Graphon Stochastic Differential Equations

• Graphon SDEs are stochastic systems where interactions are mediated by a measurable graphon kernel, replacing a common population statistic with label-dependent measures.
• They provide a framework to bridge finite network approximations with continuum models using advanced tools like FBSDEs, jump-diffusions, and mean field control.
• Applications span game theory and control problems, demonstrating propagation-of-chaos and deterministic aggregate behavior in heterogeneous network dynamics.

Graphon stochastic differential equations are stochastic systems in which interaction is mediated by a graphon kernel rather than by a single exchangeable population statistic. In the most common formulation, one studies a continuum of label-indexed processes, typically with labels $u\in I=[0,1]$, and the interaction seen by label $u$ is weighted by a measurable kernel $G(u,v)$ or $w(u,v)$. A second, distinct formulation treats the state itself as graphon-valued and derives stochastic evolutions on graphon space from large random matrices or graph processes. Across these formulations, graphon SDEs serve as continuum limits for heterogeneous network models, graphon mean field games and control problems, and graphon-valued stochastic dynamics (Bayraktar et al., 2022, Harchaoui et al., 2022, Cao et al., 26 May 2025).

1. Canonical formulations

A standard graphon mean field SDE assigns to each label $u$ a state process $X^u$ satisfying

$$dX^u_t = b^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dt + \sigma^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dW^u_t,\qquad X^u_0=\xi^u,$$

with $\mu_t^u=\mathcal L(X_t^u)$ and graphon-weighted neighborhood law

$[\mu_t]^u(dx)=\int_I G(u,v)\,\mu_t^v(dx)\,dv.$

This is the graphon analogue of a McKean–Vlasov SDE: the mean field seen by label $u$ is not a single common law, but a label-dependent family of measures weighted by the graphon (Cao et al., 26 May 2025).

In graphon mean field games and graphon FBSDE models, the same interaction is often written directly in terms of graphon-weighted averages of marginal laws. A representative limiting system is

$u$0

where $u$1 (Bayraktar et al., 2022).

In linear-quadratic continuum games, the graphon enters through a graphon aggregate

$u$2

and the state equation takes the form

$u$3

Here the graphon supplies a personalized weighted aggregate for each player $u$4, rather than the single population statistic of a standard mean field game (Aurell et al., 2021).

A different construction arises in graphon-valued stochastic analysis. There the limiting object is not a continuum of agents but a graphon trajectory $u$5, often obtained from large symmetric matrices or exchangeable edge arrays. In the reflected-diffusion framework,

$u$6

with

$u$7

This is described as a graphon analogue of the McKean–Vlasov limit for interacting diffusions, with “particles” corresponding to edges rather than vertices (Harchaoui et al., 2022).

2. Measurability, continuum noise, and deterministic aggregates

A recurring technical difficulty is that graphon interaction requires joint measurability in the sample variable and the label variable. In continuum-player models with idiosyncratic Brownian motions, the aggregate

$u$8

cannot be handled on an ordinary product space if the family of Brownian motions is not jointly measurable in $u$9. For this reason, the linear-quadratic stochastic graphon game is formulated on a Fubini extension

$G(u,v)$0

which supports an essentially pairwise independent jointly measurable family $G(u,v)$1 and preserves Fubini’s theorem for integrable functions (Aurell et al., 2021).

Within that framework, the graphon aggregate is shown to have a deterministic version. More precisely, after solving the fixed-point problem for the aggregate map, one obtains

$G(u,v)$2

with an everywhere deterministic version $G(u,v)$3. The interaction survives in the continuum limit, but the aggregate itself becomes deterministic because of the exact law of large numbers on the Fubini extension (Aurell et al., 2021).

The same measurability issue reappears in graphon systems with jumps. When the compensator of the Poisson random measure varies with the label $G(u,v)$4, direct use of label-specific noises becomes problematic. The jump-diffusion graphon game resolves this by a canonical coupling: one common Brownian motion, one common Poisson random measure on $G(u,v)$5, and a label-dependent quantile map $G(u,v)$6 encoding the varying jump law. The coupled system preserves the law of each label process and ensures measurability of $G(u,v)$7 (Amini et al., 2023).

A related simplification appears in graphon FBSDE theory: instead of placing each type on its own stochastic basis, one can place all types on a common probability space and drive them by the same Brownian motion, obtaining a one-to-one correspondence with the original family of independently driven equations. This “one Brownian motion” reformulation is used to make the map $G(u,v)$8 measurable (Bayraktar et al., 2022).

3. Well-posedness and forward-backward structure

Much of the analytical theory is formulated at the FBSDE level. In graphon mean field games, the forward equation describes the label-indexed state dynamics and the backward equation encodes optimality or equilibrium. Under monotonicity and Lipschitz assumptions, existence and uniqueness can be proved either by a contraction mapping argument or by a method of continuation. In one formulation, the solution is constructed as the fixed point of two maps,

$G(u,v)$9

with the solution obtained as the fixed point of $w(u,v)$0 (Bayraktar et al., 2022).

For controlled graphon mean field systems, the adjoint equation contains not only the usual $w(u,v)$1 term but also graphon-averaged Lions-derivative corrections. The graphon adjoint BSDE is

$w(u,v)$2

where

$w(u,v)$3

Under the stated graphon, Lipschitz, differentiability, and integrability assumptions, there exists a unique solution $w(u,v)$4 with uniform square-integrability bounds (Cao et al., 26 May 2025).

In linear-quadratic graphon games, the Hamiltonian minimization yields explicit feedback. For the continuum-player model with quadratic running and terminal costs, the minimizer is

$w(u,v)$5

and substitution produces a linear graphon-dependent FBSDE. Under the Riccati-type structural assumptions listed in the paper, this FBSDE admits a unique solution in the Fubini space (Aurell et al., 2021).

Leader-follower models generate a further layer of graphon-aggregated FBSDEs. In the leader-follower linear-quadratic stochastic graphon game, the followers satisfy

$w(u,v)$6

and the follower equilibrium is characterized by a graphon-aggregated linear FBSDE. The paper also proves a general continuation-method theorem for linear graphon-aggregated FBSDEs with monotonicity condition

$w(u,v)$7

leading to existence, uniqueness, and stability of the solution (Chen et al., 30 Jan 2026).

4. Finite-network approximation and propagation of chaos

A central justification for graphon SDEs is that they arise as limits of large network systems. In the continuum-player LQ graphon game, the $w(u,v)$8-player aggregate is

$w(u,v)$9

and the comparison error

$u$0

between finite-player equilibrium and graphon equilibrium converges to zero for $u$1-almost every sequence $u$2. If the graphon is uniformly $u$3-Hölder in its first argument, the paper proves

$u$4

and the graphon equilibrium yields an $u$5-Nash equilibrium with

$u$6

These statements are presented as a propagation-of-chaos approximation for graph-based network games (Aurell et al., 2021).

For graphon FBSDEs, stability with respect to the graphon is the key mechanism behind propagation of chaos. Under Assumption 1 and Assumption 3, if $u$7, then the finite-$u$8 interacting FBSDE converges to the graphon limit; one representative estimate is

$u$9

This yields the rate

$X^u$0

and, when $X^u$1 is continuous, convergence of the empirical law $X^u$2 to the continuum law $X^u$3 (Bayraktar et al., 2022).

The same continuum approximation principle extends to jump-diffusion graphon games. If $X^u$4 and $X^u$5 solve the finite and graphon systems with the same control, then

$X^u$6

with a stronger estimate in $X^u$7. The neighborhood empirical measures $X^u$8 converge to the graphon neighborhood law $X^u$9, and the graphon equilibrium induces an approximate Nash equilibrium with $$dX^u_t = b^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dt + \sigma^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dW^u_t,\qquad X^u_0=\xi^u,$$0; under Lipschitz regularity, the paper states

$$dX^u_t = b^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dt + \sigma^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dW^u_t,\qquad X^u_0=\xi^u,$$1

These results are proved for deterministic and Bernoulli-sampled graphons (Amini et al., 2023).

In graphon mean field control, the finite $$dX^u_t = b^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dt + \sigma^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dW^u_t,\qquad X^u_0=\xi^u,$$2-player system with neighborhood empirical measure

$$dX^u_t = b^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dt + \sigma^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dW^u_t,\qquad X^u_0=\xi^u,$$3

is approximated by the continuum graphon system. If $$dX^u_t = b^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dt + \sigma^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dW^u_t,\qquad X^u_0=\xi^u,$$4 in $$dX^u_t = b^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dt + \sigma^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dW^u_t,\qquad X^u_0=\xi^u,$$5, then

$$dX^u_t = b^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dt + \sigma^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dW^u_t,\qquad X^u_0=\xi^u,$$6

Under the sharper assumption $$dX^u_t = b^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dt + \sigma^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dW^u_t,\qquad X^u_0=\xi^u,$$7, the paper gives the explicit rate

$$dX^u_t = b^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dt + \sigma^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dW^u_t,\qquad X^u_0=\xi^u,$$8

where

$$dX^u_t = b^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dt + \sigma^u(t,X^u_t,[\mu_t]^u,\alpha^u_t)\,dW^u_t,\qquad X^u_0=\xi^u,$$9

The associated cost is also approximately optimal at the same rate (Cao et al., 26 May 2025).

Setting	Finite-to-limit conclusion	Quantitative statement
LQ graphon game	finite Nash equilibrium converges to graphon equilibrium	$\mu_t^u=\mathcal L(X_t^u)$0; $\mu_t^u=\mathcal L(X_t^u)$1
Graphon FBSDEs	particle system converges to graphon limit	error $\mu_t^u=\mathcal L(X_t^u)$2 initial discrepancy
Jump-diffusion graphon game	graphon equilibrium gives approximate Nash equilibrium	$\mu_t^u=\mathcal L(X_t^u)$3; under Lipschitz regularity, $\mu_t^u=\mathcal L(X_t^u)$4
Graphon mean field control	finite controlled system is approximated by graphon-optimal profile	rate $\mu_t^u=\mathcal L(X_t^u)$5 under $\mu_t^u=\mathcal L(X_t^u)$6

5. Control, games, and equilibrium concepts

Graphon SDEs support several control-theoretic and game-theoretic regimes. In graphon mean field control, one minimizes

$\mu_t^u=\mathcal L(X_t^u)$7

over admissible profiles $\mu_t^u=\mathcal L(X_t^u)$8. The stochastic Pontryagin principle is adapted to this setting through a label-indexed Hamiltonian

$\mu_t^u=\mathcal L(X_t^u)$9

a linear variation equation, and a Gâteaux derivative formula

$[\mu_t]^u(dx)=\int_I G(u,v)\,\mu_t^v(dx)\,dv.$0

Necessary and sufficient optimality conditions are then expressed in terms of Hamiltonian minimization or the first-order condition $[\mu_t]^u(dx)=\int_I G(u,v)\,\mu_t^v(dx)\,dv.$1 (Cao et al., 26 May 2025).

In graphon games with jumps, the interaction remains heterogeneous and label-dependent, but controls enter the drift, diffusion, and jump amplitude. The controlled continuum system is

$[\mu_t]^u(dx)=\int_I G(u,v)\,\mu_t^v(dx)\,dv.$2

A graphon equilibrium is a fixed point $[\mu_t]^u(dx)=\int_I G(u,v)\,\mu_t^v(dx)\,dv.$3 such that $[\mu_t]^u(dx)=\int_I G(u,v)\,\mu_t^v(dx)\,dv.$4 is optimal against $[\mu_t]^u(dx)=\int_I G(u,v)\,\mu_t^v(dx)\,dv.$5. Existence is proved through relaxed controls, a controlled graphon martingale problem, and the Kakutani–Fan–Glicksberg fixed point theorem; uniqueness follows from a Lasry–Lions-type monotonicity condition (Amini et al., 2023).

The linear-quadratic graphon game of a continuum of players is structurally simpler but analytically instructive. Each player minimizes a quadratic cost depending on $[\mu_t]^u(dx)=\int_I G(u,v)\,\mu_t^v(dx)\,dv.$6, and equilibrium is characterized by a forward-backward SDE derived from the Pontryagin maximum principle. Because the aggregate $[\mu_t]^u(dx)=\int_I G(u,v)\,\mu_t^v(dx)\,dv.$7 is deterministic in the continuum limit, the individual state equation is stochastic while the interaction term becomes a deterministic graphon transform of the population mean trajectory (Aurell et al., 2021).

A hierarchical extension appears in leader-follower linear-quadratic stochastic graphon games. The followers are coupled through the graphon aggregation $[\mu_t]^u(dx)=\int_I G(u,v)\,\mu_t^v(dx)\,dv.$8 in both drift and diffusion, while the leader interacts with the followers through $[\mu_t]^u(dx)=\int_I G(u,v)\,\mu_t^v(dx)\,dv.$9. For a fixed leader control, the followers compete to attain a Nash equilibrium, and the leader optimizes by anticipating that equilibrium response. The resulting Stackelberg analysis uses one graphon-aggregated FBSDE for the followers and a second augmented FBSDE for the leader, together with two Riccati equations and the row-sum condition

$u$0

which implies

$u$1

Under the stated invertibility and solvability conditions, the paper derives explicit feedback laws for a Stackelberg-Nash equilibrium (Chen et al., 30 Jan 2026).

6. Graphon-valued stochastic dynamics and terminological boundaries

Not all graphon stochastic dynamics are label-indexed interacting diffusions. One important line of work studies stochastic optimization on large symmetric matrices, embedded as step graphons, and derives graphon-valued limits. Without large additive noise, projected SGD converges to a deterministic graphon gradient flow. With Brownian noise scaled at the correct level, the limit becomes a graphon-valued reflected stochastic dynamical system: $u$2 followed, as $u$3, by an infinite exchangeable array of reflected diffusions and a graphon trajectory $u$4. In the constant-noise case, reflection contributes a boundary-density correction, so for $u$5 the graphon evolution is not a gradient flow (Harchaoui et al., 2022).

Another adjacent framework studies graph-valued stochastic processes driven by vertex-level fluctuations. There the random graph is observed through its empirical graphon, and the limiting object is a deterministic graphon process induced by the limiting type path. Under the regime $u$6, vertex-level fluctuations dominate the sample-path large deviations, and the empirical graphon satisfies an LDP with rate function

$u$7

The generalized model permits edge activation and deactivation rates $u$8 and $u$9 that depend on the current graph state, producing a self-consistent mean-field-type graphon evolution (Braunsteins et al., 2022).

A common source of confusion is the similarity of terminology with graph neural stochastic differential equations. Graph Neural SDEs inject Brownian motion into latent graph representations, for example through

$u$00

and are developed for uncertainty-aware graph representation learning, confidence prediction, and out-of-distribution detection. They are stochastic latent models on finite graphs, not graphon-weighted continuum interaction models or graphon-valued limits (Bergna et al., 2023).

Taken together, these developments show that “graphon stochastic differential equations” names a family of mathematically distinct but structurally related objects. In one branch, graphons encode heterogeneous mean field interaction among a continuum of stochastic agents; in another, graphons are themselves the evolving state variable. The shared principle is that the graphon kernel replaces a homogeneous mean field by a non-exchangeable, label-dependent interaction structure, while the stochastic analysis proceeds through SDEs, BSDEs, FBSDEs, reflected diffusions, and finite-to-continuum approximation theorems (Bayraktar et al., 2022, Harchaoui et al., 2022, Amini et al., 2023).