text:
\section{Introduction}\label{sec.intro}
Coupled dynamical systems on graphs represent many diverse models
throughout the natural sciences and technology. Examples range from
regulatory and neuronal
networks in biology \cite{LaiCho01,BenHan97, MZ12, Swi80},
to Josephson junctions and coupled lasers in physics \cite{LiErn92, PhiZan93, WatStr94},
to communication, sensor, and power networks in technology \cite{DorBul12, Med12},
to name a few. Compared to partial differential equations and lattice dynamical
systems, the analysis of networks meets a new principal challenge: the rich variety
and possible complexity of the underlying graphs. The algebraic methods of graph
theory \cite{Biggs, Chung-Spectral} have been useful in understanding the contribution
of the network topology to certain aspects of networks dynamics, especially in problems
involving synchronization \cite{Med12, MZ12}. The continuum limit of nonlocally coupled
dynamical networks is one of few analytical approaches that have a potential for elucidating
dynamics of a broad class of networks \cite{KurBat02, AbrStr06, WilStr06,
 GirHas12, OmeWol12,OmeRie12}.
In this limit, the solutions of the
initial value problems (IVPs) for evolution equations on large discrete
networks are approximated by those for the limiting  integro-differential
equations posed on continuous spatial domains. This limiting
procedure has been used to study the mechanisms
of some very interesting effects such as chimera states
\cite{KurBat02,AbrStr06},
multistability \cite{WilStr06,GirHas12},
synchronization, and the coherence-incoherence transition \cite{OmeWol12}.
However, a rigorous justification for taking the continuum limit in
nonlocally models was lacking. In this paper, we use the combination of
techniques from the theory of evolution equations \cite{EvaPDE} and the recent theory
of graph limits \cite{BorChay06, BorChay08, LovSze06, LovSze07, LovGraphLim12}
 to provide such justification for a large class of dynamical
models on deterministic graphs. In fact, some of the tools that we develop
in this work come in useful in the analysis of the  continuum limit of dynamical systems
on random graphs undertaken in a companion paper \cite{Med13a}.

To motivate the forthcoming analysis of the continuum limit
in the nonlocally coupled systems, we first review several representative
examples. In \cite{WilStr06}, Wiley, Strogatz, and Girvan
studied a nonlocally coupled system of phase oscillators
\begin{equation}\label{Kuramoto}
\dot\phi_i=\omega+{1\over n} \sum_{j=i-k}^{i+k} \sin\left(\phi_j-\phi_i\right),
\end{equation}
where $\phi_i:~{\mathbb R}^+\to {\mathbb S}^1:={\mathbb R}/2\pi{\mathbb Z},\; i\in [n]:=\{1,2,3,\dots,n\}$ is interpreted as the phase
of oscillator~$i$, $\omega$ is the intrinsic frequency, and the sum models the interactions between
oscillator~$i$ and $k$ of its nearest neighbors from each side
(cf. \cite{Kur84, KurBat02}). The oscillators are located on a ring and indexed
by integers from ${\mathbb Z}/n{\mathbb Z}$. By recasting (\ref{Kuramoto}) in uniformly rotating
frame of reference, one can absorb $\omega$. Thus, below we set $\omega=0$.

\begin{figure}
\begin{center}
{\bf a}\hspace{0.1 cm}\includegraphics[height=1.8in,width=2.0in]{newf1a.pdf}\hspace{1.0cm}
{\bf b}\hspace{0.1 cm}\includegraphics[height=1.8in,width=2.0in]{newf1b.pdf}
\end{center}
\caption{The plot of the support of the  function $W_{G_n}$ representing the adjacency matrix
of the $k$-nearest-neighbor graph $G_n$ ({\bf a}) and that of its limit $W_G$ ({\bf b}).
}
\label{f.1}
\end{figure}

It is instructive to view (\ref{Kuramoto}) as a system of differential
equations on graph $G_n=\langle V(G_n), E(G_n)\rangle$ with the
vertex set $V(G_n)=[n]$ and the edge set
$$
E(G_n)=\left\{ (i,j)\in [n]^2:~ 0<\mbox{dist}(i,j)\le k\right\},\;\mbox{where}\;
\mbox{dist}(i,j) =\min\{ |i-j|, n-|i-j|\}.
$$
Let $W_{G_n}:~I^2\to\{0,1\}$ such that
$$
W_{G_n}(x,y)=1\;\mbox{if}\; (i,j)\in E(G_n)\;\mbox{and}\; (x,y)\in [(i-1)n^{-1}, in^{-1})\times [(j-1)n^{-1}, jn^{-1}).
$$
Here and below, $I$ denotes $[0,1]$, the spatial domain of the
continuum limits considered in this paper.
The plot of the support of $W_{G_n}(x,y)$ in Fig.~\ref{f.1}a provides the  pixel picture
of the adjacency matrix of $G_n$ \cite{Biggs}.  In Fig.~\ref{f.1}a and
in similar plots throughout this paper,
we place the origin of the unit square  in the top left corner of the plot to emphasize
the correspondence between $W_{G_n}$ and the adjacency matrix of $G_n$.
As $n\to \infty$, $\{W_{G_n}\}$ converges to the $\{0,1\}$-valued function $W_G(x,y)$,
whose support is shown in Fig.~\ref{f.1}b.

In \cite{WilStr06}, the analysis of the attractors  of (\ref{Kuramoto}) employs  the continuum limit
of (\ref{Kuramoto}).
 Specifically, let $k=rn$ for some fixed $r\in (0,1]$. After interpretting the right
hand-side of (\ref{Kuramoto}) as a Riemann sum and sending $n\to\infty$,
in the uniformly rotating frame of coordinates  (\ref{Kuramoto}) formally becomes
\begin{equation}\label{cont}
{\partial\over \partial t}\phi(x,t)=\int_I W_G(x,y)
\sin\left(\phi(y,t)-\phi(x,t)\right)dy,
\end{equation}
where $\phi(x,t)$ describes the evolution of the continuum of oscillators
distributed over  $I$.
Equation (\ref{cont}) is called the continuum (thermodynamic) limit of
(\ref{Kuramoto}).\footnote{There is another form of the continuum limit
for the Kuramoto model \cite{StrMir91, Str2000, OttAnt08, Lai09}.
It is formulated in terms of the density characterizing the state of the
continuous system. We do not consider this limit in the present paper.}

The continuum equation (\ref{cont}) has a family of steady state
solutions
\begin{equation}\label{twist}
\theta^{(q)}(x,t)=2\pi qx+c, \; q\in {\mathbb Z}, \; c\in {\mathbb R},
\end{equation}
called $q-$twisted states. In \cite{WilStr06}, the stability analysis of
the continuous twisted states (\ref{twist}) was used to study their
discrete counterparts, which are the steady state solutions of
(\ref{Kuramoto}) ($\omega=0$) for finite $n$. The stability analysis
in \cite{WilStr06} can, in fact, be
completely translated into the discrete setting. However, suppose we replace
the family of $k$-nearest-neighbor graphs in (\ref{Kuramoto})
by a family of small-world graphs (see Fig.~\ref{ff.1}a). Then not only
does the continuum limit provide a convenient setting for the stability
analysis but also the twisted states, as the steady states
of the Kuramoto model, exist only in the limit
as the number of oscillators goes to infinity (see  Fig.~\ref{ff.1}b)~\cite{Med13b}.
Therefore, in this case the continuum
limit affords the analysis of the asymptotic behavior of solutions
of the Kuramoto model for large $n$, which is not otherwise feasible in the discrete
setting. The Kuramoto-Battogtokh model generating chimera states \cite{KurBat02}
is another example, where the contnuum limit seems to be critical
for  understanding the nontrivial dynamics in the discrete systems.
We will return to the discussion of chimera states in Section \ref{sec.chimera}.

\begin{figure}
\begin{center}
{\bf a}\hspace{0.1 cm}\includegraphics[height=1.8in,width=2.0in]{newf2a.pdf}\hspace{1.0cm}
{\bf b}\hspace{0.1 cm}\includegraphics[height=1.8in,width=2.0in]{newf2b.pdf}
\end{center}
\caption{ {\bf a}) The pixel picture of a small-world graph obtained from that shown in
Fig.~\ref{f.1}{\bf a}
by replacing a random set of the local connections by randomly chosen long-range ones.
{\bf b}) The pixel picture for a large small-world graph.
}
\label{ff.1}
\end{figure}

These examples lead to the following questions.
\begin{description}
\item[(A)] Does the continuum model (\ref{cont})  truly approximate the dynamics of the discrete
model (\ref{Kuramoto}) for large finite $n$? If so, in what sense the solutions of the integro-differential
equation
approximate  those of (\ref{Kuramoto}) with $\omega=0$?
\item[(B)] How big is the class of network topologies for which one can use the continuum
limit? Is it restricted to the special graphs like $k-$nearest-neighbor one on a ring?
Can it be applied, for instance, to the small world networks, the original motivation for the
analysis in \cite{WilStr06}?
\end{description}

The function $W_G$ shown in Fig.~\ref{f.1}b is the limit of the functions
$\{W_{G_n}\}$ (Fig.~\ref{f.1}a) representing the adjacency matrices of the
$k$-nearest neighbor family of graphs $\{G_n\}$. The latter  is an example
of a convergent graph sequence and $W_G$ is the corresponding graph
limit \cite{LovGraphLim12}. We will explain the meaning of the limit of
a graph sequence in Section~\ref{sec.limit}. Meanwhile, we refer to the
geometric interpretation of the adjacency matrix  for the $k$-nearest-neighbor
graph in Fig.~\ref{f.1}a, which suggests the limiting pattern of $\{W_{G_n}\}$
as $n\to\infty$ (see Fig.~\ref{f.1}b). Likewise, the pixel picture of the large
small-world graph in Fig.~\ref{ff.1}b suggests the (piecewise constant)
limit for the small-world family of graphs, which in turn can be used in the
derivation of the continuum model like (\ref{cont})  \cite{Med13b}.
These observations hint on the possible relevance of the theory of graph limits for constructing
the continuum limits for dynamical networks. We explore this
relation for dynamical systems on convergent families of deterministic graphs
in this paper and extend this approach to random networks in \cite{Med13a}.
Interestingly, in the process of justifying the continuum limit, we discovered the
link between this problem and that of convergence of several classical
numerical methods. Specifically, we show that dynamical networks on simple
and weighted graphs analyzed in Sections~\ref{sec.simple} and \ref{sec.weight}
can be interpreted as the discretizatizations of the continuum evolution equation
by the collocation method and the Galerkin method respectively. Furthermore,
the analysis of the continuum limit for networks on random graphs in \cite{Med13a}
features a similar connection with the Monte-Carlo method. Therefore, in addition to the
rigorous justification of taking the continuum limit for a large class of dynamical
networks, our results characterize convergence of these numerical methods
for solving IVPs for certain nonlinear integro-differential equations.

This paper is organized as follows.
We review the necessary background on graph limits in Section~\ref{sec.limit}.
In Section~\ref{sec.formulate}, we discuss the heat equation on graphs
and graph limits. Here, we extend a classical linear heat equation
on graphs  to allow nonlinear diffusion. This extension covers
many dynamical networks
arising in applications including  coupled oscillator models like (\ref{Kuramoto}).
In the same section, we formally define the continuum limit for
dynamical networks of a convergent sequence of dense (weighted)
graphs. In this limit, the discrete diffusion operator becomes
an integral operator with the kernel representing the limit of
the infinite family of graphs. We show that the IVP for the
limiting equation is well-posed and admits a unique solution in
$C^1({\mathbb R};L^\infty(I))$. Further, in Theorem~\ref{thm.reg}, we specify
assumptions on the kernel and the initial conditions, which guarantee
that the solutions of the IVPs remain continuous in space over
subdomains of $I$.
This result is used to characterize the attractors of the
continuum model. In particular, we apply it to study
the regions of continuity of the chimera states and attractors
of the Kuramoto equation on certain multipartite graphs (see Section~\ref{sec.examples}).
The rest of the paper is focused on  studying the relation between
the solutions of the IVPs for discrete networks and  and their continuum counterparts.
In Section~\ref{sec.simple}, for sequences
of  simple graphs converging to $\{0,1\}-$valued graphons, we  show that
the rate of convergence depends on the fractal dimension
of the boundary of the support of the graph limit.
This shows explicitly how the geometry of the graphon affects
the accuracy of the continuum limit. In Section~\ref{sec.weight},
we analyze networks on convergent
weighted graph sequences.
 The results of this paper
are illustrated with the discussion of the dynamics of two concrete
models: the Kuramoto-Battogtokh nonlocal system generating
chimera states \cite{KurBat02} and the Kuramoto equation on the
half and complete bipartite graphs (cf.~Section~\ref{sec.examples}).
The final section, Section~\ref{sec.conclusion},
contains concluding remarks.

\section{Graph limits}\label{sec.limit}
\setcounter{equation}{0}

In this section, we review several definitions and results from the theory of graph
limits that we will need later. In our brief tour through graph limits, we mainly follow
\cite{BorChay11} and \cite{Pikh10}. For the full exposition of this powerful theory
with many diverse applications, we refer an interested reader to the pioneering
papers by Lov{\' a}sz and Szegedy  \cite{LovSze06, LovSze07}, and Borgs, Chayes,
Lov{\' a}sz, S{\'o}s, and Vesztergombi \cite{BorChay06, BorChay08}; and to the
monograph \cite{LovGraphLim12}.

An undirected graph $G=\langle V(G), E(G)\rangle$
without loops and multiple edges is called simple. $V(G)$ stands for the
set of nodes and $E(G)\subset V(G)\times V(G)$ denotes the edge set.
\begin{figure}
\begin{center}
{\bf a}\hspace{0.1 cm}\includegraphics[height=1.8in,width=2.0in]{newf3a.pdf}\hspace{1.0cm}
{\bf b}\hspace{0.1 cm}\includegraphics[height=1.8in,width=2.0in]{newf3b.pdf}
\end{center}
\caption{ {\bf a}) The pixel picture of the Erd\H{o}s-R\'{e}nyi graph  $G(40,0.5)$.
The edge between a pair of distinct nodes is inserted with probability $0.5$.
{\bf b}) The pixel picture of $G(600,05)$.
}
\label{ff.2}
\end{figure}

Let $G_n=\langle V(G_n), E(G_n)\rangle, n\in\mathbb{N}$ be a sequence of dense
(simple) graphs, i.e., $|E(G_n)|=O(|V(G_n)|^2)$, where $\left|\cdot\right|$ denotes the cardinality of a set.
The convergence of the graph sequence  $\{G_n\}$ is defined in terms of the homomorphism densities
\begin{equation}\label{hdense}
t(F,G_n)={\mbox{hom}(F,G_n)\over \left|V(G_n)\right|^{|V(F)|}}.
\end{equation}
Here, $F=\langle V(F), E(F)\rangle$ is a simple graph and $\mbox{hom}(F,G_n)$ stands for
 the number of homomorphisms (i.e., adjacency preserving maps
$V(F)\to V(G_n)$).
In probabilistic terms,  (\ref{hdense}) is the likelihood of a random
map $h:~V(F)\to V(G_n)$ to be a homomorphism.

\begin{df}\label{df.convergent}\cite{LovSze06, BorChay08}
The sequence of graphs $\{G_n\}$
is called convergent if $t(F,G_n)$ is convergent for every simple graph
$F$.\footnote{In the theory of graph limits, convergence in Definition~\ref{df.convergent}
is called left-convergence. Since this is the only convergence of graph sequences  used
in this paper, we refer to the left-convergent sequences as convergent.}
\end{df}

It turns out that the limiting object can be represented by a measurable symmetric function
$W: I^2\to I$. We recall that $I$ stands for $[0,1]$.
Such functions are called graphons. The set of all graphons is
denoted by $\mathcal{W}_0$.

\begin{thm}\cite{LovSze06}
For every convergent sequence of simple graphs, there is  $W\in\mathcal{W}_0$
such that
\begin{equation}\label{t-to-t}
t(F,G_n)\to t(F,W):=\int_{I^{|V(F)|}} \prod_{(i,j)\in E(F)} W(x_i,x_j) dx
\end{equation}
for every simple graph $F$.
Moreover, for every $W\in\mathcal{W}_0$ there is a sequence of graphs
$\{G_n\}$ satisfying
(\ref{t-to-t}).
\end{thm}

The cut-norm is important for describing  the metric  properties of graphons.
For any integrable function and, in particular, for any graphon $W\in\mathcal{W}_0$,
$$
\|W\|_\square =\sup_{S,T\in \mathcal{L}_I} \left| \int_{S\times T} W(x,y) dxdy\right|
$$
is called the cut-norm of $W$. Here, $\mathcal{L}_I$ stands for the set of all Lebesgue measurable
subsets of  $I$. The cut-distance between two graphons $W$ and $U$
is defined by
$$
\delta_\square(U,W)=\inf_{\phi}\|U-W^\phi\|_\square,\;\;
$$
where $W^\phi(x,y):=W(\phi (x),\phi (y))$ and $\phi$ ranges over all measure-preserving
bijections of $I$.   The infinum over all $\phi$ is used to make the cut-distance between graphons
invariant with respect to graph isomorphisms, as well as some other transformations that do not change
the asymptotic properties of the graph sequences  (see \cite{BorChay08, LovGraphLim12} for
more details).
A graph sequence is convergent if and only if it is Cauchy in the cut-distance \cite{BorChay08}.

Graph limits are the equivalence classes of graphons
$$
[W]=\left\{U\in\mathcal{W}_0:~\delta_\square(U,W)=0\right\}.
$$
With a customary abuse of notation, we refer to both $W$ and $[W]$ as graphons.
The pseudo-metric $\delta_\square(\cdot,\cdot)$ induces the metric on
$\chi=\{[W]:~W\in\mathcal{W}_0\}$.
The metric space $(\chi,\delta_\square)$ is compact \cite{LovSze07}.

\begin{figure}
\begin{center}
{\bf a}\hspace{0.1 cm}\includegraphics[height=1.8in,width=2.0in]{newf4a.pdf}\hspace{1.0cm}
{\bf b}\hspace{0.1 cm}\includegraphics[height=1.8in,width=2.0in]{newf4b.pdf}
\end{center}
\caption{ {\bf a}) The pixel picture of the half-graph $H_{20,20}$.
{\bf b}) The limit of $\{W_{H_{n,n}}\}$.
}
\label{ff.3}
\end{figure}

We now describe a simple construction of convergent graph sequences that will be used
in the analysis of the continuum limit of dynamical networks below.
Consider a sequence of simple graphs $\{G_n\}$ on $n$ nodes. Define
\begin{equation}\label{pixel}
W_{G_n}(x,y)=\left\{ \begin{array}{ll} 1, & \mbox{if}\; (i,j)\in E(G_n)\;\mbox{and}\;
(x,y)\in \left[{i-1\over n}, {i\over n}\right)\times \left[{j-1\over n}, {j\over n}\right),\\
0,& \mbox{otherwise}.
\end{array}\right.
\end{equation}
The support of $W_{G_n}$ provides  the pixel picture of the adjacency
matrix of $G_n$ (see Fig.~\ref{ff.1}a), and  $[W_{G_n}]$ is the corresponding graphon.
Note that $[W_G]$ is invariant under relabeling the nodes of $G$
while $W_{G_n}$ is not.
The graph sequence $\{G_n\}$ is convergent if $W_{G_n}$ converge
with respect to the cut-norm. In particular, since for any integrable
function $W\in\mathcal{W}_0$
$$
\|W\|_\square\le \|W\|_{L^1(I^2)},
$$
convergence of $\{W_{G_n}\}$ in the $L^1$-norm implies convergence
of the graph sequence $\{G_n\}$. The deterministic networks
analyzed in this paper are actually convergent with respect to the stronger
$L^1$-norm. However, the convergence of graphons with
respect to the cut-norm does not in general imply that with respect to
$L^1$-norm. For instance, the sequence of Erd\H{o}s-R{\'e}nyi
graphs with edge density $p\in (0,1)$ is convergent to the constant
function $p$ on $I^2$, $\mbox{Const}(p)$ \cite{LovSze06, BorChay08},
while no sequence of $\{0,1\}$-valued graphons
can converge to $\mbox{Const}(p)$ with $p\in (0,1)$ in the $L^1$-norm.
In particular, $L^1$-estimates for graphons are insufficient for the
analysis of the continuum limits of
networks on random graphs \cite{Med13a}.

We conclude this section we several examples of convergent graph sequences.
\begin{ex}\label{ex.ER}\cite{LovSze06, BorChay08}
The Erd\H{o}s-R\'{e}nyi graphs.
Let $p\in (0,1)$ and consider a sequence of random graphs
$G(n,p)=\langle V(G(n,p)),  E(G(n,p))\rangle$,
$V(G(n,p))=[n]$ such that the probability
${\mathbb P~}\{(i,j)\in E(G(n,p))\}=p$ for any $(i,j)\in [n]^2$ (see Fig.~\ref{ff.2}a).
Then for any simple graph $F,$ $t(F, G(n,p))$ is convergent with probability
$1$ to $p^{|E(F)|}$ as $n\to\infty$ \cite{BorChay08}.
Thus, $\{G(n,p)\}$  is a convergent sequence with the limit given by the
constant graphon $p$.  The pixel picture of $W_{G(n,p)}$ in Fig.~\ref{ff.2}b
provides the intuition behind the graph limit for $\{G(n,p)\}$. Note that
for large $n$, the plot of the support of $W_{G(n,p)}$ resembles
that of the constant function if looked at from a distance.
In fact, the limiting graphon reflects  the asymptotic density
of connections in $G(n,p)$ as  $n\to\infty$. Using the strong law of large numbers,
one can show that $\|W_{G(n,p)} - p\|_\square\to 0$  as $n\to\infty$  with probability
$1$. Thus,  $\{W_{G(n,p)}\}$ is convergent in the cut-norm but not in the $L^1$-norm.
\end{ex}

\begin{ex}\label{ex.half}\cite{LovSze06} The half-graphs. Let $H_{n,n}=\langle V(H_{n,n}), E(H_{n,n})\rangle$
be a bipartite graph on $2n$ nodes such that
$$
V(H_{n,n})=\{1,2,\dots,n,1^\prime, 2^\prime,\dots, n^\prime \},\;
E(H_{n,n})=\{( i,j^\prime) \in V(H_{n,n})\times V(H_{n,n}):~ i\le j\}
$$
(see Fig.~\ref{ff.3}a).
The sequence $\{ H_{n,n}\}$ converges to the graphon $[H]$ where
$H: I^2\to I$ is the characteristic function of the set
$\{ (x,y):~ |x-y| \ge 1/2\}$ (see Fig.~\ref{ff.3}a).In this example, $\{W_{H_{n,n}}\}$
converges  to $H$ pointwise, and, by the dominated convergence theorem,
in the $L^1$-norm.
\end{ex}

\section{The formulation of the problem} \label{sec.formulate}
\setcounter{equation}{0}
\subsection{The heat equation on discrete and continuous domains}
Let $G_n=\langle V(G_n), E(G_n), W(G_n)\rangle$ be  a sequence of weighted graphs,
where $V(G_n)=[n]$ and $E(G_n)$ are the sets of nodes and edges respectively;
and $W(G_n)\!:\ [n]^2\to[-1, 1]$ is a symmetric weight matrix of the form
$$
(W(G_n))_{ij}=\left\{\begin{aligned}&w_{ij}^{(n)}, && (i, j)\in E(G_n),\\ &0,
&&{\rm otherwise.}\end{aligned}\right.
$$
If $G_n$ is a  simple  graph, $W(G_n)$ is a $\{0,1\}$-valued matrix.

By  the nonlinear heat equation on $G_n$ we mean  the system of
differential equations
\begin{equation}\label{dheat}
{d\over dt} u_i^{(n)}(t)=\lambda_i^{(n)}\sum_{j=1}^n w_{ij}^{(n)}
D\left(u^{(n)}_j-u^{(n)}_i\right),\; i\in [n],
\end{equation}
where $u^{(n)}(t)=\left( u^{(n)}_1(t),u^{(n)}_2(t),\dots,u^{(n)}_n(t)\right)^\mathsf{T}$,
and $\lambda_i^{(n)}$ are scaling coefficients.
The function $D:~{\mathbb R}\to{\mathbb R}$ is  Lipschitz continuous
\begin{equation}\label{Lip}
\left|D(u)-D(v)\right|\le L |u-v| \;\forall u,v\in {\mathbb R}.
\end{equation}
Throughout this paper, we will
use $\lambda_i^{(n)}=n^{-1}$. However, other scalings may also be used.

\begin{rem}\label{rem.generalize}
Our analysis applies to a more general class of equations
\begin{equation}\label{more-general}
{d\over dt} u_i^{(n)}(t)=\lambda_i^{(n)}\sum_{j=1}^n w_{ij}^{(n)}
D\left(u^{(n)}_j-u^{(n)}_i\right) + f_i(t,u^{(n)}),\; i\in [n],
\end{equation}
where functions $f_i(t,u), i\in [n],$ can be taken, for instance, to be
continuous in $t$ and Lipschitz continuous in $u$:
$$
\left|f_i(t,u)-f_i(t,v)\right|\le L |u-v| \;\forall u,v,t \in {\mathbb R}, \; i\in [n].
$$
To keep the presentation simple, we will restrict the  analysis to the case of
(\ref{dheat}).
It is straightforward to extend our results to cover (\ref{more-general}).
\end{rem}

If $D(u)=u$, the coupling operator on the right-hand side
of (\ref{dheat}) is the graph Laplacian, and  Equation (\ref{dheat})
becomes the linear heat equation on $G_n$. The linear heat equation
has many applications in combinatorial problems such as random walks on graphs
\cite{Chung-Spectral}, and dynamical problems, e.g., analysis of consensus
protocols \cite{Med12}. In this paper, we focus on the nonlinear heat
equation, which provides the framework for a large class of dynamical networks.
In particular, the  Kuramoto equation (\ref{Kuramoto}) is of this type.

In the remainder of this paper,  we will derive and justify
the continuum counterpart of (\ref{dheat})
\begin{equation}\label{nheat}
{\partial\over\partial t} u(x,t) =\int_{I} W(x,y) D\left( u(y,t)-u(x,t)\right) dy.
\end{equation}
The kernel $W$ will be specified separately for each class of problems that we consider
below.

\subsection{The well-posedness of the IVP}
Before setting out to study the relation between solutions of the discrete and continuous
heat equations (\ref{dheat}) and (\ref{nheat}), we first address the well-posedness
of the IVP for (\ref{nheat}).

It is  convenient to interpret the solution of the IVP for (\ref{nheat}),
$u(x,t)$, as a vector-valued map $\mathbf{u}:[0,T]\to L^\infty(I)$. Throughout this paper,
we will use the bold font to denote the vector-valued function $\mathbf{u}(t)$
corresponding to a function of two variables $u(x,t)$.

\begin{thm}\label{thm.wellposed}
Suppose $D$ is Lipschitz continuous, $W\in L^\infty(I^2)$, and $\mathbf{g}\in L^\infty(I)$.
Then for any $T>0$, there exists a unique solution of the IVP for (\ref{nheat})
$\mathbf{u}\in C^1({\mathbb R};L^\infty(I))$ subject to the initial condition $\mathbf{u}(0)=\mathbf{g}$.
\end{thm}
{\it Proof.\;}\; The proof of Theorem~\ref{thm.wellposed} is based on the contraction mapping principle
(cf.~\cite[Theorem~1.1, Chapter VII]{DalKre70}). We include the details for completeness.

Rewrite the IVP for (\ref{nheat}) as the integral equation
\begin{equation}\label{fix}
\mathbf{u}=K\mathbf{u},
\end{equation}
where
$$
[K\mathbf{u}](x,t):=\mathbf{g} +\int_0^t \int_I W(x,y) D\left( u(y,s)-u(x,s)\right)dyds.
$$
Let $M_\mathbf{g}$ be a metric subspace of $C(0,\tau; L^\infty (I))$ (where $\tau>0$ will be
specified later) consisting of functions $\mathbf{u}$ satisfying $\mathbf{u}(0)=\mathbf{g}$.
Then (\ref{fix})  is the fixed point equation for the operator $K: M_{\mathbf{g}}\rightarrow M_{\mathbf{g}}$.
We show below that $K$ is a contraction for a small $\tau>0$.

Indeed, let
\begin{equation}\label{tau}
\tau\le (4L\|W\|_{L^\infty(I^2)})^{-1},
\end{equation}
where $L$ is the Lipschitz constant of $D(\cdot)$. For any $\mathbf{u}, \mathbf{v}\in M_\mathbf{g}$
we have
$$
\left\| K\mathbf{u} - K\mathbf{v} \right\|_{M_\mathbf{g}} = \max_{t\in [0,\tau]}
\left\| K\mathbf{u}(t) - K\mathbf{v}(t) \right\|_{L^\infty (I)}
$$
$$
\le \max_{t\in [0,\tau]} \mathop{\esssup}_{x\in I}
 \int_{I\times [0,t]}   \left| W(x,y)\right|
\left|  D\left( u(y,t)-u(x,t)\right)- D\left( v(y,t)-v(x,t)\right)
\right| dy dt
$$
$$
\le \tau L \|W\|_{L^\infty(I^2)} \max_{t\in [0,\tau]}
\left\{  \int_I \left|u(y,t)-v(y,t)\right| dy+
\left\| \mathbf{u}(t)-\mathbf{v}(t)\right\|_{L^\infty (I)} \right\}
$$
$$
\le
2\tau L \|W\|_{L^\infty(I^2)}\max_{t\in [0,\tau]} \left\|\mathbf{u}(t)-\mathbf{v}(t)\right\|_{L^\infty (I)}.
$$
Thus, by (\ref{tau}) we have
\begin{equation}\label{contract}
\| K\mathbf{u}-K\mathbf{v} \|_{M_\mathbf{g}} \le {1\over 2}
\| \mathbf{u}-\mathbf{v} \|_{M_\mathbf{g}} .
\end{equation}

By the Banach contraction mapping principle, there exists a unique
solution of the IVP for (\ref{nheat}) $\bar{\mathbf{u}}\in M_\mathbf{g}\subset C(0,\tau; L^\infty(I))$.
Using $\bar{\mathbf{u}}(\tau)$ as the initial condition, the local solution
can be extended  to $[0, 2\tau]$, and, by repeating this argument,
to $[0, T]$ for any $T>0$. In a similar fashion, we can prove the existence and
uniqueness of the solution of the IVP for (\ref{nheat}) on $[-T,0]$ for any
$T>0$.
Furthermore, since the integrand
in (\ref{fix}) is continuous as a map
$L^\infty(I)\to L^\infty(I)$, $\mathbf{u}$ is continuously differentiable.
Thus, we have a classical solution
of the IVP for (\ref{nheat}) on the whole real axis.\\
$\square$

\subsection{Spatial regularity}
The classical heat equation, as  a parabolic partial differential equation, has a strong
smoothening property.
Regardless of the regularity of the initial data, the solution of the IVP
for the classical heat equation is a smooth function of the space variables
for all positive times. No such mechanism is present in the heat
equation on graph limits. Below we show that
the spatial regularity of solutions of the IVP is  determined by the
regularity of graphon $W$ and initial condition $\mathbf{u}(0)$.

\begin{thm}\label{thm.reg}
Let $D:{\mathbb R}\to{\mathbb R}$ be a Lipschitz continuous function and
$J=(\alpha,\beta)\subset I$. Suppose
for all $x\in J$ and for almost all $y\in I$,
$W\in  L^\infty(I^2)$ has a weak derivative ${\partial\over\partial x}W(x,y)$
and
\begin{equation}\label{weak-assumption}
\esssup_{y\in I} \left\|{\partial\over\partial x}W(\cdot,y)\right\|_{L^2(J)}\le C_1,
\end{equation}
for some $C_1>0$.
Then for any $0<T<\infty$, all $t\in [0,T],$ and
$\alpha<\alpha^\prime<\beta^\prime<\beta,$ the solution of the
IVP for (\ref{nheat}) satisfies\footnote{$H^1(J)$ stands
for the Sobolev space of all Lebesgue measurable functions
$f$ on an open interval $J\subset {\mathbb R}^1$ such that $f$ and
its distributional derivative $f_x$ are in $L^2(J)$ \cite{CheMil12}.}
$$
\mathbf{u}(t)\in H^1(J^\prime),\; J^\prime=(\alpha^\prime, \beta^\prime),
$$
provided $\mathbf{u}(0)\in L^\infty(I)\cap H^1(J)$.
\end{thm}
{\it Proof.\;}\;
Let $T>0$ be arbitrary but fixed, and
$$
h_0={1\over 2} \min\{ \alpha^\prime-\alpha, \beta-\beta^\prime\}.
$$
Then  for $0<h<h_0$, the difference quotient
$$
\xi(x,t)={u(x+h,t)-u(x,t)\over h}
$$
is a well-defined function on $\Omega_T=J^\prime\times [0,T]$. Further,
for $(x,t)\in \Omega_T$, $\xi(x,t)$
satisfies the following equation
\begin{eqnarray}\nonumber
{\partial \over \partial t}\xi(x,t)&=&
\int_I W(x,y) h^{-1}\left\{ D\left(u(y,t)-u(x+h,t)\right)-
D\left(u(y,t)-u(x,t)\right)\right\}dy\\
\label{xiode}
&+&\int_I D^h_x W(x,y) D\left(u(y,t)-u(x+h,t)\right) dy,
\end{eqnarray}
where
$$
D^h_x W(x,y)={W(x+h,y) -W(x,y)\over h}.
$$
By multiplying both sides of (\ref{xiode}) by $\xi(x,t)$ and
integrating both sides of the resultant equation over $J^\prime$ with respect
to $x$, we have
\begin{eqnarray}\nonumber
{1\over 2}\int_{J^\prime} {\partial \over \partial t}\xi(x,t)^2 dx &=&\int_{J^\prime \times I}
 W(x,y) h^{-1}\left\{ D\left(u(y,t)-u(x+h,t)\right)-
D\left(u(y,t)-u(x,t)\right)\right\}\xi(x,t) dxdy\\
\nonumber
&+&\int_{J^\prime\times I}
D^h_x W(x,y) D\left(u(y,t)-u(x+h,t)\right) \xi(x,t) dxdy\\
\label{re-xiode}
&=:& T_1+T_2.
\end{eqnarray}
Using $\mathbf{u}\in C(0,T;L^\infty(I))$, Lipschitz continuity of $D(\cdot)$,
and the triangle inequality, we have
\begin{equation}\label{boundD}
\max_{t\in [0,T]} \esssup_{(x,y)\in I^2} |D(u(y,t)-u(x,t))|\le 2L
\|\mathbf{u}\|_{C(0,T; L^\infty(I))}=:C_2.
\end{equation}
Furthermore,  using Fubini's theorem, (\ref{weak-assumption}), and
the standard results  for the difference quotients
(see, e.g., Theorem~5.8.3~\cite{EvaPDE}), we have
\begin{equation}\label{boundDW}
\|D_x^h W\|_{L^2(J^\prime \times I)} \le \esssup_{y\in I}\|D_x^h W\|_{L^2(J^\prime)}
 \le C_3 \esssup_{y\in I} \|{\partial\over \partial x} W(\cdot,y)\|_{L^2(J)} \le C_4,
\end{equation}
and, likewise,
\begin{equation}\label{bound-xi0}
\|{\boldsymbol\xi}(0)\|_{L^2(J^\prime)} \le C_5 \| \mathbf{u}(0)\|_{H^1(J)},
\end{equation}
where positive constants $C_4$ and $C_5$ are independent of $h\in(0,h_0)$.

Using (\ref{Lip}), we bound the first term on
the right hand side (\ref{re-xiode})
\begin{equation}\label{T1}
|T_1|\le  \|W\|_{L^\infty(I^2)} \int_{J^\prime\times I} L \xi(x,t)^2 dxdy= L \|W\|_{L^\infty(I^2)}\|{\boldsymbol\xi}(t)\|_{L^2(J^\prime)}^2.
\end{equation}
For the second term, we use (\ref{boundD}), (\ref{boundDW}),
and the Cauchy-Schwarz inequality
\begin{eqnarray}\nonumber
|T_2| &\le &  C_2  \int_{J^\prime\times I}
\left| D^h_x W  {\boldsymbol\xi}(t)  \right| dxdy
 \le C_2  \|D_x^h W \|_{L^2(J^\prime \times I)}
\|{\boldsymbol\xi}(t)\|_{L^2(J^\prime)}  \\
\label{T2}
&\le & C_2 C_4 \|{\boldsymbol\xi}(t)\|_{L^2(J^\prime)}.
\end{eqnarray}

By combining (\ref{re-xiode}), (\ref{T1}), and (\ref{T2}), we have
$$
{d\over dt} \|{\boldsymbol\xi}(t)\|^2_{L^2(J^\prime)} \le C_6\|{\boldsymbol\xi}(t)\|^2_{L^2(J^\prime)}+C_7, \;
C_6=2L \|W\|_{L^\infty(I^2)} +C_7, \; C_7=C_2 C_4,
$$
where inequality $2\|{\boldsymbol\xi}(t)\|_{L^2(J^\prime)} \le
\|{\boldsymbol\xi}(t)\|^2_{L^2(J^\prime)}+1$ was used.
Using  Gronwall's inequality, we obtain
\begin{eqnarray}\nonumber
\|{\boldsymbol\xi}(t)\|^2_{L^2(J^\prime)} &\le&
\left(\|{\boldsymbol\xi}(0)\|^2_{L^2(J^\prime)} +{ C_7 \over C_6}\right)
\exp\{C_6T\}\\
\label{bound-xi}
&\le&
\left( C_5^2 \| \mathbf{u}(0)\|^2_{H^1(J)}      +{ C_7 \over C_6}\right)
\exp\{ C_6T\}, \; t\in[0,T].
\end{eqnarray}
The last inequality  yields a uniform in $h\in (0,h_0]$ bound on the difference quotient
$\|{\boldsymbol\xi}(t)\|_{L^2(J^\prime)}$.
Using the properties of the difference quotients
(cf. Theorem~5.8.3~\cite{EvaPDE}), we conclude  that $\mathbf{u}(t)\in H^1(J^\prime)$
for all  $t\in (0,T]$.\\
$\square$

\section{Networks on simple graphs}\label{sec.simple}
\setcounter{equation}{0}
In this and in the following sections, we prove that the solution
of the IVP for appropriately chosen continuous problem (\ref{nheat})
approximates the solutions of the discrete problems (\ref{dheat})
when $n$ is sufficiently large. We prove this result for two classes
of convergent graph sequences. In this section, we consider
the case of a sequence of simple graphs converging to a
$\{0,1\}$-valued graphon, and we study a more general case
of convergent sequences of weighted
graphs\footnote{For weighted graphs, one can also define
convergence by extending the notion of the homomorphism density
for this case (see \cite{LovSze06} for details). We do not discuss this
generalization here, because for the problems that we study in this
paper a simpler (and stronger) form of convergence, convergence
in $L^1-$norm, is sufficient (see Section~5).}
in the next section.
We single out networks on $\{0,1\}$-valued graphons for two reasons.
First, many coupled oscillator models  fit into
this framework (see, e.g.,  \cite{WilStr06, GirHas12} and \S\ref{sec.half}).
Second, for this class of networks we can explicitly estimate the accuracy
of approximation of the solutions of the discrete models by those
of their continuum limits in terms of the network size and the geometry
of the graphon of the network (cf. Theorem~\ref{coloc-rate}).
This result is important, because it reveals the structural properties of the
graphs shaping the accuracy of the thermodynamic limit.

Let $W:I^2\to\{0,1\}$ be a symmetric measurable function.
We denote the support of $W$ by
$$
W^+=\{ (x,y)\in I^2:\; W(x,y)\neq 0\}
$$
and its boundary by $\partial W^+$.

For  convenience, we rewrite the IVP for (\ref{nheat})
\begin{eqnarray}\label{sheat}
{\partial\over\partial t} u(t,x)&=&\int_I W(x,y) D\left( u(y,t)-u(x,t)\right) dy,\\
\label{sheat-ic}
u(x,0) &=& g(x).
\end{eqnarray}
Throughout this section, to simplify presentation we assume that
$g(x)$
is a step function.

Next, we define a sequence of discrete problems.
To this end, we fix $n\in{\mathbb N}$,  divide $I$ into $n$ subintervals
\begin{equation}\label{partition}
 I^{(n)}_1=\left[0, {1\over n}\right),\; I^{(n)}_2=\left[{1\over n},{2\over n}\right), \dots,
I_n^{(n)}=\left[ {n-1\over n},1 \right),
\end{equation}
and define a sequence  of simple graphs $G_n=\langle V(G_n), E(G_n) \rangle$
such that
$V(G_n)=[n]$ and
$$
E(G_n)=\{(i,j)\in [n]^2:\; (I_i^{(n)}\times I_j^{(n)})\cap W^+\neq\emptyset \}.
$$

The IVP for the nonlinear heat equation on $\{G_n\}$, a discrete
counterpart of (\ref{sheat}),
is given by
\begin{eqnarray}\label{dsheat}
{d\over dt} u_i^{(n)}(t) &=& n^{-1} \sum_{j: (i,j)\in E(G_n)} D(u^{(n)}_j-u^{(n)}_i),\\
\label{dsheat-ic}
u^{(n)}_i(0)&=&g^{(n)}_i,\; i\in [n].
\end{eqnarray}
There are many ways of approximating $g(x)$ by $g_n(x)$. For concreteness,
we assign $g_i^{(n)}$ the average value of $g(x)$ on $I_i$:
\begin{equation}\label{mean-g}
g_i^{(n)}=n \int_{I^{(n)}_i} g(x) dx.
\end{equation}

To compare the solutions of the discrete and continuous models, it is
convenient to represent the discrete function $u^{(n)}=(u^{(n)}_1,
u^{(n)}_2,\dots, u^{(n)}_n)^\mathsf{T}$ as a step function on $I$ as follows
\begin{equation}\label{step-fun}
u_n(x,t) = u^{(n)}_i, \;\mbox{if} \; x\in I^{(n)}_i.
\end{equation}
Then $u_n(x,t)$ satisfies the following IVP
\begin{eqnarray}\label{coloc}
{\partial\over\partial t} u_n(t,x)&=&\int_I \hat W_n(x,y) D\left( u_n(y,t)-u_n(x,t)\right) dy,\\
\label{coloc-ic}
u(x,0) &=& g_n(x),
\end{eqnarray}
where
$$
g_n(x)=g_i^{(n)} \;\mbox{if}\;  x\in I^{(n)}_i, i\in[n].
$$
and $\hat W_n (x,y)$ is the step function such that for $(x,y)\in
I^{(n)}_i\times I^{(n)}_j,$
$(i,j)\in [n]^2,$
\begin{equation}\label{hat-W}
\hat W_n (x,y)= \left\{\begin{array}{ll} 1, & \mbox{if}\;  (I^{(n)}_i\times I^{(n)}_j)\cap W^+\neq\emptyset,\\
                                     0, & \mbox{otherwise}.
\end{array}
\right.
\end{equation}

\begin{thm}\label{coloc-rate}
Let $\mathbf{u}$ and $\mathbf{u_n}$ denote the vector-valued functions
corresponding to the solutions
of (\ref{sheat}), (\ref{sheat-ic}), and (\ref{coloc})-(\ref{mean-g})
respectively.  Denote
the upper box-counting
dimension of $\partial W^+$
by $2b=\overline{\dim}_B \partial W^+$
(cf. \S~3.1,\cite{Falc-Fractal}) and suppose
that $b\in [0.5, 1)$.
Then for any $\epsilon> 0$ and all sufficiently large $n$
\begin{equation}\label{rate}
\| \mathbf{u} -\mathbf{u_n}\|_{C(0,T;L^2(I))} \le C_1
n^{-(1-b-\epsilon)},
\end{equation}
where constant $C_1$ is independent of $n$.
\end{thm}
{\it Proof.\;}\;
Denote $\xi_n(x,t)= u_n(x,t)-u(x,t)$. By subtracting
(\ref{sheat}) from (\ref{coloc}), we have
\begin{eqnarray}\nonumber
{\partial \xi_n\over \partial t} &=& \int_I \hat W_n(x,y)\left\{ D\left(u_n(y,t)-u_n(x,t)\right)-
D\left(u(y,t)-u(x,t)\right)\right\} dy\\
&+&
\label{subtract1}
\int_I \left( \hat W_n(x,y)- W(x,y)\right) D\left(u(y,t)-u(x,t)\right)dy.
\end{eqnarray}
Next, we multiply both sides of (\ref{subtract1}) by $\xi_n(x,t)$ and
integrate over $I$
\begin{eqnarray}\nonumber
{1\over 2} \int_I {\partial\over \partial t} \xi_n(x,t)^2 dx &=& \int_{I^2} \hat W_n(x,y)\left\{ D\left(u_n(y,t)-u_n(x,t)\right)-
D\left(u(y,t)-u(x,t)\right)\right\}  \xi_n(x,t) dxdy\\
&+&
\label{dxi2a}
\int_{I^2} \left(\hat W_n(x,y)-W(x,y)\right) D\left(u(y,t)-u(x,t)\right)\xi_n(x,t) dxdy.
\end{eqnarray}
Using the Lipschitz continuity of $D(\cdot)$,  $\|\hat W\|_{L^\infty (I^2)}=1$,
the triangle inequality, and the Cauchy-Schwarz inequality,
we estimate the first term
on the right-hand side of (\ref{dxi2a})
\begin{eqnarray}\nonumber
\left|
\int_{I^2} \hat W_n(x,y)\left\{ D\left(u_n(y,t)-u_n(x,t)\right)-
D\left(u(y,t)-u(x,t)\right)\right\}  \xi_n(x,t) dxdy
\right|
\\
\label{first}
\le L \int_{I\times I} \left|\left(\xi_n(y,t)-\xi_n(x,t)\right)
  \xi_n(x,t)\right| dxdy \le
2L\|{\boldsymbol\xi}_n(t)\|_{L^2 (I^2)}^2.
\end{eqnarray}

We estimate
the second term on the right-hand side of  (\ref{dxi2a}),
using the Cauchy-Schwarz inequality and the bound on $D(\cdot)$ (cf. (\ref{boundD}))
\begin{eqnarray}\nonumber
\left|\int_{I^2} \left( \hat W_n(x,y)-W (x,y)\right) D\left(u(y,t)-u(x,t)\right)\xi_n(x,t) dxdy\right|\\
\nonumber
\le \esssup_{(x,y,t)\in I^2\times [0,T]} \left| D\left(u(y,t)-u(x,t)\right)\right|
\left|\int_{I^2} \left( \hat W_n(x,y)-W (x,y)\right)\xi_n(x,t) dxdy\right|\\
\label{second}
\le C_2 \|W-\hat W_n\|_{L^2(I^2)} \|{\boldsymbol\xi}_n\|_{L^2(I)}
\end{eqnarray}
for some constant $C_2>0$ independent of $n$.

Using (\ref{first}) and (\ref{second}), from  (\ref{dxi2a}) we have
\begin{equation}\label{insert-1}
{d\over dt} \|{\boldsymbol\xi}_n\|^2_{L^2(I)} \le 4L \|{\boldsymbol\xi}_n\|^2_{L^2(I)}+
2C_2 \|W-\hat W_n\|_{L^2(I^2)}\|{\boldsymbol\xi}_n\|_{L^2(I)}.
\end{equation}

Let $\varepsilon>0$ be arbitrary but fixed, and set
$$
\phi_\varepsilon(t)=\sqrt{\|{\boldsymbol\xi}_n\|^2_{L^2(I)}+\varepsilon}.
$$
By (\ref{insert-1}),
\begin{equation}\label{insert-2}
{d\over dt} \phi_\varepsilon(t)^2\le 4L \phi_\varepsilon(t)^2+
2C_2 \|W-\hat W_n\|_{L^2(I^2)}\| \phi_\varepsilon(t).
\end{equation}
Since $\phi_\varepsilon(t)$ is positive on $[0,T]$, from
(\ref{insert-2}), we have
$$
{d\over dt} \phi_\varepsilon(t)\le 2L \phi_\varepsilon(t)+
C_2 \|W-\hat W_n\|_{L^2(I^2)}, \; t\in [0,T].
$$
By Gronwall's inequality,
\begin{equation}\label{insert-3}
\sup_{t\in [0,T]} \phi_\varepsilon(t) \le \left( \phi_\varepsilon(0)
+{C_2\|W-\hat W_n\|_{L^2(I^2)}\over 2L}\right) \exp\{2LT\}.
\end{equation}
Since $\varepsilon>0$ is arbirtrary, (\ref{insert-3}) implies
\begin{equation}\label{Gron}
\sup_{t\in [0,T]}\|{\boldsymbol\xi}_n(t)\|_{L^2(I)}\le \left( \|\mathbf{g}-\mathbf{g}_n\|_{L^2(I)}
+{C_2\|W-\hat W_n\|_{L^2(I^2)}\over 2L}\right) \exp\{2LT\}.
\end{equation}

It remains to estimate $\|W-\hat W_n\|_{L^2(I^2)}$.
To this end, consider the set of discrete cells $I_i^{(n)} \times I_j^{(n)}$ that covers the boundary
of the support of $W$
$$
J(n)=\{(i,j)\in[n]^2:~ (I_i^{(n)}\times I_j^{(n)})\cap \partial W^+\neq\emptyset \} \;\mbox{and}\;
C(n)=\left|J(n)\right|.
$$
Using one of several equivalent definitions of the upper box-counting dimension of
a subset of ${\mathbb R}^n$, we have
$$
2b:=\overline{\dim}_B \partial W^+=
\overline{\lim_{\delta\to 0}} {\log N_\delta (\partial W^+)\over -\log\delta},
$$
where $N_\delta (\partial W^+)$ is the number of cells of a $(\delta\times\delta)$-mesh  that intersect $\partial W^+$
(see Equation (3.12)(iv) in \cite{Falc-Fractal}).
Thus, for any $\epsilon>0$ and all sufficiently large $n$, we have
$$
C(n)\le  n^{2(b+\epsilon)}.
$$

Since $W$ and $\hat W_n$ coincide on all cells $I_i^{(n)}\times I_j^{(n)}$ for which
$(i,j)\notin J(n)$, for any $\epsilon>0$ and all sufficiently large $n$, we have
\begin{equation}\label{estimW}
\|W-\hat W_n\|_{L^2(I^2)}^2=\int_{I^2} (W-\hat W_n)^2 dxdy \le C(n)n^{-2}\le  n^{-2(1-b-\epsilon)}.
\end{equation}

Finally, from (\ref{mean-g}) it is easy to see that
\begin{equation}\label{numbered}
\|\mathbf{g}-\mathbf{g}_n\|_{L^2(I)}^2=O(n^{-1})
\end{equation}
The combination of (\ref{Gron}), (\ref{estimW}), and (\ref{numbered})
implies  (\ref{rate}).\\
$\square$

\section{Networks on  weighted graphs}\label{sec.weight}
\setcounter{equation}{0}

In this section, we study a more general case of the heat equation
on convergent sequences of weighted graphs. First, we define two
graph sequences generated by a given graphon $W$ and then we
prove the convergence of the corresponding discrete problems
to the continuum limit (\ref{sheat}).

Throughout this section, we assume that $W: I^2\to [-1,1]$ is a
symmetric measurable function. Let $\mathcal{P}_n$ denote the
partition of $I$ into $n$ intervals, $\mathcal{P}_n=\{I_i^{(n)}, i\in [n]\}$
(see (\ref{partition})) and
$$
X_n=\left\{{1\over n}, {2\over n}, \dots, {n\over n}\right\}.
$$

The quotient of $W$ and $\mathcal{P}_n$,
denoted $W/\mathcal{P}_n$, is the
complete graph on $n$ nodes
$$
W/\mathcal{P}_n =\langle [n], [n]\times [n], \bar W_n\rangle,
$$
such that weights $(\bar W_n)_{ij}$ are obtained by averaging
$W$ over the sets in $\mathcal{P}_n$
\begin{equation}\label{average}
(\bar W_n)_{ij} =n^2 \int_{I_i \times I_j} W(x,y) dxdy.
\end{equation}

The second sequence of weighted graphs is obtained in a  way that
is similar to the construction of $W$-random graph (cf. \cite{LovSze06})
\begin{equation}\label{graphH}
\mathbb{H}(S_n,W)=\langle [n], [n]\times [n], \tilde W_n\rangle, \;
(\tilde W_n)_{ij}=W\left({i\over n}, {j\over n}\right).
\end{equation}

In the remainder of this section, we prove convergence of the nonlinear
heat equations on $W/\mathcal{P}_n$ and $\mathbb{H}(S_n,W)$ to the continuum
equation on the graphon $W$ (cf.~(\ref{sheat})). Furthermore, we show that
the former problems correspond to the discretizations of (\ref{sheat})
using the method of Galerkin and the collocation method respectively,
thus, relating the problem of justification of the thermodynamic
limit for dynamical networks to two well-known numerical schemes for
equations of mathematical physics.

We first consider the IVP for the heat equation on $W/\mathcal{P}_n$
\begin{eqnarray}\label{ode}
{d\over dt} u_i^{(n)} (t) &=& n^{-1}\sum_{j=1}^n (\bar W_n)_{ij} D\left(  u^{(n)}_j(t)-u^{(n)}_i (t) \right), \\
\label{ode-ic}
u_i^{(n)}(0)&=&g^{(n)}_i, \; i\in [n],
\end{eqnarray}
where
$g^{(n)}_i$ is defined in (\ref{mean-g}).

By  associating the step function $u_n(x,t)$ with $u^{(n)}(t)$ (see (\ref{step-fun})), we rewrite
(\ref{ode}) and (\ref{ode-ic}) as
\begin{eqnarray}\label{finite-int}
{\partial \over \partial t} u_n(x,t) &=&\int_{I} W_n(x,y) D\left(u_n(y,t) -u_n(x,t)\right)dy,\\
\label{finite-int-ic}
u_n(x,0) &=& g_n(x),
\end{eqnarray}
where $W_n$ and $g_n$  are the step functions
\begin{eqnarray*}
W_n(x,y) &=& \bar{W}_{ij} \;\mbox{for} \; (x,y)\in I^{(n)}_i\times I^{(n)}_j,\\
g_n(x) &=& g^{(n)}_i, \;\mbox{for} \; x\in I^{(n)}_i.
\end{eqnarray*}

\begin{rem}\label{rem.Galerkin}
It is instructive to note that (\ref{ode}) and
(\ref{ode-ic}) can be viewed as the Galerkin
approximation of the IVP (\ref{sheat}) and (\ref{sheat-ic}).
Indeed, let $H_n$ denote a finite-dimensional subspace of $L^2(I)$
$$
H_n=\mbox{span}\{\mathbf{\phi}_1, \mathbf{\phi}_2,\dots, \mathbf{\phi}_n\},
$$
where $\mathbf{\phi}_i=\chi_{I^{(n)}_i}$ is the characteristic function of
$I_i^{(n)}=[(i-1)n^{-1}, in^{-1})$.

Replacing $u(x,t)$ in (\ref{sheat}) with
$$
u_n(x,t)=\sum_{k=1}^n u^{(n)}_k(t) \phi_k (x)\in H_n
$$
and projecting the resultant equation on $H_n$, we arrive at (\ref{ode}).
\end{rem}

\begin{thm}\label{Galerkin-convergence}
Let $\mathbf{u}$ and $\mathbf{u_n}$ be the solutions
of (\ref{sheat}), (\ref{sheat-ic}), and (\ref{finite-int}), (\ref{finite-int-ic}),
respectively. Suppose $W\in L^\infty(I^2)$ and $\mathbf{g}\in L^\infty (I)$.
Then
\begin{equation}\label{converge}
\| \mathbf{u} -\mathbf{u}_n\|_{C(0,T;L^2(I))} \to 0\;\mbox{as}\; n\to \infty.
\end{equation}
\end{thm}
{\it Proof.\;}\; By following the lines of the proof of Theorem~\ref{coloc-rate}
(see (\ref{Gron})),
for $\xi_n(x,t)=u_n(x,t)-u(x,t)$
we obtain
\begin{equation}\label{Gron2}
\sup_{t\in [0,T]}\|{\boldsymbol\xi}_n(t)\|_{L^2(I)}\le \left( \|\mathbf{g}-\mathbf{g}_n\|^2_{L^2(I)}
+{C_1\|W-W_n\|_{L^2(I^2)}\over C_2}\right) \exp\{C_2 T\},
\end{equation}
where positive constants $C_1$ and $C_2$ are  independent of $n$.
By the Lebesgue differentiation theorem,
$$
W_n\to W  \;\mbox{and} \; \mathbf{g}_n\to \mathbf{g},\;\mbox{as}\; n\to\infty,
$$
almost everywhere on $I^2$ and $I$ respectively. Thus, the statement of the theorem follows
from (\ref{Gron2}).\\
$\square$

The heat equation on $\mathbb{H}(X_n,W)$ is analyzed in complete analogy to
the IVP for $W/\mathcal{P}_n$. The IVP in this case remains (\ref{finite-int})
and (\ref{finite-int-ic}) modulo the definition of the step function
\begin{equation}\label{Wntilde}
W_n(x,y) = \tilde{W}_{ij} \;\mbox{for} \; (x,y)\in I^{(n)}_i\times I^{(n)}_j.
\end{equation}
We assume that $W(x,y)$ is a bounded symmetric measurable
function that is almost everywhere continuous on $I^2$. Then using
the observation in Lemma 2.5~\cite{BorChay11},
$$
W_n(x,y) \to W(x,y),\;\mbox{as}\; n\to\infty
$$
at every point of continuity of $W$, i.e., almost everywhere.
Thus, by the dominated convergence theorem, we have
$$
\|W-W_n\|_{L^2(I^2)}\to 0\;\mbox{as}\; n\to\infty.
$$
With this observation, the proof of Theorem~\ref{Galerkin-convergence}
applies to the situation at hand. Thus, we have the following theorem.

\begin{thm}\label{collocation-convergence}
Let $\mathbf{u}$ and $\mathbf{u_n}$ be the solutions
of (\ref{sheat}), (\ref{sheat-ic}), and (\ref{finite-int}), (\ref{Wntilde}),
(\ref{finite-int-ic}),
respectively. Suppose $W\in L^\infty(I^2)$, $\mathbf{g}\in L^\infty (I)$,
and $W$ is continuous almost everywhere on $I^2$.
Then
\begin{equation}\label{converge}
\| \mathbf{u} -\mathbf{u}_n\|_{C(0,T;L^2(I))} \to 0\;\mbox{as}\; n\to \infty.
\end{equation}
\end{thm}

\section{Examples} \label{sec.examples}
\setcounter{equation}{0}
In this section, we illustrate the results of this paper
with several examples. First, we apply Theorem~\ref{thm.reg} to explain
the regions of continuity in the chimera states \cite{KurBat02}. Next, we discuss the
attractors of the system of Kuramoto oscillators on multipartite
graphs.
\begin{figure}
\begin{center}
{\bf a}\;\includegraphics[height=1.8in,width=2.0in]{chimera-ic.pdf}\qquad
{\bf b}\;\includegraphics[height=1.8in,width=2.0in]{chimera-state-a.pdf}
\end{center}
\caption{
a) The initial conditions (\ref{chimera-ic}) for the chimera state shown in b).
b) A snapshot of the chimera state generated by (\ref{chimera}).
}
\label{f.2}
\end{figure}

\subsection{Regions of continuity of chimera states} \label{sec.chimera}
Chimera states are  persistent patterns of coexisting regions of
spatially coherent and chaotic behaviors (see Fig.~\ref{f.2}b).
They were discovered  by  Kuramoto and Battogtokh
in the following  continuum limit of a system of coupled  phase oscillators \cite{KurBat02}
\begin{equation}\label{chimera}
{\partial\over \partial t} \phi(x,t) = \omega +
\int_0^1 G(x-y) \sin\left(\phi(y,t)-\phi(x,t)+\alpha\right) dy.
\end{equation}
Function $\phi:~[0,1]\times{\mathbb R}^+\to {\mathbb S}^1:={\mathbb R}/2\pi{\mathbb Z}$ describes the evolution of the
phase of  oscillator at $x\in [0,1]$. The exponential kernel $G(x)=\exp\{-\kappa|x|\}$
provides nonlocal coupling
between oscillators. Equation (\ref{chimera}) was obtained using the phase reduction from the
Ginzburg-Landau equation, which describes collective dynamics of nonlocally  coupled limit cycle
oscillators (cf. \cite{KurBat02}). The sequences of discrete problems converging to (\ref{chimera})
can be obtained using one of the schemes of Section~\ref{sec.weight}.

The Kuramoto-Battogtokh model was the first example of a system featuring robust patterns that combine
coherent and irregular dynamics. Since then chimera states were demonstrated in a variety
of computational and experimental settings \cite{LaiKar12, Show12,LarPen13}.
The precise mathematical mechanism underlying
these patterns is the subject of ongoing research \cite{Ome13}. Here,
we focus on one aspect of the chimera states: the regions of
continuity.
Specifically, we use Theorem~\ref{thm.reg} to explain why the synchronous
dynamics is restricted to the two subdomains of $I$ (see Fig.~\ref{f.2}a). We show that
this possible because of the lack of the smoothening property of the
heat equation on graph limits, which is  one important
distinction from the classical heat equation.

The numerical generation of the chimera states in (\ref{chimera}) requires a careful setup,
which we review next.
To trigger a chimera state one has to start with the appropriate initial
conditions, otherwise oscillators end up evolving
in phase. Abrams and Strogatz reported that they were unable to generate chimera
states in (\ref{chimera}) from smooth initial conditions \cite{AbrStr06}. Instead, one has to initialize the
system with the initial condition  that combines
the regions of coherent and incoherent spatial profiles.
The  following initial condition was suggested by Kuramoto (cf. \cite{AbrStr06}):
\begin{equation}\label{chimera-ic}
\phi(x_i,0)=h(x_i) r_i, \; \mbox{where} \;
h(x)=6\exp\left\{ -30\left(x_i -(1/2)\right)^2\right\},\; x_i=in^{-1},\; i\in [n],
\end{equation}
and  $r_i$ are independent random variables drawn from the uniform distribution on $(-1/2, 1/2)$
(see Fig.~\ref{f.2}a).
The values of the other parameters are $\kappa=4$, $\alpha=1.457$ (cf. \cite{AbrStr06}).
 Numerical integration of
(\ref{chimera}) and (\ref{chimera-ic}) with these parameter values yields
persistent patterns with coexisting regions of spatially coherent and chaotic dynamics.
A representative snapshot is shown in Fig.~\ref{f.2}b.

Theorem~\ref{thm.reg} explains the role of the initial conditions in generating chimera
states.
Note that function $h(x)$ in (\ref{chimera-ic}) is rapidly decaying to $0$ outside a
neighborhood of $1/2$ . Therefore, the initial conditions
in the intervals $J_1=(0,0.2)$ and $J_2=(0.8, 1)$ near the endpoints of the interval $[0,1]$
for all practical purposes can be viewed if they were produced by discretization of a function
that is smooth over $J_1$ and $J_2$ (see Fig.~\ref{f.2}b). For such initial conditions,
Theorem~\ref{thm.reg} implies that the
solution $\phi(x,t)$ will remain continuous on $J_1$ and $J_2$, because $H^1(J_{1,2})\subset C(J_{1,2})$
by the Sobolev Embedding Theorem \cite{EvaPDE}. This  explains why the spatial profile remains coherent over
$J_1$ and $J_2$ for positive times (see Fig.~\ref{f.2}a).
Theorem~\ref{thm.reg} also implies  that it is impossible to generate chimera states starting
from smooth initial data, because for such data the solution of the continuum limit remains
continuous over the entire domain for all $t>0$. This rules out regions of chaotic
behavior in large networks, because their solutions remain close to that of the continuous
system by Theorem~\ref{Galerkin-convergence} or Theorem~\ref{collocation-convergence}.
This explains  failed attempts to produce chimera states from smooth
initial conditions  in \cite{AbrStr06}.

\subsection{The Kuramoto equation on multipartite graphs}\label{sec.half}

To illustrate our results for networks on simple graphs (see Section~\ref{sec.simple}),
we discuss the Kuramoto equation on multipartite graphs.
The examples of this subsection illustrate another implication of the
lack of smoothening property of the heat equation on graph limits. This time
we show that the lack of smoothness  of the limiting graphon  may result in stable
discontinuous patterns.

Consider the Kuramoto equation on the sequence of bipartite complete graphs
\begin{equation}\label{halfK}
\dot u_i^{(n)} (t) ={ (-1)^\sigma\over n} \sum_{j:~(j,i)\in E(K_{n,n})}
\sin\left( u_j^{(n)} (t)-u_i^{(n)} (t)\right),\quad i\in [2n],
\end{equation}
where
$$
K_{n,n}=\langle [2n], E(K_{n,n})\rangle,\;\;\mbox{and}\;\;
E(K_{n,n})=\{ (i,j)\in [n]^2:~ 1\le i\le n< j\le 2n\}.
$$
The sequence $\{K_{n,n}\}$ is  convergent with the limit shown in Fig.~\ref{f.3}a.
We consider two models for $\sigma=0$
and $\sigma=1$.
As shown below, the space homogeneous (synchronous) solution is stable
for the $\sigma=0$ model and is unstable if $\sigma=1$.

\begin{figure}
\begin{center}
{\bf a}\includegraphics[height=1.8in,width=2.0in]{f3a.pdf}
{\bf b}\includegraphics[height=1.8in,width=2.0in]{half-sync.pdf}
{\bf c}\includegraphics[height=1.8in,width=2.0in]{half-step.pdf}
\end{center}
\caption{
\textbf{a})The plot of support of $W_{K_{n,n}}$.
 \textbf{b,c}) Solutions of the IVP problem for the Kuramoto equation
on the bipartite complete graphs converge to the
synchronous solution for $\sigma=0$ ({\bf b}) and to the step function for
$\sigma=1$ ({\bf c}).
}
\label{f.3}
\end{figure}

Along with (\ref{halfK}) we consider its continuum limit
\begin{equation}\label{halfC}
{\partial \over \partial t} u(x,t) = (-1)^\sigma \int_I K(x,y) \sin\left( u(y,t)-u(x,t)\right) dy,
\end{equation}
where graphon $K\in\mathcal{W}_0$ is the limit of $\{K_{n,n}\}$ (see Fig.~\ref{f.3}a).
Suppose  $u(x,0)\in C(I)$. By Theorem~\ref{thm.reg}, for any $t>0$,
$u(x,t)\in \tilde C(I)$ where
$$
\tilde C(I)=\{ u\in L^\infty(I):~ \mbox{for any open interval}~ J\subset (0,1/2)
\cup (1/2,1)~ u\left|_J\right.\in C(J)\}.
$$
Here, by $u\left|_J\right.$ we denote the restriction of $u$ to $J$.

We look for steady state solutions of (\ref{halfC}) that belong to $\tilde C(I)$.
Setting the right hand side of (\ref{halfC}) to $0$, we obtain
\begin{eqnarray}\label{H1}
\int_{1/2}^1 \sin\left( u(y,t)-u(x,t)\right) dy=0, & x\in (0,1/2),\\
\label{H2}
\int_0^{1/2} \sin\left( u(y,t)-u(x,t)\right) dy=0, & x\in (1/2, 1).
\end{eqnarray}
From (\ref{H1}) and (\ref{H2}), we find that the only piecewise constant steady state solutions
from $\tilde C(I)$ are the space homogeneous
function
$$
u^{h}(x)=c,\;\mbox{for}\; x\in[0,1],
$$
and the step function
$$
u^s(x)=\left\{ \begin{array}{cc} c_1, & x\in [0,1/2),\\
                                               c_2, & x\in [1/2,1],
\end{array}
             \right.
$$
where constants $c,c_1,c_2\in {\mathbb S}^1$ and $|c_2-c_1|=\pi$.

Next, we turn to the discrete model (\ref{halfK}). The discrete counterparts of
$u^{s}(x,t)$ and $u^{h}(x,t)$ are
$$
u^s=c\mathbf{1}_{2n}\in {\mathbb R}^{2n} \;\mbox{and}\;
u^h=(c_1\mathbf{1}_{n}^\mathsf{T}, c_2\mathbf{1}_{n}^\mathsf{T})\in {\mathbb R}^{2n},
$$
where $\mathbf{1}_{n}=(1,1,\dots,1)^\mathsf{T}\in{\mathbb R}^n$.

The linearization of (\ref{halfK}) about $u=u^h$ yields
\begin{equation}\label{lin-half}
\dot \xi = { (-1)^{\sigma+1}\over n} \mathbf{L} \xi.
\end{equation}
Matrix $\mathbf{L}$ is the Laplacian of $K_{n,n}$
\begin{equation}\label{matrixL}
\mathbf{L}=\begin{pmatrix} nI_n & -J_n\\ -J_n & nI_n \end{pmatrix}, \;
\end{equation}
where $I_n$ is the $n\times n$ identity matrix and $J_n=\mathbf{1}_n \mathbf{1}_n^\mathsf{T}$.
As a graph Laplacian of an undirected connected graph, $\mathbf{L}$ is a symmetric
positive  semi-definite matrix with a simple eigenvalue $0$ \cite{Fied73}.
Thus, the space homogeneous solution $u^h$ is stable
for $\sigma=0$ and is unstable when $\sigma=1$.\footnote{The simple
zero eigenvalue in the spectrum of the linearized problem reflects
the translational invariance of (\ref{halfK}), which does not affect the
stability.}
The linearization of (\ref{halfK}) about $u^s$ yields
$$
\dot \xi ={(-1)^{\sigma+1}\over n}\mathbf{L}\xi,
$$
which, up to a sign, coincides with (\ref{lin-half}).
Thus, $u^s$ is unstable if $\sigma=0$ and is stable for $\sigma=1$.

The discrete model (\ref{halfK}) has many other piecewise constant
steady state solutions besides $u^h$ and $u^s$. But the latter are the only two
that approximate functions in $\tilde C(I)$ and, therefore, only these solutions
can be attractors of the discrete system for large $n$ (cf. Theorem~\ref{thm.reg}).
This  is consistent with the numerical simulations shown in Fig.~\ref{f.3}b,c.
Numerical experiments show that the synchronous state is the attractor for
the Kuramoto model with $\sigma=0$, while the step function is the attractor
for the model with $\sigma=1$ (see Fig.~\ref{f.3}b,c).

\begin{rem}\label{rem.half}
The  Kuramoto model on the family of half-graphs (cf.~Example~\ref{ex.half})
also exhibits exhibits stable step-like patterns, whose
analysis follows the lines  of that for the complete bipartite graphs.
\end{rem}
\begin{figure}
\begin{center}
{\bf a}\includegraphics[height=1.8in,width=2.0in]{multipartite-matrix.pdf}\qquad
{\bf b}\includegraphics[height=1.8in,width=2.0in]{multipartite-trace.pdf}
\end{center}
\caption{
{\bf a}) The block structure of $A(G_{nm})$.  {\bf b}) A stable multistep pattern
generated by the Kuramoto model on a multipartite graph.
}
\label{f.last}
\end{figure}
In conclusion, we briefly discuss how the Kuramoto model on $\{K_{n,n}\}$ can be generalized
to produce stable patterns with arbitrary number of steps.
To this end, let $C_n=\langle V(C_n), E(G_n)\rangle$ be an $n$-cycle, i.e.,
$V(C_n)=[n]$ and $E(C_n)=\{(i,j)\in [n]^2:~ \mbox{dist}(i,j)=1\}$.
Recall  $\mbox{dist}(i,j):=\min\{ |i-j|, n-|i-j|\}$.
The adjacency matrix of $C_n$ is given by
\begin{equation}\label{A(C)}
A(C_n)=\begin{pmatrix} 0 & 1&  0 &  \dots & 0&1\\ 1 &0 &  1 &  \dots
&0  &0\\ & & & \dots &  &\\ 1 & 0&  0 & \dots & 1& 0 \end{pmatrix}.
\end{equation}

Let $K_m$ denote the complete graph on $m$ nodes. Define graph
$C_{n,m}=C_n\otimes K_m$
on $nm$ nodes by replacing each node of $C_n$ with a copy of the complete graph
$K_m$. The adjacency matrix of the resultant graph
is the Kronecker product of $A(C_n)$ and $A(K_m)$
$$
A(C_{n,m})=A(C_n)\otimes A(K_m).
$$
The block structure of $A(C_{n,m})$ is shown in Fig.~\ref{f.last}a.

The Kuramoto model (\ref{halfK}) with $K_{n,n}$ replaced by $C_{n,m}$
generates stable patterns with $n$ steps like those shown in
Fig.~\ref{f.last}b.  In computational  neuroscience, such patterns
have been sought in the context of modeling memory.
The stability analysis of these multistep patterns, which can be done in analogy
to the analysis in this subsection, will be
presented elswhere.

\section{Conclusion}\label{sec.conclusion}
\setcounter{section}{0}
The heat equation is a fundamental equation of mathematical
physics. On Euclidean domains, the heat operator is used to model
phenomena involving diffusion, propagation, and pattern formation
in diverse problems of physics and biology. On Riemannian manifolds,  the
heat equation has been a powerful tool for studying the topology of the underlying
manifold \cite{Rosenberg-Laplacian}.  Its discrete counterpart, the heat equation
on graphs plays an important role in the spectral graph theory \cite{Chung-Spectral}.

Motivated by the dynamics  large networks, in this paper
we have studied the nonlinear heat equation on dense graphs.
We identified two classes of convergent graph sequences, for which the dynamics
of large coupled networks is approximated by the heat equation on the graph
limit. The latter is a nonlinear evolution equation with an integral operator
that describes nonlocal spatial interactions. The nonlocal heat equation differs
from its partial differential equation counterpart in several respects. First,
the IVP for the heat equation on a graph limit is well-posed in both forward
and backward time. Second, the solutions of the IVPs for the nonlocal heat equation
lack the smoothening property, i.e., the spatial regularity of solutions for positive times
is determined by the initial data and the regularity of the graph limit. In particular,
the heat equation on a graph limit can have attractors that are  piecewise continuous
in space (see Subsection~\ref{sec.half}), or  combine regions with qualitatively
distict dynamics like in chimera states (see Subsection~\ref{sec.chimera}).

Our analysis highlights the properties of the convergent graph sequences that
are necessary for supporting the continuum limit for coupled dynamical systems.
Note that for convergent sequences of simple graphs analyzed in Section~\ref{sec.simple},
we require that the graph limit is a $\{0,1\}$-valued graphon. For such sequences,
we are able to represent the discrete problems using the step functions $\{\hat W_{G_n}\}$
(cf.~(\ref{hat-W})), which are convergent in the $L^1$-norm. This construction does
not work for an arbitrary sequence of simple graphs. For instance,  a sequence
of Paley graphs converges to the constant graphon equal to $1/2$,
$\mbox{Const}~(1/2)$ \cite{BorChay08}.
However, the corresponding
continuum limit (\ref{sheat}) does not approximate the dynamics of the discrete problems.
On the other hand, the analysis in \cite{Med13a} shows that the heat equation on
the sequence of the Erd\H{o}s-R'{e}nyi graphs  (which is also a sequence of
simple albeit random graphs converging to $\mbox{Const}(1/2)$) has a well-defined
continuum limit. In contrast to the present work, the analysis of the continuum limit in
\cite{Med13a} does not rely on the $L^1$-norm for graphons, but effectively uses
the cut-norm.

Our results for networks on convergent sequences of simple graphs also reveal
what properties of  graphs affect the accuracy of the continuum limit.
Specficially, the rate of convergence estimate in Theorem~\ref{coloc-rate} shows that
the accuracy of approximation of the solutions of the discrete problems by their
continuous counterparts depends on the regularity of the boundary of support
of the graph limit. In particular, the convergence may slow down
significantly if the Hausdorff dimension of the boundary is close to $2$. It is interesting to compare this
result with the rate of convergence estimate for random networks in \cite{Med13a}.
For random networks, the rate is determined by the Central Limit Theorem and
is independent of the regularity of the underlying graphon.

The theory of graph limits provides a useful set of tools for studying dynamics
of large networks \cite{LovGraphLim12}. On  one hand, known graph limits for various
convergent sequences like that of half graphs or Erd\H{o}s-R\'{e}nyi graphs
suggest  continuum limits for the corresponding networks.
On the other hand, this rich theory offers many useful ideas and analytical
results that can be applied to the analysis dynamical networks. In this paper, we analyzed
two families of networks on convergent sequences of deterministic graphs.
In \cite{Med13a, Med13b} a similar approach is used to study networks on convergent
sequences of random graphs. Therefore, the results of this paper and in
\cite{Med13a} justify the continuum limit for a broad class of networks.

\vskip 0.2cm
\noindent
{\bf Acknowledgements.} The  author thanks A.~Grinshpan and
D.~Kaliuzhnyi-Verbovetskyi for useful discussions and
valuable comments on the manuscript. This work was supported in part by
the NSF grant DMS 1109367.

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  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}

redpajama_set_name: RedPajamaArXiv


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To best prepare for the rollover event, users of GPS equipment who are concerned should update their firmware, or contact their equipment manufacturer to ensure their equipment is ready for this event.
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Recent devices have likely been designed to handle the rollover event.
Older GPS receivers, however, or receivers that have not been provided manufacturer updates, may be affected by the rollover. The impact might occur in April, or could affect such equipment at a later date. The date might revert back to August 1999, or may revert to another date. Since this issue does not affect the other parts of the GPS navigation message (it only affects the date), the receiver's ability to calculate the position and to display the exact time of day should not be impacted.
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I am the Phazon, and the Monado, my name is PhazonMonado, overlord of the undead.
Nah, I'm just messing with you, nothing too bad is going to happen because of me.
The concept of "past, present, future" means nothing to me, it is all a single event.


text:
An A-Z of theory Althusser (part II)
In his latest 'In Theory' column, political theorist Andy Robinson introduces a new entry in his "A to Z of theory". This week: the second and final part on the French theorist Louis Althusser.
In Theory, New in Ceasefire - Posted on Friday, February 25, 2011 0:00 - 2 Comments
Tags: althusser, deleuze, Guattari, in theory, marx, philosophy
By Andrew Robinson
Ideological Apparatuses
Althusser locates the emergence of ideology in an account of the functioning of capitalism. As a social system, capitalism has to not only produce, but also reproduce its conditions of production, not simply in terms of each firm but across the system as a whole, to make it an endless chain. The point of view of reproduction is obscured by the obviousness of production. As well as reproducing means of production, capitalism needs to reproduce labour power – in other words, to keep turning human beings into 'workers', and to reproduce relations of production, meaning relations of exploitation (the subordination of workers to capitalists).
The reproduction of the relations of production is carried out mainly by ideology. The ISAs carry out this reproduction behind the "shield" provided by repression. The repressive apparatus is portrayed as forcing the ISAs to base their practices on the ideas and themes rendered dominant by the ruling class, and as acting against a residue of 'bad subjects' who have not been correctly subjectified by the ISAs. Between them, the ideological and repressive apparatuses sustain the reproduction of capitalism.
The autonomy of the infrastructures allows them to express clashes in the class struggle as well as expressing the imperatives of capital. In the last instance, however, the state uses repressive power to make sure the ISAs are functional. Think of measures such as OFSTED inspections and school closures, Parenting Orders, D-Notices, superinjunctions and so on, the Egyptian state shutdown of the internet, or the speed with which much of the internet lost its "neutrality" in the case of Wikileaks. Think, also, of periodic scares directed against certain Muslim clerics deemed not to reproduce the dominant system.
Althusser analyses ideology as the product of a particular set of social institutions, or "apparatuses". Capitalism is structured into three levels: the economic infrastructure and two superstructures, the political state and ideology. Althusser rather confusingly refers to all social institutions outside of direct economic production (including religion and the media) as State apparatuses. This follows from an unconventional use of the term 'state' derived from certain passages in Gramsci, and traceable to Hegel, in which "state" means something akin to "dominant social system" (rather than something closer to "government" or "public sector"). Althusser claims that apparently private institutions are part of the state because the public-private divide is itself internal to bourgeois law. Whether they are state apparatuses is defined by their function rather than their formal status.
State apparatuses can thus be divided into repressive and ideological state apparatuses. The political state is primarily repressive, using violence in the interests of capital (though it also uses ideology, for instance in the legal and party systems). Ideology is also for Althusser created by state apparatuses, but of a different kind from the repressive apparatuses, a kind which functions mainly by ideology (though also using repression in forms such as discipline).
The ideological state apparatuses (ISA's), unlike the repressive state apparatuses, appear to be distinct, autonomous institutions; they include institutions such as religion, education, the family, culture, trade unions, the media, the political party system, sport, and the legal system. While diffuse, they are unified by their subordination to ruling-class ideology – they are unified functionally rather than organisationally. They are also, however, sites of struggle, as their diffusion leaves them open to contestation. They vary in importance over time: whereas the Church was the main ISA in medieval Europe, in capitalism the main ISA is educational. In spite of this, their basic role is of a single kind: they promote ideology.
Social institutions are thus analysed functionally. The education system, for instance, teaches future workers the rules of the class system (such as submission to the boss), and skills used at work (such as literacy). It reproduces submission to the ruling ideology. Althusser seems especially to be thinking of the hidden curriculum, referring to the teaching of docility to students turned out at lower levels and arrogance to those turned out at higher levels. Today, the hidden curriculum is often in contradiction with the explicit content of education.
The school is treated as natural and neutral, and the majority of teachers have no idea that the system, 'which is bigger than they are and crushes them', is forcing or persuading them to do its work, often with all their heart. Only a few turn the school back against the system.
Althusser's theory of ideological apparatuses is, of course, controversial. Critics within sociology question whether these institutions perform an integrated ideological function. While there are reasons to be unhappy with a purely structural derivation of ideology, the functioning of institutions like the media and education as adjuncts of capitalism and the state is empirically beyond question. Keep in mind for instance the coverage of the student protests and the actions of Nottingham University as an adjunct of the security state.
That there is a 'dominant ideology' is fairly clear from the homogeneity of certain prevalent themes, for example the 'moral breakdown' discourse involved in moral panics. This said, it is unclear from Althusser's account how and why the apparatuses convey capitalist/statist ideology in spite of their organisational autonomy. What is needed beyond Althusser – and provided in part by authors such as Foucault, Illich and scholars of "news values" – is a clearer sense of how capitalist imperatives are translated into specific institutional functioning where no direct command relation exists. It seems that, in many cases, an organisation can pursue its own autonomous logic and function for capitalism, because it is applying standards of capitalist origin: performance and discipline criteria in schools, criteria of newsworthiness which encourage scaremongering, and so on.
One of the strange things about Althusser's theory is that, in relation to much of the welfare state it has almost become uncontroversial at the same time as being rejected. In Althusser's day, the only people arguing that universities were an extension of capitalism were Marxists. Today, the idea of such a link is openly professed by policymakers. It is as if ideology critique is no longer necessary – the system now declares openly what it's doing! It's almost as if capitalism has suddenly stopped pretending that schooling, university, families and so on should be about anything other than producing conformity to the system. Universities, schools, the benefit system and so on are openly treated as apparatuses for serving capitalism or inculcating capitalist values.
Yet there are other fields, notably the ideological legitimation of state repression, which are still very much concealed, even if it only takes a find-and-replace switching "dominant system" for "society" to reveal the ideological nature of such discourses. People who carry out authoritarian parenting or schooling, or support draconian anti-"crime" crackdowns, think they are preventing moral decay or interpersonal harms, not functioning as supplementary assemblages of capitalism.
In fact they are functioning as supplementary assemblages, responding to a sense of their own interpellation as capitalist subjects, but the transmission of commands or "hailings" from the system to its authoritarian supporters is largely unconscious: they directly feel the pull of the call for more discipline in families, the latent fear mobilised by the latest scare, the sense of being responsible for moral decay if they don't act in the prescribed way, and the desire to identify with the good subjects rather than feel the risk of being labelled among the bad ones.
Yet they don't realise this is what's happening, they think they are spontaneously arriving at such conclusions as a result of empirically perceived realities, that they are simply preparing their children for the world or protecting them from drugs or crime. Ideology is borne indirectly by the way such people perceive society, subjectivity, moral values, and so on. It is the reason they often say things they think are their own opinion, but which are copied directly from slogans in the media and politics.
This cannot be the whole account, however, because it does not explain why some people react in this way and some do not. Nor does it explain the overwhelming fear which drives people into authoritarian attachments. For this, Althusser would need to be supplemented with theorists such as Wilhelm Reich.
Determination and Autonomy
Does this mean that there is nothing but ideology, that everyone is necessarily trapped within ideology, and that ideology is inescapable? Does the theory of ISAs mean the economic base determines how institutions such as education operate? This is not exactly Althusser's position, but the articulation of his basic structuralism with other aspects of his theory is tenuous at best. For Althusser, ideology has no outside on its own terms, but from the viewpoint of science, it is nothing but outside. Science consists of a standpoint which is objective, and therefore escapes the contours of the formation of the subject. Althusser does not primarily mean empirical science, which he says gives rise to a "spontaneous ideology of the scientists", but to Marxist theory (if done in a structuralist and psychoanalytic way).
Revolutionary activity must be directed from science against ideology. Since the way people spontaneously think and see themselves is taken to be an effect of ideology, Althusser is opposed to ideas of self-activity which imply spontaneity or self-presence. This can give his theory an authoritarian bent. For instance, he is a big fan of the practice of 'self-criticism' in orthodox Leninist parties, through which people are required by the collective to renounce their past positions. Self-criticism is taken to transcent ideology towards science.
To complicate matters further, Althusser also sees ideology and the ISAs as fields of struggle. Sometimes, subordinate classes alter the functioning of ideology through such struggle. This suggests that subordinate classes have some source of agency (probably in their objective life-conditions) which sometimes counteracts the influence of ideology. Yet if people relate to their real conditions through the frame of ideology, it is unclear how such an alternative locus could emerge Then there is the question of 'bad subjects'. Most subjects function within the system by themselves, but some, 'bad subjects', are instead attacked by the repressive apparatuses, precisely because they have not been successfully subjectified. There must, therefore, be some point within the otherwise closed system at which interpellation can fail.
Indeed, Althusser doesn't really see the various social apparatuses as obeying the dictates of the economic base at all, though he also does not see them as entirely independent. This is expressed in the two concepts of 'relative autonomy' and 'determination in the last instance'. In general, Althusser treats the different levels as de facto autonomous, and believes influences occur between all the levels. Some of the superstructures are 'relatively' autonomous, and function by their own logics (they have, in Althusser's terms, their own 'essences').
But in the last instance, they are determined by the economic base, the 'determinant instance'. The repressive state apparatus is not autonomous as it has its own central command, but the absence of command integration among ISAs renders them relatively autonomous. Another aspect of relative autonomy is that the repressive apparatuses restrict the autonomous development of the ISAs (see above), though this does not explain how the former are kept in check by the base.
The view of relative autonomy and determination in the last instance is counterposed to the idea of a simple essence-appearance or dichotomy, in which the economy is the essence or 'truth' of the superstructures. It is counterposed to the view of the economic base as the sole determining factor. Althusser claims that the last instance predominates over time, but that it is never actually found in any single moment or instance (it 'never comes'). It always coexists with other forces.
The ideologies promoted through the ISAs, and those which oppressed groups manage to defend in or against them, 'come from elsewhere'; they are not born in the ISAs but in class struggle (which as a site of underlying antagonism, is arguably for Althusser equivalent to the Lacanian Real, the fundamental void created as a residue or ground of subject-formation). This would seem to suggest there is some force acting on individuals aside from their ideological subject-positions via the economic base, viewed by Althusser as conditions of existence, experiences of struggle, etc.
In relation to later discourse theorists such as Foucault and Spivak, this seems rather problematic: how can a person who experiences her or himself as an ideological subject, also experience struggle and life-conditions in a frame not resulting from this subjectivity? It is also possible that Althusser conceives of class struggle as an internal rather than external rupture: it expresses the contradiction within capitalism between the forces and relations of production.
This leaves it rather ambiguous how a revolutionary rupture could occur. In addition, given that it seems to be the integration of the overall system rather than the base in particular which integrates the system overall, the 'last instance' seems largely unnecessary to the overall theory. Why is the base still determinant in the last instance? One might suggest, rather, than ideological constructs actually determine transformations of economics and form the 'last instance' determining the base, or that reproduction is actually primary over production. It seems the system, as a functional totality, can be reconceived in any of these ways.
Revolution and Overdetermination
Althusser's theory of relative autonomy is also relevant to his theory of revolution. For Althusser, revolutionary breaks occur at the point where contradictions are most intense, as in Lenin's idea of the "weakest link". It is not just a question of the contradiction between capital and labour, but of the instability of the entire context, its ideological and repressive apparatuses for example. While the capital-labour contradiction is sufficient to create a generally revolutionary situation, it is not enough to bring about revolutions.
Revolutions come from a 'ruptural unity' of forces drawn from an accumulation of different circumstances and currents. These forces are heterogeneous, yet group themselves together as a kind of accumulation of contradictions. One might think here of an anti-capitalist protest, where ecologists, labour groups, anarchists and single-issue campaigners might be found in struggle together, or an event such as the upheavals in Egypt, where "leftists and liberals and Muslim Brotherhood discuss and sing and eat together".
The whole of a situation is reflected or concentrated in a vivid moment, as each specific contradiction or issue is displaced onto a single event. Such unification across causes has been frequently observed, and is often analysed by in terms of 'alliance' politics, political opportunities, or 'hegemony'. Althusser believes, however, that such unification is more than a marriage of convenience or even a common single cause. He believes that it is driven by the force of the underlying capital-labour contradiction and the resultant fact that a particular historical period is revolutionary.
In the summit protest wave, this might be conceived as the end of the Washington Consensus; in the North African wave, as the terminal crisis of a series of regimes born of post-war Arab nationalism, which have lost both their affirmative force as mobilising doctrines and their ability to maintain clientelist relations – the crisis of a particular regional insertion of capitalist power.
For Althusser, the diverse specific contradictions all derive ultimately from the capital-labour contradiction, but indirectly, through the relatively autonomous spheres: the superstructures, the relations of production, the international conjuncture and so on. Their unity in the revolutionary rupture reveals their real common basis or 'nature' as inseparable from the contradictions of the total situation. Specific issues are always 'overdetermined' by the last-instance contradiction.
The term 'overdetermination' is borrowed from psychoanalysis, and means that, even if something is sufficiently determined by its overt content, it also carries an excess energy derived from the latent content of the unconscious. It means that something can be explained in several ways, each of which would be sufficient in itself, but all of which are needed to make sense of it. For instance, the fact that someone is angry with their boss can be sufficiently explained by their actual work-conditions, and/or by their alienation in capitalism mapped onto its local agent, and/or by their projecting onto their boss an earlier rebellion against parental authority. Any one of the reasons would be enough, but all three are at work simultaneously.
'Overdetermination' is counterposed to simple determination (for instance, the view of superstructures as simply reflecting or determined by the base). Althusser rejects what he takes to be the Hegelian view of the social system as a closed system, an 'organic totality', in which a single internal principle is actualised in each of its specific institutions, or 'determinations'. The basic contradiction is always 'overdetermined' by the effects of various relatively autonomous fields.
The contradiction between capital and labour is not a simple contradiction, but necessarily goes by way of specific historical forms and circumstances, particularly those of the ideological and repressive apparatuses – the "superstructure", and the historical legacy and current formation of a particular national context and of the global situation (such as effects of uneven development) – the "national and international conjuncture". These fields are largely autonomous, with their own internal dynamics. Overdetermination can either block the expression of a contradiction or push it to a revolutionary rupture. Overdetermination has the effect of introducing a distance between the abstract case of a capital-labour conflict and all actual historical events. Hence, revolutions and other events are always 'exceptions', in relation to the abstract idea of a basic antagonism.
Furthermore, a seizure of state power or an economic transformation does not fully bring down the old order, because the ISAs and the conjuncture may remain the same as before. This said, the idea of an underlying contradiction expresses a certain unifying force which renders the combinations of issues non-arbitrary. In interpreting situations, Althusser assumes a certain essential unity of the situation around the "last instance". Difference can ultimately be reduced to sameness, though only with an awful lot of mediations. This is a major difference between Althusser's theory and those of related scholars such as Ernesto Laclau, for whom the unification of struggles is necessarily contingent.
The Althusserian theory of the 'last instance' is deeply problematic, in that this 'instance' seems to lack specificity relative to other forces. It is true, however, that revolts often concentrate different forces with different specific issues or demands. The unification is not entirely arbitrary, but amounts to recognition of the causality of the social system in relation to each of the problems. It is often the case that dissidents recognise some kind of dominant system which they are fighting against.
Yet Althusser goes too far in theorising this system as an integrated totality, treating it as a closed system with no outside except of its own creation (As argued above, in Althusser's theory, labour is not necessarily an "outside", but an internal contradiction, a bearer of the "forces of production" in excess over the "relations of production", or a kind of interior lack which resubjectifies people as bad subjects). This way of conceiving social power exaggerates drastically the ability of a dominant system to actually capture or reshape the full range of subjectivities and desires.
Althusser Today
What implications and problems arise from Althusser's theory in relation to the contemporary context? One of Althusser's most important contributions was to focus attention on the importance of reproduction in capitalism. The emphasis on reproduction is taken further by autonomous feminists and post-development theorists, and has inspired important developments in theorising the relationship of capitalism to emerging issues in feminism, ecology, post-development and other 'new social movements'.
The 60s rebellions were in many ways focused on resisting the reproduction of capitalism, forcing changes which loosened the rigidity of apparatuses such as the family. Capitalism was to initiate a backlash against these changes, yet certain effects are still being felt. Today there is a crisis of reproduction of labour-power, as wages have fallen below levels which sustain reproduction. There is a sustained population decline in core countries and global cities which is only counteracted by migration from marginal areas with subsistence economies. In short, capitalism seems to be failing to meet its own reproduction needs, relying on marginally-integrated "outsides" to do so.
One of the difficulties with synchronic theory is that it exaggerates the unity between different social logics within the dominant system. The role of the state tends to be simplified, because it is viewed as serving the needs of the capitalist system in general. It is unclear how the needs of capital are conveyed to the state, and it seems to simplify the state's own interests and dynamics to treat it in this way. It makes more sense to see the state as an autonomous social logic, but one which capitalism needs (or at least uses) to perform certain functions. This has unhelpful political implications. Not viewing the state as having its own social logic supports the view that the state can simply be seized and then transformed by oppressed classes.
If anything, the view that ideological apparatuses and subjectivities will persist "after the revolution" increases the role of the state in a post-revolutionary society. If the state turns out, however, to be irreducibly alienated as a social logic, its reproduction and expanded use will only intensify alienation, even if it destroys capitalism specifically.
Theorists drawing on Althusser also tend to have difficulty dealing with the status of what I term "reactive networks" – groups orienting to hierarchical ideologies, but often counterposed to the dominant system organisationally (e.g. ethnonationalist and religious movements). These groups often reproduce elements of the dominant ideology (particularly its authoritarian elements), but articulate such elements with discourses of resistance (for instance, sharp opposition to the current power-holders and the current composition of capital), and with an organisational form which has more in common with autonomous social movements than with states (though usually with a hierarchical core).
Althusserian theory seems to offer us three ways to conceptualise such movements – either they are simply ISA's (implausible given their antagonistic non-integration), or they are simply resistance movements (implausible given their reproduction of the dominant ideology), or they are terrains of contestation between the two (implausible given their ideological rigidity and the high degree of integration of the pro- and anti-system discursive tropes). In order to make sense of such movements, it is necessary to recognise that something can have a social form other than that of either the dominant system or its radical negation.
There is another worrying effect which may have arisen from Althusser's overly extensive view of the totalising power of the dominant system. Many of the Althusserians – such as Laclau and Castells – transmuted into Third Way authors in the 1980s. On the surface, this seems strange: of all the Western Marxist theorists, Althusser is one of the most sectarian. But perhaps Althusser's sectarianism was a way of warding off the conservative and managerial implications built into his theory. Althusser's theory could link to the Third Way, because of the way it assumes a priority of ideology over individuals and because of its elision of resistance.
If capitalism cannot be overthrown, and we cannot escape our interpellation as subjects, why not make the most of the situation by using one's structural awareness to play the system? Why not, indeed, use the mechanisms of interpellation and ideology to create a better-functioning version of the system with oneself at the top? Why not explicitly use the whole of society as a set of apparatuses led by the state, in order to realise capitalist goals such as national productivity? One need not be constrained in this by individual rights, since the individual is simply a delusional effect of ideology; nor should existing institutions or relations be sacrosanct, since they are simply apparatuses of power which can be assessed by their functionality. In other words, by taking Althusser and stripping out his belief in revolution, one is left with a project of managerial social engineering – which is exactly what Third Way theory amounts to.
Andrew Robinson is a political theorist and activist based in the UK. His book Power, Resistance and Conflict in the Contemporary World: Social Movements, Networks and Hierarchies (co-authored with Athina Karatzogianni) was published in Sep 2009 by Routledge. His 'In Theory' column appears every other Friday.
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Alain Badiou: The Excluded Part and the Evental Site | Ceasefire Magazine
Sep 24, 2014 12:38
[…] but is excluded from it for this very reason. This is a development of Althusser's idea of determination in the last instance. At this point, the occurrence of change seems to be a matter of chance for Badiou. Only the […]
Louis Althusser – On the reproduction of capital | onderwijs filosofie
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redpajama_set_name: RedPajamaCommonCrawl


Elias Nikolaus Ammerbach was a German organist and arranger of organ music of the Renaissance. The Passamezzo Antico refers to a chord progression that was popular during the Renaissance and is still in use today. Your brass quintet will enjoy this moderately challenging but very approachable interpretation of a set of dances originally written for the Renaissance organ.


The Secularization of Martin Luther King Jr.
Below is the fully printed text of the recent post: "The Secularization of Martin Luther King Jr.", written by Lee Habeeb. There was something funky going on with the original web link so I simply printed the article here for your convenience. Should you want to try and access the original link you can click here. This article was also referenced by the Washington Times. You can access the Times article by click here.
It's a story you won't hear anywhere this week as the life of Martin Luther King is celebrated: the story of the battle for the soul of a nation between King and his rival, Malcolm X. The battle between a Baptist preacher and a Nation of Islam disciple.
And if you listen carefully, what you'll hear is deafening silence about the one element of King's life without which none of his efforts would have been possible: his devotion to his faith and his God — his devotion to Jesus Christ.
Rev. Martin Luther King thought differently. Indeed, he found the Bible so compelling that his undergraduate degree was in Bible studies and his Ph.D. was in theology. To King, the Bible wasn't a software license. It was software code — a deep, mysterious code authored by God for man's eternal soul.
But that won't stop the media from redacting any and all references to the source of King's inspiration. You'll hear endless references to Dr. Martin Luther King this week — but never to Reverend King. The clips you'll hear, the videos you'll see, will be King's stirring secular rhetoric. What you will not hear are the parts of the speeches filled with references to God. Or the book from which sprang the source of this man's devotion to justice: the Bible.
You won't hear any mention of any of this in the media this week. That's how deep the antipathy toward all things Christian runs.
King didn't leave the church unscathed in this speech.
King was referring to the white churches in the South that did little to correct the injustice of racial segregation. But in the end, King understood that sin — and man coming up short of the calling of God — was not new. It's as old as the Old Testament.
In the week ahead, you will hear endless comments about King's courage. But he was courageous because he was faithful. He was great because he was godly. That's what made him so dangerous to segregationists. And that's why totalitarians in every era always go after believers first. They fear the power of God's love most of all.
It was love at the center of King's message. King always preached nonviolence, and always carried himself with dignity no matter the circumstances, and he did so because his God commanded it. He commanded that King love his neighbor as himself — all of his neighbors, not just those who treated him well.
Not everyone agreed with King's Christian approach. A young African-American Muslim named Malcolm X had a very different vision. A brilliant public speaker, Malcolm was a member of the Nation of Islam. He believed King's talk of love and mercy was weak, and often accused King of being an Uncle Tom.
What a story it would be for the media to tell, the two competing visions for the heart and soul of America: Martin's vs. Malcolm's. Luckily for all of us, Rev. Martin Luther King Jr.'s vision prevailed, and Malcolm X's failed.
You won't hear any of this on TV or the radio this week. The media will ignore all of King's God talk — the God he loved to talk about right up until his tragic death.
King was on fire in the speech, and the audience was, too. They didn't know — they couldn't know — he would be assassinated the very next day at the Lorraine Hotel.
Listen to the stories of Reverend Martin Luther King Jr. this week, and what you'll hear is the sound of a secular media hard at work stripping the animating spirit out of one of America's great men: the spirit of his Lord Jesus Christ.
The one enduring fact — the one enduring truth — the media can never rewrite, as hard as they might, is this: Martin Luther King Jr.'s desire to serve the God he loved changed forever the country he loved.
Lee Habeeb is VP of content for Salem Radio Network and host of "Our American Stories." He lives in Oxford, Mississippi, with his wife, Valerie, and his daughter, Reagan.


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