Realistic analysis of percolation in wireless network under omnidirectional transmission environment

In this work, we deal with the conditions of when the network percolates. Unlike the previous work, where establishing connections are limited to only four adjacent sub-squares (the left, right, upper and lower sub-squares), we allow the transmission connection can be established with all eight adjacent sub-squares of the sub-square of interest. To prove the condition of percolation, we find the lower bound of the probability of having a crossing path from the left of the right of a network. That lower bound is not easy to obtained. In this work, we define the small structures to help reach that lower bound. The small structures in this work are the most influential ways of connection. With the help of these structures, the critical probability of having an open sub-edge is derived, which is 0.3118. Under that critical probability, the smallest transmission radius required to achieve percolation is 1.52d/k. The simulation results have confirmed accuracy of the derived results.