In , we consider a numerical scheme for Hamilton–Jacobi equations based on a direct discretization of the Lax–Oleinik semi–group. We prove that this method is convergent with respect to the time and space stepsizes provided the solution is Lipschitz, and give an error estimate. Moreover, we prove that the numerical scheme is a geometric integrator satisfying a discrete weak–KAM theorem which allows to control its long time behavior. Taking advantage of a fast algorithm for computing min–plus convolutions based on the decomposition of the function into concave and convex parts, we show that the numerical scheme can be implemented in a very efficient way.

In  we consider exact sampling from the stationary distribution of a closed queueing network with finite capacities. In a recent work a compact representation of sets of states was proposed that enables exact sampling from the stationary distribution without considering all initial conditions in the coupling from the past (CFTP) scheme.This representation reduces the complexity of the one-step transition in the CFTP algorithm to O($K{M}^2$), where $K$ is the number of queues and M the total number of customers; while the cardinality of the state space is exponential in the number of queues. In this paper, we extend these previous results to the multiserver case. The main focus and the contribution of this work is on the algorithmic and the implementation issues. We propose a new representation, that leads to one-step transition complexity of the CFTP algorithm that is in $O\left(KM\right)$. We provide a detailed description of our matrix-based implementation. Matlab toolbox Clones (CLOsed queueing Networks Exact Sampling) can be downloaded at http://www.di.ens.fr/~rovetta/Clones

In  we consider queueing networks which are made from servers exchanging their positions on a graph. When two servers exchange their positions, they take their customers with them. Each customer has a fixed destination. Customers use the network to reach their destinations, which is complicated by movements of the servers. We develop the general theory of such networks and establish the convergence of the symmetrized version of such a network to some nonlinear Markov process.

This work, reported in , concerns design of control systems for Demand Dispatch to obtain ancillary services to the power grid by harnessing inherent flexibility in many loads. The role of “local intelligence” at the load has been advocated in prior work, randomized local controllers that manifest this intelligence are convenient for loads with a finite number of states. The present work introduces two new design techniques for these randomized controllers: (i) The Individual Perspective Design (IPD) is based on the solution to a one-dimensional family of Markov Decision Processes, whose objective function is formulated from the point of view of a single load. The family of dynamic programming equation appears complex, but it is shown that it is obtained through the solution of a single ordinary differential equation. (ii) The System Perspective Design (SPD) is motivated by a single objective of the grid operator: Passivity of any linearization of the aggregate input-output model. A solution is obtained that can again be computed through the solution of a single ordinary differential equation. Numerical results complement these theoretical results.

In our previous research  , it was argued that loads can provide most of the ancillary services required today and in the future. Through load-level and grid-level control design, high-quality ancillary service for the grid is obtained without impacting quality of service delivered to the consumer. This approach to grid regulation is called demand dispatch: loads are providing service continuously and automatically, without consumer interference. In  work we investigate what intelligence is required at the grid-level. In particular, does the grid-operator require more than one-way communication to the loads? Our main conclusion: risk is not great in lower frequency ranges, e.g., PJM's RegA or BPA's balancing reserves. In particular, ancillary services from refrigerators and pool-pumps can be obtained successfully with only one-way communication. This requires intelligence at the loads, and much less intelligence at the grid level.

Nowadays, telecommunication infrastructures are composed of property hardware operated by a single entity to offer communication services to their final users. While this architecture simplifies the design and optimization of the network equipment for specific tasks, its low degree of flexibility represents the main limitation for the evolution of the network infrastructure. For this reason, network operators and equipment manufacturers have started the standardization process of a plethora of virtualization solutions that have been individually developed in recent years for enabling the sharing of general-purpose resources and increasing the flexibility of their network architectures. Such a process has led to the specification of the Network Functions Virtualization (NFV) technology, which promises to bring about several benefits, such as reduced CAPEX and OPEX (CAPital and OPerational EXpenditure), low time-tomarket for new network services, higher flexibility to scale up and down the services according to users' demand, simple and cheap testing of new services. Nevertheless, the consolidation of the virtualization technology represents one of the main challenging problems for its success and widespread utilization in telecommunication infrastructures, which still consist of a huge set of property hardware appliances and software systems. Indeed, the sharing of the physical infrastructure among multiple virtual operators as well as the simple configuration of network services require the design of complex management mechanisms for the orchestration of the network equipment, with the final goal of dynamically adapting the infrastructure to the resource utilization.

In particular, spatio-temporal correlation of traffic demands and computational loads can result in high congestion and low network performance for virtual operators, thus leading to service level agreement breaches. In , we propose novel orchestration mechanisms to optimally control and mitigate the resource congestion of a physical infrastructure based on the NFV paradigm. More specifically, we analyze the congestion resulting from the sharing of the physical infrastructure and propose innovative orchestration mechanisms based on both centralized and distributed approaches, aimed at unleashing the potential of the NFV technology. In particular, we first formulate the network functions composition problem as a non-linear optimization model to accurately capture the congestion of physical resources. To further simplify the network management, we also propose a dynamic pricing strategy of network resources, proving that the resulting system achieves a stable equilibrium in a completely distributed fashion, even when all virtual operators independently select their best network configuration. Numerical results show that the proposed approaches consistently reduce resource congestion. Furthermore, the distributed solution well approaches the performance that can be achieved using a centralized network orchestration system.

Content Delivery Networks (CDNs) have been identified as one of the relevant use cases where the emerging paradigm of Network Functions Virtualization (NFV) will likely be beneficial. In fact, virtualization fosters flexibility, since on-demand resource allocation of virtual CDN nodes can accommodate sudden traffic demand changes. However, there are cases where physical appliances should still be preferred, therefore we envision a mixed architecture in between these two solutions, capable to exploit the advantages of both of them. Motivated by these reasons, in  we formulate a two-stage stochastic planning model that can be used by CDN operators to compute the optimal long-term network planning decision, deploying physical CDN appliances in the network and/or leasing resources for virtual CDN nodes in data centers. Key findings demonstrate that for a large range of pricing options and traffic profiles, NFV can significantly save network costs spent by the operator to provide the content distribution service.

The radio frequency (RF) spectrum is a scarce resource that has recently become particularly critical with the increased wireless demand. For this reason, the Federal Communications Commission (FCC) has recently allowed for opportunistic access to the unused spectrum in the TV bands (also called “white space”). With opportunistic access, however, there is a need to deploy enhanced channel allocation and power control techniques to mitigate interference, including Adjacent-Channel Interference (ACI). TV White Space (TVWS) spectrum access is often investigated without taking into account ACI between the transmissions of TV Bands Devices (TVBDs) and licensed TV stations. Guard Bands (GBs) can be used to protect data transmissions and mitigate the ACI problem. Therefore, in we consider a spectrum database that is administrated by a database operator, and an opportunistic secondary system, in which every TVBD is equipped with a single antenna that can be tuned to a subset of licensed channels. This can be done, for example, through adaptive channel aggregation or bonding techniques.

We investigate the distributed spectrum management problem in opportunistic TVWS systems using a game theoretical approach that accounts for adjacent channel interference and spatial reuse. TVBDs compete to access idle TV channels and select channel “blocks” that optimize an objective function. This function provides a tradeoff between the achieved rate and a cost factor that depends on the interference between TVBDs. We consider practical cases where contiguous or non-contiguous channels can be accessed by TVBDs, imposing realistic constraints on the maximum frequency span between the aggregated/bonded channels. We show that under general conditions, the proposed TVWS management games admit a potential function. Accordingly, a “best response” strategy allows us to determine the spectrum assignment of all players. This algorithm is shown to converge in a few iterations to a Nash Equilibrium (NE). Furthermore, we propose an effective algorithm based on Imitation dynamics, where a TVBD probabilistically imitates successful selection strategies of other TVBDs in order to improve its objective function. Numerical results show that our game theoretical framework provides a very effective tradeoff (close to optimal, centralized spectrum allocations) between efficient TV spectrum use and reduction of interference between TVBDs.

5G is envisioned to support scalable networks and improved user experience with virtually zero latency and ultra broad-band service. Supporting unlimited seamless mobility is one of the key issues and also for network resource utilization efficiency. In , we focus on mobility management and user equipment (UE) speed class estimation, also known as mobility state estimation (MSE). We propose a method for estimating the UE mobility which is compliant with UE history information specifications by 3GPP (3rd Generation Partnership Project). We also exploit the impact of the environment on the UE trajectory and speed when determining UE mobility state. We evaluate the effectiveness of our algorithm using realistic mobility traces and network topology of the city of Cologne in Germany provided by the Kolntrace project. Results show that the speed classification of UEs can be achieved with much higher accuracy compared to existing legacy 3GPP LTE MSE procedures.

Estimating mobile user speed is a problematic issue which has significant impacts to radio resource management and also to the mobility management of Long Term Evolution (LTE) networks. In  introduces two algorithms that can estimate the speed of mobile user equipments (UE), with low computational requirement, and without modification of neither current user equipment nor 3GPP standard protocol. The proposed methods rely on uplink (UL) sounding reference signal (SRS) power measurements performed at the eNodeB (eNB) and remain efficient with large sampling period (e.g., 40 ms or beyond). We evaluate the effectiveness of our algorithms using realistic LTE system data provided by the eNB Layer1 team of Alcatel-Lucent. Results show that the classification of UE's speed required by LTE can be achieved with high accuracy. In addition, they have minimal impact to the central processing unit (CPU) and the memory of eNB modem. We see that they are very practical to today's LTE networks and would allow a continuous and real-time UE speed estimation

In small cell networks, high mobility of users results in frequent handoff and thus severely restricts the data rate for mobile users. To alleviate this problem, in  we propose to use heterogeneous, two-tier network structure where static users are served by both macro and micro base stations, whereas the mobile (i.e., moving) users are served only by macro base stations having larger cells; the idea is to prevent frequent data outage for mobile users due to handoff. We use the classical two-tier Poisson network model with different transmit powers (cf  ), assume independent Poisson process of static users and doubly stochastic Poisson process of mobile users moving at a constant speed along infinite straight lines generated by a Poisson line process. Using stochastic geometry, we calculate the average downlink data rate of the typical static and mobile (i.e., moving) users, the latter accounted for handoff outage periods. We consider also the average throughput of these two types of users defined as their average data rates divided by the mean total number of users co-served by the same base station. We find that if the density of a homogeneous network and/or the speed of mobile users is high, it is advantageous to let the mobile users connect only to some optimal fraction of BSs to reduce the frequency of handoffs during which the connection is not assured. If a heterogeneous structure of the network is allowed, one can further jointly optimize the mean throughput of mobile and static users by appropriately tuning the powers of micro and macro base stations subject to some aggregate power constraint ensuring unchanged mean data rates of static users via the network equivalence property (see  ).

In  we consider location-dependent opportunistic bandwidth sharing between static and mobile downlink users in a cellular network. Each cell has some fixed number of static users. Mobile users enter the cell, move inside the cell for some time and then leave the cell. In order to provide higher data rate to mobile users, we propose to provide higher bandwidth to the mobile users at favourable times and locations, and provide higher bandwidth to the static users in other times. We formulate the problem as a long run average reward Markov decision process (MDP) where the per-step reward is a linear combination of instantaneous data volumes received by static and mobile users, and find the optimal policy. The transition structure of this MDP is not known in general. To alleviate this issue, we propose a learning algorithm based on single timescale stochastic approximation. Also, noting that the unconstrained MDP can be used to solve a constrained problem, we provide a learning algorithm based on multi-timescale stochastic approximation. The results are extended to address the issue of fair bandwidth sharing between the two classes of users. Numerical results demonstrate performance improvement by our scheme, and also the trade-off between performance gain and fairness.

In  we develop Gibbs sampling based techniques for learning the optimal content placement in a cellular network. A collection of base stations are scattered on the space, each having a cell (possibly overlapping with other cells). Mobile users request for downloads from a finite set of contents according to some popularity distribution. Each base station can store only a strict subset of the contents at a time; if a requested content is not available at any serving base station, it has to be downloaded from the backhaul. Thus, there arises the problem of optimal content placement which can minimize the download rate from the backhaul, or equivalently maximize the cache hit rate. Using similar ideas as Gibbs sampling, we propose simple sequential content update rules that decide whether to store a content at a base station based on the knowledge of contents in neighbouring base stations. The update rule is shown to be asymptotically converging to the optimal content placement for all nodes. Next, we extend the algorithm to address the situation where content popularities and cell topology are initially unknown, but are estimated as new requests arrive to the base stations. Finally, improvement in cache hit rate is demonstrated numerically.

This work contributes to the line of research on the development of analytic tools for the QoS evaluation and dimensioning of operator cellular networks which is the subject of long-term collaboration between TREC/DYOGENE and Orange Labs (cf Section ). Our focus in  is to explicitly characterize the disparity of quality of service (QoS) metrics between base stations in large heterogeneous wireless cellular networks. The considered QoS metrics are cell load, users' number, and user throughput. The spatial disparity of these metrics is due to the irregularity of the cells' geometry. In order to consider these irregularities, we assume a Poisson point process of base station locations, random transmission powers, and log-normal shadowing. The interdependency between the performances of the base stations is characterized by a system of load equations. The typical cell simulation model consists in resolving this system in order to find the loads and then deduce the remaining characteristics for each cell of the network. Using stochastic geometric and queueing theoretic techniques, we define the QoS averages, variances, and distributions. Inspired by the analysis of the typical cell model, several investigations lead us to propose a fully analytic approach, called mean cell model, that approximates the averages, variances, and distributions of these QoS metrics. Numerical experiments show a good agreement between the proposed approximations, simulation results, and real-life network measurements.

This work contributes to the line of research on Poisson convergence in wireless networks with strong shadowing initiated in  , . More recently, Keeler, Ross and Xia derived in  approximation and convergence results, which imply that the point process formed from the signal strengths received by an observer in a wireless network under a general statistical propagation model can be modeled by an inhomogeneous Poisson point process on the positive real line. The basic requirement for the results to apply is that there must be a large number of transmitters with a small proportion having a strong signal. The aim of  is to apply some of the main results of  in a less general but more easily applicable form, to illustrate how the results can apply to functions of the point process of signal strengths, and to gain intuition on when the Poisson model for transmitter locations is appropriate. A new and useful observation is that it is the stronger signals that behave more Poisson, which supports recent experimental work.

A number of real-world systems consisting of interacting agents can be usefully modelled by graphs, where the agents are represented by the vertices of the graph and the interactions by the edges. Such systems can be as diverse and complex as social networks (traditional or online), protein-protein interaction networks, internet, transport network and inter-bank loan networks. One important question that arises in the study of these networks is: to what extent, the local statistics of a network determine its global topology. This problem can be approached by constructing a random graph constrained to have some of the same local statistics as those observed in the graph of interest. One such random graph model is configuration model, which is constructed in such a way that a uniformly chosen vertex has a given degree distribution. This is the random graph which provides the underlying framework for the problems considered in the PhD thesis . As our first problem, we consider propagation of influence on configuration model, where each vertex can be influenced by any of its neighbours but in its turn, it can only influence a random subset of its neighbours. Our (enhanced) model is described by the total degree of the typical vertex and the number of neighbours it is able to influence. We give a tight condition, involving the joint distribution of these two degrees, which allows with high probability the influence to reach an essentially unique non-negligible set of the vertices, called a big influenced component, provided that the source vertex is chosen from a set of good pioneers. We explicitly evaluate the asymptotic relative size of the influenced component as well as of the set of good pioneers, provided it is non-negligible. Our proof uses the joint exploration of the configuration model and the propagation of the influence up to the time when a big influenced component is completed, a technique introduced in Janson and Luczak  . Our model can be seen as a generalization of the classical Bond and Node percolation on configuration model, with the difference stemming from the oriented conductivity of edges in our model. We illustrate these results using a few examples which are interesting from either theoretical or real-world perspective. The examples are, in particular, motivated by the viral marketing phenomenon in the context of social networks. Next, we consider the isolated vertices and the longest edge of the minimum spanning tree of a weighted configuration model. Using Stein-Chen method, we compute the asymptotic distribution of the number of vertices which are separated from the rest of the graph by some critical distance, say alpha. This distribution gives the scaling of the length of the longest edge of the nearest neighbour graph with the size of the graph. We then use the results of Fountoulakis  on percolation to prove that after removing all the edges of length greater than alpha, the subgraph obtained is connected but for the isolated vertices. This leads us to conclude that the longest edge of the minimal spanning tree and that of the nearest neighbour graph coincide with high probability. Finally, we investigate a more general question, that is, whether some ordering based on local statistics of the graph would lead to an ordering of the global topological properties, so that the bounds for more complex graphs could be obtained from their simplified versions. To this end, we introduce a convex order on random graphs and discuss some implications, particularly how it can lead to the ordering of percolation probabilities in certain situations.

In  we consider compressed sensing reconstruction from $M$ measurements of $K$-sparse structured signals which do not possess a writable correlation model. Assuming that a generative statistical model, such as a Boltzmann machine, can be trained in an unsupervised manner on example signals, we demonstrate how this signal model can be used within a Bayesian framework of signal reconstruction. By deriving a message-passing inference for general distribution restricted Boltzmann machines, we are able to integrate these inferred signal models into approximate message passing for compressed sensing reconstruction. Finally, we show for the MNIST dataset that this approach can be very effective, even for $M<K$.

In , we consider the sparse stochastic block model in the case where the degrees are uninformative. The case where the two communities have approximately the same size has been extensively studied and we concentrate here on the community detection problem in the case of unbalanced communities. In this setting, spectral algorithms based on the non-backtracking matrix are known to solve the community detection problem (i.e. do strictly better than a random guess) when the signal is sufficiently large namely above the so-called Kesten Stigum threshold. In this regime and when the average degree tends to infinity, we show that if the community of a vanishing fraction of the vertices is revealed, then a local algorithm (belief propagation) is optimal down to Kesten Stigum threshold and we quantify explicitly its performance. Below the Kesten Stigum threshold, we show that, in the large degree limit, there is a second threshold called the spinodal curve below which, the community detection problem is not solvable. The spinodal curve is equal to the Kesten Stigum threshold when the fraction of vertices in the smallest community is above $p^{*}=\frac{1}{2}-\frac{1}{2\sqrt{3}}$, so that the Kesten Stigum threshold is the threshold for solvability of the community detection in this case. However when the smallest community is smaller than $p^{*}$, the spinodal curve only provides a lower bound on the threshold for solvability. In the regime below the Kesten Stigum bound and above the spinodal curve, we also characterize the performance of best local algorithms as a function of the fraction of revealed vertices. Our proof relies on a careful analysis of the associated reconstruction problem on trees which might be of independent interest. In particular, we show that the spinodal curve corresponds to the reconstruction threshold on the tree.

presents a novel spectral algorithm with additive clustering, designed to identify overlapping communities in networks. The algorithm is based on geometric properties of the spectrum of the expected adjacency matrix in a random graph model that we call stochastic blockmodel withoverlap (SBMO). An adaptive version of the algorithm, that does not require the knowledge of the number of hidden communities, is proved to be consistent under the SBMO when the degrees in the graph are (slightly more than) logarithmic. The algorithm is shown to perform well on simulated data and on real-world graphs with known overlapping communities.

The threshold model is widely used to study the propagation of opinions and technologies in social networks. In this model individuals adopt the new behavior based on how many neighbors have already chosen it. In  we study cascades under the threshold model on sparse random graphs with community structure to see whether the existence of communities affects the number of individuals who finally adopt the new behavior. Specifically, we consider the permanent adoption model where nodes that have adopted the new behavior cannot change their state. When seeding a small number of agents with the new behavior, the community structure has little effect on the final proportion of people that adopt it, i.e., the contagion threshold is the same as if there were just one community. On the other hand, seeding a fraction of population with the new behavior has a significant impact on the cascade with the optimal seeding strategy depending on how strongly the communities are connected. In particular, when the communities are strongly connected, seeding in one community outperforms the symmetric seeding strategy that seeds equally in all communities.

In  We consider the problem of grouping items into clusters based on few random pairwise comparisons between the items. We introduce three closely related algorithms for this task: a belief propagation algorithm approximating the Bayes optimal solution, and two spectral algorithms based on the non-backtracking and Bethe Hessian operators. For the case of two symmetric clusters, we conjecture that these algorithms are asymptotically optimal in that they detect the clusters as soon as it is information theoretically possible to do so. We substantiate this claim for one of the spectral approaches we introduce.

Let P be a simple, stationary, clustering point process on the $d$-dimensional Euclidean space, in the sense that its correlation functions factorize up to an additive error decaying exponentially fast with the separation distance. Let $P_n$ be its restriction to a hypercube windows of volume $n$. We consider statistics of $P_n$ admitting the representation as sums of spatially dependent terms $H_n={\sum }_{x\in P_n}\xi (x,P_n)$, where $\xi (x,P_n)$ is a real valued (score) function, representing the interaction of $x$ with $P_n$. When the score function depends locally on $P_n$ in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics, and central limit theorems for $H_n$ as the volume n of the window goes to infinity.

This gives the limit theory for non-linear geometric statistics (such as clique counts, the number of Morse critical points, intrinsic volumes of the Boolean model, and total edge length of the k-nearest neighbor graph) of determinantal point processes with fast decreasing kernels, including the $\alpha$-Ginibre ensembles. It also gives the limit theory for geometric U-statistics of permanental point processes as well as the zero set of Gaussian entire functions. This extends the existing literature treating the limit theory of sums of stabilizing scores of Poisson and binomial input. In the setting of clustering point processes, it also extends the results of Soshnikov  as well as work of Nazarov and Sodin  .

The proof of the central limit theorem relies on a factorial moment expansion originating in Blaszczyszyn  to show clustering of mixed moments of the score function. Clustering extends the cumulant method to the setting of purely atomic random measures, yielding the asymptotic normality of $H_n$.

In  we consider a family of Boolean models, indexed by integers $n\ge 1$, where the $n$-th model features a Poisson point process in ${ℝ}^n$ of intensity $e^{n{\rho }_n}$ with ${\rho }_n\to \rho$ as $n\to \infty$, and balls of independent and identically distributed radii distributed like ${\overline{X}}_n\sqrt{n}$, with ${\overline{X}}_n$ satisfying a large deviations principle. It is shown that there exist three deterministic thresholds: ${\tau }_d$ the degree threshold; ${\tau }_p$ the percolation threshold; and ${\tau }_v$ the volume fraction threshold; such that asymptotically as $n$ tends to infinity, in a sense made precise in the paper: (i) for $\rho <{\tau }_d$, almost every point is isolated, namely its ball intersects no other ball; (ii) for ${\tau }_d<\rho <{\tau }_p$, almost every ball intersects an infinite number of balls and nevertheless there is no percolation; (iii) for ${\tau }_p<\rho <{\tau }_v$, the volume fraction is 0 and nevertheless percolation occurs; (iv) for ${\tau }_d<\rho <{\tau }_v$, almost every ball intersects an infinite number of balls and nevertheless the volume fraction is 0; (v) for $\rho >{\tau }_v$, the whole space covered. The analysis of this asymptotic regime is motivated by related problems in information theory, and may be of interest in other applications of stochastic geometry.