A necessary condition for connected wireless CSMA multi-hop networks

Connectivity is one of the most fundamental properties of wireless multi-hop networks. In a wireless network with many concurrent transmissions, signals transmitted at the same time may mutually interfere with each other. In this paper we consider the impact of interference on the connectivity of CSMA networks using the SINR model. On the basis of our earlier work in which we give a sufficient condition, i.e. an upper bound, on the critical transmission power required for a CSMA network with a total of n nodes i.i.d. on a √n × √n square following a uniform distribution to be a.a.s. connected as n → ∞ under the SINR model, in this paper we continue to study the necessary condition for the above CSMA network to be a.a.s. connected. A lower bound is obtained on the critical transmission power required for the above CSMA network to be a.a.s. connected under any scheduling scheme satisfying the carrier-sensing constraint. The lower bound differs from the upper bound by a constant factor only. Compared with previous literature assuming a unit disk model, it is shown that the critical transmission power for a CSMA network under the SINR model to be a.a.s. connected is within a constant factor of that required for a network under the unit disk model, which does not consider the impact of interference, to be a.a.s. connected. That is, transmission power only needs to be increased by a constant factor to combat interference and maintain connectivity. This result is also in sharp contrast with previous results considering the connectivity of ALOHA networks under the SINR model.