We study delay tolerant networking (DTN) and in particular, its capacity to store, carry and forward messages so that the messages eventually reach their final destinations. We approach this broad question in the framework of percolation theory. To this end, we assume an elementary mobility model, where nodes arrive to an infinite plane according to a Poisson point process, move a certain distance L, and then depart. In this setting, we characterize the mean density of nodes required to support DTN style networking. In particular, under the given assumptions, we show that DTN is feasible when the mean node degree is greater than 4 e(g), where parameter g=L d is the ratio of the distance L to the transmission range d, and e(g) is the critical reduced number density of tilted cylinders in a directed continuum percolation model. By means of Monte Carlo simulations, we give numerical values for e(g). The asymptotic behavior of e(g) when g tends to infinity is also derived from a fluid flow analysis.
In contrast, in @cite_5 , we assumed and showed that opportunistic content dissemination schemes, such as the floating content, can be analyzed by using a three-dimensional continuum percolation model. In this paper, we adapt the same approach, but instead of approximating the process by undirected percolation model, we consider the characterizing the dissemination of messages by exactly in a DTN network. There are less results available for the directed percolation models than there are for the normal undirected percolation, and results for directed continuum models are even more scarce. The basic directed cases are the bond and site percolation models, where, e.g., bonds have fixed directions (e.g., up and right in a square lattice) @cite_15 @cite_14 . Directed models have been also studied in the context of scale-free networks @cite_6 , which arise in Internet, social networks etc.
@cite_5 @cite_15 @cite_14 @cite_6
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We consider the critical percolation threshold for aligned cylinders, which provides a lower bound for the required node degree for the permanence of information in opportunistic networking. The height of a cylinder corresponds to the time a node is active in its current location. By means of Monte Carlo simulations, we obtain an accurate numerical estimate for the critical reduced number density, ηc ≈ 0.3312(1) for constant height cylinders. This threshold is the same for all ratios of the height to the diameter of the base, and corresponds to the mean node degree of 1.3248 in opportunistic networking, which is clearly below the percolation threshold of 4.51 above which a gigantic connected component emerges in the network. A method for conversion of directed site animals to their bond counterparts is used to significantly extend the number of such configurations in 2, 3, 4 dimensions, obtain their loop partition and study the singularity structure (for both dominant and subdominant singularities) of the corresponding dilute polymer transitions. A special property of bond directed configurations (equivalence lence of loop and bond perimeter partition) is exploited to obtain longer susceptibility series in 3 and 4 dimensions and the gap exponent estimate is refined into the intervals TANO Many complex networks in nature have directed links, a property that affects the network’s navigability and large-scale topology. Here we study the percolation properties of such directed scale-free networks with correlated in and out degree distributions. We derive a phase diagram that indicates the existence of three regimes, determined by the values of the degree exponents. In the first regime we regain the known directed percolation mean field exponents. In contrast, the second and third regimes are characterized by anomalous exponents, which we calculate analytically. In the third regime the network is resilient to random dilution, i.e., the percolation threshold is pc!1.