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Delay-capacity tradeoffs for mobile networks have been analyzed through a number of research works. However, Le&#x0301;vy mobility known to closely capture human movement patterns has not been adopted in such work. Understanding the delay-capacity tradeoff for a network with Le&#x0301;vy mobility can provide important insights into understanding the performance of real mobile networks governed by human mobility. This paper analytically derives an important point in the delay-capacity tradeoff for Le&#x0301;vy mobility, known as the critical delay. The critical delay is the minimum delay required to achieve greater throughput than what conventional static networks can possibly achieve (i.e., O(1/&#x221A;n) per node in a network with n nodes). The Le&#x0301;vy mobility includes Le&#x0301;vy flight and Le&#x0301;vy walk whose step-size distributions parametrized by &#x03B1; &#x2208; (0,2] are both heavy-tailed while their times taken for the same step size are different. Our proposed technique involves: 1) analyzing the joint spatio-temporal probability density function of a time-varying location of a node for Le&#x0301;vy flight, and 2) characterizing an embedded Markov process in Le&#x0301;vy walk, which is a semi-Markov process. The results indicate that in Le&#x0301;vy walk, there is a phase transition such that for &#x03B1; &#x2208; (0,1), the critical delay is always &#x0398;(n[1/2]), and for &#x03B1; &#x2208; [1,2] it is &#x0398;(n[(&#x03B1;)/2]). In contrast, Le&#x0301;vy flight has the critical delay &#x0398;(n[(&#x03B1;)/2]) for &#x03B1; &#x2208; (0,2].

On the Critical Delays of Mobile Networks Under Lévy Walks and Lévy Flights