The goal of this thesis defended in 2015 is to develop tools and methods for the evaluation of the QoS (Quality of Service) perceived by users, as a function of the traffic demand, in modern wireless cellular networks. This complex problem, directly related to network dimensioning, involves modeling dynamic processes at several time-scales, which due to their randomness are amenable to probabilistic formalization. Firstly, on the ground of information theory, we capture the performance of a single link between a base station and a user in the context of a cellular network with orthogonal channels and MIMO technology. We prove and use some lower bounds of the information-theoretic ergodic capacity of such a link, which account also for the fast channel variability caused by multi-path propagation. These bounds give robust basis for further user QoS evaluation. Next, one considers several (possibly mobile) users, arriving in the network and requesting some service from it. We consider variable (elastic) bit-rate services, in which transmissions of some amounts of data are realized in a best-effort manner, or constant bit-rate services, in which a certain transmission rate needs to be maintained during requested times. On the ground of queuing theory, one captures this traffic demand and service process using appropriate (multi-class) processor sharing (PS) or loss models. In this thesis, we adapt existing PS models and develop a new loss model for wireless streaming traffic, in which the aforementioned information-theoretic capacities of single links describe the instantaneous user service rates. The multi-class models are used to capture the spatial heterogeneity of user channels, which depends on the user geographic locations and propagation shadowing phenomenon. Finally, on top of the queueing-theoretic processes, one needs to consider a multi-cellular network, whose base stations are not necessarily regularly placed, and whose geometry is further perturbed by the shadowing phenomenon. We address this randomness aspect by using some models from stochastic geometry, notably Poisson point processes and Palm formalism applied to the typical cell of the network. Applying the above three-fold approach, supposed to represent all crucial mechanisms and engineering parameters of cellular networks (such as LTE), we establish some macroscopic relations between the traffic demand and the user QoS metrics for some elastic and constant bit-rate services. These relations are mostly obtained in a semi-analytic way, i.e., they only involve static simulations of a Poisson point process (modeling the locations of base stations) in order to evaluate its characteristics which are not amenable to analytic expressions. More precisely, regarding the data traffic (the elastic bit-rate service), we capture the inter-cell interference, making the PS queue models of individual cells dependent, via some system of cell-load equations. These equations allow one to determine the mean user throughput, the mean number of users and the mean cell load in a large network, as a function of the traffic demand. The spatial distribution of these QoS metrics in the network is also studied. We validate our approach by comparing the obtained results with those measured from live-network traces. We observe a remarkably good agreement between the model predictions and the statistical data collected in several deployment scenarios. Regarding constant bit-rate services, we propose a new stochastic model to evaluate the frequency and the number of interruptions during real-time streaming calls in function of user radio conditions. Despite some fundamental similarities with the classical Erlang loss model, a more adequate model was required for in this case, where the denial of service is not definitive for a given call: it takes the form of, hopefully short, interruptions or outage periods. Our model allows one to take into account realistic implementations of the considered streaming service. We use it to study the quality of service metrics in function of user radio conditions in LTE networks. All established results contribute to the development of network dimensioning methods and are currently used in Orange internal tools for network capacity calculations.

In  we propose two analytically tractable stochastic-geometric models of interference in ad-hoc networks using pure (non-slotted) Aloha as the medium access. In contrast the slotted model, the interference in pure Aloha may vary during the transmission of a tagged packet. We develop closed form expressions for the Laplace transform of the empirical average of the interference experienced during the transmission of a typical packet. Both models assume a power-law path-loss function with arbitrarily distributed fading and feature configurations of transmitters randomly located in the Euclidean plane according to a Poisson point process. Depending on the model, these configurations vary over time or are static. We apply our analysis of the interference to study the Signal-to-Interference-and-Noise Ratio (SINR) outage probability for a typical transmission in pure Aloha. The results are used to compare the performance of non-slotted Aloha to the slotted one, which has almost exclusively been previously studied in the same context of mobile ad-hoc networks.

In  we study, as a basic model, a stationary Poisson pattern of nodes on a line embedded in an independent planar Poisson field of interfering nodes. Assuming slotted Aloha and the signal-to-interference-and-noise ratio capture condition, with the usual power-law path loss model and Rayleigh fading, we explicitly evaluate several local and end-to-end performance characteristics related to the nearest-neighbor packet relaying on this line, and study their dependence on the model parameters (the density of relaying and interfering nodes, Aloha tuning and the external noise power). Our model can be applied in two cases: the first use is for vehicular ad-hoc networks, where vehicles are randomly located on a straight road. The second use is to study a typical route traced in a (general) planar ad-hoc network by some routing mechanism. The approach we have chosen allows us to quantify the non-efficiency of long-distance routing in pure ad-hoc networks and evaluate a possible remedy for it in the form of additional fixed relaying nodes, called road-side units in a vehicular network. It also allows us to consider a more general field of interfering nodes and study the impact of the clustering of its nodes the routing performance. As a special case of a field with more clustering than the Poison field, we consider a Poisson-line field of interfering nodes, in which all the nodes are randomly located on random straight lines. The comparison to our basic model reveals a paradox: clustering of interfering nodes decreases the outage probability of a single (typical) transmission on the route, but increases the mean end-to-end delay.

Based on a stationary Poisson point process, a wireless network model with random propagation effects (shadowing and/or fading) is considered in  in order to examine the process formed by the signal-to-interference-plus-noise ratio (SINR) values experienced by a typical user with respect to all base stations in the down-link channel. This SINR process is completely characterized by deriving its factorial moment measures, which involve numerically tractable, explicit integral expressions. This novel framework naturally leads to expressions for the k-coverage probability, including the case of random SINR threshold values considered in multi-tier network models. While the k-coverage probabilities correspond to the marginal distributions of the order statistics of the SINR process, a more general relation is presented connecting the factorial moment measures of the SINR process to the joint densities of these order statistics. This gives a way for calculating exact values of the coverage probabilities arising in a general scenario of signal combination and interference cancellation between base stations. The presented framework consisting of mathematical representations of SINR characteristics with respect to the factorial moment measures holds for the whole domain of SINR and is amenable to considerable model extension.

In  we propose a model for heterogeneous cellular networks assuming a space-time Poisson process of call arrivals, independently marked by data volumes, and served by different types of base stations (having different transmission powers) represented by the superposition of independent Poisson processes on the plane. Each station applies a processor sharing policy to serve users arriving in its vicinity, modeled by the Voronoi cell perturbed by some random signal propagation effects (shadowing). Users' peak service rates depend on their signal-to-interference-and-noise ratios (SINR) with respect to the serving station. The mutual-dependence of the cells (due to the extra-cell interference) is captured via some system of cell-load equations impacting the spatial distribution of the SINR. We use this model to study in a semi-analytic way (involving only static simulations, with the temporal evolution handled by the queuing theoretic results) network performance metrics (cell loads, mean number of users) and the quality of service perceived by the users (mean throughput) served by different types of base stations. Our goal is to identify macroscopic laws regarding these performance metrics, involving averaging both over time and the network geometry. The reveled laws are validated against real field measurement in an operational network.

Geographic locations of cellular base stations sometimes can be well fitted with spatial homogeneous Poisson point processes. In  we make a complementary observation: In the presence of the log-normal shadowing of sufficiently high variance, the statistics of the propagation loss of a single user with respect to different network stations are invariant with respect to their geographic positioning, whether regular or not, for a wide class of empirically homogeneous networks. Even in perfectly hexagonal case they appear as though they were realized in a Poisson network model, i.e., form an inhomogeneous Poisson point process on the positive half-line with a power-law density characterized by the path-loss exponent. At the same time, the conditional distances to the corresponding base stations, given their observed propagation losses, become independent and log-normally distributed, which can be seen as a decoupling between the real and model geometry. The result applies also to Suzuki (Rayleigh-log-normal) propagation model. We use Kolmogorov-Smirnov test to empirically study the quality of the Poisson approximation and use it to build a linear-regression method for the statistical estimation of the value of the path-loss exponent.

In  we propose an analytic approach to the frequency bandwidth dimensioning problem, faced by cellular network operators who deploy/upgrade their networks in various geographical regions (countries) with an inhomogeneous urbanization. We present a model allowing one to capture fundamental relations between users' quality of service parameters (mean downlink throughput), traffic demand, the density of base station deployment, and the available frequency bandwidth. These relations depend on the applied cellular technology (3G or 4G impacting user peak bit-rate) and on the path-loss characteristics observed in different (urban, sub-urban and rural) areas. We observe that if the distance between base stations is kept inversely proportional to the distance coefficient of the path-loss function, then the performance of the typical cells of these different areas is similar when serving the same (per-cell) traffic demand. In this case, the frequency bandwidth dimensioning problem can be solved uniformly across the country applying the mean cell approach proposed in [Blaszczyszyn et al. WiOpt2014]. We validate our approach by comparing the analytical results to measurements in operational networks in various geographical zones of different countries.

In  we consider the problem of an optimal geographic placement of content in wireless cellular networks modelled by Poisson point processes. Specifically, for the typical user requesting some particular content and whose popularity follows a given law (e.g. Zipf), we calculate the probability of finding the content cached in one of the base stations. Wireless coverage follows the usual signal-to-interference-and noise ratio (SINR) model, or some variants of it. We formulate and solve the problem of an optimal randomized content placement policy, to maximize the user's hit probability. The result dictates that it is not always optimal to follow the standard policy "cache the most popular content, everywhere". In fact, our numerical results regarding three different coverage scenarios, show that the optimal policy significantly increases the chances of hit under high-coverage regime, i.e., when the probabilities of coverage by more than just one station are high enough.

In  we consider a model of cellular networks where the base station locations constitute a Poisson point process and each base station is equipped with three sectorial antennas is proposed. This model permits to study the spatial distribution of the SINR in the downlink. In particular, this distribution is shown to be insensitive to the distribution of antenna azimuths. Moreover, the effect of horizontal sectorisation is shown to be equivalent to that of shadowing. Assuming ideal vertical antenna pattern, an explicit expression of the Laplace transform of the inverse of SINR is given. The model is validated by comparing its results to measurements in an operational network. It is observed numerically that, in the case of dense urban regions where interference is preponderant, one may neglect the effect of the vertical sectorization when calculating the distribution of the SINR, which provides considerable tractability. Combined with queuing theory results, the SINR’s distribution permits to express the user’s quality of service as function of the traffic demand. This permits in particular to operators to predict the required investments to face the continual increase of traffic demand.

The objective of  is to establish a theoretical expression of the link performance in the downlink of a multiple input multiple output (MIMO) cellular network and compare it to the real Long-Term Evolution (LTE ) performance. In order to account for the interference, we prove that the worst additive noise process in the MIMO context is the white Gaussian one. Based on this theoretical result, we build an analytic expression of the link performance in LTE cellular networks with MIMO. We study also the minimum mean square error (MMSE) scheme currently implemented in the field, as well as its improvement MMSE-SIC (successive interference cancellation) known to achieve the MIMO capacity. Comparison to simulation results as well as to measurements in the field shows that the theoretical expression predicts well practical link performance of LTE cellular networks. This theoretical expression of link performance is the basis of a global analytic approach to the evaluation of the quality of service perceived by the users in the long run of their arrivals and departures.

In a paper accepted for publication in the Journal of Applied Probability, F. Baccelli and V. Anantharam consider a family of Boolean models, indexed by integers $n\ge 1$. The $n$-th model features a Poisson point process in ${ℝ}^n$ of intensity $e^{n{\rho }_n}$ and balls of independent and identically distributed radii distributed like ${\overline{X}}_n\sqrt{n}$. Assume that ${\rho }_n\to \rho$ as $n\to \infty$, and that ${\overline{X}}_n$ satisfies a large deviations principle. It is shown that there then exist three deterministic thresholds: ${\tau }_d$ the degree threshold; ${\tau }_p$ the percolation probability threshold; and ${\tau }_v$ the volume fraction threshold, such that asymptotically as $n$ tends to infinity, we have the following features. (i) For $\rho <{\tau }_d$, almost every point is isolated, namely its ball intersects no other ball; (ii) for ${\tau }_d<\rho <{\tau }_p$, the mean number of balls intersected by a typical ball converges to infinity and nevertheless there is no percolation; (iii) for ${\tau }_p<\rho <{\tau }_v$, the volume fraction is 0 and nevertheless percolation occurs; (iv) for ${\tau }_d<\rho <{\tau }_v$, the mean number of balls intersected by a typical ball converges to infinity 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 problems in information theory, but it could be of independent interest in stochastic geometry. The relations between these three thresholds and the Shannon–Poltyrev threshold are discussed.

In an Infocom'15 paper, F. Baccelli and X. Zhang (Qualcomm) have introduced an analytically tractable stochastic geometry model for urban wireless networks, where the locations of the nodes and the shadowing are highly correlated and different path loss functions can be applied to line-of-sight (LOS) and non-line-of-sight (NLOS) links.

Using a distance-based LOS path loss model and a blockage (shadowing)-based NLOS path loss model, one can derive the distribution of the interference observed at a typical location and the joint distribution at different locations of the network. When applied to cellular networks, this model leads to tractable coverage probabilities (SINR distribution) expressions. This model captures important features of urban wireless networks, which were difficult to analyze using existing models.

This model was lately extended in a joint work by the same authors and Robert Heath (UT Austin) in a paper presented at IEEE Globecom'15 where it received the best paper award.

In a paper to be published in IEEE Transactions of Information Theory, F. Baccelli, N. Lee and Robert Heath consider large random wireless networks where transmit-and-receive node pairs communicate within a certain range while sharing a common spectrum. By modeling the spatial locations of nodes as Poisson point processes, analytical expressions for the ergodic spectral efficiency of a typical node pair are derived as a function of the channel state information available at a receiver (CSIR) in terms of relevant system parameters: the density of communication links, the number of receive antennas, the path loss exponent, and the operating signal-to-noise ratio. One key finding is that when the receiver only exploits CSIR for the direct link, the sum spectral efficiency increases linearly with the density, provided the number of receive antennas increases as a certain super-linear function of the density. When each receiver exploits CSIR for a set of dominant interfering links in addition to that of the direct link, the sum spectral efficiency in creases linearly with both the density and the path loss exponent if the number of antennas is a linear function of the density. This observation demonstrates that having CSIR for dominant interfering links provides an order gain in the scaling law. It is also shown that this linear scaling holds for direct CSIR when incorporating the effect of the receive antenna correlation, provided that the rank of the spatial correlation matrix scales super-linearly with the density. These scaling laws are derived from integral representations of the distribution of the Signal to Interference and Noise Ratio, which are of independent interest and which in turn derived from stochastic geometry and more precisely from the theory of Shot Noise fields.

In a joint work with Mir-Omid Haji-Mirsadeghi, Sharif University, Department of Mathematics, F. Baccelli studied a class of non-measure preserving dynamical systems on counting measures called point-maps. This research introduced two objects associated with a point map $f$ acting on a stationary point process $\Phi$:

Two papers on the matter available. The first one is under revision for Annals of Probability.

In recent years, wearable devices and wireless body area networks have gained momentum as a means to monitor people’s behavior and simplify their interaction with the surrounding environment, thus representing a key element of the body-to-body networking (BBN) paradigm. Within this paradigm, several transmission technologies, such as 802.11 and 802.15.4, that share the same unlicensed band (namely, the industrial, scientific, and medical band) coexist, dramatically increasing the level of interference and, in turn, negatively affecting network performance. In this paper, we analyze the cross-technology interference (CTI) caused by the utilization of different transmission technologies that share the same radio spectrum. We formulate an optimization model that considers internal interference, as well as CTI to mitigate the overall level of interference within the system, explicitly taking into account node mobility. We further develop three heuristic approaches to efficiently solve the interference mitigation problem in large-scale network scenarios. Finally, we propose a protocol to compute the solution that minimizes CTI in a distributed fashion. Numerical results show that the proposed heuristics represent efficient and practical alternatives to the optimal solution for solving the CTI mitigation (CTIM) problem in large-scale BBN scenarios.

The ongoing evolution of wireless technologies has fostered the development of innovative network paradigms like the Internet of Things (IoT). Wireless Body Area Networks, and more specifically Body-to-Body Area Networks (BBNs), are emerging solutions for the monitoring of people's behavior and their interaction with the surrounding environment. These networks represent a key building block of the IoT paradigm. In BBNs several transmission technologies like 802.11 and 802.15.4 that share the same unlicensed band (namely the industrial, scientific and medical (ISM) radio band) coexist, increasing dramatically the level of interference and, in turn, negatively affecting network’s performance. In , we investigate the Cross-Technology Interference Mitigation (CTIM) problem caused by the utilization of different transmission technologies that share the same radio spectrum, from a centralized and distributed point of view, respectively.

In , we compute the actual worst-case end-to-end delay for a flow in a feed-forward network of FIFO-multiplexing service curve nodes, where flows are shaped by piecewise-affine concave arrival curves, and service curves are piecewise affine and convex. We show that the worst-case delay problem can be formulated as a mixed integer-linear programming problem, whose size grows exponentially with the number of nodes involved. Furthermore, we present approximate solution schemes to find upper and lower delay bounds on the worst-case delay. Both only require to solve just one linear programming problem, and yield bounds which are generally more accurate than those found in the previous work, which are computed under more restrictive assumptions.

Computing deterministic performance guarantees is a defining issue for systems with hard real-time constraints, like reactive embedded systems. In , we use burst-rate constrained arrivals and rate-latency servers to deduce tight worst-case delay bounds in tandem networks under arbitrary multiplexing. We present a constructive method for computing the exact worst-case delay, which we prove to be a linear function of the burstiness and latencies; our bounds are hence symbolic in these parameters. Our algorithm runs in quadratic time in the number of servers. We also present an application of our algorithm to the case of stochastic arrivals and server capacities. For a generalization of the exponentially bounded burstiness (EBB) model, we deduce a polynomial-time algorithm for stochastic delay bounds that strictly improve the state-of-the-art separated flow analysis (SFA) type bounds.

Renewable energy sources such as wind and solar power have a high degree of unpredictability and time-variation, which makes balancing demand and supply challenging. One possible way to address this challenge is to harness the inherent flexibility in demand of many types of loads. Introduced in is a technique for decentralized control for automated demand response that can be used by grid operators as ancillary service for maintaining demand-supply balance. A randomized control architecture is proposed, motivated by the need for decentralized decision making, and the need to avoid synchronization that can lead to large and detrimental spikes in demand. An aggregate model for a large number of loads is then developed by examining the mean field limit. A key innovation is a linear time-invariant (LTI) system approximation of the aggregate nonlinear model, with a scalar signal as the input and a measure of the aggregate demand as the output. This makes the approximation particularly convenient for control design at the grid level.

describes a new way of thinking about demand-side resources to provide ancillary services to control the grid. It is shown that loads can be classified based on the frequency bandwidth of ancillary service that they can offer. If demand response from loads respects these frequency limitations, it is possible to obtain highly reliable ancillary service to the grid, while maintaining strict bounds on the quality of service (QoS) delivered by each load. It is argued that automated demand response is required for reliable control. Moreover, some intelligence is needed at demand response loads so that the aggregate will be reliable and controllable.

concerns state estimation problems in a mean field control setting. In a finite population model, the goal is to estimate the joint distribution of the population state and the state of a typical individual. The observation equations are a noisy measurement of the population. The general results are applied to demand dispatch for regulation of the power grid, based on randomized local control algorithms. In prior work by the authors it has been shown that local control can be carefully designed so that the aggregate of loads behaves as a controllable resource with accuracy matching or exceeding traditional sources of frequency regulation. The operational cost is nearly zero in many cases. The information exchange between grid and load is minimal, but it is assumed in the overall control architecture that the aggregate power consumption of loads is available to the grid operator. It is shown that the Kalman filter can be constructed to reduce these communication requirements, and to provide the grid operator with accurate estimates of the mean and variance of quality of service (QoS) for an individual load.

In , we consider open Jackson networks with losses with mixed finite and infinite queues and analyze the efficiency of sampling from their exact stationary distribution. We show that perfect sampling is possible, although the underlying Markov chain may have an infinite state space. The main idea is to use a Jackson network with infinite buffers (that has a product form stationary distribution) to bound the number of initial conditions to be considered in the coupling from the past scheme. We also provide bounds on the sampling time of this new perfect sampling algorithm for acyclic or hyper-stable networks. These bounds show that the new algorithm is considerably more efficient than existing perfect samplers even in the case where all queues are finite. We illustrate this efficiency through numerical experiments. We also extend our approach to variable service times and non-monotone networks such as queueing networks with negative customers.

The maximum independent set (MIS) problem is a well-studied combinatorial optimization problem that naturally arises in many applications, such as wireless communication, information theory and statistical mechanics. MIS problem is NP-hard, thus many results in the literature focus on fast generation of maximal independent sets of high cardinality. One possibility is to combine Gibbs sampling with coupling from the past arguments to detect convergence to the stationary regime. This results in a sampling procedure with time complexity that depends on the mixing time of the Glauber dynamics Markov chain. We propose in an adaptive method for random event generation in the Glauber dynamics that considers only the events that are effective in the coupling from the past scheme, accelerating the convergence time of the Gibbs sampling algorithm.

In , we consider a dynamic matching model with random arrivals. In prior work, authors have proposed policies that are stabilizing, and also policies that are approximately finite-horizon optimal. This paper considers the infinite-horizon average-cost optimal control problem. A relaxation of the stochastic control problem is proposed, which is found to be a special case of an inventory model, as treated in the classical theory of Clark and Scarf. The optimal policy for the relaxation admits a closed-form expression. Based on the policy for this relaxation, a new matching policy is proposed. For a parameterized family of models in which the network load approaches capacity, this policy is shown to be approximately optimal, with bounded regret, even though the average cost grows without bound.

In we present an exact sampling method for multiclass closed queuing networks. We consider networks for which stationary distribution does not necessarily have a product form. The proposed method uses a compact representation of sets of states, that is used to derive a bounding chain with significantly lower complexity of one-step transition in the coupling from the past scheme. The coupling time of this bounding chain can be larger than the coupling time of the exact chain, but it is finite in expectation. Numerical experiments show that coupling time is close to that of the exact chain. Moreover, the running time of the proposed algorithm outperforms the classical algorithm.

In this paper, we revisit the problem of constructing a near-optimal rank $k$ approximation of a matrix $M\in {[0,1]}^{m\times n}$ under the streaming data model where the columns of $M$ are revealed sequentially. We present SLA (Streaming Low-rank Approximation), an algorithm that is asymptotically accurate, when $k{s}_{k+1}\left(M\right)=o\left(\sqrt{mn}\right)$ where $s_{k+1}\left(M\right)$ is the $(k+1)$-th largest singular value of $M$. This means that its average mean-square error converges to 0 as $m$ and $n$ grow large (i.e., $\parallel {\widehat{M}}^{\left(k\right)}-M^{\left(k\right)}{\parallel }_F^2=o\left(mn\right)$ with high probability, where ${\widehat{M}}^{\left(k\right)}$ and $M^{\left(k\right)}$ denote the output of SLA and the optimal rank $k$ approximation of $M$, respectively). Our algorithm makes one pass on the data if the columns of $M$ are revealed in a random order, and two passes if the columns of $M$ arrive in an arbitrary order. To reduce its memory footprint and complexity, SLA uses random sparsification, and samples each entry of $M$ with a small probability $\delta$. In turn, SLA is memory optimal as its required memory space scales as $k(m+n)$, the dimension of its output. Furthermore, SLA is computationally efficient as it runs in $O\left(\delta kmn\right)$ time (a constant number of operations is made for each observed entry of $M$), which can be as small as $O(klog{\left(m\right)}^{4}n)$ for an appropriate choice of $\delta$ and if $n\ge m$.

investigates stochastic and adversarial combinatorial multi-armed bandit problems. In the stochastic setting under semi-bandit feedback, we derive a problem-specific regret lower bound, and discuss its scaling with the dimension of the decision space. We propose ESCB, an algorithm that efficiently exploits the structure of the problem and provide a finite-time analysis of its regret. ESCB has better performance guarantees than existing algorithms, and significantly outperforms these algorithms in practice. In the adversarial setting under bandit feedback, we propose CombEXP, an algorithm with the same regret scaling as state-of-the-art algorithms, but with lower computational complexity for some combinatorial problems.

A non-backtracking walk on a graph is a directed path such that no edge is the inverse of its preceding edge. The non-backtracking matrix of a graph is indexed by its directed edges and can be used to count non-backtracking walks of a given length. It has been used recently in the context of community detection and has appeared previously in connection with the Ihara zeta function and in some generalizations of Ramanujan graphs. In , we study the largest eigenvalues of the non-backtracking matrix of the Erdos-Renyi random graph and of the Stochastic Block Model in the regime where the number of edges is proportional to the number of vertices. Our results confirm the "spectral redemption" conjecture that community detection can be made on the basis of the leading eigenvectors above the feasibility threshold.

In , we address the problem of content replication in large distributed content delivery networks, composed of a data center assisted by many small servers with limited capabilities and located at the edge of the network. The objective is to optimize the placement of contents on the servers to offload as much as possible the data center. We model the system constituted by the small servers as a loss network, each loss corresponding to a request to the data center. Based on large system / storage behavior, we obtain an asymptotic formula for the optimal replication of contents and propose adaptive schemes related to those encountered in cache networks but reacting here to loss events, and faster algorithms generating virtual events at higher rate while keeping the same target replication. We show through simulations that our adaptive schemes outperform significantly standard replication strategies both in terms of loss rates and adaptation speed.

In , we consider the problem of partially recovering hidden binary variables from the observation of (few) censored edge weights, a problem with applications in community detection, correlation clustering and synchronization. We describe two spectral algorithms for this task based on the non-backtracking and the Bethe Hessian operators. These algorithms are shown to be asymptotically optimal for the partial recovery problem, in that they detect the hidden assignment as soon as it is information theoretically possible to do so.

In the ordinary stochastic block model, all degrees in a cluster have the same expected degree. The Degree-Corrected Stochastic Block Models (DC-SBM) is a generalization of the former where the expected degrees of individual nodes follow a prescribed degree-sequence. We consider community detection in the DC-SBM in a paper currently in preparation . We perform spectral clustering on a suitably normalized adjacency matrix. This leads to consistent recovery of the block-membership of all but a vanishing fraction of nodes, in the regime where the lowest degree is of order log$\left(n\right)$ or higher. The main contributions of this paper are $\left(i\right)$ the fact that recovery succeeds for very heterogeneous degree-distributions and $\left(ii\right)$ a clean analysis for the DC-SBM, which is a messy model.

In a paper currently in preparation , we consider a degree-corrected planted-partition model: a random graph on $n$ nodes with two equal-sized clusters. The model parameters are two constants $a,b>0$ and an i.i.d. sequence ${\left({\phi }_i\right)}_{i=1}^n$, with finite second moment ${\Phi }^2$. Vertices $i$ and $j$ are joined by an edge with probability $\frac{{\phi }_i{\phi }_j}{n}a$ whenever they are in the same class and with probability $\frac{{\phi }_i{\phi }_j}{n}b$ otherwise. We prove that the underlying community structure cannot be accurately recovered from observations of the graph when ${(a-b)}^2{\Phi }^2\le 2(a+b)$.

In , we consider a class of nonlinear mappings ${𝖥}_{A,N}$ in ${ℝ}^N$ indexed by symmetric random matrices $A\in {ℝ}^{N\times N}$ with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Bolthausen [Comm. Math. Phys. 325 (2014) 333-366]. Within information theory, they are known as "approximate message passing" algorithms. We study the high-dimensional (large $N$) behavior of the iterates of $𝖥$ for polynomial functions $𝖥$, and prove that it is universal; that is, it depends only on the first two moments of the entries of $A$, under a sub-Gaussian tail condition. As an application, we prove the universality of a certain phase transition arising in polytope geometry and compressed sensing. This solves, for a broad class of random projections, a conjecture by David Donoho and Jared Tanner.

In , we consider a threshold epidemic model on a clustered random graph with overlapping communities. In other words, our epidemic model is such that an individual becomes infected as soon as the proportion of her infected neighbors exceeds the threshold q of the epidemic. In our random graph model, each individual can belong to several communities. The distributions for the community sizes and the number of communities an individual belongs to are arbitrary. We consider the case where the epidemic starts from a single individual, and we prove a phase transition (when the parameter q of the model varies) for the appearance of a cascade, i.e. when the epidemic can be propagated to an infinite part of the population. More precisely, we show that our epidemic is entirely described by a multi-type (and alternating) branching process, and then we apply Sevastyanov's theorem about the phase transition of multi-type Galton-Watson branching processes. In addition, we compute the entries of the matrix whose largest eigenvalue gives the phase transition.

In , we study the impact of random exponential edge weights on the distances in a random graph and, in particular, on its diameter. Our main result consists of a precise asymptotic expression for the maximal weight of the shortest weight paths between all vertices (the weighted diameter) of sparse random graphs, when the edge weights are i.i.d. exponential random variables.