[Paper Review] Long time dynamics for interacting oscillators on graphs

F. Coppini|arXiv (Cornell University)|Aug 5, 2019

Nonlinear Dynamics and Pattern Formation32 references4 citations

TL;DR

This paper establishes a deterministic condition on sequences of graphs—based on Grothendieck’s Inequality and $/document/ ext{cut-norm} o ext{concentration in }ackslash ext{ell}_{ackslash ext{infty}}\toackslash ext{ell}_{1}$—that ensures the empirical measure of the stochastic Kuramoto model remains close to the McKean-Vlasov solution over long times. The condition guarantees synchronization persists up to exponentially long times, even in subcritical and supercritical regimes, under diverging average degree.

ABSTRACT

The stochastic Kuramoto model defined on a sequence of graphs is analyzed: the emphasis is posed on the relationship between the mean field limit, the connectivity of the underlying graph and the long time behavior. We give an explicit deterministic condition on the sequence of graphs such that, for any finite time and any initial condition, even dependent on the network, the empirical measure of the system stays close to the solution of the McKean-Vlasov equation associated to the classical mean field limit. Under this condition, we study the long time behavior in the subcritical and in the supercritical regime: in both regimes, the empirical measure stays close to the (possibly degenerate) manifold of stable stationary solutions, up to times which can diverge as fast as the exponential of the size of the system, before Large Deviation phenomena take over. The condition on the sequence of graphs is derived by means of Grothendieck's Inequality and expressed through a concentration in $\\ell\\_{\\infty}\ o \\ell\\_1$ norm. It is shown to be satisfied by a large class of graphs, random and deterministic, provided that the average number of neighbors per site diverges, as the size of the system tends to infinity.

Motivation & Objective

• To identify a minimal, deterministic graph condition ensuring the mean field limit holds for interacting oscillators on graphs over finite and long time intervals.
• To analyze long-time synchronization dynamics in both subcritical and supercritical regimes of the Kuramoto model on general graph sequences.
• To show that the empirical measure remains close to the manifold of stable stationary solutions for times scaling exponentially with system size.
• To demonstrate that the graph condition is satisfied by diverse random and deterministic graphs, including Erdős-Rényi with diverging average degree.
• To bridge the gap between mean field limits and long-time behavior in complex networks, particularly under weak assumptions on initial conditions and graph structure.

Proposed method

• Derives a deterministic condition on graph sequences using Grothendieck’s Inequality, expressed as concentration in the $ackslash ext{ell}_{ackslash ext{infty}}\toackslash ext{ell}_{1}$ operator norm.
• Analyzes the stochastic Kuramoto model on graphs with adjacency matrices $ackslash\xi^{(n)}$ and dilution parameter $p_n$, with interaction scaled as $1/(np_n)$.
• Applies the theory of self-normalized processes and spectral analysis of the Fokker-Planck operator $L_\psi$ to control the dynamics in $H_{-1}$ and $H_{-2}$ spaces.
• Uses a variation of Gronwall’s lemma (Lemma B.4) to control the growth of deviations between the empirical measure and the McKean-Vlasov solution.
• Establishes semigroup estimates for $e^{tL_\psi}$ and $e^{tL_\psi^*}$ in weighted Sobolev spaces, with bounds on operator norms in $H_{-1}$ and $H_{-2}$.
• Applies duality and eigenfunction analysis to bound the $L^\infty$ norm of eigenfunctions of $L_\psi^*$, ensuring regularity of the semigroup.

Experimental results

Research questions

• RQ1 What deterministic condition on a sequence of graphs ensures the empirical measure of the Kuramoto model stays close to the McKean-Vlasov solution over finite time intervals?
• RQ2 How does the long-time behavior of the system depend on the graph structure, particularly in subcritical and supercritical synchronization regimes?
• RQ3 Can the system remain synchronized for times that grow exponentially with system size, and what graph conditions enable this?
• RQ4 Is the mean field limit robust to weak initial dependence and non-i.i.d. initial conditions under general graph sequences?
• RQ5 What is the minimal graph condition—expressed via functional norms—that guarantees convergence to the mean field limit and long-time stability?

Key findings

• The paper identifies a deterministic graph condition based on $ackslash ext{ell}_{ackslash ext{infty}}\toackslash ext{ell}_{1}$ concentration, derived from Grothendieck’s Inequality, that ensures the empirical measure stays close to the McKean-Vlasov solution for any finite time and any initial condition.
• The condition is satisfied by a large class of graphs, including Erdős-Rényi random graphs with diverging average degree, as long as $np_n \to \infty$.
• In both subcritical and supercritical regimes, the empirical measure remains close to the (possibly degenerate) manifold of stable stationary solutions up to times that can grow as fast as $\exp(n)$, before Large Deviation effects dominate.
• The analysis establishes a quantitative bound on the deviation of the empirical measure from the McKean-Vlasov solution using a modified Gronwall inequality, ensuring stability over long times.
• The spectral analysis of the Fokker-Planck operator $L_\psi$ yields eigenvalue bounds $\lambda_l \in [l^2/C, Cl^2]$ and uniform $L^\infty$ bounds on eigenfunctions, crucial for controlling the semigroup behavior.
• The semigroup $e^{tL_\psi}$ satisfies $\|e^{tL_\psi}u\|_{-1} \leq C(1 + t^{-1/2})\|u\|_{-2}$, which enables control of the dynamics in negative Sobolev norms and supports long-time estimates.