Anaïs Vergne

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 C̆ech 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.

A note on the simulation of the Ginibre point process

The Ginibre point process (GPP) is one of the main examples of determinantal point processes on the complex plane. It is a recurring distribution of random matrix theory as well as a useful model in applied mathematics. In this paper we briefly overview the usual methods for the simulation of the GPP. Then we introduce a modified version of the GPP which constitutes a determinantal point process more suited for certain applications, and we detail its simulation. This modified GPP has the property of having a fixed number of points and having its support on a compact subset of the plane. See Decreusefond et al. (2013) for an extended version of this paper.

Reduction algorithm for simplicial complexes

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.

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 paper, 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.

Construction of the Generalized Czech Complex

In this paper, we introduce a centralized algorithm which constructs the generalized Cech complex. The generalized Cech complex represents the topology of a wireless network whose cells are different in size. This complex is useful to address a wide variety of problems in wireless networks such as: boundary holes detection, disaster recovery or energy saving. We have shown that our algorithm constructs the minimal generalized Cech complex, which satisfies the requirements of these applications, in polynomial time.

Simplicial homology based energy saving algorithms for wireless networks

Energy 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 robust 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 robust algorithm gives a local optimal solution. Our simulations show that this local optimal solution also approaches the global optimal one. Our algorithms can save at most 65% of systems maximal consumption power in polynomial time. The probability density function of the optimized radii of cells is also analyzed and discussed.

Homology based algorithm for disaster recovery in wireless networks

Considering a damaged wireless network, presenting coverage holes or disconnected components, we propose a disaster recovery algorithm repairing the network. It provides the list of locations where to put new nodes to patch the coverage holes and mend the disconnected components. In order to do this we first consider the simplicial complex representation of the network, then the algorithm adds supplementary nodes in excessive number, and afterwards runs a reduction algorithm in order to reach a unimprovable result. One of the novelty of this work resides in the proposed method for the addition of nodes. We use a determinantal point process: the Ginibre point process which has inherent repulsion between vertices, which simulation is new in wireless networks application. We compare both the determinantal point process addition method with other vertices addition methods, and the whole disaster recovery algorithm to the greedy algorithm for the set cover problem.

Disaster recovery in wireless networks: A homology-based algorithm

In this paper, we present an algorithm for the recovery of wireless networks after a disaster. Considering a damaged wireless network, presenting coverage holes or/and many disconnected components, we propose a disaster recovery algorithm which repairs the network. It provides the list of locations where to put new nodes in order to patch the coverage holes and mend the disconnected components. In order to do this we first consider the simplicial complex representation of the network, then the algorithm adds supplementary vertices in excessive number, and afterwards runs a reduction algorithm in order to reach an optimal result. One of the novelty of this work resides in the proposed method for the addition of vertices. We use a determinantal point process: the Ginibre point process which has inherent repulsion between vertices, and has never been simulated before for wireless networks representation. We compare both the determinantal point process addition method with other vertices addition methods, and the whole disaster recovery algorithm to the greedy algorithm for the set cover problem.

Building a Coverage Hole-free Communication Tree

Wireless networks are present everywhere but their management can be tricky since their coverage may contain holes even if the network is fully connected. In this paper we propose an algorithm that can build a communication tree between nodes of a wireless network with guarantee that there is no coverage hole in the tree. We use simplicial homology to compute mathematically the coverage, and Prims algorithm principle to build the communication tree. Some simulation results are given to study the performance of the algorithm and compare different metrics. In the end, we show that our algorithm can be used to create coverage hole-free communication groups with a limited number of hops.

Optimize wireless networks for energy saving by distributed computation of Čech complex

In this paper, we introduce a distributed algorithm to compute the Čech complex. This algorithm is aimed at solving the coverage problems in self organized wireless networks. The complexity to compute the minimal Čech complex that gives information about coverage and connectivity of the network is O(n2), where n is the average number of neighbors of each cell. An application based on the distributed computation of the Čech complex, which is aimed at optimizing the wireless network for energy saving, is also proposed. This application also has polynomial complexity. The performance of the proposed algorithm and its application are evaluated. The simulation results show that the distributed computation of the Čech complex provides a consistent outcome with the one obtained by the centralized computation that is introduced in [6], while requires a much shorter calculation time. The optimized coverage saves 65% of the total transmission power, while also keeps the maximal coverage for the network.