LQG Graphon Mean Field Games

This paper formulates infinite dimensional graphon dynamical systems to study the limit problem of linear quadratic Gaussian graphon mean field game (LQG-GMFG). The limit problem is characterized by forward and backward graphon dynamical system equations. Based on this representation, sufficient conditions on the existence of a unique fixed point are established. Furthermore, under the finite rank assumption on the underlying graphon, exact numerical solution methods are presented.

[1]  Peter E. Caines,et al.  Spectral Representations of Graphons in Very Large Network Systems Control , 2019, 2019 IEEE 58th Conference on Decision and Control (CDC).

[2]  Fabio Camilli,et al.  Stationary Mean Field Games Systems Defined on Networks , 2015, SIAM J. Control. Optim..

[3]  Peter E. Caines,et al.  Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle , 2006, Commun. Inf. Syst..

[4]  Alain Bensoussan,et al.  Representation and Control of Infinite Dimensional Systems, 2nd Edition , 2007, Systems and control.

[5]  Roland P. Malhamé,et al.  Collective Stochastic Discrete Choice Problems: A Min-LQG Game Formulation , 2016 .

[6]  Peter E. Caines,et al.  Social Optima in Mean Field LQG Control: Centralized and Decentralized Strategies , 2012, IEEE Transactions on Automatic Control.

[7]  Peter E. Caines,et al.  Optimal and Approximate Solutions to Linear Quadratic Regulation of a Class of Graphon Dynamical Systems , 2019, 2019 IEEE 58th Conference on Decision and Control (CDC).

[8]  François Delarue,et al.  MEAN FIELD GAMES: A TOY MODEL ON AN ERDOS-RENYI GRAPH , 2017 .

[9]  Peter E. Caines,et al.  The NCE (Mean Field) Principle With Locality Dependent Cost Interactions , 2010, IEEE Transactions on Automatic Control.

[10]  Hendrik J. Kuiper,et al.  Strong Solutions for Infinite-Dimensional Riccati Equations Arising in Transport Theory , 1980 .

[11]  Olivier Guéant.  Existence and Uniqueness Result for Mean Field Games with Congestion Effect on Graphs , 2011, 1110.3442.

[12]  Mathieu Lauriere,et al.  Stochastic Graphon Games: I. The Static Case , 2019, Math. Oper. Res..

[13]  V. Sós,et al.  Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing , 2007, math/0702004.

[14]  Alejandro Ribeiro,et al.  The Graphon Fourier Transform , 2020, ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[15]  Bal'azs Szegedy,et al.  The automorphism group of a graphon , 2014 .

[16]  Sriram Vishwanath,et al.  Sequential Decomposition of Graphon Mean Field Games , 2020, ArXiv.

[17]  Georgi S. Medvedev,et al.  The Nonlinear Heat Equation on W-Random Graphs , 2013, 1305.2167.

[18]  M. Bálek,et al.  Large Networks and Graph Limits , 2022 .

[19]  Peter E. Caines,et al.  Subspace Decomposition for Graphon LQR: Applications to VLSNs of Harmonic Oscillators , 2021, IEEE Transactions on Control of Network Systems.

[20]  Timoteo Carletti,et al.  Random Walks on Dense Graphs and Graphons , 2019, SIAM Journal on Applied Mathematics.

[21]  Georgi S. Medvedev,et al.  The Nonlinear Heat Equation on Dense Graphs and Graph Limits , 2013, SIAM J. Math. Anal..

[22]  Peter E. Caines,et al.  Graphon Mean Field Games and the GMFG Equations , 2018, 2018 IEEE Conference on Decision and Control (CDC).

[23]  V. Sós,et al.  Convergent Sequences of Dense Graphs II. Multiway Cuts and Statistical Physics , 2012 .

[24]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[25]  Shuang Gao,et al.  The control of arbitrary size networks of linear systems via graphon limits: An initial investigation , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[26]  Thorsten Gerber,et al.  Semigroups Of Linear Operators And Applications To Partial Differential Equations , 2016 .

[27]  Hendrik J. Kuiper.  Generalized Operator Riccati Equations , 1985 .

[28]  Georgi S. Medvedev,et al.  The mean field analysis of the Kuramoto model on graphs Ⅰ. The mean field equation and transition point formulas , 2016, Discrete & Continuous Dynamical Systems - A.

[29]  Shuang Gao,et al.  Graphon Control of Large-Scale Networks of Linear Systems , 2018, IEEE Transactions on Automatic Control.

[30]  Santiago Segarra,et al.  Centrality Measures for Graphons: Accounting for Uncertainty in Networks , 2017, IEEE Transactions on Network Science and Engineering.

[31]  Shuang Gao,et al.  Grapbon Linear Quadratic Regulation of Large-Scale Networks of Linear Systems , 2018, 2018 IEEE Conference on Decision and Control (CDC).

[32]  Francesca Parise,et al.  Graphon Games , 2018, EC.

[33]  Minyi Huang,et al.  Large-Population Cost-Coupled LQG Problems With Nonuniform Agents: Individual-Mass Behavior and Decentralized $\varepsilon$-Nash Equilibria , 2007, IEEE Transactions on Automatic Control.

[34]  R. Showalter.  Monotone operators in Banach space and nonlinear partial differential equations , 1996 .

[35]  Federica Garin,et al.  Graphon-Based Sensitivity Analysis of SIS Epidemics , 2020, IEEE Control Systems Letters.