Efficient collection of sensor data via a new accelerated random walk

Motivated by the problem of efficiently collecting data from wireless sensor networks via a mobile sink, we present an accelerated random walk on random geometric graphs (RGG). Random walks in wireless sensor networks can serve as fully local, lightweight strategies for sink motion that significantly reduce energy dissipation but introduce higher latency in the data collection process. In most cases, random walks are studied on graphs like Gn,p and grid. Instead, we here choose the RGG model, which abstracts more accurately spatial proximity in a wireless sensor network. We first evaluate an adaptive walk (the random walk with inertia) on the RGG model; its performance proved to be poor and led us to define and experimentally evaluate a novel random walk that we call γ-stretched random walk. Its basic idea is to favour visiting distant neighbours of the current node towards reducing node overlap and accelerate the cover time. We also define a new performance metric called proximity cover time that, along with other metrics such as visit overlap statistics and proximity variation, we use to evaluate the performance properties and features of the various walks

Bootstrap Percolation in the Random Geometric Graph

Bootstrap percolation on a graph is a process which models the spread of an infection given an initially infected set of vertices of the graph. To state the problem more precisely, suppose G is a graph, k is a natural number, and I_0 is a set of initially infected vertices. Then for any discrete time t, we define I_t to be I_{t-1} along with any vertex outside of I_{t-1} which has at least k edges to vertices of I_{t-1}. This type of process may be used to model the spread of a disease by taking people as vertices, interactions between people as edges, and assuming a rate at which the infection spreads. That is, if the rate of spread is 10%, we would expect that if a person is in contact with 10 infected people, then the person will become infected. Other applications of this type of model involve the spread of rumors and the fault tolerance for distributed computing. The random geometric graph is formed by fixing an r value and choosing n points from the unit square uniformly at random. We then join a pair of these points by an edge if their distance is less than r. This kind of random graph seems particularly relevant in the current socially distant world in which people attempt to only interact with others when necessary (i.e. distance-based edges roughly model this kind of relative isolation). Random geometric graphs have been well-studied. Bootstrap percolation on random geometric graphs has been examined although prior results in this direction cover limited regimes of the parameters. In our project, we extend previous work to study other ranges of values of the parameters. Along the way we use similar ideas to identify the threshold for connectivity in the random geometric graph which is a problem of independent interest.https://digitalcommons.winthrop.edu/sureposters/1000/thumbnail.jp

Scaling limits of random walks and their related parameters on critical random trees and graphs

In this thesis we study random walks in random environments, a major area in Probability theory. Within this broad topic, we are mainly focused in studying scaling limits of random walks on random graphs at criticality, that is precisely when we witness the emergence of a giant component that has size proportional to the number of vertices of the graph. Critical random graphs of interest include critical Galton-Watson trees and maximal components that belong to the Erd}os- R_enyi universality class. The first part of the thesis expands upon using analytic and geometric properties of those random graphs to establish distributional convergence of certain graph parameters, such as the blanket time. Our contribution refines the previous existing results on the order of the mean blanket time. The study of this problem can be seen as a stepping stone to deal with the more delicate problem of establishing convergence in distribution of the rescaled cover times of the discrete-time walks in each of the applications of our main result. Relying on powerful resistance techniques developed in recent years, another part of the thesis investigates random walks in random enviroments on tree-like spaces and their scaling limits in a certain regime, that is when the potential of the random walk in random environment converges. Results include novel scaling continuum limits of a biased random walk on large critical branching random walk and a self-reinforced discrete process on size-conditioned critical Galton-Watson trees. In both cases the diffusions that are not on natural scale are identified as Brownian motions on a continuum random fractal tree with its natural metric replaced by a distorted resistance metric

The acquaintance time of (percolated) random geometric graphs

In this paper, we study the acquaintance time \AC(G) defined for a connected graph $G$. We focus on \G(n,r,p), a random subgraph of a random geometric graph in which $n$ vertices are chosen uniformly at random and independently from $[0,1]^2$, and two vertices are adjacent with probability $p$ if the Euclidean distance between them is at most $r$. We present asymptotic results for the acquaintance time of \G(n,r,p) for a wide range of $p=p(n)$ and $r=r(n)$. In particular, we show that with high probability \AC(G) = \Theta(r^{-2}) for G \in \G(n,r,1), the "ordinary" random geometric graph, provided that $\pi n r^2 - \ln n \to \infty$ (that is, above the connectivity threshold). For the percolated random geometric graph G \in \G(n,r,p), we show that with high probability \AC(G) = \Theta(r^{-2} p^{-1} \ln n), provided that p n r^2 \geq n^{1/2+\eps} and p < 1-\eps for some \eps>0

Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

We study the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs. We prove that such graphs have small separators. Next, we present efficient $(1+\varepsilon)$-approximation algorithms for these graphs, for Independent Set, Set Cover, and Dominating Set problems, among others. We also prove corresponding hardness of approximation for some of these optimization problems, providing a characterization of their intractability in terms of density