Interacting Diffusions on Random Graphs with Diverging Average Degrees: Hydrodynamics and Large Deviations

Abstract

We consider systems of mean-field interacting diffusions, where the pairwise interaction structure is described by a sparse (and potentially inhomogeneous) random graph. Examples include the stochastic Kuramoto model with pairwise interactions given by an Erdős–Rényi graph. Our problem is to compare the bulk behavior of such systems with that of corresponding systems with dense nonrandom interactions. For a broad class of interaction functions, we find the optimal sparsity condition that implies that the two systems have the same hydrodynamic limit, which is given by a McKean–Vlasov diffusion. Moreover, we also prove matching behavior of the two systems at the level of large deviations. Our results extend classical results of dai Pra and den Hollander and provide the first examples of LDPs for systems with sparse random interactions.

Appendices

Appendix: An Approximation Result

In this subsection we prove the existence of a good approximation as in Lemma 4.

Lemma 8

Let $\phi :\mathbb {R}^3\rightarrow \mathbb {R}$ differentiable. Suppose that there is a constant $M \in \mathbb {R}$ such that $\left\| \phi \right\| _{\infty }\le M$ and $\left\| \nabla \phi \right\| _{op,\infty }\le M$. Let $N_3 \in \mathbb {R}^3$ be a normal random variable with mean zero and covariance matrix identity $Id_{3\times 3}$. For any $\varepsilon \in (0,1]$ and for each $\mathbf {x} \in \mathbb {R}^3$ define $\phi _\varepsilon (\mathbf {x})=\mathbb {E}\left[ \phi (\mathbf {x}+\varepsilon N_3)\right] .$ Then

• 1. $\phi _{\varepsilon } \in C^\infty (\mathbb {R}^3).$

• 2. $\left\| \phi _\varepsilon \right\| _{\infty }\le \left\| \phi \right\| _{\infty }.$

• 3. $\left\| \phi -\phi _\varepsilon \right\| _{\infty }\le \varepsilon \left\| \nabla \phi \right\| _{op,\infty }\mathbb {E}\left[ |N_3|\right] .$

• 4. $\left\| \nabla \phi _\varepsilon \right\| _{op,\infty }\le \left\| \nabla \phi \right\| _{op,\infty }.$

Proof

Let $\gamma$ be the density of $N_3$ with respect to the Lebesgue measure. By definition

$$\begin{aligned} \phi _\varepsilon (\mathbf {x})= & {} \int _{\mathbb {R}^3}\phi (\mathbf {x}+\varepsilon \mathbf {y})\gamma (\mathbf {y})d\mathbf {y} \\= & {} \int _{\mathbb {R}^3}\phi (\mathbf {z})\gamma \left( \dfrac{\mathbf {z}-\mathbf {x}}{\varepsilon ^3}\right) d\mathbf {y}. \end{aligned}$$

Therefore, applying the Convergence Dominated Theorem we can show that

$$\begin{aligned} \dfrac{\partial \phi _\varepsilon }{\partial x_i}(\mathbf {x})= & {} \int _{\mathbb {R}^3}\phi (\mathbf {z})\dfrac{\partial \gamma }{\partial x_i}\left( \dfrac{\mathbf {z}-\mathbf {x}}{\varepsilon ^3}\right) d\mathbf {y} \end{aligned}$$

and the same is true for all higher derivatives. Therefore, $\gamma \in C^{\infty }(\mathbb {R}^3)$ implies $\phi _\varepsilon \in C^{\infty }(\mathbb {R}^3).$

Again by the Convergence Dominated Theorem, using that $\phi$ has one derivative

$$\begin{aligned} \dfrac{\partial \phi _\varepsilon }{\partial x_i}(\mathbf {x})= & {} \int _{\mathbb {R}^3}\dfrac{\partial \phi }{\partial x_i}(\mathbf {x}+\varepsilon \mathbf {y})\gamma (\mathbf {y})d\mathbf {y}. \end{aligned}$$

This implies that $\left\| \nabla \phi _\varepsilon \right\| _{op,\infty }\le \left\| \nabla \phi \right\| _{op,\infty }$. For the third claim we write

$$\begin{aligned} \phi _\varepsilon (x)-\phi (x)=\mathbb {E}\left[ \phi (x+\varepsilon N)-\phi (x)\right] \end{aligned}$$

to see that the Mean Value Theorem implies $\left\| \phi _\varepsilon -\phi \right\| _{\infty }\le \varepsilon \left\| \nabla \phi \right\| _{op,\infty }\mathbb {E}\left[ |N|\right] .$$\square$

The next Lemma implies Lemma 8 in the main text. We will need a bump function $\xi$, that is, a $C^{\infty }$ function such that

• $\left\| \xi \right\| _{\infty }\le 1$.

• $\left\| \xi '\right\| _{\infty }\le C_1$ (a constant that does not depend in any parameter).

• $\xi \equiv 1$ in $[-1,1]$.

• $\xi \equiv 0$ in $[-2,2]^c$.

Lemma 9

Consider $\phi$ and $\phi _\varepsilon$ as in Lemma 8. Define also for all $R\ge 1$ and $\mathbf {x} \in \mathbb {R}^3$

$$\begin{aligned} \phi _{\varepsilon ,R}(\mathbf {x})=\phi _\varepsilon (\mathbf {x})\xi \left( \dfrac{\left\| \mathbf {x}\right\| _{2}^2}{R^2}\right) . \end{aligned}$$

Then

• 1. $\phi _{\varepsilon ,R} \in C^\infty (\mathbb {R}^3).$

• 2. supp.$\phi _{\varepsilon ,R} \subset B_{2R}(\mathbf {0}).$

• 3. $\left\| \phi _{\varepsilon ,R}\right\| _{\infty }\le \left\| \phi \right\| _{\infty }.$

• 4. $\left\| \nabla \phi _{\varepsilon ,R}\right\| _{op,\infty }\le \left\| \nabla \phi \right\| _{op,\infty }+\left\| \xi '\right\| _{\infty }\left\| \phi \right\| _{\infty }.$

• 5. $\left\| \phi _{\varepsilon ,R}-\phi \right\| _{L^{\infty }(B_{R}(\mathbf {0}))}\le \varepsilon \left\| \nabla \phi \right\| _{op,\infty }\mathbb {E}\left[ |N_3|\right] .$

In this way we choose

$$\begin{aligned} M=\max \{\left\| \phi \right\| _{\infty },\left\| \nabla \phi \right\| _{op,\infty }+\left\| \xi '\right\| _{\infty }\left\| \phi \right\| _{\infty },\left\| \nabla \phi \right\| _{op,\infty }\mathbb {E}\left[ |N_3|\right] \} \end{aligned}$$

and write $\phi ^{\epsilon ,R}:=\phi _{\epsilon /M,R}$ to state Lemma 4.

Proof

Items 1–3 are immediate from the definition of $\xi$ and $\phi _\varepsilon .$ To check item 4 we apply the product rule to obtain

$$\begin{aligned} \dfrac{\partial \phi _{\varepsilon ,R}}{\partial x_i}(x)=\dfrac{\partial \phi _\varepsilon }{\partial x_i}(x)\xi \left( \dfrac{x}{R}\right) +\phi _\varepsilon (x)\xi '\left( \dfrac{\left\| \mathbf {x}\right\| _{2}^2}{R^2}\right) \dfrac{2x_i}{R^2}. \end{aligned}$$

For the first term on the right hand side remember that $\xi \le 1$ and $\left\| \nabla \phi _\varepsilon \right\| _{op,\infty }\le \left\| \nabla \phi \right\| _{op,\infty }$. The second term vanishes when $\left\| \mathbf {x}\right\| _{2}\ge R$ since supp.$\xi \subset [-1,1]$. In the case $\left\| \mathbf {x}\right\| _{2}< R$ we have that

$$\begin{aligned} \left| \dfrac{2x_i}{R^2}\right| \le 1. \end{aligned}$$

To finish item 4 remember that $\left\| \mathbf {x}\right\| _{2}\le \left\| \mathbf {x}\right\| _{1}$ in such way that we just need to sum the last bounds.

To check item 5 we just need to note that $\phi _{\varepsilon ,R}=\phi _\varepsilon$ in $B_R(\mathbf {0})$ and use item 3 of Lemma 8. $\square$

Appendix: Extension of the “Dense” LDP

In this Appendix we check that the same large deviations result and McKean–Vlasov limit obtained by dai Pra and den Hollander [25] holds in our slightly more general setting. More specifically, we wish to sketch a proof of the following result.

Theorem 3

Consider the sequence of empirical measures $\{\overline{L}_n\}_{n\in \mathbb {N}}$ under Assumptions 1, 2 and 3. Then $\{\overline{L}_n\}_{n\in \mathbb {N}}$ satisfies a Large Deviations Principle with the rate function I in Definition 3, which has a unique McKean–Vlasov diffusion as minimizer.

We review the points we discussed in Remark 2. The trajectories in $\overline{\theta }^{(n)}$ evolve according to the Hamiltonian

$$\begin{aligned} \overline{H}_n(x^{(n)},\omega ^{(n)}):= \frac{1}{2n}\sum _{i,j=1}^n\,\overline{f}(x^{(n)}_i-x^{(n)}_j,\omega _i^{(n)},\omega _j^{(n)}) + \sum _{i=1}^n\,g(x^{(n)}_i,\omega _i^{(n)}). \end{aligned}$$

This is the same kind of Hamiltonian in Reference [25], except that f is replaced by $\overline{f}$.

Our assumptions on the measures $\mu$ and $\lambda$ are the same as in Reference [25]. The assumptions on $\overline{f}$ and g are nearly the same as in Reference [25], but we only assume $\overline{f}',\overline{f}'',g',g''$ are bounded Lipschitz, whereas Reference [25] also requires that $\overline{f},g$ be bounded.

We now explain how to adapt the proofs of Lemma 1, Theorem 1 and Theorem 2 in Reference [25] to our slightly weaker assumption. One important point is that $\overline{f}=\overline{f}(x,\omega ,\pi )$ and $g=g(x,\omega )$ are L-Lipschitz in the first variable, with a constant $L>0$ that does not depend on $\omega$ or $\pi$. In particular, Lemma 1 in their paper, which describes the law of $\overline{L}_n$ as an exponential tilt, works exactly the same way as in their paper, via Girsanov’s Theorem and Itô’s Formula.

$$\begin{aligned} P_N(\cdot )=\int d(W^{\otimes N}\otimes \mu ^{\otimes N})\exp \left( NF(L_N) \right) \mathbb {I}_{\{L_N \in \cdot \}} \end{aligned}$$

(54)

Theorem 1 uses the exponential tilting argument to derive a LDP for $\overline{L}_n$. This requires a slight amount of care, as the tilting functional F is unbounded in our setting. However, the fact that f, g are Lipschitz implies:

$$\begin{aligned} |F(L_N)|\le K\,\left( 1+\int \,|x_T-x_0|\,L_N(dx_{[0,T]}d\omega )\right) \end{aligned}$$

for some constant $K>0$. Thus the exponential integrability conditions in Varadhan’s Lemma (cf. [13, Theorem 4.3.1]) apply and allow us to conclude the proof.

For Theorem 2, the main body of the proof follows in the same way from Itô’s Formula. The only change is in the argument for uniqueness in Appendix A. More specifically, what we need to do (in their notation) is show that the density of $Q_*$ at time t conditionally on $\omega$ satisfies a bound:

$$\begin{aligned} q_t^{\omega }(z)\le B_T\,t^{-\alpha } \end{aligned}$$

with $0\le \alpha <1/2$ and B independent of $\omega$ (but may depend on T).

To obtain this, the [25] uses the boundedness of f and g when they claim that the drift $\beta _t^{\omega ,\varPi _tQ_*}$ is the bounded derivative of a bounded function. In our case the drift is a bounded derivative of a Lipschitz function. Therefore, for any event $E\subset C([0,T],\mathbb {R})\times \mathbb {R}$,

$$\begin{aligned} Q_*(E) = \int _{A}\,Z_T\,W_\lambda \otimes \mu (dx_{[0,T]}d\omega ) \end{aligned}$$

where $|\log Z_T|\le K\,(1+|x(T)-x(0)|)$ and $W_\lambda$ is the law of Brownian motion started from measure $\lambda$. Now if E takes the form:

$$\begin{aligned} E:= \{(x_{[0,T]},\omega )\,:\, x(T)\in A,\omega \in B\}, \end{aligned}$$

then:

$$\begin{aligned} Q_*(E) \le \left( \int _{\mathbb {R}^2} e^{K\,(1+|x|)}\,\mathbb {I}_{A}(x+y)\rho _{t}(x)\phi (y)\,dx\,dy\right) \times \mu (B), \end{aligned}$$

where $\rho _{t}$ is the density of a N(0, t) random variable and $\phi$ is the density of the initial measure $\lambda$. Using the notation of their paper, we obtain:

$$\begin{aligned} q_t^{\omega }(z)\le \int _\mathbb {R}\,e^{K\,(1+|z-y|)}\,\phi (y)\,\rho _t(z-y)\,dy. \end{aligned}$$

We may apply Hölder’s inequality as in their proof to obtain:

$$\begin{aligned} q_t^{\omega }(z)\le \Vert \phi \Vert _{L^p}\,\left( \int _\mathbb {R}\,e^{K\,q(1+|z-y|)}\,\rho _t(z-y)^q\,dy\right) ^{\frac{1}{q}}\le B\,t^{(1/2 -q/2)}. \end{aligned}$$