In this paper we propose a distributed iterated hard thresholding algorithm termed DiFIGHT over a network that is built on the diffusion mechanism and also propose a modification of the proposed algorithm, termed MoDiFIGHT, that has low complexity in terms of communication in the network. We additionally propose four different strategies termed RP, RNP, RGP$_r$, and RGNP$_r$ that are used to randomly select a subset of nodes that are subsequently activated to take part in the distributed algorithm, so as to reduce the mean number of communications during the run of the distributed algorithm. We present theoretical estimates of the long run communication per unit time for these different strategies, when used by the two proposed algorithms. Also, we present analysis of the two proposed algorithms and provide provable bounds on their recovery performance with or without using the random node selection strategies. Finally we use numerical studies to show that both when the random strategies are used as well as when they are not used, the proposed algorithms display performances far superior to distributed IHT algorithm using consensus mechanism .