Fundamentals of Large Sensor Networks: Connectivity, Capacity, Clocks and Computation

Sensor networks potentially feature large numbers of nodes that can sense their environment over time, communicate with each other over a wireless network, and process information. They differ from data networks in that the network as a whole may be designed for a specific application. We study the theoretical foundations of such large scale sensor networks, addressing four fundamental issues- connectivity, capacity, clocks and function computation. To begin with, a sensor network must be connected so that information can indeed be exchanged between nodes. The connectivity graph of an ad-hoc network is modeled as a random graph and the critical range for asymptotic connectivity is determined, as well as the critical number of neighbors that a node needs to connect to. Next, given connectivity, we address the issue of how much data can be transported over the sensor network. We present fundamental bounds on capacity under several models, as well as architectural implications for how wireless communication should be organized. Temporal information is important both for the applications of sensor networks as well as their operation.We present fundamental bounds on the synchronizability of clocks in networks, and also present and analyze algorithms for clock synchronization. Finally we turn to the issue of gathering relevant information, that sensor networks are designed to do. One needs to study optimal strategies for in-network aggregation of data, in order to reliably compute a composite function of sensor measurements, as well as the complexity of doing so. We address the issue of how such computation can be performed efficiently in a sensor network and the algorithms for doing so, for some classes of functions.Comment: 10 pages, 3 figures, Submitted to the Proceedings of the IEE

Information aggregation in sensor networks

In many sensor network applications, one is interested only in computing some relevant \textit{function} of the sensor measurements. In this thesis, we study optimal strategies for in-network computation and communication in such wireless sensor networks. We begin by considering a directed graph $G = (\mathcal{V},\mathcal{E})$ on the sensor nodes, with a designated collector node, where the goal is to characterize the rate region in $\mathbf{R}^{|\mathcal{E}|}$, i.e., the set of vector rates for which there exist feasible encoders and decoders which achieve zero-error computation for large enough block length. For directed tree graphs, we determine a necessary and sufficient condition for each edge that yields the optimal alphabet, from which we then calculate the minimum worst case and average case complexity. For general directed acyclic graphs, we provide an outer bound on the rate region by finding the disambiguation requirements for each cut, and describe examples where this outer bound is tight. Next, we consider undirected tree networks, where each node has an integer measurement, and all nodes want to compute a symmetric Boolean function. For a class of functions called sum-threshold functions, we derive an optimal strategy which minimizes the worst-case number of bits exchanged on each edge. In the case of general graphs, we present a cut-set lower bound, and an achievable scheme based on aggregation along trees. For complete graphs, the complexity of this scheme is no more than twice that of the optimal scheme. We then turn to a collocated network of nodes, where each node has a Boolean measurement and we wish to compute a symmetric Boolean function of these measurements with zero error. Our objective is to determine the minimum worst-case total number of bits to be communicated to perform the desired computation. We define three classes of functions, namely threshold functions, delta functions and interval functions. We provide exactly optimal strategies for the first two classes, and an order-optimal strategy with optimal preconstant for interval functions. Using these results, we can characterize the complexity of computing percentile type functions. The results also extend to the case of integer measurements and certain integer-valued functions. We use lower bounds from communication complexity theory, and provide an achievable scheme using information theoretic tools. In the collocated network scenario, minimizing the average case complexity presents a variety of interesting problems. We show that the average case complexity of computing a Boolean threshold function of i.i.d. measurements, with threshold $\theta$, is $O(\theta)$, in comparison to the worst case complexity of $\Omega(\log n)$, where $n$ is the number of nodes. In the case of independent but not identically distributed measurements, we show that the optimal order of transmissions is determined by a surprisingly simple rule that depends in a trivial way on the values of the previously transmitted bits and the ordering, but not the exact values of the marginal probabilities. The approach presented can be generalized to the case of block computation, and to alternate models of communication. We also determine the optimal strategy when the number of bits to be communicated is fixed, and one wants to minimize the conditional entropy of the parity function. Finally, we consider the problem of determining the connectivity of a random graph by sequentially sampling edges. We present optimal strategies for determining connectivity in series graphs, parallel graphs, series-parallel graphs and parallel-series graphs. In the case of general graphs, we consider the related problem of finding a certificate for connectivity, and conjecture the optimal strategy

A Discrete Parameter Stochastic Approximation Algorithm for Simulation Optimization

The authors develop a two-timescale simultaneous perturbation stochastic approximation algorithm for simulation-based parameter optimization over discrete sets. This algorithm is applicable in cases where the cost to be optimized is in itself the long-run average of certain cost functions whose noisy estimates are obtained via simulation. The authors present the convergence analysis of their algorithm. Next, they study applications of their algorithm to the problem of admission control in communication networks. They study this problem under two different experimental settings and consider appropriate continuous time queuing models in both settings. Their algorithm finds optimal threshold-type policies within suitable parameterized classes of these. They show results of several experiments for different network parameters and rejection cost. The authors also study the sensitivity of their algorithm with respect to its parameters and step sizes. The results obtained are along expected lines

Zero-error function computation in sensor networks

Abstract — We consider the problem of data harvesting in wireless sensor networks. A designated collector node seeks to compute a function of the sensor measurements. For a directed graph G = (V,E) on the sensor nodes, we wish to determine the optimal encoders on each edge which achieve zero-error block computation of the function at the collector node. Our goal is to characterize the rate region in R |E |. We start with the two node problem, and determine a necessary and sufficient condition for the encoder that yields the optimal alphabet, from which we then calculate the minimum worst case and average case complexity. We then extend this result to trees and derive a necessary and sufficient condition for the encoder on each edge. The further extension of these results to directed acyclic graphs is not immediate. We provide an outer bound on the rate region by finding the disambiguation requirements for each cut, and describe examples where this outer bound is tight. Finally, we consider a collocated network of nodes with a specified order of transmission. We determine a necessary and sufficient condition for each encoder which is based on the transmissions received, and show that the average case complexity of computing a type-threshold function is Θ(1), in comparison to the worst case complexity of Θ(logn). I

Optimal ordering of transmissions for computing boolean threshold functions

Abstract—We address a sequential decision problem that arises in the computation of symmetric Boolean functions of distributed data. We consider a collocated network, where each node’s transmissions can be heard by every other node. Each node has a Boolean measurement and we wish to compute a given Boolean function of these measurements. We suppose that the measurements are independent and Bernoulli distributed. Thus, the problem of optimal computation becomes the problem of optimally ordering nodes ’ transmissions so as to minimize the total expected number of bits. We solve the ordering problem for the class of Boolean threshold functions. The optimal ordering is dynamic, i.e., it could potentially depend on the values of previously transmitted bits. Further, it depends only on the ordering of the marginal probabilites, but not on their exact values. This provides an elegant structure for the optimal strategy. For the case where each node has a block of measurements, the problem is significantly harder, and we conjecture the optimal strategy. I