Degrees of freedom of sparsely connected wireless networks

We investigate how the network connectivity can affect the degrees of freedom (DoF) of wireless networks. We consider a network of n source-destination (SD) pairs and assume that any two nodes are connected with a positive probability p, independent of other node pairs. We show that, for any arbitrarily small p, a constant DoF is achievable for every SD pair with probability approaching one as n tends to infinity. The achievability is based on the two-hop transmission with decode-and-forward relaying and over each-hop we adopt interference alignment. Considering that an achievable per-user DoF for direct or one-hop transmission can be arbitrarily small as the connectivity probability p decreases, our result shows that, somewhat surprisingly, two-hop transmission is enough to guarantee non-vanishing per-user DoF for any p showing that sparsely connected networks can still provide non-vanishing per-user DoF.