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Delay-capacity tradeoffs for mobile networks have been analyzed through anumber of research work. However, L\'{e}vy mobility known to closely capturehuman movement patterns has not been adopted in such work. Understanding thedelay-capacity tradeoff for a network with L\'{e}vy mobility can provideimportant insights into understanding the performance of real mobile networksgoverned by human mobility. This paper analytically derives an important pointin the delay-capacity tradeoff for L\'{e}vy mobility, known as the criticaldelay. The critical delay is the minimum delay required to achieve greaterthroughput than what conventional static networks can possibly achieve (i.e.,$O(1/\sqrt{n})$ per node in a network with $n$ nodes). The L\'{e}vy mobilityincludes L\'{e}vy flight and L\'{e}vy walk whose step size distributionsparametrized by $\alpha \in (0,2]$ are both heavy-tailed while their timestaken for the same step size are different. Our proposed technique involves (i)analyzing the joint spatio-temporal probability density function of atime-varying location of a node for L\'{e}vy flight and (ii) characterizing anembedded Markov process in L\'{e}vy walk which is a semi-Markov process. Theresults indicate that in L\'{e}vy walk, there is a phase transition such thatfor $\alpha \in (0,1)$, the critical delay is always $\Theta (n^{1/2})$ and for$\alpha \in [1,2]$ it is $\Theta(n^{\frac{\alpha}{2}})$. In contrast, L\'{e}vyflight has the critical delay $\Theta(n^{\frac{\alpha}{2}})$ for$\alpha\in(0,2]$.

On the Critical Delays of Mobile Networks under L\'{e}vy Walks and L\'{e}vy Flights