Detection Performance in Balanced Binary Relay Trees with Node and Link Failures

We study the distributed detection problem in the context of a balanced binary relay tree, where the leaves of the tree correspond to $N$ identical and independent sensors generating binary messages. The root of the tree is a fusion center making an overall decision. Every other node is a relay node that aggregates the messages received from its child nodes into a new message and sends it up toward the fusion center. We derive upper and lower bounds for the total error probability $P_N$ as explicit functions of $N$ in the case where nodes and links fail with certain probabilities. These characterize the asymptotic decay rate of the total error probability as $N$ goes to infinity. Naturally, this decay rate is not larger than that in the non-failure case, which is $\sqrt N$. However, we derive an explicit necessary and sufficient condition on the decay rate of the local failure probabilities $p_k$ (combination of node and link failure probabilities at each level) such that the decay rate of the total error probability in the failure case is the same as that of the non-failure case. More precisely, we show that $\log P_N^{-1}=\Theta(\sqrt N)$ if and only if $\log p_k^{-1}=\Omega(2^{k/2})$

Submodularity and Optimality of Fusion Rules in Balanced Binary Relay Trees

We study the distributed detection problem in a balanced binary relay tree, where the leaves of the tree are sensors generating binary messages. The root of the tree is a fusion center that makes the overall decision. Every other node in the tree is a fusion node that fuses two binary messages from its child nodes into a new binary message and sends it to the parent node at the next level. We assume that the fusion nodes at the same level use the same fusion rule. We call a string of fusion rules used at different levels a fusion strategy. We consider the problem of finding a fusion strategy that maximizes the reduction in the total error probability between the sensors and the fusion center. We formulate this problem as a deterministic dynamic program and express the solution in terms of Bellman's equations. We introduce the notion of stringsubmodularity and show that the reduction in the total error probability is a stringsubmodular function. Consequentially, we show that the greedy strategy, which only maximizes the level-wise reduction in the total error probability, is within a factor of the optimal strategy in terms of reduction in the total error probability

Dissipation of information in channels with input constraints

One of the basic tenets in information theory, the data processing inequality states that output divergence does not exceed the input divergence for any channel. For channels without input constraints, various estimates on the amount of such contraction are known, Dobrushin's coefficient for the total variation being perhaps the most well-known. This work investigates channels with average input cost constraint. It is found that while the contraction coefficient typically equals one (no contraction), the information nevertheless dissipates. A certain non-linear function, the \emph{Dobrushin curve} of the channel, is proposed to quantify the amount of dissipation. Tools for evaluating the Dobrushin curve of additive-noise channels are developed based on coupling arguments. Some basic applications in stochastic control, uniqueness of Gibbs measures and fundamental limits of noisy circuits are discussed. As an application, it shown that in the chain of $n$ power-constrained relays and Gaussian channels the end-to-end mutual information and maximal squared correlation decay as $\Theta(\frac{\log\log n}{\log n})$, which is in stark contrast with the exponential decay in chains of discrete channels. Similarly, the behavior of noisy circuits (composed of gates with bounded fan-in) and broadcasting of information on trees (of bounded degree) does not experience threshold behavior in the signal-to-noise ratio (SNR). Namely, unlike the case of discrete channels, the probability of bit error stays bounded away from $1\over 2$ regardless of the SNR.Comment: revised; include appendix B on contraction coefficient for mutual information on general alphabet

Distributed Detection and Estimation in Wireless Sensor Networks

In this article we consider the problems of distributed detection and estimation in wireless sensor networks. In the first part, we provide a general framework aimed to show how an efficient design of a sensor network requires a joint organization of in-network processing and communication. Then, we recall the basic features of consensus algorithm, which is a basic tool to reach globally optimal decisions through a distributed approach. The main part of the paper starts addressing the distributed estimation problem. We show first an entirely decentralized approach, where observations and estimations are performed without the intervention of a fusion center. Then, we consider the case where the estimation is performed at a fusion center, showing how to allocate quantization bits and transmit powers in the links between the nodes and the fusion center, in order to accommodate the requirement on the maximum estimation variance, under a constraint on the global transmit power. We extend the approach to the detection problem. Also in this case, we consider the distributed approach, where every node can achieve a globally optimal decision, and the case where the decision is taken at a central node. In the latter case, we show how to allocate coding bits and transmit power in order to maximize the detection probability, under constraints on the false alarm rate and the global transmit power. Then, we generalize consensus algorithms illustrating a distributed procedure that converges to the projection of the observation vector onto a signal subspace. We then address the issue of energy consumption in sensor networks, thus showing how to optimize the network topology in order to minimize the energy necessary to achieve a global consensus. Finally, we address the problem of matching the topology of the network to the graph describing the statistical dependencies among the observed variables.Comment: 92 pages, 24 figures. To appear in E-Reference Signal Processing, R. Chellapa and S. Theodoridis, Eds., Elsevier, 201

Decentralized detection in resource-limited sensor network architectures

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (leaves 201-207).We consider the problem of decentralized binary detection in a network consisting of a large number of nodes arranged as a tree of bounded height. We show that the error probability decays exponentially fast with the number of nodes under both a Neyman-Pearson criterion and a Bayesian criterion, and provide bounds for the optimal error exponent. Furthermore, we show that under the Neyman-Pearson criterion, the optimal error exponent is often the same as that corresponding to a parallel configuration, implying that a large network can be designed to operate efficiently without significantly affecting the detection performance. We provide sufficient, as well as necessary, conditions for this to happen. For those networks satisfying the sufficient conditions, we propose a simple strategy that nearly achieves the optimal error exponent, and in which all non-leaf nodes need only send 1-bit messages. We also investigate the impact of node failures and unreliable communications on the detection performance. Node failures are modeled by a Galton-Watson branching process, and binary symmetric channels are assumed for the case of unreliable communications. We characterize the asymptotically optimal detection performance, develop simple strategies that nearly achieve the optimal performance, and compare the performance of the two types of networks. Our results suggest that in a large scale sensor network, it is more important to ensure that nodes can communicate reliably with each other(e.g.,by boosting the transmission power) than to ensure that nodes are robust to failures. In the case of networks with unbounded height, we establish the validity of a long-standing conjecture regarding the sub-exponential decay of Bayesian detection error probabilities in a tandem network. We also provide bounds for the error probability, and show that under the additional assumption of bounded Kullback-Leibler divergences, the error probability is (e cnd ), for all d> 1/2, with c c(logn)d being a positive constant. Furthermore, the bound (e), for all d> 1, holds under an additional mild condition on the distributions. This latter bound is shown to be tight. Moreover, for the Neyman-Pearson case, we establish that if the sensors act myopically, the Type II error probabilities also decay at a sub-exponential rate.(cont.) Finally, we consider the problem of decentralized detection when sensors have access to side-information that affects the statistics of their measurements, and the network has an overall cost constraint. Nodes can decide whether or not to make a measurement and transmit a message to the fusion center("censoring"), and also have a choice of the transmission function. We study the tradeoff in the detection performance with the cost constraint, and also the impact of sensor cooperation and global sharing of side-information. In particular, we show that if the Type I error probability is constrained to be small, then sensor cooperation is not necessary to achieve the optimal Type II error exponent.by Wee Peng Tay.Ph.D

Network Information Flow with Correlated Sources

In this paper, we consider a network communications problem in which multiple correlated sources must be delivered to a single data collector node, over a network of noisy independent point-to-point channels. We prove that perfect reconstruction of all the sources at the sink is possible if and only if, for all partitions of the network nodes into two subsets S and S^c such that the sink is always in S^c, we have that H(U_S|U_{S^c}) < \sum_{i\in S,j\in S^c} C_{ij}. Our main finding is that in this setup a general source/channel separation theorem holds, and that Shannon information behaves as a classical network flow, identical in nature to the flow of water in pipes. At first glance, it might seem surprising that separation holds in a fairly general network situation like the one we study. A closer look, however, reveals that the reason for this is that our model allows only for independent point-to-point channels between pairs of nodes, and not multiple-access and/or broadcast channels, for which separation is well known not to hold. This ``information as flow'' view provides an algorithmic interpretation for our results, among which perhaps the most important one is the optimality of implementing codes using a layered protocol stack.Comment: Final version, to appear in the IEEE Transactions on Information Theory -- contains (very) minor changes based on the last round of review

Improving Multicast Communications Over Wireless Mesh Networks

In wireless mesh networks (WMNs) the traditional approach to shortest path tree based multicasting is to cater for the needs of the poorest performingnode i.e. the maximum permitted multicast line rate is limited to the lowest line rate used by the individual Child nodes on a branch. In general, this meansfixing the line rate to its minimum value and fixing the transmit power to its maximum permitted value. This simplistic approach of applying a single multicast rate for all nodes in the multicast group results in a sub-optimal trade-off between the mean network throughput and coverage area that does not allow for high bandwidth multimedia applications to be supported. By relaxing this constraint and allowing multiple line rates to be used, the mean network throughput can be improved. This thesis presents two methods that aim to increase the mean network throughput through the use of multiple line rates by the forwarding nodes. This is achieved by identifying the Child nodes responsible for reducing the multicast group rate. The first method identifies specific locations for the placement of relay nodes which allows for higher multicast branch line rates to be used. The second method uses a power control algorithm to tune the transmit power to allow for higher multicast branch line rates. The use of power control also helps to reduce the interference caused to neighbouring nodes.Through extensive computer simulation it can be shown that these two methods can lead to a four-fold gain in the mean network throughput undertypical WMN operating conditions compared with the single line rate case