Simplicial homology of random configurations

Given a Poisson process on a d-dimensional torus, its random geometric simplicial complex is the complex whose vertices are the points of the Poisson process and simplices are given by the u Cech complex associated to the coverage of each point. We compute explicitly the mean number of k-simplices as well as the mean of the Euler characteristic. Then, by means of Malliavin calculus, we show that the number of any connected geometric simplicial complex converges to the Gaussian law when the intensity of the Poisson point process tends to infinity. We use a concentration inequality to find bounds for the for the distribution of the Betti number of first order in such simplicial complex. 1. Motivation As technology goes on (1, 2, 3), one can expect a wide expansion of the so-called sensor networks. Such networks represent the next evolutionary step in building, utilities, industrial, home, agriculture, defense and many other contexts (4). These networks are built upon a multitude of small and cheap sensors which are devices with limited transmission capabilities and power. Each sensor monitors a region around itself by measuring some environmental quantities (e.g., temperature, hu- midity), detecting intrusion, etc, and broadcasts its collected information to other sensors or to a central node. Two questions are of paramount importance: can information be shared among the whole network, is the whole region totally moni- tored ? Several researches have recently been dedicated to this problem considering a variety of situations. One can distinguish three main scenarios: those where it is possible to choose the position of each sensor, those where sensors are arbitrarily deployed in the target region with the control of a central station and those where the sensor locations are random in a decentralized system. In many cases, placing the sensors is impossible or implies a high cost. Sometimes this impossibility comes from the fact that the cost of placing each sensor is too large and sometimes the network has an inherent random behavior (like in the ad-hoc case, where users move). In addition, this policy cannot take into account the configuration of the network in the case of failure of some sensor. The drawback of the second scenario is a higher cost of sensors, since each one has to communicate with the central station. Besides, the central station itself increases the cost of the whole system. Moreover, if sensors are supposed to know their positions, an absolute positioning system has to be included in each sensor, making their hardware even more complex and then more expensive. It is thus important to investigate the third scenario: randomly located sensors, no central station. Actually, if we can predict some characteristics of the topology of a random network, the number of sensors (or, as well, the power supply of them) can be a priori determined such that a given network may operate with high probability. For instance, we can choose the mean number of sensors