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The study in mobile ad-hoc networks (MANET) is facing challenges brought by recent discovery of non-exponential behavior of the inter-contact time distribution of mobile nodes. In this paper, we analyze various characteristics of the relative mobility of a random pair of nodes in MANET to show that they produce inter-contact time with different aging properties. First, by fixing one node and resorting to the random walks on directed graphs, we mathematically prove that under four classes of stochastic mobility patterns, the resulting inter-contact times have constant/decreasing/increasing failure rate and new-better-than-used property. Then, we consider the case when both nodes are mobile and use simulation results to uncover the aging property of their inter-contact times under random waypoint models and random walk mobility models. This aging property tells us how to correctly relate the past experience of mobile nodes with their future behavior, thereby allowing tremendous opportunities brought by the memory structure in the non-exponential inter contact time, which would be impossible under the widely assumed exponentially distributed (memoryless) inter-contact time. As an application of our results, we establish for the first time that the approach based on exponential inter-contact time assumption can either under-estimate or over-estimate the actual system performance, under different stochastic mobility patterns indexed by their aging properties. Our results on aging properties also provide theoretic guidelines on how to exploit the memory structure toward better design of protocols under general mobility.