In this paper, a novel resource allocation scheme based on discrete Cournot-Nash equilibria and optimal transport theory is proposed. The originality of this framework lies in the joint optimization of downlink bandwidth allocation and cooperation between base stations. A tractable formalization is given in the form of a quadratic optimization problem. A low complexity approximate solution is derived and theoretically characterized. Simulations highlight the existence of an optimal working point, that maximizes user satisfaction ratio and network load. The impact of the network deployment on the optimum is numerically investigated, thanks to the β-Ginibre model. Indeed, base stations are assumed to be drawn according to β-Ginibre point processes. Numerical analysis shows that the network performance increases with β going to one.

In this paper, we introduce a distributed load balancing algorithm for cellular networks. Traffic load in cellular networks is sometimes unbalanced. Some cells are overloaded, while others remain free. Simplicial homology is a tool from algebraic topology that allows to compute the coverage of a network by using only simple matrix computations. Our algorithm, which is based on simplicial homology, controls the transmission power of each cell in the network, not only to satisfy the coverage constraint, but also to redirect users from the overloaded cells to the underloaded ones. As a result, the traffic load of the cellular network is more balanced. The simulation results show that this algorithm improves the capacity of the whole network by 2.3% when the user demand is fast varying.

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.

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 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.

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 Čech complex and Rips complex. Čech 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 Čech 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.

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.

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.