Parsimonious Flooding in Geometric Random-Walks

We study the information spreading yielded by the \emph{(Parsimonious) $1$-Flooding Protocol} in geometric Mobile Ad-Hoc Networks. We consider $n$ agents on a convex plane region of diameter $D$ performing independent random walks with move radius $\rho$. At any time step, every active agent $v$ informs every non-informed agent which is within distance $R$ from $v$ ($R>0$ is the transmission radius). An agent is only active at the time step immediately after the one in which has been informed and, after that, she is removed. At the initial time step, a source agent is informed and we look at the \emph{completion time} of the protocol, i.e., the first time step (if any) in which all agents are informed. This random process is equivalent to the well-known \emph{Susceptible-Infective-Removed ($SIR$}) infection process in Mathematical Epidemiology. No analytical results are available for this random process over any explicit mobility model. The presence of removed agents makes this process much more complex than the (standard) flooding. We prove optimal bounds on the completion time depending on the parameters $n$, $D$, $R$, and $\rho$. The obtained bounds hold with high probability. We remark that our method of analysis provides a clear picture of the dynamic shape of the information spreading (or infection wave) over the time.