Description du sujet de thèse

Research context:
This PhD position is part of a 4-year research project, funded by ANR (the French national science foundation) and devoted to "Continuous Methods for the Control of Large Networks” (acronym: COCOON).
Networks constitute an essential paradigm to describe a huge variety of large complex systems, including social networks, epidemics networks, transportation networks, supply chains, power networks, multi-agent systems, and collective motion and navigation. These socio-technical networks are the environment of complex dynamics that (depending on each case) may take the form of dynamics of flows, opinions, contagion, queues, or positions. The theory of Automatic Control needs substantial advancements to manage these network dynamics, because achieving control and estimation objectives using standard methods is made intractable by the ever-growing network size. Instead, these large networks and the dynamics therein require adapted tools for modeling, learning, monitoring, and control.
For this reason, the COCOON project advocates a scalable approach to large networks that is based on continuous network models instead of the usual (discrete) graphs. Towards this broad objective, this project aims at concurrently developing and cross-fertilising two promising methods to define continuous dynamics that approximate large-network dynamics: (1) Using graph limit objects such as graphons; (2) Defining analog approximations through a continuation process that replaces a large systems of ordinary differential equations with a single partial differential equation. These methods can be beneficial in a multitude of potential applications: the project will address three distinct applications with potentially high societal impact: epidemic models, electro-mobility networks and, with a bigger thrust, multimodal mobility networks.

PhD topic description:
Graphons, whose theoretical foundations are described in [1], are continuous limits of convergent sequences of graphs as the number of their nodes goes to infinity. Their usage is supported by a growing body of literature and specifically by a well-developed approximation theory that dictates how large the number of nodes N must be for the approximation to be accurate [2]. Graphons have become a popular tool to describe large networks in machine learning, but have not been so much used in control theory, despite the ground-breaking work [3]. The latter has laid down a theory of the approximate control of complex network systems by the use of graphon theory and the theory of infinite dimensional systems. By this theory, graphon dynamical system models can be formulated in an appropriate infinite dimensional space, in order to represent arbitrary-size networks of linear dynamical systems, and consistency is defined as the convergence of sequences of network systems to the graphon system.
This PhD thesis will have the objective to develop control-theoretic tools for large networks represented by graphons. These objectives cover a broad range of questions about modelling, estimation, and control:
1. Enhancing the qualitative analysis to better understand how accurately the continuous approximations describe the dynamical properties of the original system. Notice that conditions that are defined on the continuous systems can be tested at a computational cost that is independent of the size of the discrete system. Very relevant properties are, for instance, the equilibria of the dynamics and their stability.
2. Developing control methods with approximation guarantees. The key idea is the following: given a graph-based model, first derive the corresponding continuous model. Then, design a control for the continuous model and finally discretize it back to the graph model. The design being performed on the continuous system, its computational cost is independent of the size of the discrete system: hence we have a fully scalable design method. Crucially, we need to guarantee that the discretized controller will retain the desired good properties (stabilization, optimality) that were designed in the continuous model.
3. Developing estimation methods to infer the correct continuous model from data. Graphon approximation requires to estimate the graphon from partial observations of finite graph samples.
4. Case study: Epidemics. Besides their success along the decades, epidemic models have recently risen to even higher prominence due to the concerns from the COVID-19 pandemics. The DANCE team has already worked on network epidemic models, producing both theoretical results and realistic simulations, both at the level of workplaces (hundreds of individuals) and at the city level (tens of thousands of individuals), as well as some theoretical works on larger networks [4,5], which set the ground for this PhD work.

Candidate profile:
The candidate will have a MS degree in Applied Mathematics, Mathematics, Automatic Control or related disciplines

Research team:
This work will be carried out in the DANCE team (Dynamics and Control of Networks), a research team of GIPSA-Lab research center in Grenoble, France. The team's research concerns modeling, estimation and control of network systems, with a broad spectrum of theoretical and applied topics including traffic networks, intelligent vehicles, social dynamics, and analysis of large-scale complex networks.

Bibliography:
[1] L. Lovasz. Large networks and graph limits. American Mathematical Society, 2012.
[2] M. Avella-Medina, F. Parise, M. T. Schaub, and S. Segarra. Centrality measures for graphons: Accounting for uncertainty in networks. IEEE Trans. Network Science Engineering, 7(1):520–537, 2020.
[3] S. Gao and P. Caines. Graphon control of large-scale networks of linear systems. IEEE Trans. Automatic Control, 65(10):4090–4105, 2019
[4] R. Vizuete, P. Frasca, and F. Garin. Graphon-based sensitivity analysis of SIS epidemics. IEEE Control Systems Letters, 4(3):542–547, 2020. arXiv:1912.10330
[5] J.-F.Delmas, P. Frasca, A. Velleret, F. Garin, V. C. Tran, P.-A. Zitt, Individual based SIS models on (not so) dense large random networks, arXiv:2302.13385

Contexte de travail

The Gipsa-lab is a joint research laboratory of the CNRS, Grenoble-INP -UGA and the University of Grenoble Alpes. It is under agreement with Inria and the Observatory of Sciences of the Universe of Grenoble. He conducts theoretical and applied research on AUTOMATICS, SIGNAL, IMAGES, SPEECH, COGNITION, ROBOTICS and LEARNING.
Multidisciplinary and at the interface between the human, the physical and digital worlds, our research is confronted with measurements, data, observations from physical, physiological and cognitive systems. They focus on the design of methodologies and algorithms for processing and extracting information, decisions, actions and communications that are viable, efficient and compatible with physical and human reality. Our work is based on mathematical and computer theories for the development of models and algorithms, validated by hardware and software implementations.
By relying on its platforms and partnerships, Gipsa-lab maintains a constant link with applications in a wide variety of fields: health, environment, energy, geophysics, embedded systems, mechatronics, processes and industrial systems, telecommunications, networks, transport and vehicles, operational safety and security, human-computer interaction, linguistic engineering, physiology and biomechanics, etc.
Due to the nature of its research, Gipsa-lab is in direct and constant contact with the economic environment and society.
Its potential as teacher-researchers and researchers is invested in training at the level of universities and engineering schools on the Grenoble site (Grenoble Alpes University).
Gipsa-lab develops its research through 16 teams or themes organized into 4 divisions:
• Automatic and Diagnosis (PAD)
• Data Science (PSD)
• Speech and Cognition (PPC)
• Geometries, Learning, Information and Algorithms (GAIA).
The staff supporting research (38 engineers and technicians) is distributed in the common services distributed within 2 divisions:
• The Administrative and Financial Pole
• The Technical Pole
Gipsa-lab has around 150 permanent staff, including 70 teacher-researchers and 41 researchers. It also welcomes guest researchers and post-docs.
Gipsa-lab supervises nearly 150 theses, including around 50 new ones each year. All the theses carried out in the laboratory are financed and supervised by teacher-researchers and researchers, including 50 holders of an HDR.
Finally, around sixty Master's trainees come each spring to swell the ranks of the laboratory.

Le poste se situe dans un secteur relevant de la protection du potentiel scientifique et technique (PPST), et nécessite donc, conformément à la réglementation, que votre arrivée soit autorisée par l'autorité compétente du MESR.

Contraintes et risques

Defence Security :
This position is likely to be situated in a restricted area (ZRR), as defined in Decree No. 2011-1425 relating to the protection of national scientific and technical potential (PPST). Authorisation to enter an area is granted by the director of the unit, following a favourable Ministerial decision, as defined in the decree of 3 July 2012 relating to the PPST.
An unfavourable Ministerial decision in respect of a position situated in a ZRR would result in the cancellation of the appointmen

Informations complémentaires

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