The mean field equation for the Kuramoto model on graph sequences with non-Lipschitz limit

TL;DR

This paper rigorously justifies the mean field equation for the Kuramoto model on graph sequences with non-Lipschitz limits, including practical graph types like small-world and nearest-neighbor graphs.

Contribution

It extends Neunzert's method to prove the mean field approximation for the Kuramoto model on complex graph sequences with non-Lipschitz limits.

Findings

Established rigorous justification of the mean field equation on non-Lipschitz graph limits.

Applied the method to graphs like small-world and nearest-neighbor networks.

Provided a mathematical foundation for analyzing synchronization on complex networks.

Abstract

The Kuramoto model (KM) of coupled phase oscillators on graphs provides the most influential framework for studying collective dynamics and synchronization. It exhibits a rich repertoire of dynamical regimes. Since the work of Strogatz and Mirollo, the mean field equation derived in the limit as the number of oscillators in the KM goes to infinity, has been the key to understanding a number of interesting effects, including the onset of synchronization and chimera states. In this work, we study the mathematical basis of the mean field equation as an approximation of the discrete KM. Specifically, we extend the Neunzert's method of rigorous justification of the mean field equation to cover interacting dynamical systems on graphs. We then apply it to the KM on convergent graph sequences with non-Lipschitz limit. This family of graphs includes many graphs that are of interest in applications, e.g., nearest-neighbor and small-world graphs.

Equations

$${\dot{u}}_{ni}={\omega }_i+\frac{K}{n}\sum _{j=1}^n\sin \left(u_{nj}-u_{ni}+\alpha \right),i\in [n]:=\{1,2,\dots ,n\}.$$

$${\dot{u}}_{ni}={\omega }_i+\frac{K}{n}\sum _{j:j\sim i}\sin \left(u_{nj}-u_{ni}+\alpha \right),i\in [n],$$

$$X_n=\{x_{n1},x_{n2},\dots ,x_{nn}\},x_{ni}=i/n,i\in [n],$$

$$W_{n,ij}:=n^2{\int }_{I_{n,i}\times I_{n,j}}W(x,y)dxdy,I_{n,i}:=[x_{n(i-1)},x_{ni}),i,j\in [n].$$

$$E({\Gamma }_n)=\left\{\{i,j\}:W_{n,ij}\ne 0,i,j\in [n]\right\}.$$

$${\dot{u}}_{ni}={\omega }_i+\frac{K}{n}\sum _{j=1}^n{W}_{n,ij}\sin \left(u_{nj}-u_{ni}+\alpha \right),i\in [n].$$

$$\mathbb{P}\left(\{i,j\}\in E({\Gamma }_n)\right)=W_{n,ij},$$

$${\dot{\bar{u}}}_{ni}={\omega }_i+K{n}^{-1}\sum _{j=1}^n{e}_{nij}\sin ({\bar{u}}_{nj}-{\bar{u}}_{ni}),i\in [n],$$

$$\mathbb{P}(e_{nij}=1)=W_{n,ij},$$

$$\underset{n\to \infty }{\lim }\underset{t\in [0,T]}{\sup }{\Vert u_n(t)-{\bar{u}}_n(t)\Vert }_{1,n}=0\text{mbox}a.s.,$$

$$\Vert u_n{\Vert }_{1,n}=\sqrt{n^{-1}\sum _{i=1}^n{u}_{ni}^2}$$

$$W_{p,\mathit{h}}(x,y)=\left\{\begin{array}[]{ll}1-p,&d_{\SS}(2\pi x,2\pi y)\leq 2\pi\mathit{h},\\ p,&\mbox{otherwise},\end{array}\right.$$

$$d_{\text{SS}}(\theta ,{\theta }^{\prime })=\min \{|\theta -{\theta }^{\prime }|,2\pi -|\theta -{\theta }^{\prime }|\}$$

$${\dot{u}}_{n,i}$$

$$u_{n,i}(0)$$

$$\underset{(x,y)\in I^2}{\sup }|W(x,y)|\le 1,\underset{u\in \text{SS}}{\max }|D(u)|\le 1,$$

$$|D(u)-D(v)|\le |u-v|\forall u,v\in \text{SS}.$$

$$\underset{\delta \to 0}{\lim }{\int }_I|W(x+\delta ,y)-W(x,y)|dy=0\forall x\in I.$$

$$\frac{\partial }{\partial t}\rho (t,u,x)+\frac{\partial }{\partial u}\left\{V(t,u,x)\rho (t,u,x)\right\}=0,$$

$$V(t,u,x)={\int }_I{\int }_{\text{SS}}W(x,y)D(v-u)\rho (t,v,y)dvdy.$$

$$\rho (0,u,x)={\rho }^0(u,x)\in L^1(G)$$

$${\mu }_t^n(A)=n^{-1}\sum _{i=1}^n{1}_A((u_{ni}(t),x_{ni}))(A),A\in \mathcal{B}(G),$$

$${\mu }_t(A)=\stackrel{[}{A}]\int \int \rho (t,u,x)dudx,A\in \mathcal{B}(G).$$

$$d(\mu ,\eta )=\underset{f\in \mathcal{L}}{\sup }\left|{\int }_{\text{SS}}f(v)\left(d\mu (v)-d\eta (v)\right)\right|,$$

$$\bar{\mathcal{M}}:=\left\{\bar{\mu }:I\to \mathcal{M}\right\}.$$

$$\bar{d}(\bar{\mu },\bar{\eta })={\int }_{I}d({\mu }^x,{\eta }^x)dx.$$

$$\bar{d}({\mu }_{n_k},{\mu }_{n_{k+1}})={\int }_{I}d({\mu }_{n_k}^x,{\mu }_{n_{k+1}}^x)dx<\frac{1}{2^{k+1}},k=1,2,\dots .$$

$$\sum _{k=1}^{\infty }d({\mu }_{n_k}^x,{\mu }_{n_{k+1}}^x)$$

$$\sum _{k=1}^{\infty }{\int }_{I}d({\mu }_{n_k}^x,{\mu }_{n_{k+1}}^x)dx={\int }_{I}f(x)dx.$$

$$d({\mu }_{n_i}^x,{\mu }_{n_j}^x)\le \sum _{k=i}^{j-1}d({\mu }_{n_k}^x,{\mu }_{n_{k+1}}^x),$$

$${\mu }^x=\underset{k\to \infty }{\lim }{\mu }_{n_k}^x,a.e.x\in I.$$

Full text

1 Introduction

The KM of coupled phase oscillators provides a useful framework for studying collective behavior in large ensembles of interacting dynamical systems. It is derived from a weakly coupled system of nonlinear oscillators, which are described by autonomous systems of ordinary differential equations possessing a stable limit cycle [7]. Originally, Kuramoto considered all-to-all coupled systems, in which each oscillator interacts with all other oscillators in exactly the same way. In this case, the KM has the following form

[TABLE]

Here, $u_{ni}:{\mathbb{R}}^{+}\to\SS:={\mathbb{R}}/2\pi{\mathbb{Z}}$ stands for the phase of oscillator $i$ as a function of time, $\omega_{i}$ is its intrinsic frequency, $K$ is the coupling strength, and $\alpha$ is a parameter defining the type of interactions.

Despite its simple form, the KM (1.1) features a rich repertoire of interesting dynamical effects. For the purpose of this review, we mention the onset of synchronization in (1.1) with randomly distributed intrinsic frequencies $\omega_{i}$ (Fig. 1a, b) (cf. [19]) and chimera states, interesting spatio-temporal patterns combining coherent and incoherent behaviors [9, 1]. The mathematical analysis of these and many other dynamical regimes uses the mean field equation, derived in the limit when the number of oscillators goes to infinity [20]. The mean field equation is a partial differential equation for the probability density describing the distribution of the phases on $\SS$. We discuss the mean field equation in more detail below.

Recently, there has been a growing interest in the dynamics of coupled dynamical systems on graphs [18]. In the KM on a graph, each oscillator is placed at a node of an undirected graph $\Gamma_{n}=\langle V(\Gamma_{n}),E(\Gamma_{n})\rangle$. Here, $V(\Gamma_{n})=[n]$ stands for the node set of $\Gamma_{n},$ and $E(\Gamma_{n})$ denotes its edge set. The oscillator $i$ interacts only with the oscillators at the adjacent nodes:

[TABLE]

where $j\sim i$ is a shorthand for $\{i,j\}\in E(\Gamma_{n})$.

Clearly, one can not expect limiting behavior of solutions of (1.2) as $n\to\infty$, unless the graph sequence $\{\Gamma_{n}\}$ is convergent in the appropriate sense. In the present paper, we use the following construction of the convergent sequence $\{\Gamma_{n}\}$. Let $W$ be a symmetric measurable function on the unit square $I^{2}:=[0,1]$. $W$ is called a graphon. It will be used to define the asymptotic behavior of $\{\Gamma_{n}\}$. Further, let

[TABLE]

and

[TABLE]

The weighted graph $\Gamma_{n}=G(W,X_{n})$ on $n$ nodes is defined as follows. The vertex set is $V(\Gamma_{n})=[n]$ and the edge set is

[TABLE]

Each edge $\{i,j\}\in E(\Gamma_{n})$ is supplied with the weight $W_{n,ij}$111There are several possible ways of defining the weights $W_{n,ij},i,j\in[n]$ (see Remark 3.5)..

The KM on $\Gamma_{n}=G(W,X_{n})$ has the following form

[TABLE]

For different $W$ (1.6) implements the KM on a variety of simple and weighted graphs. Moreover, it provides an effective approximation of the KM on random graphs. Indeed, let $\bar{\Gamma}_{n}=G_{r}(X_{n},W)$ be a random graph on $n$ nodes, whose edge set is defined as follows:

[TABLE]

assuming the range of $W$ is $[0,1]$. The decision for each pair $\{i,j\}$ is made independently from the decisions on other pairs. $\bar{\Gamma}_{n}=G_{r}(X_{n},W)$ is called a W-random graph [11].

The KM on the W-random graph $\bar{\Gamma}_{n}=G_{r}(X_{n},W)$ has the following form:

[TABLE]

where $e_{nij},1\leq i\leq j\leq n$ are independent Bernoulli RVs:

[TABLE]

and $e_{nij}=e_{nji}.$

The following lemma shows that the deterministic model (1.6) approximates the KM on the random graph ${\bar{\Gamma}}_{n}$ (1.8).

Lemma 1.1**.**

[3]** Let $u_{n}(t)$ and $\bar{u}_{n}(t)$ denote solutions of the IVP for (1.6) and (1.8) respectively. Suppose that the initial data for these problems coincide $u_{n}(0)=\bar{u}_{n}(0).$ Then

[TABLE]

where $u_{n}=(u_{1},u_{2},\dots,u_{n}),$ ${\bar{u}}_{n}=({\bar{u}}_{1},{\bar{u}}_{2},\dots,{\bar{u}}_{n}),$ and

[TABLE]

is a discrete $L^{2}$-norm.

Example 1.2**.**

A few examples are in order.

Let $W(x,y)\equiv p\in(0,1)$. Then $\bar{\Gamma}_{n}=G_{r}(X_{n},W)$ is an Erdős-Rényi graph (Fig. 2a). 2. 2.

Let

[TABLE]

where $p,\mathit{h}\in(0,1/2)$ are two parameters and

[TABLE]

is the distance on $\SS$. Then $\bar{\Gamma}_{n}=G_{r}(X_{n},W_{p,\mathit{h}})$ is a W-small-world graph **[15]** (Fig. 2b). 3. 3.

$\Gamma_{n}=G(X_{n},W_{1,\mathit{h}})$* is a $k$-nearest-neighbor graph (Fig. 2c).*

For more examples, we refer an interested reader to [3].

Remark 1.3*.*

For simplicity, we restrict the presentation to the KM on dense graphs. The KM on W-random graphs (1.8) easily extends to sparse graphs like scale-free graphs (see [8] for details).

Below, we will focus on the deterministic model (1.6). All results for this model can be extended to the KM on random graphs via Lemma 1.1. Furthermore, from now on we will assume that all intrinsic frequencies in (1.6) are the same $\omega_{i}=\omega$, $i\in[n]$, and, thus, $\omega$ can be set to [math] by switching to the rotating frame of coordinates. Extending the analysis in the main part of this paper to models with distributed frequencies $\omega_{i}$ is straightforward (see, e.g., [3]), but it complicates the presentation. We will comment on the adjustments in the analysis that are necessary to cover the distributed intrinsic frequencies case in Section 4. Until then we consider the following system of $n$ coupled oscillators on $\Gamma_{n}=G(W,X_{n})$:

[TABLE]

where $D$ is a Lipschitz continuous $2\pi$-periodic function.

Without loss of generality, we assume

[TABLE]

and

[TABLE]

In addition, we assume that the graphon $W$ satisfies the following condition:

[TABLE]

Having defined the KMs on deterministic and random graphs (1.6) and (1.8) respectively, we will now turn to the mean field limit:

[TABLE]

where

[TABLE]

The initial condition

[TABLE]

is a probability density function on $\SS$ for every $x\in I$. It approximates the distribution of the initial conditions (1.14).

In the continuum limit as $n\to\infty$, the nodes of $\Gamma_{n}$ fill out $I$. Thus, heuristically, $\rho(t,u,x)$ in (1.18) stands for the density of the probability distribution of the phase of the oscillator at $x\in I$ on $\SS$ at time $t\geq 0$. As we will see below, this probability distribution is indeed continuous for $t>0$, provided that the initial conditions for the discrete problem (1.13), (1.14) converge weakly to the probability distribution with density (1.20). In fact, in [3] it is shown that in this case, the empirical measure on the Borel subsets of $G:=\SS\times I$, $\mathcal{B}(G)$,

[TABLE]

converges weakly to the absolutely continuous measure

[TABLE]

The analysis in [3], which extends the analysis of the all-to-all coupled KM (1.1) by Lancellotti [10], relies on the Lipschitz continuity of $W$. This is the essential assumption of the Neunzert’s fixed point argument that lies at the heart of the method used in [10, 3]. This puts the KM on such common graphs as the small-world and $k$-nearest-neighbor ones out of the scope of applications of [3] (see Example 1.2). It is the goal of the present paper to fill this gap. Specifically, we extend the Neunzert’s method to the KM on convergent families of graphs with non-Lipschitz limits. Our results apply to a general model of $n$ interacting particles on a graph (cf. [6]). However, for concreteness and in view of the diverse applications of the KM, in this paper, we present our method in the context of the KM of coupled phase oscillators.

The organization of this paper is as follows. In the next section, we revise the Neunzert’s fixed point theory to adapt it to the KM on convergent graph sequences. This includes a careful choice of the underlying metric space in Subsection 2.1, setting up the fixed point equation in Subsection 2.2, proving existence and uniqueness of solution of the fixed point equation in Subsection 2, and showing continuous dependence on the initial data in Subsection 2.4 and on the graphon $W$ in Subsection 2.5. In Section 3, we apply the fixed point theory to the KM on graphs. To this end, we first apply it to an auxiliary problem and then show that this problem approximates the original KM on graphs. We conclude with a brief discussion of our results in Section 4.

2 The fixed point equation

2.1 The metric space

Let ${\mathcal{M}}$ denote the space of Borel probability measures on $\SS$. The bounded Lipschitz distance on ${\mathcal{M}}$ is given by

[TABLE]

where $\mathcal{L}$ stands for the class of Lipschitz continuous functions on $\SS$ with Lipschitz constant at most $1$ (cf. [4]). $\langle{\mathcal{M}},d\rangle$ is a complete metric space.

Consider the set of measurable ${\mathcal{M}}$-valued functions222 $\bar{\mu}:I\to{\mathcal{M}}$ is called measurable if the preimage of an open set in ${\mathcal{M}}$ is a Lebesgue measurable subset of $I$.

$\bar{\mu}:x\mapsto\mu^{x}\in{\mathcal{M}}$

[TABLE]

Equip $\bar{\mathcal{M}}$ with the metric

[TABLE]

Lemma 2.1**.**

$\langle\bar{\mathcal{M}},\bar{d}\rangle$* is a complete metric space.*

Proof.

Since $d$ is a metric, it is straightforward that $\bar{d}$ is a metric as well. In order to prove the completeness of $\langle\bar{\mathcal{M}},\bar{d}\rangle$, take a Cauchy sequence $\{\bar{\mu}_{n}\}$ in $\bar{{\mathcal{M}}}$. Then there is an increasing sequence of indices $n_{k}$ such that

[TABLE]

By B. Levi’s theorem, the series

[TABLE]

converges for a.e. $x\in I$ to some measurable function $f(x)$, and

[TABLE]

Since, for every $i,j$ with $j>i$,

[TABLE]

the sequence $\{\mu^{x}_{n_{k}}\}$ is Cauchy for a.e. $x\in I$. Since the metric space $\langle{\mathcal{M}},d\rangle$ is complete, there exists the limit

[TABLE]

Extending the definition of $\mu^{x}$ in an arbitrary way to all of $I$, we obtain a function

[TABLE]

which is measurable as an a.e. pointwise limit of measurable functions. Thus $\bar{\mu}\in\bar{{\mathcal{M}}}$. Next, for every $i,j$ with $j>i$, we have

[TABLE]

We also have that, for all $j>i$, $d(\mu^{x}_{n_{i}},\mu^{x}_{n_{j}})\leq f(x)$ a.e. $x\in I$ and the function $f$ is Lebesque integrable on $I$. By the Lebesque Dominated Convergence Theorem, letting $j\to\infty$, we obtain that

[TABLE]

Since the subsequence $\{\bar{\mu}_{n_{i}}\}$ of the Cauchy sequence $\{\bar{\mu}_{n}\}$ converges to $\bar{\mu}$ in $\bar{{\mathcal{M}}}$, we conclude that the sequence $\{\bar{\mu}_{n}\}$ converges to $\bar{\mu}$ as well. Thus $\langle\bar{{\mathcal{M}}},\bar{d}\rangle$ is a complete metric space. $\Box$

Let $T>0$ be arbitrary but fixed and denote $\mathcal{T}=[0,T]$. We define ${\mathcal{M}}_{\mathcal{T}}=C(\mathcal{T},\bar{\mathcal{M}})$, the space of continuous $\bar{\mathcal{M}}$-valued functions.

For any $\alpha>0,$ the following is a metric on $\bar{\mathcal{M}}_{\mathcal{T}}$:

[TABLE]

2.2 The equation of characteristics

Recall that $\mathcal{T}:=[0,T]$, where $T>0$ is arbitrary but fixed. For a given $\bar{\mu}_{.}\in{\mathcal{M}}_{\mathcal{T}},$ consider the following equation of characteristics

[TABLE]

where

[TABLE]

Lemma 2.2**.**

For every $\bar{\mu}_{.}\in\bar{\mathcal{M}}_{\mathcal{T}}$, $V[W,\bar{\mu}_{.}](u,x,t)$ is Lipschitz continuous in $u$ and continuous in $x$ and $t$.

Proof.

The proof follows from the following estimates. First, using (1.15) and (1.16), we have

[TABLE]

Using the bound on $D$ (cf. (1.15)), we obtain

[TABLE]

The continuity of $V[W,\bar{\mu}_{.}]$ in $x$ follows from (2.5) and (1.17).

Finally,

[TABLE]

$\Box$

Similarly to the derivation of the last inequality, we prove the following lemma.

Lemma 2.3**.**

[TABLE]

Consider the initial value problem (IVP) for (2.3) subject to the initial condition at time $s\in\mathcal{T},$ $u(s)=u_{s}$. By Lemma 2.2, for every $x\in I$ and $u_{s}\in\SS,$ there exists a unique solution of the IVP for (2.3). Since $V[W,\bar{\mu}_{.}](u,x,t)$ is uniformly Lipschitz in $u$, $u(t)$ can be extended to $t\in\mathcal{T}$. Thus, the equation of characteristics (2.3) generates the flow on $\SS:$

[TABLE]

For every $x\in I$, $T^{x}_{t,s}[W,\bar{\mu}_{.}],t,s\in\mathcal{T},$ is a two-parameter family of one-to-one transformations of $\SS$ to itself depending continuously on $x$:

[TABLE]

2.3 Existence of solution of the fixed point equation

In the remainder of this section, we will study the following fixed point equation. For a given $\bar{\mu}_{0}\in\bar{\mathcal{M}}$, consider the pushforward measure

[TABLE]

which is interpreted as

[TABLE]

First, we address existence and uniqueness of solution of (2.8).

Theorem 2.4**.**

For every $\bar{\mu}_{0}\in\bar{\mathcal{M}}$, the fixed point equation (2.8) has a unique solution $\bar{\mu}_{.}\in\bar{\mathcal{M}}_{\mathcal{T}}$.

For the proof of Theorem 2.4, we will need a variant of the Gronwall’s inequality, which we formulate below for convenience.

Lemma 2.5**.**

Let $\phi(t)$ and $a(t)$ be continuous functions on $[0,T]$ and

[TABLE]

where $A\geq 0$. Then

[TABLE]

Proof.

Denote the right-hand side of (2.10) by $\psi(t)$. Differentiating $\psi(t)$ and using (2.10), we have

[TABLE]

Thus,

[TABLE]

and

[TABLE]

Finally,

[TABLE]

$\Box$

Proof of Theorem 2.4.

Given $\bar{\mu}_{0}\in\bar{\mathcal{M}},$ consider $A:\bar{\mathcal{M}}_{\mathcal{T}}\to\bar{\mathcal{M}}_{\mathcal{T}}$ defined by

[TABLE]

Below we show that $A$ is a contraction on $(\bar{\mathcal{M}}_{\mathcal{T}},d_{\alpha})$ with $\alpha>2$. To this end,

[TABLE]

The change of variables formula used in (2.15) is explained in [12, §6.1]. Using (2.3) and (2.6), we obtain

[TABLE]

Using Gronwall’s inequality (cf. Lemma 2.5), from (2.16) we obtain

[TABLE]

Combining (2.15), (2.16), and (2.17), we have

[TABLE]

and

[TABLE]

We conclude the proof with using the contraction mapping principle to establish a unique solution of (2.8). $\Box$

2.4 Continuous dependence on initial data

Lemma 2.6**.**

Let $\bar{\mu}_{.},\bar{\eta}_{.}\in\bar{\mathcal{M}}_{\mathcal{T}}$ be two solutions of (2.8) corresponding to initial conditions $\bar{\mu}_{0},\bar{\eta}_{0}\in\bar{\mathcal{M}}$ respectively. Then

[TABLE]

Proof.

For every $t\in\mathcal{T}$, by the triangle inequality, we have

[TABLE]

Exactly in the same way as in (2.15), we estimate the first term on the right hand side of (2.21) as follows

[TABLE]

Similarly, repeating the steps in (2.16)

[TABLE]

Using Gronwall’s inequality, from (2.23) we obtain

[TABLE]

Next, we turn to the second term on the right–hand side of (2.21):

[TABLE]

The combination of the (2.21), (2.22), (2.24), and (2.25) yields

[TABLE]

Denote $\phi(t):=\bar{d}(\bar{\mu}_{t},\bar{\eta}_{t})e^{-t}$ and rewrite (2.26) as

[TABLE]

Arguing as in the proof of Lemma 2.5, we derive

[TABLE]

Further,

[TABLE]

and, thus,

[TABLE]

Recalling, the definition of $\phi(t)$, we arrive at

[TABLE]

from which (2.20) follows. $\Box$

2.5 Continuous dependence on the kernel

In this subsection, we study how the solution of the fixed point equation (2.8) changes under the perturbation of the kernel $W$. To this end, let $W$ and $U$ be two bounded measurable functions on $I^{2}$ satisfying (1.17). Then for a given $\mu_{0}\in{\mathcal{M}}$ each of the fixed point equations

[TABLE]

has a unique solution in ${\mathcal{M}}_{\mathcal{T}}$, which we denote by $\bar{\mu}_{t}$ and $\bar{\nu}_{t}$ respectively.

Lemma 2.7**.**

[TABLE]

Proof.

We proceed along the lines of the proof of Lemma 2.6. Replicating the steps in (2.15), we derive

[TABLE]

Further,

[TABLE]

We estimate the first term on the right–hand side of (2.33), using Lemma 2.3:

[TABLE]

For the second term, we have

[TABLE]

where we used $|D(u)|\leq 1$ to get the latter inequality. Finally, to estimate the third term, we use Lipschitz continuity of $V[\cdot,\cdot](u,x,t)$ in $u$:

[TABLE]

Plugging (2.34)-(2.36) into (2.33), we obtain

[TABLE]

By Gronwall’s inequality (cf. Lemma 2.5),

[TABLE]

The combination of (2.32) and (2.38) yields

[TABLE]

Denote $\phi(t):=e^{-t}\bar{d}(\bar{\mu}_{t},\bar{\nu}_{t})$ and rewrite (2.39) as

[TABLE]

Using Lemma 2.5 again, we have

[TABLE]

Recalling, the definition of $\phi(t)$, we finally get

[TABLE]

$\Box$

3 Application to coupled systems

In this section, we apply the fixed point theory developed in the previous section to the proof of convergence of solutions of the KM on graphs (1.2).

3.1 The initial value problem

We begin by addressing the wellposedness of the IVP for (1.18), i.e., review the notion of the weak solution of the mean field equation (1.18) and then prove the existence and uniqueness of the weak solution of the IVP (1.18)-(1.20). The following definition of the weak solution of (1.18) is adapted from [17].

Definition 3.1**.**

A measurable function $\rho:\mathcal{T}\times G\to{\mathbb{R}}$ is called a weak solution of (1.18)-(1.20) if the following conditions hold a.e. $x\in I$.

$\rho(t,u,x)$* is weakly continuous in $t\in\mathcal{T}$, i.e., $t\mapsto\int_{\SS}\rho(t,u,x)f(u)du$ is a continuous map for every $f\in C(\SS)$.* 2. 2.

The following identity holds

[TABLE]

for every $w\in C^{1}(\mathcal{T}\times\SS)$ with support in $(0,T]\times\SS$.

Theorem 3.2**.**

Suppose $W\in L^{1}(I^{2})$ satisfies (1.17) and $\rho_{0}\in L^{1}(\SS\times I)$. Then there is a unique weak solution to the IVP (1.18)-(1.20).

Proof.

Recall that $T_{0,t}^{x}:=T_{0,t}^{x}[W,\bar{\mu}_{.}]$ is the flow generated by the equation of characteristics (2.3) on $\SS$ (see (2.7)). By Theorem 2.4, for $\mu^{x}_{0}$, a family of measures on $(\SS,\mathcal{B}(\SS))$ with densities $\rho_{0}(\cdot,x)$, there is a unique solution of the fixed point equation (2.8).

For a.e. $x\in I$ and $t\in\mathcal{T}$, $T_{0,t}^{x}$ is one-to-one and Lipschitz continuous. As an absolutely continuous function, $T_{0,t}^{x}$ is differentiable a.e. on $\SS$, and has an essentially bounded weak derivative. Using (2.9) and the change of variables formula for Lipschitz maps (cf. [5, Theorem 2, §3.3.3]), we have

[TABLE]

for any Borel $A\subset\SS$. In the last integral of (3.2), ${\partial\over\partial u}T_{0,t}^{x}u$ is understood as a weak derivative. Thus, for $t\in\mathcal{T}$ and a.e. $x\in I$, $\mu_{t}^{x}$ is an absolutely continuous measure with density

[TABLE]

To show that $\rho(t,u,x)$ is a weak solution of (1.18)-(1.20), as in the proof of Thorem 1 in [17], we set

[TABLE]

and compute

[TABLE]

Further, using the chain rule and (3.4), we have

[TABLE]

where $\partial_{1,2}$ stand for the partial derivatives with respect to the first and second argument respectively. The combination of (3.5) and (3.6) yields

[TABLE]

$\Box$

3.2 Approximation

We continue by collecting several results on approximation, which will be used later in this section.

Let $W\in L^{2}(I^{2})$ and consider a step function $\bar{W}_{n}:I\to{\mathbb{R}}$, which on each cell $I_{n,i}\times I_{n,j}$, $i,j\in[n]$, is equal to the average value

[TABLE]

Lemma 3.3**.**

$\bar{W}_{n}\to W$* a.e. and in $L^{2}(I^{2})$ as $n\to\infty.$*

Proof.

Consider a probability space $\left(I^{2},\mathcal{B}(I^{2}),\lambda\right)$ with sample space $I^{2}$, Borel $\sigma$–algebra of subsets of $I^{2}$, and the Lebesgue measure as the probability $\lambda$. On this probability space, $(\bar{W}_{n})$ forms a sequence of $L^{2}$-bounded martingales [22]. Indeed, let $\mathcal{A}_{n}$ be a $\sigma$–algebra of subsets of $I^{2}$ generated by $I_{n,i}\times I_{n,j}$, $i,j\in[n]$. Then $\bar{W}_{n}$ is the conditional expectation

[TABLE]

Since $W\in L^{2}(I^{2})$, the statement of the lemma follows from the $L^{p}$–Martingale Convergence Theorem [22, Theorem 12.1]. $\Box$

Let $\Gamma_{n}=\langle[n],E(\Gamma_{n}),W_{n}\rangle$ be a weighted graph on $n$ nodes. Here, $[n]$ is the set of vertices,

[TABLE]

is the edge set. Each edge $\{i,j\}$ is equipped with a real weight $W_{n,ij}$, the $ij$th entry of the $n\times n$ weight matrix $W_{n}$.

On $\Gamma_{n}$ consider a coupled system:

[TABLE]

By $u_{n}=(u_{n,1},u_{n,2},\dots,u_{n,n})^{\mathsf{T}}\in{\mathbb{R}}^{n}$ we denote a solution of the coupled system (3.7). Along with (3.7), consider the coupled system on a weighted graph $\tilde{\Gamma}_{n}=\langle[n],E(\Gamma_{n}),\tilde{W}_{n}\rangle$, whose solution is denoted by $\tilde{u}_{n}$.

Recall the discrete $L^{2}$-norm defined in (1.10). We will also need its analog for discretizations of functions on $I^{2}$:

[TABLE]

Lemma 3.4**.**

[3, Lemma 4.1]** Let $u_{n}(t)$ and $\tilde{u}_{n}(t)$ denote solutions of the IVPs for the coupled system (3.7) on weighted graphs $\Gamma_{n}=\langle[n],E(\Gamma_{n}),W_{n}\rangle$ and $\tilde{\Gamma}_{n}=\langle[n],E(\Gamma_{n}),\tilde{W}_{n}\rangle$ respectively. Suppose that the initial data for these problems coincide

[TABLE]

Then for any $T>0$,

[TABLE]

where $C_{1}=\sqrt{Te^{5T}}>0.$

Remark 3.5*.*

Note that (3.10) implies that the solutions of any two discrete models (1.6) with weights given by $W_{n}$ and $\tilde{W}_{n}$ will be close, provided that $W_{n}$ and $\tilde{W}_{n}$ converge to $W$ in $L^{2}$.

3.3 The auxiliary problem

Our next goal is to establish convergence to the mean field limit for an auxiliary discrete model. To this end, recall that $W_{n}:I^{2}\to[-1,1]$ is a step function taking constant value

[TABLE]

on each cell $I_{n,i}\times I_{n,j},$ $i,j\in[n],$ (cf. (1.4)).

Let $N=nm$ for some $m\in{\mathbb{N}}$ and consider a coupled system

[TABLE]

We are going to describe the solution of (3.11), (3.12) in terms of the empirical measure

[TABLE]

We will show that in the large $m$ limit, the behavior of solutions of the discrete model (3.11), (3.12), is effectively approximated by the IVP for the following integro-differential equation:

[TABLE]

where

[TABLE]

and subject to the initial condition

[TABLE]

About the initial condition $\rho^{0}(u,x)$ we assume the following. For each $x\in I$, $\rho^{0}(\cdot,x)$ is a probability density function, i.e.,

[TABLE]

Furthermore, $x\mapsto\int_{\SS}f(u)\rho^{0}(u,x)du$ is Riemann integrable for every $f\in C(\SS)$.

By Theorem 3.2, there is a unique weak solution of the IVP (3.14), (3.16), which defines a family of absolutely continuous measures on $(\SS,\mathcal{B}(\SS))$:

[TABLE]

Theorem 3.6**.**

Let $n\in{\mathbb{N}}$ be arbitrary but fixed and suppose that the the initial values $v_{N,j}^{0},j\in[N],$ are independent random variables, whose distributions have densities $\rho^{0}(\cdot,jN^{-1})$, $j\in[N]$. Then

[TABLE]

The proof of Theorem 3.6 follows from two lemmas, which we prove next.

Lemma 3.7**.**

The empirical measure $\mu_{n,m,t}^{x}$ and the absolutely continuous measure $\mu_{n,t}^{x}$ satisfy the following estimate

[TABLE]

Proof.

The proof follows from the fact that the empirical measure $\mu_{n,m,t}^{x}$ and the absolutely continuous measure $\mu_{n,t}^{x}$ satisfy the fixed point equation

[TABLE]

for $\bar{\mu}_{0}:=\bar{\mu}_{n,m,0}$ and $\bar{\mu}_{0}:=\bar{\mu}_{n,0}$ respectively.

For the continuous measures $\mu_{n,t}^{x}$ this follows from the proof of Theorem 3.2. Thus, it remains to verify (3.20) for the discrete measures $\mu_{n,m,t}^{x}$. By plugging (3.13) into (3.15), for every $x\in I_{n,k}$ we have

[TABLE]

The right hand side of (3.21) with $u:=u_{N,(k-1)m+l}(t)$ yields the velocity field acting on the oscillator $((k-1)m+l)$ of the discrete system (3.11). Therefore, by construction of the empirical measure (3.13), for every $x\in I$ we have

[TABLE]

Finally, (3.19) follows from Lemma 2.6. $\Box$

Lemma 3.8**.**

[TABLE]

Proof.

[TABLE]

Let $i\in[n]$ and $y\in I_{n,i}$ be arbitrary but fixed. Using the definition of the metric $d(\cdot,\cdot)$, we have

[TABLE]

Since $\mathcal{L}$ is a subset of the separable space of continuous functions on $\SS$, we can choose a countable dense (with respect to $L^{\infty}$–norm) subset of $\mathcal{L}$, $\{f_{l}\}$.

Fix $l\in{\mathbb{N}}$ and for $y\in I_{n,i}$ consider

[TABLE]

Random variables $Y^{l}_{N,im+j},j\in[m],$ are independent and uniformly bounded. Further,

[TABLE]

By the Strong Law of Large Numbers,

[TABLE]

Thus,

[TABLE]

By density of $\{f_{l}\}$ in $\mathcal{L}$,

[TABLE]

In other words, $\forall i\in[n]$ (recall $\mu_{n,m,0}^{y}$ is constant in $y$ over $I_{n,i}$),

[TABLE]

The combination of (3.23) and (3.28) yields (3.22). $\Box$

3.4 The main result

We now turn to the original model (1.13), (1.14). In analogy to how it was done for the auxiliary problem in the previous subsection, for given $n,m\in{\mathbb{N}}$ and $N=nm,$ we define the empirical measure

[TABLE]

where $u_{N}$ is the solution of the IVP (1.13),(1.14). Likewise, using the solution of the IVP for the continuum limit (1.18), (1.20), we define the continuous measure

[TABLE]

The well-posedness of (1.18), (1.20) on $\mathcal{T}$ is established via the same argument used for the auxiliary problem in the proof of Theorem 3.6.

Our goal is to show that for large $n,m\in{\mathbb{N}}$, the continuos measure (3.30) approximates the empirical measure (3.29), and, thus, describes the behavior of solutions of the discrete model (1.13) with $N=nm$. This is achieved in the following theorem.

Theorem 3.9**.**

For a given $\epsilon>0$ there exist $n_{1},m_{1}\in{\mathbb{N}}$ such that for $n\geq n_{1}$, $m\geq m_{1}$ and $N=nm$, for the empirical measure (3.29) and the continuous measure (3.30), we have

[TABLE]

Proof.

Recall that $W_{n}$ stands for the step function taking on each cell $I_{n,i}\times I_{n,j},$ $i,j\in[n],$ a constant value $n^{2}\int_{I_{n,i}\times I_{n,j}}W(x,y)dxdy$, the average value of $W$. By Lemma 3.3,

[TABLE]

where $c_{1}:=\left(2(\sqrt{Te^{5T}}+e^{T})\right)^{-1}.$ Let $n\geq n_{1}$ be fixed for the remainder of the proof. 2. 2.

By Theorem 3.6,

[TABLE] 3. 3.

By Lemma 2.7

[TABLE] 4. 4.

We estimate

[TABLE]

Further, by the Schwarz inequality followed by the application of Lemma 3.4, we continue the string of estimates in (3.35) as follows

[TABLE]

where we used (3.32) to derive the last inequality. 5. 5.

By combining the estimates in 1.-4., we have

[TABLE]

holding for every $t\in\mathcal{T}$.

$\Box$

4 Discussion

The choice of the KM in this paper was motivated by its role in the theory of synchronization [19] and analytical convenience. The analysis in the previous sections can be naturally extended to other models. First, by extending the phase space to include $\omega$ it applies easily to the KM with distributed intrinsic frequencies (1.6). Specifically, let $G=\SS\times{\mathbb{R}}$ be an extended phase space, ${\mathcal{M}}_{G}$ be the space of Borel probability measures on $G$, $\bar{\mu}:x\in I\mapsto\mu^{x}\in{\mathcal{M}}_{G}$ be a ${\mathcal{M}}_{G}$-measurable function, and ${\bar{\mu}}_{t}:t\in\mathcal{T}\mapsto{\bar{\mu}}^{x}_{t}$ be a weakly continuous function. Then we rewrite the equation of characteristics in the following form:

[TABLE]

By assigning initial condition $(u(0),\phi(0))=(u_{i}^{0},\omega_{i})$, the right hand side of the top equation in (4.38) yields the velocity field acting on oscillator $i$ with the intrinsic frequency $\omega_{i}$. Since the newly added second component of the vector field in (4.38) is trivial, all necessary estimates for $V[W,\bar{\mu}_{.}](u,\phi,x,t)$ are as before. With the equation of characteristics (4.38) in hand, one can set up the fixed point equation and then analyze it exactly in the same way as it was done in Sections 2 and 3. We refer an interested reader to [3], where the KM with distributed frequencies was analyzed albeit for Lipschitz graph limits.

Likewise, there are no principal difficulties in applying our analysis to other models of interacting dynamical systems on graphs, such as Cucker-Smale model of flocking [21, 16], consensus protocols [14], as well as neuronal networks [2]. The multidimensionality of the phase space can be handled in the same way as explained above for the KM with distributed frequencies. The treatment of the coupling term, the key ingredient in the analysis, remains the same as in the present paper. On the other hand, the weighted graph model, which we adopted in this paper (see (1.3)-(1.5)), provides a simple unified treatment of interacting dynamical systems on a variety of deterministic and random graphs, including Erdős-Rényi, small-world, and certain approximations of power law graphs [13, 8].

Acknowledgements. GM thanks Carlo Lancellotti for useful discussions. This work was supported in part by the NSF DMS 1412066 (GM).