Cover Time in Edge-Uniform Stochastically-Evolving Graphs

We define a general model of stochastically-evolving graphs, namely the \emph{Edge-Uniform Stochastically-Evolving Graphs}. In this model, each possible edge of an underlying general static graph evolves independently being either alive or dead at each discrete time step of evolution following a (Markovian) stochastic rule. The stochastic rule is identical for each possible edge and may depend on the past $k \ge 0$ observations of the edge's state. We examine two kinds of random walks for a single agent taking place in such a dynamic graph: (i) The \emph{Random Walk with a Delay} (\emph{RWD}), where at each step the agent chooses (uniformly at random) an incident possible edge, i.e., an incident edge in the underlying static graph, and then it waits till the edge becomes alive to traverse it. (ii) The more natural \emph{Random Walk on what is Available} (\emph{RWA}) where the agent only looks at alive incident edges at each time step and traverses one of them uniformly at random. Our study is on bounding the \emph{cover time}, i.e., the expected time until each node is visited at least once by the agent. For \emph{RWD}, we provide a first upper bound for the cases $k = 0, 1$ by correlating \emph{RWD} with a simple random walk on a static graph. Moreover, we present a modified electrical network theory capturing the $k = 0$ case. For \emph{RWA}, we derive some first bounds for the case $k = 0$, by reducing \emph{RWA} to an \emph{RWD}-equivalent walk with a modified delay. Further, we also provide a framework, which is shown to compute the exact value of the cover time for a general family of stochastically-evolving graphs in exponential time. Finally, we conduct experiments on the cover time of \emph{RWA} in Edge-Uniform graphs and compare the experimental findings with our theoretical bounds

Community computation

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Materials Science and Engineering, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 171-186).In this thesis we lay the foundations for a distributed, community-based computing environment to tap the resources of a community to better perform some tasks, either computationally hard or economically prohibitive, or physically inconvenient, that one individual is unable to accomplish efficiently. We introduce community coding, where information systems meet social networks, to tackle some of the challenges in this new paradigm of community computation. We design algorithms, protocols and build system prototypes to demonstrate the power of community computation to better deal with reliability, scalability and security issues, which are the main challenges in many emerging community-computing environments, in several application scenarios such as community storage, community sensing and community security. For example, we develop a community storage system that is based upon a distributed P2P (peer-to-peer) storage paradigm, where we take an array of small, periodically accessible, individual computers/peer nodes and create a secure, reliable and large distributed storage system. The goal is for each one of them to act as if they have immediate access to a pool of information that is larger than they could hold themselves, and into which they can contribute new stuff in a both open and secure manner. Such a contributory and self-scaling community storage system is particularly useful where reliable infrastructure is not readily available in that such a system facilitates easy ad-hoc construction and easy portability. In another application scenario, we develop a novel framework of community sensing with a group of image sensors. The goal is to present a set of novel tools in which software, rather than humans, examines the collection of images sensed by a group of image sensors to determine what is happening in the field of view. We also present several design principles in the aspects of community security. In one application example, we present community-based email spain detection approach to deal with email spams more efficiently.by Fulu Li.Ph.D

Mobility Problems in Distributed Search and Combinatorial Games

This thesis examines a collection of topics under the general notion of mobility of agents. We examine problems where a set of entities, perceived as robots or tokens, navigate in some given (discrete or continuous) environment to accomplish a goal. The problems we consider fall under two main research fields. First, Distributed Search where the agents cooperate to explore their environment or search for a specific target location within it. Second, Combinatorial Games, in the spirit of Pursuit-Evasion, where the agents are now divided into two groups with complementary objectives competing against each other. More specifically, we consider three distinct problems: disk evacuation, exploration of dynamic graphs and eternal domination. In Disk Evacuation, two robots with different speeds aim to discover an unknown exit lying on the boundary of a unit disk. For a wide range of speeds, we provide matching upper and lower bounds. In Dynamic Graph Exploration, we analyze the exploration time for a randomly-walking agent wishing to visit all the vertices of a stochastically-evolving graph. In Eternal Domination, we consider rectangular grid graphs and upper bound the amount of guard agents needed to perpetually defend the vertices against an attacker