Simplicial Homology for Future Cellular Networks

Simplicial homology is a tool that provides a mathematical way to compute the connectivity and the coverage of a cellular network without any node location information. In this article, we use simplicial homology in order to not only compute the topology of a cellular network, but also to discover the clusters of nodes still with no location information. We propose three algorithms for the management of future cellular networks. The first one is a frequency auto-planning algorithm for the self-configuration of future cellular networks. It aims at minimizing the number of planned frequencies while maximizing the usage of each one. Then, our energy conservation algorithm falls into the self-optimization feature of future cellular networks. It optimizes the energy consumption of the cellular network during off-peak hours while taking into account both coverage and user traffic. Finally, we present and discuss the performance of a disaster recovery algorithm using determinantal point processes to patch coverage holes

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{C}ech complex associated to the coverage of each point. By means of Malliavin calculus, we compute explicitly the n$th$ order moment of the number of $k$-simplices. The two first order moments of this quantity allow us to find the mean and the variance of the Euler caracteristic. Also, 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 and the Euler characteristic in such simplicial complex

Topologie algébrique appliquée aux réseaux de capteurs

Simplicial complex representation gives a mathematical description of the topology of a wireless sensor network, i.e., its connectivity and coverage. In these networks, sensors are randomly deployed in bulk in order to ensure perfect connectivity and coverage. We propose an algorithm to discover which sensors are to be switched off, without modification of the topology, in order to reduce energy consumption. Our reduction algorithm can be applied to any type of simplicial complex and reaches an optimum solution. For random geometric simplicial complexes, we find boundaries for the number of removed vertices, as well as mathematical properties for the resulting simplicial complex. The complexity of our reduction algorithm boils down to the computation of the asymptotical behavior of the clique number of a random geometric graph. We provide almost sure asymptotical behavior for the clique number in all three percolation regimes of the geometric graph. In the second part, we apply the simplicial complex representation to cellular networks and improve our reduction algorithm to fit new purposes. First, we provide a frequency auto-planning algorithm for self-configuration of SON in future cellular networks. Then, we propose an energy conservation fot the self-optimization of wireless networks. Finally, we present a disaster recovery algorithm for any type of damaged wireless network. In this last chapter, we also introduce the simulation of determinantal point processes in wireless networks.Simplicial complex representation gives a mathematical description of the topology of a wireless sensor network, i.e., its connectivity and coverage. In these networks, sensors are randomly deployed in bulk in order to ensure perfect connectivity and coverage. We propose an algorithm to discover which sensors are to be switched off, without modification of the topology, in order to reduce energy consumption. Our reduction algorithm can be applied to any type of simplicial complex and reaches an optimum solution. For random geometric simplicial complexes, we find boundaries for the number of removed vertices, as well as mathematical properties for the resulting simplicial complex. The complexity of our reduction algorithm boils down to the computation of the asymptotical behavior of the clique number of a random geometric graph. We provide almost sure asymptotical behavior for the clique number in all three percolation regimes of the geometric graph. In the second part, we apply the simplicial complex representation to cellular networks and improve our reduction algorithm to fit new purposes. First, we provide a frequency auto-planning algorithm for self-configuration of SON in future cellular networks. Then, we propose an energy conservation fot the self-optimization of wireless networks. Finally, we present a disaster recovery algorithm for any type of damaged wireless network. In this last chapter, we also introduce the simulation of determinantal point processes in wireless networks.La représentation par complexes simpliciaux fournit une description mathématique de la topologie d’un réseau de capteurs, c’est-à-dire sa connectivité et sa couverture. Dans ces réseaux, les capteurs sont déployés aléatoirement en grand nombre afin d’assurer une connectivité et une couverture parfaite. Nous proposons un algorithme qui permet de déterminer quels capteurs mettre en veille, sans modification de topologie, afin de réduire la consommation d’énergie. Notre algorithme de réduction peut être appliqué à tous les types de complexes simpliciaux, et atteint un résultat optimal. Pour les complexes simpliciaux aléatoires géométriques, nous obtenons des bornes pour le nombre de sommets retirés, et trouvons des propriétés mathématiques pour le complexe simplicial obtenu. En cherchant la complexité de notre algorithme, nous sommes réduits à calculer le comportement asymptotique de la taille de la plus grande clique dans un graphe géométrique aléatoire. Nous donnons le comportement presque sûr de la taille de la plus grande clique pour les trois régimes de percolation du graphe géométrique. Dans la deuxième partie, nous appliquons la représentation par complexes simpliciaux aux réseaux cellulaires, et améliorons notre algorithme de réduction pour répondre à de nouvelles demandes. Tout d’abord, nous donnons un algorithme pour la planification automatique de fréquences, pour la configuration automatique des réseaux cellulaires de la nouvelle génération bénéficiant de la technologie SON. Puis, nous proposons un algorithme d’économie d’énergie pour l’optimisation des réseaux sans fil. Enfin, nous présentons un algorithme pour le rétablissement des réseaux sans fil endommagés après une catastrophe. Dans ce dernier chapitre, nous introduisons la simulation des processus ponctuelsdéterminantaux dans les réseaux sans fil

A case study on regularity in cellular network deployment

This paper 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

Value of ultrasonography as a marker of early response to abatacept in patients with rheumatoid arthritis and an inadequate response to methotrexate: results from the APPRAISE study

Objectives: To study the responsiveness of a combined power Doppler and greyscale ultrasound (PDUS) score for assessing synovitis in biologic-naïve patients with rheumatoid arthritis (RA) starting abatacept plus methotrexate (MTX). Methods: In this open-label, multicentre, single-arm study, patients with RA (MTX inadequate responders) received intravenous abatacept (∼10 mg/kg) plus MTX for 24 weeks. A composite PDUS synovitis score, developed by the Outcome Measures in Rheumatology–European League Against Rheumatism (OMERACT–EULAR)-Ultrasound Task Force, was used to evaluate individual joints. The maximal score of each joint was added into a Global OMERACT–EULAR Synovitis Score (GLOESS) for bilateral metacarpophalangeal joints (MCPs) 2–5 (primary objective). The value of GLOESS containing other joint sets was explored, along with clinical efficacy. Results: Eighty-nine patients completed the 24-week treatment period. The earliest PDUS sign of improvement in synovitis was at week 1 (mean change in GLOESS (MCPs 2–5): −0.7 (95% CIs −1.2 to −0.1)), with continuous improvement to week 24. Early improvement was observed in the component scores (power Doppler signal at week 1, synovial hyperplasia at week 2, joint effusion at week 4). Comparable changes were observed for 22 paired joints and minimal joint subsets. Mean Disease Activity Score 28 (C reactive protein) was significantly reduced from weeks 1 to 24, reaching clinical meaningful improvement (change ≥1.2) at week 8. Conclusions: In this first international prospective study, the composite PDUS score is responsive to abatacept. GLOESS demonstrated the rapid onset of action of abatacept, regardless of the number of joints examined. Ultrasound is an objective tool to monitor patients with RA under treatment. Trial registration number: NCT00767325

Algebraic topology for wireless sensor networks

La représentation par complexes simpliciaux fournit une description mathématique de la topologie d’un réseau de capteurs, c’est-à-dire sa connectivité et sa couverture. Dans ces réseaux, les capteurs sont déployés aléatoirement en grand nombre afin d’assurer une connectivité et une couverture parfaite. Nous proposons un algorithme qui permet de déterminer quels capteurs mettre en veille, sans modification de topologie, afin de réduire la consommation d’énergie. Notre algorithme de réduction peut être appliqué à tous les types de complexes simpliciaux, et atteint un résultat optimal. Pour les complexes simpliciaux aléatoires géométriques, nous obtenons des bornes pour le nombre de sommets retirés, et trouvons des propriétés mathématiques pour le complexe simplicial obtenu. En cherchant la complexité de notre algorithme, nous sommes réduits à calculer le comportement asymptotique de la taille de la plus grande clique dans un graphe géométrique aléatoire. Nous donnons le comportement presque sûr de la taille de la plus grande clique pour les trois régimes de percolation du graphe géométrique. Dans la deuxième partie, nous appliquons la représentation par complexes simpliciaux aux réseaux cellulaires, et améliorons notre algorithme de réduction pour répondre à de nouvelles demandes. Tout d’abord, nous donnons un algorithme pour la planification automatique de fréquences, pour la configuration automatique des réseaux cellulaires de la nouvelle génération bénéficiant de la technologie SON. Puis, nous proposons un algorithme d’économie d’énergie pour l’optimisation des réseaux sans fil. Enfin, nous présentons un algorithme pour le rétablissement des réseaux sans fil endommagés après une catastrophe. Dans ce dernier chapitre, nous introduisons la simulation des processus ponctuelsdéterminantaux dans les réseaux sans fil.Simplicial complex representation gives a mathematical description of the topology of a wireless sensor network, i.e., its connectivity and coverage. In these networks, sensors are randomly deployed in bulk in order to ensure perfect connectivity and coverage. We propose an algorithm to discover which sensors are to be switched off, without modification of the topology, in order to reduce energy consumption. Our reduction algorithm can be applied to any type of simplicial complex and reaches an optimum solution. For random geometric simplicial complexes, we find boundaries for the number of removed vertices, as well as mathematical properties for the resulting simplicial complex. The complexity of our reduction algorithm boils down to the computation of the asymptotical behavior of the clique number of a random geometric graph. We provide almost sure asymptotical behavior for the clique number in all three percolation regimes of the geometric graph. In the second part, we apply the simplicial complex representation to cellular networks and improve our reduction algorithm to fit new purposes. First, we provide a frequency auto-planning algorithm for self-configuration of SON in future cellular networks. Then, we propose an energy conservation fot the self-optimization of wireless networks. Finally, we present a disaster recovery algorithm for any type of damaged wireless network. In this last chapter, we also introduce the simulation of determinantal point processes in wireless networks.Simplicial complex representation gives a mathematical description of the topology of a wireless sensor network, i.e., its connectivity and coverage. In these networks, sensors are randomly deployed in bulk in order to ensure perfect connectivity and coverage. We propose an algorithm to discover which sensors are to be switched off, without modification of the topology, in order to reduce energy consumption. Our reduction algorithm can be applied to any type of simplicial complex and reaches an optimum solution. For random geometric simplicial complexes, we find boundaries for the number of removed vertices, as well as mathematical properties for the resulting simplicial complex. The complexity of our reduction algorithm boils down to the computation of the asymptotical behavior of the clique number of a random geometric graph. We provide almost sure asymptotical behavior for the clique number in all three percolation regimes of the geometric graph. In the second part, we apply the simplicial complex representation to cellular networks and improve our reduction algorithm to fit new purposes. First, we provide a frequency auto-planning algorithm for self-configuration of SON in future cellular networks. Then, we propose an energy conservation fot the self-optimization of wireless networks. Finally, we present a disaster recovery algorithm for any type of damaged wireless network. In this last chapter, we also introduce the simulation of determinantal point processes in wireless networks

Simplicial homology based energy saving algorithms for wireless networks

International audienceEnergy saving is one of the most investigated problems in wireless networks. In this paper, we introduce two homology based algorithms: a simulated annealing one and a downhill one. These algorithms optimize the energy consumption at network level while maintaining the maximal coverage. By using simplicial homology, the complex geometrical calculation of the coverage is reduced to simple matrix computation. The simulated annealing algorithm gives a solution that approaches the global optimal one. The downhill algorithm gives a local optimal solution. The simulated annealing algorithm and downhill algorithm converge to the solution with polynomial and exponential rate, respectively. Our simulations show that this local optimal solution also approaches the global optimal one. Our algorithms can save at most 65% of system's maximal consumption power in polynomial time. The probability density function of the optimized radii of cells is also analyzed and discussed

Stochastic Geometry: Boolean model and random geometric graphs

This paper collects the four contributions which were presented during the session devoted to Stochastic Geometry at the journées MAS 2014. It is focused in particular on several questions related to the transmission of information in a general sense in different random media. The underlying models include the Boolean model, simplicial complexes or geometric random graphs induced by a point process