k}X_{k}}{R^{2k}}\), where \(X_{k}\) represent the existing propagation effects in the wireless channel (such as path loss \(X_{k}=1\) or log-normal fading \(X_{k}=e^{2\pi G}\), for \(G\) being the normally distributed random variable of zero mean and unit variance). Moreover, \(R\) is the distance and \(b\) amplitude loss exponent.