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.

💡 Research Summary

The paper addresses the fundamental limitation of traditional static‑graph models for wireless ad‑hoc networks by introducing a dynamic multi‑digraph that captures the time‑varying set of transmitters under slotted ALOHA with half‑duplex radios. Nodes are assumed to be distributed as a Poisson point process of density λ, each attempting transmission independently with probability p in every slot. When a node transmits, directed edges are created toward all receivers within a communication radius r that successfully decode the packet; success depends on the SINR threshold and, consequently, on the presence of simultaneous interferers.

The authors first derive the exact in‑degree and out‑degree distributions of the dynamic digraph. In the noise‑limited regime (interference ignored) the success probability is essentially one, and both in‑ and out‑degrees follow a Poisson distribution with mean λπr²p. When interference is accounted for, the success probability becomes p·(1‑q), where q is the probability that a concurrent transmission within the interference region causes a decoding failure. The degree distribution remains Poisson but with a reduced mean λπr²p(1‑q).

A central contribution is the definition of “path formation time” – the minimum number of slots required for a causal directed path to appear between a source and a destination. This metric is equivalent to a first‑passage percolation (FPP) problem on a time‑evolving random graph. By invoking percolation theory, the authors show that if the transmission probability p exceeds a critical value p_c (which depends on λ, r, and the interference model), a giant component emerges in the infinite‑plane limit. Within this giant component, the expected first‑passage time between two nodes separated by Euclidean distance d scales linearly:

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