New Results on the Connectivity in Wireless Ad Hoc Networks

We first investigate when it is possible for two nodes in a wireless network to communicate with each other. Based on the result from bond percolation in a two-dimensional lattice, as long as the probability that a sub-square is close is less than 0.5 and each sub-square contains at least four nodes, percolation occurs. Following that, we establish the conditions for full connectivity in a network graph. How two adjacent sub-squares are connected differentiates this work from others. Two adjacent sub-squares are connected if there exists a communicating path between them instead of a direct communication link. The full connectivity occurs almost surely if each sub-square contains at least one node and the probability of having an open sub-edge is no less than 0.3822. Simulations are conducted to validate the proposed conditions for percolation and full connectivity. We also apply the results to SINR model [1]. We find each node can tolerate more interference than that stated in [1]. Last, we extend the results to the case of unreliable transmission. Under this situation, increasing nodes per sub-square or maintaining the probability of successful transmission above a certain threshold seems to be two possible approaches to achieve percolation and full connectivity.