Analysis of Individual Pair and Aggregate Intercontact Times in Heterogeneous Opportunistic Networks

Foundational work in the area of opportunistic networks has shown that the distribution of intercontact times between pairs of nodes has a key impact on the network properties, for example, in terms of convergence of forwarding protocols. Specifically, forwarding protocols may yield infinite expected delay if the intercontact time distributions present a particularly heavy tail. While these results hold for the distributions of intercontact times between individual pairs, most of the literature uses the aggregate distribution, i.e., the distribution obtained by considering the samples from all pairs together, to characterize the properties of opportunistic networks. In this paper, we provide an analytical framework that can be used to check when this approach is correct and when it is not, and we apply it to a number of relevant cases. We show that the aggregate distribution can be way different from the distributions of individual pair intercontact times. Therefore, using the former to characterize properties that depend on the latter is not correct in general, although this is correct in some cases. We substantiate this finding by analyzing the most representative distributions characterizing real opportunistic networks that can be obtained from reference traces. We review key cases for opportunistic networking, where the aggregate intercontact time distribution presents a heavy tail with or without exponential cutoff. We show that, when individual pairs follow Pareto distributions, the aggregate distribution consistently presents a heavy tail. However, heavy tail aggregate distributions can also emerge in networks where individual pair intercontact times are not heavy tailed, for example, exponential or Pareto with exponential cutoff distributions. We show that an exponential cutoff in the aggregate appears when the average intercontact times of individual pairs are finite. Finally, we discuss how to use our analytical model to know whether collecting aggregate information about intercontact times is sufficient or not, to decideâin practiceâwhich type of routing protocols to use.