Bootstrap Percolation on Random Geometric Graphs

Bootstrap percolation has been used effectively to model phenomena as diverse as emergence of magnetism in materials, spread of infection, diffusion of software viruses in computer networks, adoption of new technologies, and emergence of collective action and cultural fads in human societies. It is defined on an (arbitrary) network of interacting agents whose state is determined by the state of their neighbors according to a threshold rule. In a typical setting, bootstrap percolation starts by random and independent "activation" of nodes with a fixed probability $p$, followed by a deterministic process for additional activations based on the density of active nodes in each neighborhood ($\theta$ activated nodes). Here, we study bootstrap percolation on random geometric graphs in the regime when the latter are (almost surely) connected. Random geometric graphs provide an appropriate model in settings where the neighborhood structure of each node is determined by geographical distance, as in wireless {\it ad hoc} and sensor networks as well as in contagion. We derive bounds on the critical thresholds $p_c', p_c"$ such that for all $p > p"_c(\theta)$ full percolation takes place, whereas for $p < p'_c(\theta)$ it does not. We conclude with simulations that compare numerical thresholds with those obtained analytically

Secure Communication in Dynamic Wireless Ad hoc Networks

We consider a wireless ad hoc network in the presence of eavesdroppers (EDs), where the nodes are distributed according to independent Poisson point processes (PPPs). The legitimate nodes follow the half-duplex mode of operation employing the slotted ALOHA protocol for transmission. For such a network, a novel communication scheme that induces a time-varying secure connectivity graph (SCG) is proposed, and the connectivity behavior of this induced SCG is studied. In particular, for a legitimate node in the network, we analyze (i) the average number of incoming edges and the average number of outgoing edges; (ii) the time to nearest-neighbor secure connectivity; and (iii) a condition on the EDs' density that allows information percolation, {\ie}, a condition for the existence of a `giant' component. The average time for secure connectivity among the nodes in this giant component is shown to scale linearly with the Euclidean distance. Further, we show that by splitting the packets into two sub-packets and routing each sub-packet along paths that are sufficiently far apart can (a) potentially improve secure connectivity and (b) reduce the overall delay incurred in exchanging packets between any two legitimate nodes in the giant component.Comment: An abridged version of this paper was submitted to IEEE GLOBECOM 201

Directed Percolation in Wireless Networks with Interference and Noise

Previous studies of connectivity in wireless networks have focused on undirected geometric graphs. More sophisticated models such as Signal-to-Interference-and-Noise-Ratio (SINR) model, however, usually leads to directed graphs. In this paper, we study percolation processes in wireless networks modelled by directed SINR graphs. We first investigate interference-free networks, where we define four types of phase transitions and show that they take place at the same time. By coupling the directed SINR graph with two other undirected SINR graphs, we further obtain analytical upper and lower bounds on the critical density. Then, we show that with interference, percolation in directed SINR graphs depends not only on the density but also on the inverse system processing gain. We also provide bounds on the critical value of the inverse system processing gain

Analytical Lower Bounds on the Critical Density in Continuum Percolation

Percolation theory has become a useful tool for the analysis of large-scale wireless networks. We investigate the fundamental problem of characterizing the critical density for d-dimensional Poisson random geometric graphs in continuum percolation theory. By using a probabilistic analysis which incorporates the clustering effect in random geometric graphs, we develop a new class of analytical lower bounds for the λ (d) c in d-dimensional Poisson random geometric graphs. The lower bounds are the tightest known to date. In particular, for the two-dimensional case, the analytical lower bound is improved to λ (2) c ≥ 0.7698.... For the three-dimensional case, we obtain λ (3) c ≥ 0.4494...