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, and, recently, infectious diseases.

We collaborate with many industrial partners. Our current industrial relations involve EDF, 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, neural networks, and power grids.

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

Our scientific interests keep evolving. Research areas which received the most of our attention in 2021 are summarized in the following sections.

Over the past years, with several researchers and collaborations with we have looked at mathematical models of the evolution of the epidemics, in particular Covid 19. This year we studied the role of geographic mobility on the propagation of epidemics using point process techniques, see .

Theory and algorithms for distributed control of networks with applications to the stabilization of power grids subject to high volatility of renewable energy production are being developed by A. Busic in collaboration with Sean Meyn [Prof. at University of Florida and Inria International Chair] and Prabir Barooah [University of Florida]. We extended the Kullback-Leibler-quadratic optimal control approach to the case of stochastic disturbance. Also new advances have been obtained for thermostatically controlled loads. Within our collaboration with Vito (Belgium) that started in 2019 the main focus in 2021 has been on the Generalized Nash Equilibrium models for P2P markets.

This year we revisited our previous line of thought on probabilistic modeling of geographic caching in wireless networks, by contributing a chapter to the book with editors V. Poor [Princeton University] and W. Chen [Tsinghua University], . In a more fundamental aspect, in we addressed the problem of spacial scaling of general probabilistic policies (e.g. of caching), going beyond Gibbs framework. Also we continued to work on cellular network dimensioning toolbox in a long-term collaboration between TREC/DYOGENE represented by B. Błaszczyszyn, and Orange Labs, represented by M. K. Karray. Furthemore, a collaboration with the Standardization and Research Lab at Nokia Bell Labs and ERC NEMO led by F. Baccelli, started in 2019 and led to several joint publications including New directions started on vehicular networks and particularly on V2X in collaborations with Chang Sik Choi (Hongik University, Korea) and Nithin Ramesan (UT Austin).

We computed information theoretic bounds for unsupervised and semi-supervised learning and proved complexity bounds for distributed optimization of convex functions using a network of computing units.

We obtained new variance reduction results on stochastic Bregman gradient methods . We introduced the "continuized" Nesterov acceleration, a close variant of Nesterov acceleration whose variables are indexed by a continuous time parameter , and obtained the first rigorous acceleration of asynchronous gossip algorithms. We also obtained concentration results of non-isotropic random tensors and studied their pplications to learning and empirical risk minimization .

Point processes and stochastic geometry offer a mathematical framework for the analysis of various random structures and dynamics embedded in Euclidean spaces. In particular, it has enabled efficient analysis of truly deployed wireless networks. Behind this are mathematical asymptotic results (involving growing number of nodes subject to random processes) often leading to ”Poissonisation” of network architectures, and enabling probabilistic analysis. More generally, the analysis of large, homogeneous networks can rely on the key concept of a typical network element (user, base station, link) and its local relationships representing the overall behavior of the network. Indeed, the framework of the stochastic geometry, equipped with Palm calculus for stationary models, and its analog framework for random unimodular graphs, essentially relate to the properties of a typical network element through the so-called ”mass transport” principle. This paradigm was extended beyond the Euclidean case, based on the theory of unimodular networks, see . We developed the probabilistic machinery and also studied applications in a variery of domains: statistical physics, combinatorial optimization, communications, particle gradient descent model for point process generation, processes on Delaunay neighbors in the Poisson-Voronoi tessellation, Dirichlet measures, stochastic games, etc. We collaborated on the matter with V. Anantharam (EECS at UC Berkeley), Ch. Hirsch (University of Groningen), S. Mallat (ENS/Flatiron Institute) and S. Zhang (IRIT-SC).

In order to understand the fundamental processes taking place in very large networks, it is clear that we sometimes have to abandon the representation of local details and focus on the more macroscopic properties of higher scales. Indeed, the main idea of Mean Field Theory is to replace all detailed body-to-body interactions with a typical element of the network guided by an average or effective interaction. This year, we have several results in this area related to neural networks, opinion dynamics and car-sharing systems.

Internet, wireless, mobile, cellular networks, transportation networks, distributed systems (cloud, call centers). In collaboration with Nokia Bell Labs and Orange Labs.

Social interactions, human communities, economic networks.

Energy networks. In collaboration with EDF and Vito (Belgium).

5. A Generalized Nash Equilibrium analysis of the interaction between a peer-to-peer financial market and the distribution grid
We consider the interaction between the distribution grid (physical level) managed by the distribution system operator (DSO), and a financial market in which prosumers optimize their demand, generation, and bilateral trades in order to minimize their costs subject to local constraints and bilateral trading reciprocity coupling constraints. We model the interaction problem between the physical and financial levels as a noncooperative generalized Nash equilibrium problem. We compare two designs of the financial level prosumer market: a centralized design and a peer-to-peer fully distributed design. We prove the Pareto efficiency of the equilibria under homogeneity of the trading cost preferences. In addition, we prove that the pricing structure of our noncooperative game does not permit free-lunch behavior. Finally, in the numerical section we provide additional insights on the efficiency loss with respect to the different levels of agents' flexibility and amount of renewables in the network. We also quantify the impact of the prosumers' pricing on the noncooperative game social cost.

7. A product form for the general stochastic matching model
We consider a stochastic matching model with a general compatibility graph, as introduced by Mairesse and Moyal (2016). We show that the natural necessary condition of stability of the system is also sufficient for the natural ‘first-come, first-matched’ matching policy. To do so, we derive the stationary distribution under a remarkable product form, by using an original dynamic reversibility property related to that of Adan, Bušić, Mairesse, and Weiss (2018) for the bipartite matching model.

36. Pair Replica Mean-Field Neural Networks
Replica-mean-field models have been proposed to decipher the activity of otherwise analytically intractable neural networks via a multiply-and-conquer approach. In this approach, one considers limit networks made of infinitely many replicas with the same basic neural structure as that of the network of interest, but exchanging spikes in a randomized manner. The key point is that these replica-mean-field networks are tractable versions that retain important features of the finite structure of interest. To date, the replica framework has been discussed for first-order models, whereby elementary replica constituents are single neurons with independent Poisson inputs. In , we extend this replica framework to allow elementary replica constituents to be composite objects, namely, pairs of neurons. As they include pairwise interactions, these pair-replica models exhibit nontrivial dependencies in their stationary dynamics, which cannot be captured by first-order replica models. Our contributions are two-fold: (i) We analytically characterize the stationary dynamics of a pair of intensity-based neurons with independent Poisson input. This analysis involves the reduction of a boundary-value problem related to a two-dimensional transport equation to a system of Fredholm integral equations—a result of independent interest. (ii) We analyze the set of consistency equations determining the full network dynamics of certain replica limits. These limits are those for which replica constituents, be they single neurons or pairs of neurons, form a partition of the network of interest. Both analyses are numerically validated by computing input/output transfer functions for neuronal pairs and by computing the correlation structure of certain pair-dominated network dynamics.

Collaborative research in the area of demand dispatch of flexible loads. PI: A. Bušić.

Contract with Huawei Technologies France started in 2018 and finished in 2021 for the co-advising by B. Błaszczyszyn of a PhD student Antoine Brochard. The PhD has been extended by Inria, who have hired the student until February 2022.

NEMO, NEtwork MOtion , is an ERC Advanced Grant (2019 – 2024, PI François Baccelli). It is an inter-disciplinary proposal centered on network dynamics. The inter-disciplinarity spans from communication engineering to mathematics, with an innovative interplay between the two. NEMO’s aim is to introduce dynamics in stochastic geometry. General mathematical tools combining stochastic geometry, random graph theory, and the theory of dynamical systems will be developed. NEMO will leverage interactions of Inria with Ecole Normale Supérieure on the mathematical side, and with Nokia Bell Labs and Orange on the engineering side. This year we hired post-doc Ke Feng.

Members of Dyogene participate in Research Group GeoSto (Groupement de recherche, GdR 3477) on Stochastic Geometry led by and David Coupier [Université de Valenciennes].

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.

Members of Dyogene participate in GdR-RO (Recherche Opérationelle; GdR CNRS 3002), , working group COSMOS (Stochastic optimization and control, modeling and simulation), lead by A. Busic and E. Hyon (LIP 6);

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.

In 2021, A. Khezeli organized the weekly DYOGENE seminar. B. Roy-Chowdhury organized the NEMO reading group:

PhD defended:

PhD in progress:

Les réseaux de communications à l'INRIA. Rapport établi par F. Baccelli, I. Chrisment, J.M. Gorce et P. Mussi à la demande de B. Sportisse. Novembre 2021.

Rapport de l'académie des sciences sur les réseaux de communications du futur, établi par S. Abiteboul, D. Andler, F. Baccelli, C. Bréchignac, G. Berry, S. Candel, M. Fink et E. Moulines, juillet 2021.