Anaïs Vergne

Bio: Anaïs Vergne is an academic researcher from Télécom ParisTech. The author has contributed to research in topics: Wireless network & Simplicial homology. The author has an hindex of 11, co-authored 34 publications receiving 355 citations. Previous affiliations of Anaïs Vergne include French Institute for Research in Computer Science and Automation & Université Paris-Saclay.

Simplicial homology of random configurations

Abstract: 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 Cech complex associated to the coverage of each point. By means of Malliavin calculus, we compute explicitly the three first-order moments of the number of k-simplices, and provide a way to compute higher-order moments. Then we derive the mean and the variance of the Euler characteristic. Using the Stein method, we estimate the speed of convergence of the number of occurrences of any connected subcomplex as it converges towards the Gaussian law when the intensity of the Poisson point process tends to infinity. We use a concentration inequality for Poisson processes to find bounds for the tail distribution of the Betti number of first order and the Euler characteristic in such simplicial complexes.

Homology-based distributed coverage hole detection in wireless sensor networks

Abstract: Homology theory provides new and powerful solutions to address the coverage problems in wireless sensor networks (WSNs). They are based on algebraic objects, such as Cech complex and Rips complex. Cech complex gives accurate information about coverage quality, but requires a precise knowledge of the relative locations of nodes. This assumption is rather strong and hard to implement in practical deployments. Rips complex provides an approximation of Cech complex. It is easier to build and does not require any knowledge of nodes location. This simplicity is at the expense of accuracy. Rips complex cannot always detect all coverage holes. It is then necessary to evaluate its accuracy. This work proposes to use the proportion of the area of undiscovered coverage holes as performance criteria. Investigations show that it depends on the ratio between communication and sensing radii of a sensor. Closed-form expressions for lower and upper bounds of the accuracy are also derived. For those coverage holes that can be discovered by Rips complex, a homology-based distributed algorithm is proposed to detect them. Simulation results are consistent with the proposed analytical lower bound, with a maximum difference of 0.5%. Upper-bound performance depends on the ratio of communication and sensing radii. Simulations also show that the algorithm can localize about 99% coverage holes in about 99% cases.

Homology-based Distributed Coverage Hole Detection in Wireless Sensor Networks

Abstract: Homology theory provides new and powerful solutions to address the coverage problems in wireless sensor networks (WSNs). They are based on algebraic objects, such as Cech complex and Rips complex. Cech complex gives accurate information about coverage quality but requires a precise knowledge of the relative locations of nodes. This assumption is rather strong and hard to implement in practical deployments. Rips complex provides an approximation of Cech complex. It is easier to build and does not require any knowledge of nodes location. This simplicity is at the expense of accuracy. Rips complex can not always detect all coverage holes. It is then necessary to evaluate its accuracy. This work proposes to use the proportion of the area of undiscovered coverage holes as performance criteria. Investigations show that it depends on the ratio between communication and sensing radii of a sensor. Closed-form expressions for lower and upper bounds of the accuracy are also derived. For those coverage holes which can be discovered by Rips complex, a homology-based distributed algorithm is proposed to detect them. Simulation results are consistent with the proposed analytical lower bound, with a maximum difference of 0.5%. Upper bound performance depends on the ratio of communication and sensing radii. Simulations also show that the algorithm can localize about 99% coverage holes in about 99% cases.

A Case Study on Regularity in Cellular Network Deployment

Abstract: This letter aims to validate the $\beta$ -Ginibre point process as a model for the distribution of base station locations in a cellular network. The $\beta$ -Ginibre is a repulsive point process in which repulsion is controlled by the $\beta$ parameter. When $\beta$ tends to zero, the point process converges in law towards a Poisson point process. If $\beta$ equals to one it becomes a Ginibre point process. Simulations on real data collected in Paris, France, show that base station locations can be fitted with a $\beta$ -Ginibre point process. Moreover, we prove that their superposition tends to a Poisson point process as it can be seen from real data. Qualitative interpretations on deployment strategies are derived from the model fitting of the raw data.

Reduction algorithm for simplicial complexes

Abstract: In this paper, we aim at reducing power consumption in wireless sensor networks by turning off supernumerary sensors. Random simplicial complexes are tools from algebraic topology which provide an accurate and tractable representation of the topology of wireless sensor networks. Given a simplicial complex, we present an algorithm which reduces the number of its vertices, keeping its homology (i.e. connectivity, coverage) unchanged. We show that the algorithm reaches a Nash equilibrium, moreover we find both a lower and an upper bounds for the number of vertices removed, the complexity of the algorithm, and the maximal order of the resulting complex for the coverage problem. We also give some simulation results for classical cases, especially coverage complexes simulating wireless sensor networks.

An Introduction To The Theory Of Point Processes

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The Ginibre Point Process as a Model for Wireless Networks With Repulsion

Abstract: The spatial structure of transmitters in wireless networks plays a key role in evaluating mutual interference and, hence, performance. Although the Poisson point process (PPP) has been widely used to model the spatial configuration of wireless networks, it is not suitable for networks with repulsion. The Ginibre point process (GPP) is one of the main examples of determinantal point processes that can be used to model random phenomena where repulsion is observed. Considering the accuracy, tractability, and practicability tradeoffs, we introduce and promote the $\beta$ -GPP, which is an intermediate class between the PPP and the GPP, as a model for wireless networks when the nodes exhibit repulsion. To show that the model leads to analytically tractable results in several cases of interest, we derive the mean and variance of the interference using two different approaches: the Palm measure approach and the reduced second-moment approach, and then provide approximations of the interference distribution by three known probability density functions. In addition, to show that the model is relevant for cellular systems, we derive the coverage probability of a typical user and find that the fitted $\beta$ -GPP can closely model the deployment of actual base stations in terms of coverage probability and other statistics.

Determinantal Point Processes

Abstract: In this survey we review two topics concerning determinantal (or fermion) point processes. First, we provide the construction of diffusion processes on the space of configurations whose invariant measure is the law of a determinantal point process. Second, we present some algorithms to sample from the law of a determinantal point process on a finite window. Related open problems are listed.

Central limit theorems for $U$-statistics of Poisson point processes

Abstract: A $U$-statistic of a Poisson point process is defined as the sum $\sum f(x_1,\ldots,x_k)$ over all (possibly infinitely many) $k$-tuples of distinct points of the point process. Using the Malliavin calculus, the Wiener-Ito chaos expansion of such a functional is computed and used to derive a formula for the variance. Central limit theorems for $U$-statistics of Poisson point processes are shown, with explicit bounds for the Wasserstein distance to a Gaussian random variable. As applications, the intersection process of Poisson hyperplanes and the length of a random geometric graph are investigated.

Statistical Modeling and Probabilistic Analysis of Cellular Networks With Determinantal Point Processes

Abstract: Although the Poisson point process (PPP) has been widely used to model base station (BS) locations in cellular networks, it is an idealized model that neglects the spatial correlation among BSs. This paper proposes the use of the determinantal point process (DPP) to take into account these correlations, in particular the repulsiveness among macro BS locations. DPPs are demonstrated to be analytically tractable by leveraging several unique computational properties. Specifically, we show that the empty space function, the nearest neighbor function, the mean interference, and the signal-to-interference ratio (SIR) distribution have explicit analytical representations and can be numerically evaluated for cellular networks with DPP-configured BSs. In addition, the modeling accuracy of DPPs is investigated by fitting three DPP models to real BS location data sets from two major U.S. cities. Using hypothesis testing for various performance metrics of interest, we show that these fitted DPPs are significantly more accurate than popular choices such as the PPP and the perturbed hexagonal grid model.