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Replica Symmetry Breaking , Scaling Theory , and Sensor Networks

Device and sensor networks are shaping many activities in our society. These networks are being deployed in a growing number of applications as diverse as agricultural management, industrial controls, crime watch, and military applications. Indeed, sensor networks can be considered as a promising technology with a wide range of potential future markets [1]. Still, for all the promise, it is often difficult to integrate the individual components of a sensor network in a smart way. Although we see many breakthroughs in component devices, advanced software, and power managements, system-level understanding of the emerging technology is still weak. It requires a shift in our notion of “what to look for”. It requires a study of collective behavior and resulting trade-offs. This is the issue that we address in this work. We demonstrate the usefulness of adopting new approaches by considering the following scenario. Consider that a data center is interested in the data sequence, {X(t)}t=1, which cannot be observed directly. Therefore, the data center deploys a bunch of L sensors which each independently encodes its noisy observation of the sequence, {Yi(t)}t=1, without sharing any information, i.e., the sensors are not permitted to communicate and decide what to send to the data center beforehand. The data center collects separate samples from all the L sensors and uses them to recover the original sequence. However, since {X(t)}t=1 is not the only pressing matter which the data center must consider, the combined data rate R at which the sensors can communicate with it is strictly limited. A formulation of decentralized communication with estimation task, the “CEO problem”, was first proposed by Berger and Zhang [2], providing a new theoretical framework for large scale sensing systems. In this outstanding work, some interesting properties of such systems have been revealed. If the sensors were permitted to communicate on the basis of their pooled observations, then they would be able to smooth out their independent observation noises entirely as L goes to infinity. Therefore, the data center can achieve an arbitrary fidelity D(R), where D(·) denotes the distortion rate function of {X(t)}. In particular, the data center recovers almost complete information if R exceeds the entropy rate of {X(t)}. However, if the sensors are not allowed to communicate with each other, there does not exist a finite value of R for which even infinitely many sensors can make D arbitrarily small [2]. In this work, we introduce a new analytical model for a massive sensing system with a finite data rate R. More specifically, we assume that the sensors use LDPC codes for rate distortion coding [3], while the data center recovers the original sequence by using optimal “majority vote” estimation [4]. We consider the distributed sensing problem of deciding the optimal number of sensors L given the combined data rate R. Our asymptotic analysis successfully provides the performance of the whole sensing system when L goes to infinity, where the data rate for an individual sensor information vanishes [5]. Here, we exploit statistical methods which have recently been developed in the field of disordered statistical systems, in particular, the spin glass theory. These techniques have turned out to offer new tools and viewpoints for the understanding of information systems, including the rate distortion trade-offs which arise in the LDPC-type coding problem [6]. Surprisingly, making use of the ideas of traditional scaling theory, one can find a systematic description of the optimal distributed sensing problem. In particular, for a given value of R, there exists a threshold noise level p at which the optimal number of sensors changes in a discontinuous manner [Fig.1]. Moreover, on the basis of heuristics proposed in [7], the LDPC rate distortion codes can be shown to approach the theoretical bound in practice. These heuristics are also based on concepts from statistical mechanics [8]. In short, this work will provide a system-level insight into the optimization of distributed sensing in sensor networks equipped with state-of-the-art coding techniques.