Dynamic Connectivity in ALOHA Ad Hoc Networks
  In a wireless network the set of transmitting nodes changes frequently because of the MAC scheduler and the traffic load. Previously, connectivity in wireless networks was analyzed using static geometric graphs, and as we show leads to an overly constrained design criterion. The dynamic nature of the transmitting set introduces additional randomness in a wireless system that improves the connectivity, and this additional randomness is not captured by a static connectivity graph. In this paper, we consider an ad hoc network with half-duplex radios that uses multihop routing and slotted ALOHA for the MAC contention and introduce a random dynamic multi-digraph to model its connectivity. We first provide analytical results about the degree distribution of the graph. Next, defining the path formation time as the minimum time required for a causal path to form between the source and destination on the dynamic graph, we derive the distributional properties of the connection delay using techniques from first-passage percolation and epidemic processes. We consider the giant component of the network formed when communication is noise-limited (by neglecting interference). Then, in the presence of interference, we prove that the delay scales linearly with the source-destination distance on this giant component. We also provide simulation results to support the theoretical results. 
RadhaKrishna Ganti and Martin Haenggi