A Study of the Percolation Threshold for k-Collaborative Wireless Networks

How to efficiently use the air interface is a crucial issue in wireless networks. In order to improve the performance, mechanisms have been proposed to improve the reach and the connectivity of nodes in a wireless network. One such mechanism is to use joint, synchronized transmission from a cluster of several nodes to reach nodes outside of the transmission range of any of the transmitting nodes in the cluster. We adopt a simplified model of collaboration where the power at the receiver is the sum of the transmitted power at the sender, and study the performance of such system with clusters of arbitrary size k. We compute theoretical bounds on the gain achieved using collaboration as a function of the cluster size. Our key result is to show that, in a percolation framework, the critical node density for the infinite connectedness of the network is significantly reduced by the use of k-cooperation: for large k, it is reduced by a factor which we show to converge towards alpharadiczeta(alpha)², where zeta(.) is the Riemann zeta function. The previously known best bound on the gain of cooperation was 5/4 = 1.25 for alpha = 2 and pairwise cooperation, while our results yield a provable gain of at least pi²/6 = 1.64. We provide some simulations to display the gain of cooperation in a finite size network as well.