New results on connectivity in wireless network

• Extension of ways of connection to 13 compared to 4 possible ways in prior arts.
• Uni-direction connection is considered instead of bi-direction connection.
• Novel small structures are proposed to help derive desired theorems.
• New conditions for percolation is established.
• New conditions for full connection is established.

In this work, we discuss when two users are able to exchange multimedia content and when that exchange is always possible. Two thresholds are of interest, a threshold for percolation and a threshold for full connectivity. To derive these thresholds, when discussing the bond percolation, we consider not only four directions, upper, right, left, and lower, but also more directions like upper-right, lower-right, upper-left, lower-left. In addition, small structures are proposed to help obtain the necessary conditions for percolation and full connectivity. We find that if the probability of four directions being open is greater than 0.3118 or the transmission radius of every node is longer than 1.52d/k1.52d/k, we can find a crossing path in our network and the percolation ensues. Furthermore, if that is greater than 0.3799 or the transmission radius of every node is longer than 1.579d/k1.579d/k, a fully-connected network graph exists for sure.