The general scientific focus of DYOGENE is on the development of network mathematics. The following theories lie within our research interest: dynamical systems, queuing theory, optimization and control, information theory, stochastic processes, random graphs, stochastic geometry.

Our theoretical developments are motivated by and applied in the context of communication networks (Internet, wireless, mobile, cellular, peer-to-peer), social and economic networks, power grids.

We collaborate with many industrial partners. Our current industrial relations involve EDF, Google, Huawei, Microsoft, Nokia, Orange, Safran.

More specifically, the scientific focus of DYOGENE defined in 2013 was on geometric network dynamics arising in communications. By geometric networks we understand networks with a nontrivial, discrete or continuous, geometric definition of the existence of links between the nodes. In stochastic geometric networks, this definition leads to random graphs or stochastic geometric models.

A first type of geometric network dynamics is the one where the nodes or the links change over time according to an exogeneous dynamics (e.g. node motion and geometric definition of the links). We will refer to this as dynamics of geometric networks below. A second type is that where links and/or nodes are fixed but harbor local dynamical systems (in our case, stemming from e.g. information theory, queuing theory, social and economic sciences). This will be called dynamics on geometric networks. A third type is that where the dynamics of the network geometry and the local dynamics interplay. Our motivations for studying these systems stem from many fields of communications where they play a central role, and in particular: message passing algorithms; epidemic algorithms; wireless networks and information theory; device to device networking; distributed content delivery; social and economic networks, power grids.

The following research axes have been defined in 2013 when the project-team was created.

Algorithms for network performance analysis, led by A. Bouillard and A. Busic.

Stochastic geometry and information theory for wireless network, led by B. Blaszczyszyn and F. Baccelli.

The cavity method for network algorithms, led by M. Lelarge.

Our scientific interests keep evolving. Research areas which received the most of our attention in 2017 are summarized in the following sections.

Foundation of an entirely new science for distributed control of networks with applications to the stabilization of power grids subject to high volatility of renewable energy production is being developed A. Busic in collaboration with A. Bouillard and Sean Meyn [University of Florida].

A comprehensive approach involving information theory, queueing and stochastic geometry to model and analyze the performance of large cellular networks, validated and implemented by Orange is being led by B. Blaszczyszyn in collaboration with F. Baccelli and M. K. Karray [Orange Labs]

Community detection and non-regular ramanujan graphs sole a conjecture on the optimality of non-backtracking spectral algorithm for community dectection in sparse stochastic block model graphs, as has been proved by M. Lelarge and L. Massoulié in collaboration with C. Bordenave [IMT Toulouse].

Internet, wireless, mobile, cellular networks.

Social interactions, human communities, economic networks.

Energy networks.

Publication of a monograph *Stochastic Geometry Analysis of
Cellular Networks* by Cambridge University
Press that presents latest analytic techniques and results from stochastic geometry for modelling of heterogeneous cellular networks.

Our paper “ Optimal Algorithms for Non-Smooth Distributed Optimization in Networks” by K. Scaman, F. Bach, S. Bubeck, Y.T. Lee and L. Massoulié won a best paper award at the NeurIPS 2018 conference.

Vehicle sharing systems are becoming an urban mode of transportation, and launched in many cities, as Velib' and Autolib' in Paris. Managing such systems is quite difficult. One of the major issues is the availability of the resources: vehicles or free slots. These systems became a hot topic in Operation Research and the importance of stochasticity on the system behavior leads us to propose mathematical stochastic models. The aim is to understand the system behavior and how to manage these systems in order to improve the allocation of both resources to users.

To improve BSS (bike-sharing systems), two types of policies can be deployed: incentives to the users to choose a better station, called *natural* or *green* regulation, or redistribution by trucks, called *active* regulation. In a simple mathematical model, we proved the efficiency of the 2-choice incentive policy for BSS (bike-sharing systems). The drawback of the model is that it ignores the geometry of the system, where the choice is only local. The purpose of this first work is to deal with this policy in real systems.

We use data trip data obtained from JCDecaux and reports on station status collected as open data, to test local choice policy. Indeed we designed and tested a new policy relying on a local small change in user behaviors, by adapting their trips to resource availability around their departure and arrival stations, based on 2-choice policy. Results show that, even with a small user collaboration, the proposed method increases significantly the global balance of the bike sharing system and therefore the user satisfaction. This is done using trip data sets and detecting spatial outliers, stations having a behavior significantly different from their spatial neighbors, in a context where neighbors are heavily correlated. For that we proposed an improved version of the well-known Moran scatterplot method, using a robust distance metric called Gower similarity. Using this new version of Moran scatterplot, we show that, for the occupancy data set obtained by modifiying trips, the number of spatial outliers drastically decreases. We generalize this study with W. Ghanem and L. Massoulié testing incentive and redistribution policies on a simulator, where the tradeoff between the number of frustrated trips and the penalty for the users can be measured. We propose new versions of these policies including prediction.

A model of N queues, with a local choice policy, is studied. Each one-server queue has a Poissonian arrival of customers. When a customer arrives
at a queue, he joins the least loaded queue between this queue and the next one, ties solved
at random. Service times have exponential distribution. The system is stable if the arrival-to-service
rate ratio, also called load, is less than one. When the load tends to zero, we derive the first
terms of the expansion in this parameter for the stationary probabilities that a queue has few customers. Then we provide explicit asymptotics, as the load tends to zero, for the
stationary probabilities of the queue length. We used the analyticity of the stationary probabilities as a function of the load. It shows the
behavior difference between this local choice policy and the 2-choice policy (*supermarket model*).

We consider models of content delivery networks in which the servers are constrained by two main resources: memory and bandwidth. In such systems, the throughput crucially depends on how contents are replicated across servers and how the requests of specific contents are matched to servers storing those contents. In this paper, we first formulate the problem of computing the optimal replication policy which if combined with the optimal matching policy maximizes the throughput of the caching system in the stationary regime. It is shown that computing the optimal replication policy for a given system is an NP-hard problem. A greedy replication scheme is proposed and it is shown that the scheme provides a constant factor approximation guarantee. We then propose a simple randomized matching scheme which avoids the problem of interruption in service of the ongoing requests due to re-assignment or repacking of the existing requests in the optimal matching policy. The dynamics of the caching system is analyzed under the combination of proposed replication and matching schemes. We study a limiting regime, where the number of servers and the arrival rates of the contents are scaled proportionally, and show that the proposed policies achieve asymptotic optimality. Extensive simulation results are presented to evaluate the performance of different policies and study the behavior of the caching system under different service time distributions of the requests.

This is a joint work with Aukosh Jagannath and Patrick Lopatto. We study the statistical limits of testing and estimation for a rank one deformation of a Gaussian random tensor. We compute the sharp thresholds for hypothesis testing and estimation by maximum likelihood and show that they are the same. Furthermore, we find that the maximum likelihood estimator achieves the maximal correlation with the planted vector among measurable estimators above the estimation threshold. In this setting, the maximum likelihood estimator exhibits a discontinuous BBP-type transition: below the critical threshold the estimator is orthogonal to the planted vector, but above the critical threshold, it achieves positive correlation which is uniformly bounded away from zero.

This is a joint work with Andrea Montanari.
The Lasso is a popular regression method for high-dimensional problems in which the number of parameters

Here, we consider a standard random design model and prove exponential concentration of
its empirical distribution around the prediction provided by the Gaussian denoising model. Crucially, our results are uniform with respect to

Our proofs make use of Gaussian comparison inequalities, and in particular of a version of Gordon's minimax theorem developed by Thrampoulidis, Oymak, and Hassibi, which controls the optimum value of the Lasso optimization problem. Crucially, we prove a stability property of the minimizer in Wasserstein distance, that allows to characterize properties of the minimizer itself.

We study here the so-called spiked Wigner and Wishart models, where one observes a low-rank matrix perturbed by some Gaussian noise. These models encompass many classical statistical tasks such as sparse PCA, submatrix localization, community detection or Gaussian mixture clustering. The goal of these notes is to present in a unified manner recent results (as well as new developments) on the information-theoretic limits of these spiked matrix/tensor models. We compute the minimal mean squared error for the estimation of the low-rank signal and compare it to the performance of spectral estimators and message passing algorithms. Phase transition phenomena are observed: depending on the noise level it is either impossible, easy (i.e. using polynomial-time estimators) or hard (information-theoretically possible, but no efficient algorithm is known to succeed) to recover the signal.

We study the problem of minimizing a sum of smooth and strongly convex functions split over the nodes of a network in a decentralized fashion. We propose the algorithm

Group synchronization requires to estimate unknown elements

We focus on the case in which the graph

We consider the Degree-Corrected Stochastic Block Model (DC-SBM): a random
graph on n nodes, having i.i.d. weights

We propose and analyze a family of information processing systems, where a finite set of experts or servers are employed to extract information about a stream of incoming jobs. Each job is associated with a hidden label drawn from some prior distribution. An inspection by an expert produces a noisy outcome that depends both on the job's hidden label and the type of the expert, and occupies the expert for a finite time duration. A decision maker's task is to dynamically assign inspections so that the resulting outcomes can be used to accurately recover the labels of all jobs, while keeping the system stable. Among our chief motivations are applications in crowd-sourcing, diagnostics, and experiment designs, where one wishes to efficiently learn the nature of a large number of items, using a finite pool of computational resources or human agents. We focus on the capacity of such an information processing system. Given a level of accuracy guarantee, we ask how many experts are needed in order to stabilize the system, and through what inspection architecture. Our main result provides an adaptive inspection policy that is asymptotically optimal in the following sense: the ratio between the required number of experts under our policy and the theoretical optimal converges to one, as the probability of error in label recovery tends to zero.

In this work, we consider the distributed optimization of non-smooth convex
functions using a network of computing units. We investigate this problem
under two regularity assumptions: (1) the Lipschitz continuity of the global
objective function, and (2) the Lipschitz continuity of local individual
functions. Under the local regularity assumption, we provide the first
optimal first-order decentralized algorithm called multi-step primal-dual
(MSPD) and its corresponding optimal convergence rate. A notable aspect of
this result is that, for non-smooth functions, while the dominant term of
the error is in

There are two well known Stochastic Approximation techniques that are known to have optimal rate of convergence (measured in terms of asymptotic variance): the Ruppert-Polyak averaging technique, and stochastic Newton-Raphson (SNR)(a matrix gain algorithm that resembles the deterministic Newton-Raphson method). The Zap algorithms, introduced by Devraj and Meyn in 2017, are a version of SNR designed to behave more closely like their deterministic cousin. It is found that estimates from the Zap Q-learning algorithm converge remarkably quickly, but the per-iteration complexity can be high.
In , we introduce a new class of stochastic approximation algorithms based on matrix momentum. For a special choice of the matrix momentum and gain sequences, it is found in simulations that the parameter estimates obtained from the algorithm couple with those obtained from the more complex stochastic Newton-Raphson algorithm. Conditions under which coupling is guaranteed are established for a class of linear recursions. Optimal finite-

Consider a stochastic process

A stationary version of the process is constructed, that is coupled with a stationary version of the Markov chain

with an explicit formula for the vector

For any

An explicit formula for the function

The results are illustrated using a version of the timing channel of Anantharam and Verdu.

A new approach to computation of optimal policies for MDP (Markov decision process) models is introduced in , published in SICON this year. The main idea is to solve not one, but an entire family of MDPs, parameterized by a scalar

Inexpensive energy from the wind and the sun comes with unwanted volatility, such as ramps with the setting sun or a gust of wind. Controllable generators manage supply-demand balance of power today, but this is becoming increasingly costly with increasing penetration of renewable energy. It has been argued since the 1980s that consumers should be put in the loop: “demand response” will help to create needed supply-demand balance. However, consumers use power for a reason and expect that the quality of service (QoS) they receive will lie within reasonable bounds. Moreover, the behavior of some consumers is unpredictable, while the grid operator requires predictable controllable resources to maintain reliability.

Flexibility of energy consumption can be harnessed for the purposes of grid-level ancillary services. In particular, through distributed control of a collection of loads, a balancing authority regulation signal can be tracked accurately, while ensuring that the quality of service (QoS) for each load is acceptable on average. Subject to distributed control approaches advocated in recent research, the histogram of QoS is approximately Gaussian, and consequently, each load will eventually receive poor service. In , published this year in IEEE Transactions on Smart Grid, statistical techniques are developed to estimate the mean and variance of QoS as a function of the power spectral density of the regulation signal. It is also shown that additional local control can eliminate risk. The histogram of QoS is truncated through this local control, so that strict bounds on service quality are guaranteed. While there is a tradeoff between the grid-level tracking performance (capacity and accuracy) and the bounds imposed on QoS, it is found that the loss of capacity is minor in typical cases.

The previous designs for distributed control of TCLs ensure that the indoor temperature remains within a pre-specified bound, but other QoS metrics, especially the frequency of turning on and off was not limited. In , presented at ACM BuildSys 2018, we propose a more advanced control architecture that reduces the cycling rate of TCLs. We show through simulations that the proposed controller is able to reduce the cycling of individual TCLs compared to the previous designs with little loss in tracking of the grid-supplied reference signal.

Energy storage revenue estimation is essential for analyzing financial feasibility of investment in batteries. In , we quantify the cycles of operation considering depth-of-discharge (DoD) of operational cycles and provide an algorithm to calculate equivalent 100% DoD cycles. This facilitates in comparing cycles of different DoDs. The battery life is frequently defined as a combination of cycle and calendar life. We propose a battery capacity degradation model based on the cycle and the calendar life and operational cycles. Using equivalent 100% DoD cycles and revenue generated, we calculate the dollars per cycle revenue of storage performing electricity price based arbitrage and ancillary services for load balancing in real time. Using PJM’s (a regional transmission organization in the United States) real data we calculate short term and long term financial potential for the year of 2017. We observe that participating in ancillary services is significantly more beneficial for storage owners compared to participating in energy arbitrage.

Battery life is often described a combination of cycle life and calendar life. In , we propose a mechanism to limit the number of cycles of operation over a time horizon in an optimal arbitrage algorithm proposed in our previous work. The cycles of operation have to be tuned based on price volatility to maximize the battery life and arbitrage gains.

The model of First Come First Served infinite bipartite matching was introduced in Caldentey, Kaplan and Weiss, 2009. In this model, there is a sequence of items that are chosen i.i.d. from a finite set 𝒞 and an independent sequence of items that are chosen i.i.d. from a finite set 𝒮, and a bipartite compatibility graph G between 𝒞 and 𝒮. Items of the two sequences are matched according to the compatibility graph, and the matching is FCFS, meaning that each item in the one sequence is matched to the earliest compatible unmatched item in the other sequence. In Adan and Weiss, 2012, a Markov chain associated with the matching was analyzed, a condition for stability was derived, and a product form stationary distribution was obtained. In , we present several new results that unveil the fundamental structure of the model. First, we provide a pathwise Loynes’ type construction which enables to prove the existence of a unique matching for the model defined over all the integers. Second, we prove that the model is dynamically reversible: we define an exchange transformation in which we interchange the positions of each matched pair, and show that the items in the resulting permuted sequences are again independent and i.i.d., and the matching between them is FCFS in reversed time. Third, we obtain product form stationary distributions of several new Markov chains associated with the model. As a by-product, we compute useful performance measures, for instance the link lengths between matched items.

18-month contract titled “Mathematical Modeling of 5G Ultra Dense Wireless Networks” between Inria represented by B. Blaszczyszyn (PI) and F. Baccelli, and Huawei comes to an end in December 2018. It aimed at investigating obstacle-based shadowing fields in the spatial models of cellular networks and efficient scheduling policies. Paul Keeler was hired by Inria as a research engineer thanks to this contract. The publication is one of the deliverable of this contract.

Contract with Nokia started in 2015 for the co-advising by B. Blaszczyszyn of a PhD student of Nokia, Dalia-Georgiana Herculea came to an end in December 2018. Dalia-Georgiana Herculea has successfully defended her PhD Thesis in November 2018.

Contract with Orange started in 2017 and continued in 2018 for the co-advising by B. Blaszczyszyn of a PhD student of Orange, Quentin Le Gall.

Dyogene participates in LINCS https://www.lincs.fr/, a research centre co-founded by Inria, Institut Mines- Télécom, UPMC and Alcatel-Lucent Bell Labs (currently Nokia Bell Labs) dedicated to research and innovation in the domains of future information and communication networks, systems and services. S. Meyn [Unversity of Florida] was invited professor by LINCS and ENS from July to December 2018.

Dyogene participates to the PGMO (Gaspard Monge Program for Optimization, operations research, and their interactions with data science) via the project a 2 year project “Distributed control of flexible loads” funded through the ICODE/IROE call. This is a collaborative project between University Paris-Sud (PI: Gilles Stoltz) and Inria (PI: Ana Busic). Post-doc Cheng Wan was financed by this project from Feb-Nov 2018.

Members of Dyogene participate in Research Group GeoSto
(Groupement de recherche, GdR 3477)
http://

This is a collaboration framework for all French research teams
working in the domain of spatial stochastic modeling, both on theory
development and in applications. This year DYOGENE has co-organized
yearly conference of the GdR *Stochastic Geometry Days 2018*
14–18 mai 2018 Paris (France); https://

Members of Dyogene participate in GdR-RO (Recherche Opérationelle;
GdR CNRS 3002), http://

Probabilistic Approach for Renewable Energy Integration: Virtual Storage from Flexible Loads. The project started in January 2017. PI — A. Bušić. This project is motivated by current and projected needs of a power grid with significant renewable energy integration. Renewable energy sources such as wind and solar have a high degree of unpredictability and time variation, which makes balancing demand and supply challenging. There is an increased need for ancillary services to smooth the volatility of renewable power. In the absence of large, expensive batteries, we may have to increase our inventory of responsive fossil-fuel generators, negating the environmental benefits of renewable energy. The proposed approach addresses this challenge by harnessing the inherent flexibility in demand of many types of loads. The objective of the project is to develop decentralized control for automated demand dispatch, that can be used by grid operators as ancillary service to regulate demand-supply balance at low cost. We call the resource obtained from these techniques virtual energy storage (VES). Our goal is to create the necessary ancillary services for the grid that are environmentally friendly, that have low cost and that do not impact the quality of service (QoS) for the consumers. Besides respecting the needs of the loads, the aim of the project is to design local control solutions that require minimal communications from the loads to the centralized entity. This is possible through a systems architecture that includes the following elements: i) local control at each load based on local measurements combined with a grid-level signal; ii) frequency decomposition of the regulation signal based on QoS and physical constraints for each class of loads.

B. Błaszczyszyn and Yogeshwaran D. from Indian Statistical Institute (ISI), Bangalore, have got in 2018 the approval from Indo-French Centre for Applied Mathematics (IFCAM), for their joint project on “Geometric statistics of stationary point processes" for the period 2018–2021. B. Błaszczyszyn was visiting ISI Bangalore for two weeks in November–December 2018.

University of Florida: collaborations with Prof Sean Meyn (ECE), Associate Prof Prabir Barooah (MAE), and the PhD students: A. Devraj (ECE), A. Coffman (MAE), N. Cammardella (ECE), J. Mathias (ECE).

D. Yogeshwaran [Indian Statistical Institute, Bangalore, India]

S. Meyn [University of Florida, USA] was invited Prof at ENS and LINCS, July - December 2018

Master Probabilités et Modèles aléatoires UPMC, Walid Ghanem, *Hydrodynamic limit of a network with moving servers*, 04-07/2018, encadrant Christine Fricker.

Master MASH (Mathématiques appliquées aux sciences humaines) ENS-Paris Dauphine University, *Using customer oriented policies based on
probabilistic methods to enhance the Bike Sharing
System Velib’*, 08-011/2018, encadrants Christine Fricker et Laurent Massoulié.

Akshay Goel [Kyushu University, Fukuoka, Japan] Mars 2018,

Tokuyama Kiichi [Tokyo Tech, Tokyo, Japan], April 2018,

B. Błaszczyszyn was visiting Yogeshwaran D. at the Indian Statistical Institute Bangalore for two weeks in November–December 2018 (IFCAM project).

A. Busic was a long-term participant (March-Mai 2018) of the Real-Time Decision Making program, Simons
Institute, UC Berkeley, USA; https://

B. Błaszczyszyn was in the Organizing Commitee of
*Stochastic Geometry Days 2018*
14–18 mai 2018 Paris (France); https://

A. Busic was in the Organizing Commitee of *ALEA Days 2018* 12–16 March 2018 at CIRM, Luminy (France);
https://

M. Lelarge: IEEE’s Transactions on Network Science and Engineering, Bernoulli Journal, Queueing Systems.

Analysis of Algorithms 2018, Uppsala (Sweden), June 2018.

European Conference on Queueing Theory (ECQT) 2018, Jerusalem, July 2018.

Rencontres de Probabilités de Rouen 2018, September 2018, invited talk.

Societal Networks, RTDM program, Simons Institute, UC Berkeley, USA, March 2018, invited talk; video: https://

Advances in Modelling and Control for Power Systems of the Future (CAESARS 2018), EDF Palaiseau, September 2018, invited talk.

Licence: B. Blaszczyszyn (Cours) **Théorie de l'information et du codage** 24 heqTD, L3, ENS, France.

Licence: A. Busic (Cours) and S. Samain (TD) Structures et algorithmes aléatoires 60heqTD, L3, ENS, France.

Master: B. Blaszczyszyn (Cours)
**Processus ponctuels, graphes aléatoires et
géeométrie stochastique** 39heqTD, M2
Probabilités et Modèles Aléatoires, UPMC, France

Master: A. Busic (Cours) and L. Stephan (TD) **Modéles et algorithmes de réseaux** 60heqTD, M1, ENS, France.

Master: A. Busic (Cours) **Fondements de la modélisation des réseaux** 18 heqTD, M2 MPRI,
France.

Master: M. Lelarge (Cours) **Deep Learning Do it Yourself**, M1, ENS

Doctorat: A. Busic (Cours) **Markov chains and exact sampling**, 7heqTD, Ecole thématique CNRS, MathExp 2018 "Mathématiques expérimentales: méthodes et pratiques", 21 mai-1 juin 2018, Saint Flour, France.
https://

PhD: Dalia-Georgiana Herculea “Les Hyperfractales pour la Modelisation des Reseaux sans Fil” since October 2016, defence 21 November 2018; PhD CIFRE co-advised by B. Błaszczyszyn, and Ph. Jacquet.

PhD: Alexandre Hollocou, defense December 19 2018, Nouvelles approches pour le partitionnement de graphes, co-advised by M. Lelarge and T. Bonald (Telecom ParisTech)

PhD in progress: Léo Miolane, since 2016, High dimensional statistics, advised by M. Lelarge

PhD in progress: Alexis Galland, since 2017, Deep Learning on Graphs, advised by M. Lelarge

PhD in progress: Quentin Le Gall “Crowd networking : modélisation de la connectivité D2D” since October 2017; PhD CIFRE co-advised by B. Błaszczyszyn and E. Cali (Orange).

PhD in progress: Antoine Brochard “Signal processing for point processes and statistical learning for telecommunications”, since September 2018; PhD CIFRE co-advised by B. Błaszczyszyn and Georgios Paschos (Huawei).

PhD in progress: Md Umar Hashmi, Decentralized control for renewable integration in smartgrids, since from December 2015, advised by A. Busic.

PhD in progress: Sébastien Samain, Monte Carlo methods for performance evaluation and reinforcement learning, since November 2016, advised by A. Busic.

PhD in progress: Arnaud Cadas, Dynamic matching models, since October 2017, supervised by A. Busic.

B. Błaszczyszyn, member of the PhD defense jury of Dalia-Herculea Popescu; 21 November 2018.

A. Busic, member of the PhD defense jury of J. Horta (Télécom ParisTech, France); 16 Avril 2018.

C. Fricker, PhD defense of Farah Slim (Orange- Bretagne Loire University), 13/03/2018.

B. Błaszczyszyn is an ENS adjunct professor since September 2018.

B. Błaszczyszyn and A. Busic are members of the ENS Computer Science Department Board (Conseil du Laboratoire).

A. Busic is member of the Committee for the Technological Development, Inria Paris

C. Fricker: member of the jury of *Agrégation de Mathématiques*.