# Mean-field limits for non-linear Hawkes processes with inhibition on a
Erdős-Rényi-graph

by J. Stiefel
_Albert-Ludwigs University Freiburg_

###### Abstract

We study a multivariate, non-linear Hawkes process $Z^{N}$ on a $q$-Erdős-
Rényi-graph with $N$ nodes. Each vertex is either excitatory (probability $p$)
or inhibitory (probability $1-p$). If $p\neq\tfrac{1}{2}$, we take the mean-
field limit of $Z^{N}$, leading to a multivariate point process $\bar{Z}$. We
rescale the interaction intensity by $N$ and find that the limit intensity
process solves a deterministic convolution equation and all components of
$\bar{Z}$ are independent. The fluctuations around the mean field limit
converge to the solution of a stochastic convolution equation. In the critical
case, $p=\tfrac{1}{2}$, we rescale by $N^{1/2}$ and discuss difficulties, both
heuristically and numerically.

†† _AMS 2000 subject classification._ 60G55 (Primary) , 60F05 (Secondary).††
_Keywords and phrases._ Multivariate Hawkes process; Volterra equation; spike
train; Erdős-Rényi-graph

##  1 Introduction

In [14, 15], Hawkes processes were introduced as self-excitatory point
processes. Today, they are used in various fields of applications including
seismology [22, 11], interactions in social networks [28, 20], finance [6, 13]
and neuroscience [12, 17]. In the classical univariate, linear Hawkes process,
the firing rate at time $t$ is a linear function of
$I_{t}:=\sum_{i}\varphi(t-T_{i})$, where the $T_{i}$’s are previous jump
times. In this case, since rates cannot become negative, $\varphi\geq 0$ is
required, leading to a self-excitatory process.

Our main motivation to study Hawkes processes comes from the neurosciences. In
a graph, vertices model neurons, whereas the (directed) edges are synapses
linking the neurons. A point process indexed by the vertices models action
potentials or spike trains of electrical impulses. Communication via synapses
leads to correlated point processes such that each spike in one neuron
influences the rate by which a neighboring vertex fires. From this point of
view, the firing rate should include two important features. First, it is
known that neurons cannot only excite others, but inhibition is another
important factor (see e.g. [18]). This is why $\varphi\leq 0$ can occur as
well, and consequently the firing rate at time $t$ has to be a non-linear
function of $I_{t}$. Second, as not all neurons are connected, the firing rate
at some vertex should depend only on the spikes of connected vertices. The
Erdős-Rényi-model [7] is one of the first and simplest models for random
graphs, where the existence of edges between any two vertices is indicated by
independent Bernoulli random variables with common probability $q$. In [25],
mean-field limits for the multivariate, non-linear Hawkes process with
excitation and inhibition on a complete graph are derived. The main goal of
this paper is to generalize these limit results to a $q$-Erdős-Rényi-Graph
graph.

Nonlinear multivariate Hawkes processes have been studied to some extent in
the past decades, a summary can be found in the introduction of [25]. In a
standard mean-field setting, all components of the Hawkes process share the
same firing rate, see e.g. [4]. But even if models include features leading to
different rates at different vertices/neurons, the methods may be adapted to
derive mean-field results, see e.g. [5] for a multi-class setting. Other works
extend the standard firing rate of Hawkes processes by an age-dependence, i.e.
the rate at some vertex depends on the time since the last spike at this
vertex. It is still possible to derive mean-field limits [3], central limit
theorems [2], even in the critical case, where excitation and inhibition are
balanced [8].

In the present paper, we extend the model from [25] by a parameter $q$, which
denotes the fraction of open connections/edges between neurons/vertices. We
assume that each vertex/neuron is either excitatory or inhibitory, i.e.
excites or inhibits all of its connected neighbors. We will denote by $p$ the
fraction of excitatory vertices/neurons and distinguish the critical case
$p=1/2$ from the non-critical one. For the latter, we obtain in Theorem 1 a
classical mean-field result, i.e. by rescaling the interaction intensity by
$N$, a deterministic limit of the intensity (Theorem 1.1) and independent
point processes driven by this intensity (Theorem 1.2) arises. We also provide
a central limit result for the intensity (Theorem 1.3). In the critical case,
the methods from [25] cannot be applied. We describe the difficulties and
present simulation results to visualize features of possible mean-field
limits.

In many fields of applications, the requirement of a common connection
probability in the Erdős-Rényi model is too stringent, as edges or vertices
may have heterogeneous attributes. Some examples of Hawkes processes on more
complex graphs deal with estimation of the model parameters, [21], [24], [26],
or perform simulations, [27], both for a fixed size of the graph. In the
present paper we focus on the Erdős-Rényi model, as it allows to derive
rigorous mathematical results in the mean-field setting, i.e. when the size of
the graph tends to infinity. Of course, the generalisation of these results to
more realistic, and consequently more complex graph models is an interesting
topic for future research.

##  2 Model and assumptions

We use the following general model for a non-linear Hawkes process:

######  Definition 2.1 (Multi-variate, non-linear Hawkes process).

Let $\mathbb{G}=(\mathbb{V},\mathbb{E})$ be some finite, directed graph, and
write $ji\in\mathbb{E}$ if $j\to i$ is an edge in $\mathbb{G}$. Consider a
family of measurable, real-valued functions $(\varphi_{ji})_{ji\in\mathbb{E}}$
and a family of real-valued, non-negative functions
$(h_{i})_{i\in\mathbb{V}}$. Then, a point process $Z=(Z^{i})_{i\in\mathbb{V}}$
(with state space $\mathbb{N}_{0}^{\mathbb{V}}$) is a multi-variate non-linear
Hawkes process with interaction kernels $(\varphi_{ji})_{ji\in\mathbb{E}}$ and
transfer functions $(h_{i})_{i\in\mathbb{V}}$, if $Z^{i},Z^{j}$ do not jump
simultaneously for $i\neq j$, almost surely, and the compensator of $Z^{i}$
has the form $(\int_{0}^{t}\lambda_{s}^{i}ds)_{t\geq 0}$ with

$\lambda_{t}^{i}:=\lambda_{t}^{i}(Z_{s<t}):=h_{i}\Big{(}\sum_{j:ji\in\mathbb{E}}\int_{0}^{t-}\varphi_{ji}(t-s)dZ_{s}^{j}\Big{)},\qquad
i\in\mathbb{V}.$

We need some mimimal conditions such that the multivariate, non-linear Hawkes
process is well-defined (i.e. exists). As mentioned in [4], Remark 5, the law
of the non-linear Hawkes process is well-defined, provided that the following
assumption holds.

######  Assumption 2.2.

All interaction kernels $\varphi_{ji},ji\in\mathbb{E}$ are locally integrable,
and all transfer functions $(h_{i})_{i\in\mathbb{V}}$ are Lipschitz
continuous.

######  Remark 2.3 (Interpretation and initial condition).

  1. 1.

If $dZ_{s}^{j}=1$, we call $\varphi_{ji}(t-s)dZ_{s}^{j}$ the influence of the
point at time $s$ in vertex $j$ on vertex $i$.

  2. 2.

Consider the case of monotonically increasing transfer functions. If
$\varphi_{ji}\leq 0$, we then say that vertex $j$ inhibits $i$, since any
point in $Z^{j}$ decreases the jump rate of $Z^{i}$. Otherwise, if
$\varphi_{ji}\geq 0$, we say that $j$ excites $i$.

  3. 3.

In our formulation, we have $Z_{0}^{i}=0,i\in\mathbb{V}$, with the consequence
that the $dZ_{u}^{j}$-integral in (4.2) could also be extended to $-\infty$
without any change. We note that it would also be possible to use some initial
condition, i.e. some (fixed) $(Z_{t}^{i})_{i\in\mathbb{V},t\leq 0}$, and
extend the integral to the negative reals.

Let us now come to the mean-field model, where we fix some basic assumptions.
Note that we will show convergence for large graphs, i.e. all processes come
with a scaling parameter $N$, which determines the size of the graph.

######  Assumption 2.4 (Mean-field setting).

Let

  1. 1.

$\mathbb{G}_{N}=(\mathbb{V}_{N},\mathbb{E}_{N})$ be the $q$-Erdős-Rényi graph
on $N$ vertices, i.e. $\mathbb{V}_{N}=\\{1,...,N\\}$, and for independent
Bernoulli random variables $(V_{ji})_{j,i\in\mathbb{V}_{N}}$ with parameter
$q\in[0,1]$, $ji\in\mathbb{E}_{N}$ if and only if $V_{ji}=1$;

  2. 2.

$h_{i}=h$ for all $i\in\mathbb{V}_{N}$ where $h\geq 0$ is bounded, $h$ and
$\sqrt{h}$ are Lipschitz with constant $h_{Lip}$;

  3. 3.

$\varphi_{ji}=\theta_{N}U_{j}\varphi$ for all $j,i\in\mathbb{V}_{N}$, where
$\theta_{N}\in\mathbb{R}$ and

     * •

$U_{1},U_{2},...$ are iid with $\mathbb{P}(U_{1}=1)=1-\mathbb{P}(U_{1}=-1)=p$,

     * •

$\varphi\in\mathcal{C}_{b}^{1}([0,\infty))$, the set of bounded continuously
differentiable functions with bounded derivative.

The form of $\varphi_{ji}$ implies that node $j$ is exciting all other nodes
with probability $p$, and inhibiting all other nodes with probability $1-p$.
Additionally $V_{ji}$ indicates if there actually is a connection from node
$j$ to node $i$. Assumption 2.4 leads to the intensity

$\lambda_{t}^{i}=h\Big{(}\theta_{N}\sum_{j=1}^{N}\int_{0}^{t-}U_{j}V_{ji}\varphi(t-s)dZ_{s}^{j}\Big{)}$

at node $i\in\mathbb{V}_{N}$.

##  3 Results on the mean-field model

Our main goal is to give a limit result on the family $Z^{N,i}$, the
multivariate, non-linear Hawkes process on the graph $\mathbb{G}_{N}$ with
interaction kernels and transfer functions as given in Assumption 2.4.

###  3.1 The non-critical case

In the case $p\neq\tfrac{1}{2}$ the limit compensator is given by
$(\int_{0}^{t}h(I_{s})ds)_{t\geq 0}$, where $I$ is the weak limit of $I^{N,i}$
(see (3.1) and (3.2)). We have that $(I_{t})_{t\geq 0}$ follows a linear,
deterministic convolution equation, and all components of the limit of
$Z^{N,i}$ are independent (Theorem 1.2). The fluctiation around the limit
converges to a stochastic convolution equation, and the correlation of the
limiting fluctuations at different vertices depends on the connectivity $q$ of
the graph (Theorem 1.3). Below, we denote by $\Rightarrow$ weak convergence in
$\mathcal{D}_{\mathbb{R}^{n}}([0,\infty))$, the space of cadlag paths, which
is equipped with the Skorohod topology; see e.g. Chapter 3 in [10]. The proof
of the following result can be found in Section 4.4.

######  Theorem 1 (Mean-field limit of multi-variate non-linear Hawkes
processes, $p\neq\tfrac{1}{2}$).

Let Assumption 2.4 hold with $p\neq\tfrac{1}{2}$ and
$\theta_{N}=\tfrac{1}{N}$. Let $Z^{N}=(Z^{N,1},...,Z^{N,N})$ be the
multivariate, non-linear Hawkes process from Definition 2.1, and

$\displaystyle
I^{N,i}_{t}:=\frac{1}{N}\sum_{j=1}^{N}\int_{0}^{t-}U_{j}V_{ji}\varphi(t-s)dZ_{s}^{N,j}.$
(3.1)

  1. 1.

Then, $I^{N,i}\xrightarrow{N\to\infty}I$ almost surely, uniformly on compact
time intervals and uniformly in $i\leq N$, where $I=(I_{t})_{t\geq 0}$ is the
unique solution of the integral equation

$\displaystyle I_{t}=(2p-1)q\int_{0}^{t}\varphi(t-s)h(I_{s})ds.$ (3.2)

  2. 2.

For all $n=1,2,...$,
$(Z^{N,1},...,Z^{N,n})\xRightarrow{N\to\infty}(\bar{Z}^{1},...,\bar{Z}^{n}),$
where $\bar{Z}^{1},...,\bar{Z}^{n}$ are independent and $\bar{Z}^{i}$ is a
simple point process with intensity at time $t$ given by $h(I_{t})$,
$i=1,...,n$. It is possible to build $\bar{Z}^{1},...,\bar{Z}^{n}$, such that
the convergence is almost surely (in Skorohod-distance).

  3. 3.

Assume that $h\in\mathcal{C}^{1}(\mathbb{R})$ and that $h^{\prime}$ is bounded
and Lipschitz. Define $K^{N,k}=\sqrt{N}(I^{N,k}-I)$, the fluctuation around
the limit at vertex $k$ as well as the mean fluctuation,
$\overline{K}\,^{N}:=\frac{1}{N}\sum_{i=1}^{N}K^{N,i}$. Then, for all
$n=1,2,...$,

$\displaystyle\Big{(}\overline{K}^{N},K^{N,1},...,K^{N,n}\Big{)}\xRightarrow{N\to\infty}\Big{(}\overline{K},K^{1},...,K^{n}\Big{)}$

where $\overline{K}_{t}=\int_{0}^{t}\varphi(t-s)d\overline{G}_{s}$,
$K^{k}_{t}=\int_{0}^{t}\varphi(t-s)dG^{k}_{s}$ and

$\displaystyle\overline{G}_{t}$
$\displaystyle=q\Big{(}\int_{0}^{t}Wh(I_{s})+(2p-1)h^{\prime}(I_{s})\overline{K}_{s}ds+\int_{0}^{t}\sqrt{h(I_{s})}dB_{s}\Big{)},$
(3.3) $\displaystyle G^{k}_{t}$
$\displaystyle=\overline{G}_{t}+\sqrt{q(1-q)}\Big{(}\int_{0}^{t}\widetilde{W}^{k}h(I_{s})ds+\int_{0}^{t}\sqrt{h(I_{s})}d\widetilde{B}^{k}_{s}\Big{)}.$

Here, $B,\widetilde{B}^{1},...,\widetilde{B}^{n}$ are independent Brownian
motions, and $W,\widetilde{W}^{1},...,\widetilde{W}^{n}$ are independent
Normal distributions, $W\sim N(0,4p(1-p))$, $\widetilde{W}^{i}\sim N(0,1)$ for
each $i=1,...,n$.

######  Remark 3.1.

This generalizes [25, Theorem 1]. There it holds that $q=1$ and the
intensities at different vertices $I^{N,i}$ are the same. Obviously 1. and 2.
coincide with 1. and 2. from [25, Theorem 1]. In 3., observe that the
fluctuations around $I$ are the same at different vertices $k$,
$K^{N,k}=\overline{K}^{N}$, for each $k\in\mathbb{V}$. In (3.3) we have
$q(1-q)=0$, whence, for any $k\in\mathbb{V}$,

$\displaystyle
G^{k}_{t}=\overline{G}_{t}=\int_{0}^{t}Wh(I_{s})+(2p-1)h^{\prime}(I_{s})K^{k}_{s}ds+\int_{0}^{t}\sqrt{h(I_{s})}dB_{s}.$

This is equation (3.3) in [25].

######  Remark 3.2.

The Brownian motions $B,B^{1},...,B^{n}$ arise as limit of a rescaled sum over
compensated point processes, while the normal distributions
$W,\widetilde{W}^{1},...,\widetilde{W}^{n}$ arise as limit of rescaled sums
over the synaptic weights $(U_{j})_{j},(V_{ji})_{ji}$. See sections 4.2 and
4.3 for details.

######  Remark 3.3.

The form of (3.2) tells us that $I$ follows a linear Volterra convolution
equation, [1]. Turning into a differential equation, we write, using Fubini,

$\displaystyle\frac{dI}{dt}$
$\displaystyle=\frac{d}{dt}(2p-1)q\Big{(}\int_{0}^{t}\int_{s}^{t}\varphi^{\prime}(r-s)h(I_{s})drds+\int_{0}^{t}\varphi(0)h(I_{s})ds\Big{)}$
$\displaystyle=(2p-1)q\Big{(}\int_{0}^{t}\varphi^{\prime}(t-s)h(I_{s})ds+\varphi(0)h(I_{t})\Big{)}.$

In particular, the special choice of $\varphi(s)=e^{-\lambda s}$ gives

$\displaystyle\frac{dI}{dt}$
$\displaystyle=-\lambda(2p-1)q\Big{(}\int_{0}^{t}\varphi(t-s)h(I_{s})ds+h(I_{t})\Big{)}=-\lambda
I_{t}+(2p-1)qh(I_{t})),$

i.e. $I$ follows some ordinary differential equation in this case.

While Theorem 1 is concerned with convergence of the limit intensity of the
multivariate, non-linear Hawkes process, we are also in the situation to study
convergence of the average intensity of $Z^{N,i}$. The proof of the next
corollary is found in Section 4.4.

######  Corollary 3.4.

Let $Z^{N}=(Z^{N,1},...,Z^{N,N})$ be as in Theorem 1, and
$\bar{Z}=(\bar{Z}^{1},\bar{Z}^{2},...)$ be as in Theorem 1.2. Then,

$\displaystyle\frac{1}{N}\sum_{j=1}^{N}\Big{(}Z^{N,j}-\int_{0}^{.}h(I^{N,j}_{s})ds\Big{)}\xrightarrow{N\to\infty}0$
(3.4) and
$\displaystyle\frac{1}{N}\sum_{j=1}^{N}\Big{(}Z^{N,j}-\bar{Z}^{j}\Big{)}\xrightarrow{N\to\infty}0$
(3.5)

in probability, uniformly on compact intervals. Moreover,

$\displaystyle\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\Big{(}Z^{N,j}-\int_{0}^{.}h(I^{N,j}_{s})ds\Big{)}\xRightarrow{N\to\infty}\int_{0}^{.}\sqrt{h(I_{s})}dB^{0}_{s}$
(3.6)

for some Brownian motion $B^{0}$, and (writing $h(I):=(h(I_{t}))_{t\geq 0}$)

$\displaystyle\sqrt{N}\big{(}h(I^{N,i})-h(I)\big{)}\xRightarrow{N\to\infty}h^{\prime}(I)K^{i}.$
(3.7)

Let $B^{U},B^{0}$ be correlated Brownian motions with
$\mathbf{E}[B^{U}_{t}B^{0}_{t}]=(2p-1)t,t\geq 0$. Use $B^{U}$ in the
definition of $\overline{K}$ in Theorem 1.3. Then,

$\displaystyle\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\Big{(}Z^{N,j}-\int_{0}^{.}h(I_{s})ds\Big{)}\xRightarrow{N\to\infty}\int_{0}^{.}h^{\prime}(I_{s})\overline{K}_{s}ds+\int_{0}^{.}\sqrt{h(I_{s})}dB^{0}_{s}.$
(3.8)

######  Remark 3.5 (Correlation between $B^{0}$ and $B$).

Let us briefly discuss the correlated Brownian motions appearing in (3.8).
Clearly, the left hand side of (3.8) can be built from the left hand sides of
(3.6) and (3.7) by taking the mean value over the vertices $i$ in (3.7). The
limits $\int_{0}^{.}\sqrt{h(I_{s})}d\widetilde{B}^{0}_{s}$ and $\overline{K}$
appearing on the right hand sides of (3.6) and (3.7) are weak limits of sums
of compensated point processes. While in (3.6), we sum over all point
processes in the system, $I^{N,i}$ in (3.7) distinguishes between nodes with
different signs $U_{j}$. Hence, the correlation is positive for the proportion
$p$ of point processes with positive sign, and negative for the proportion
$1-p$ of point processes with negative sign, summing to $p-(1-p)=2p-1$. For
more details, see Lemma 4.4.

###  3.2 The critical case

For $p=\frac{1}{2}$, excitation and inhibition are balanced. If we rescale the
interaction kernels with $\theta_{N}=\frac{1}{N}$ as in Theorem 1, we can read
off

$\frac{1}{N}\sum_{j=1}^{N}\int_{0}^{t-}U_{j}V_{ji}\varphi(t-s)dZ_{s}^{j,N}\to
0$

from Theorem 1.1, and the limiting point processes $\bar{Z}^{i}$ have constant
intensity $h(0)$ (Theorem 1.2). In this critical case is natural to upscale by
$\sqrt{N}$ in order to obtain non-trivial limits. This has been done under
assumption 2.4, but with $q=1$, in [25, Theorem 2], and for a similar model in
[8], [9]. After upscaling, the intensity at vertex $i$ and time $t$ is given
by $h(I^{N,i}_{t})$, where

$\displaystyle
I^{N,i}_{t}=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\int_{0}^{t-}U_{j}V_{ji}\varphi(t-s)dZ_{s}^{N,j}.$

In order to obtain limit results we compensate $dZ_{t}^{N,j}$ by
$h(I^{N,j})_{t}dt$ and apply a martingale central limit theorem. More
precisely

$\displaystyle I^{N,i}_{t}$
$\displaystyle=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}V_{ji}\Big{(}\int_{0}^{t-}\varphi(t-s)dZ_{s}^{N,j}-\int_{0}^{t}\varphi(t-s)h(I^{N,j}_{s})ds\Big{)}$
(3.9)
$\displaystyle+\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}V_{ji}\int_{0}^{t}\varphi(t-s)h(I^{N,j}_{s})ds.$

The predictable quadratic covariation of the first line is given by
$\frac{1}{N}\sum_{j=1}^{N}V_{ji}\int_{0}^{t}\varphi(t-s)h(I^{N,j}_{s})ds$, and
we a-priori need a limit result on this covariation as well as the second line
in (3.9) to derive a limit of $I^{N,i}$. This is feasible in [25, Theorem 2],
where $q=1$ and therefore $V_{ji}=1$, i.e. the intensities $I^{N,i}$ at
different vertices $i$ coincide. In contrast, [8] derive a mean-field limit
for Hawkes processes with different intensities at different vertices. A
significant difference to our model is that the influence of vertex $j$ on all
other vertices, $U_{j}$, is not fixed over time, i.e. whenever
$dZ_{t}^{N,j}=1$, $U_{j}(t)$ are centered random variables, independent over
the jump times $t$. Consequently, the process

$\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\int_{0}^{t-}U_{j}(s)dZ^{N,j}_{s}\quad\text{is
a martingale with
covariation}\quad\frac{\sigma^{2}}{N}\sum_{j=1}^{N}\int_{0}^{t}h(I^{N,j}_{s})ds,$

where $\sigma^{2}$ is the second moment of $U_{j}$. It suffices to a-priori
derive a result on the limit of the empirical distributions of intensities at
different vertices, where the exchangeability of the system can be used, to
derive a limit of $I^{N,i}$. Compared to this approach, we face two major
difficulties in our model: First, in the covariation of the compensated
process we have to deal with the empirical distribution over the subset of
vertices connected to vertex $i$. Here, one could first try to focus on the
average over different vertices $i$ to replace $V_{ji}$ by its mean $q$.
Second, and more challenging, we would need an a-priori result on the
compensator $\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}V_{ji}dh(I^{N,j}_{t})$, i.e.
an a-priori CLT-type result for the intensities at different vertices.
Simulation
We may simulate a multivariate Hawkes-process on a finite, but large graph
using Lewis’ thinning algorithm [19], [23]. Assume the graph consists of
$N=500$ neurons and choose the parameters $p=q=0.5$. For simplicity, choose am
exponential interaction kernel, $\varphi(t)=e^{-\lambda t}$ for some
$\lambda>0$, and transfer function $h(x)=1+\tfrac{2}{\pi}\arctan(x)$. Note
that $\varphi$ and $h$ satisfy assumption 2.4. As the interaction kernel is
exponential, the Intensity at vertex is given by

$\displaystyle I^{N,i}_{t}$
$\displaystyle=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}V_{ji}Z_{t-}^{N,j}-\lambda\int_{0}^{t}I^{N,i}_{s}ds$
$\displaystyle=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}V_{ji}\Big{(}Z_{t-}^{N,j}-\int_{0}^{t}h(I^{N,j}_{s})ds\Big{)}+\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}V_{ji}\int_{0}^{t}h(I^{N,j}_{s})ds-\lambda\int_{0}^{t}I^{N,i}_{s}ds.$

We split up the martingale part,

$\displaystyle M^{i}$
$\displaystyle:=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}V_{ji}\Big{(}Z_{t-}^{N,j}-\int_{0}^{t}h(I^{N,j}_{s})ds\Big{)}$
$\displaystyle=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}(V_{ji}-q)\Big{(}Z_{t-}^{N,j}-\int_{0}^{t}h(I^{N,j}_{s})ds\Big{)}+q\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}\Big{(}Z_{t-}^{N,j}-\int_{0}^{t}h(I^{N,j}_{s})ds\Big{)}$
$\displaystyle=:\widetilde{M}^{i}+M.$

Figure 1: Approximation of the covariance $\langle M^{k},M^{k}\rangle$ in
(3.10), from left to right.

As the jump size of the martingale part tends to zero, the convergence is
determined by the predictable covariation process. Our simulations suggest
that

$\displaystyle\langle M^{k},M^{l}\rangle$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}(V_{jk}-q)(V_{jl}-q)h(I^{N,j})\approx\frac{1}{N}\sum_{j=1}^{N}(V_{jk}-q)(V_{jl}-q)\overline{h(I)}\approx
1_{k=l}q(1-q)\overline{h(I)},$ (3.10) $\displaystyle\langle M^{k},M\rangle$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}(V_{jk}-q)h(I^{N,j})\approx\frac{1}{N}\sum_{j=1}^{N}(V_{jk}-q)\overline{h(I)}\approx
0,$ $\displaystyle\langle M,M\rangle$ $\displaystyle\approx
q^{2}\overline{h(I)},$

where the approximation in the first line is visualized in figure 1. By
application of a standard martingale central limit theorem, this implies

$\displaystyle\widetilde{M}^{i}_{t}$
$\displaystyle\approx\sqrt{q(1-q)}\int_{0}^{t}\sqrt{\overline{h(I_{s})}}dB^{i}_{s},\quad\text{
as well as }\quad M\approx q\int_{0}^{t}\sqrt{\overline{h(I_{s})}}dB_{s}$
(3.11)

where $B,B^{1},...,B^{N}$ are independent Brownian motions and
$\overline{h(I)}=\frac{1}{N}\sum_{j=1}^{N}h(I^{N,j})$. This should be compared
to Theorem 1.3 and lemma 4.4, where we obtain a similar convergence result.
There, note that $\overline{h(I)}\approx h(I)$.

(a) Drift

(b) Martingale part

Figure 2: Drift and martingale part of the intensity at vertex 0, 1,
respectively.

As described above, the drift
$\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}V_{ji}\int_{0}^{t}h(I^{N,j}_{s})ds$ is
more difficult to analyse, even numerically. We expect that the value of the
drift depends on the configuration of the graph in some way. To see how it
depends on the rescaled input of the graph to node $i$,
$W^{N,i}=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}V_{ji}$, manually set

$\sum_{j=1}^{N}V_{j0}=\sum_{j=1}^{N}V_{j1}=\frac{N}{2},\sum_{j=1}^{N}V_{j0}V_{j1}=0,$

i.e. the vertices $0,1$ receive input from complementary parts of the graph.
Then choose $U_{1},...,U_{N}=\pm 1$, such that $\sum_{j}U_{j}\approx 0$ and
$W^{N,0}=W^{N,1}$, i.e. excitation and inhibition are balanced and the mean
input to node 0 and 1 are the same. As precicted above, the difference of the
martingale parts $M^{0}$ and $M^{1}$ from the mean martingale part are
negatively correlated , see figure 2b, as
$\frac{1}{N}\sum_{j=1}^{N}(V_{j0}-q)(V_{j1}-q)=-q(1-q)$.

But the drifts at vertices 0 and 1 significantly differ from each other,
figure 2a. We expect the drift to depend on higher powers of the adjacency
matrix $V=(V_{ij})_{j,i=1,...,N}$, together with multiplication with
$U=(U_{i})_{i=1,...,N}$ and applications of the necessarily non-linear
transfer function $h$. A precise analysis of this dependence in the mean-field
setting $N\to\infty$ is very complex.

##  4 Proofs

We start off in Subsection 4.1 with a result on convolution equations. We
proceed in 4.2 with a reformulation of the multivariate linear Hawkes process
using a time-change equation, and in 4.3 with results on different types of
mean values of the synaptic weights. Then, we prove Theorem 1 in Subsection
4.4.

###  4.1 Convolution lemma

In order to bound the value of a convolution equation by its integrator we
need the following

######  Lemma 4.1.

Let $J$ be the sum of an Itô process with bounded coefficients and a càdlàg
pure-jump process, and let $\varphi\in\mathcal{C}_{b}^{1}([0,\infty))$. Then

$\displaystyle\sup_{0\leq s\leq
t}\Big{(}\int_{0}^{s}\varphi(s-r)dJ_{r}\Big{)}^{2}\leq
C(\varphi,t)\cdot\sup_{0\leq s\leq t}J_{s}^{2}.$ (4.1)

###### Proof.

Wlog, we have $J_{0}=0$. By [1, Theorem 4.A] and Fubini’s theorem for
Lebesgue-integrals we can apply the Stochastic Fubini Theorem to $J$, hence

$\displaystyle\sup_{0\leq s\leq
t}\Big{(}\int_{0}^{s}\varphi(s-r)dJ_{r}\Big{)}^{2}$ $\displaystyle=\sup_{0\leq
s\leq
t}\Big{(}\int_{0}^{s}\Big{(}\varphi(0)+\int_{r}^{s}\varphi^{\prime}(s-u)du\Big{)}dJ_{r}\Big{)}^{2}$
$\displaystyle\leq||\varphi||\cdot\sup_{0\leq s\leq t}J_{s}^{2}+\sup_{0\leq
s\leq
t}\Big{(}\int_{0}^{s}\int_{0}^{u}\varphi^{\prime}(s-u)dJ_{r}du\Big{)}^{2}$
$\displaystyle=||\varphi||\cdot\sup_{0\leq s\leq t}J_{r}^{2}+\sup_{0\leq s\leq
t}\Big{(}\int_{0}^{s}\varphi^{\prime}(s-u)J_{u}du\Big{)}^{2}$
$\displaystyle\leq(||\varphi||+t||\varphi^{\prime}||)\cdot\sup_{0\leq s\leq
t}J_{s}^{2}.$

∎

###  4.2 Reformulation of Hawkes processes

Alternative descriptions of non-linear Hawkes processes have been given in the
literature. Above all, the construction using a Poisson random measures is
widely used; see e.g. Proposition 3 in [4]. Here, we rely on the following
construction using time-change equations (see e.g. Chapter 6 of [10]), which
we give here without proof.

######  Lemma 4.2.

Let $\mathbb{G}=(\mathbb{V},\mathbb{E})$, $(\varphi_{ji})_{ji\in\mathbb{E}}$
and $(h_{i})_{i\in\mathbb{V}}$ be as in Definition 2.1, and let Assumption 2.2
hold. A point process $Z=(Z^{i})_{i\in\mathbb{V}}$ is a multivariate, non-
linear Hawkes process with with interaction kernels
$(\varphi_{ji})_{ji\in\mathbb{E}}$ and transfer functions
$(h_{i})_{i\in\mathbb{V}}$, if and only if it is the weak solution of the
time-change equations

$\displaystyle
Z_{t}^{i}=Y_{i}\Big{(}\int_{0}^{t}\lambda_{s}^{i}ds\Big{)}=Y_{i}\Big{(}\int_{0}^{t}h_{i}\Big{(}\sum_{j:ji\in\mathbb{E}}\int_{0}^{s-}\varphi_{ji}(s-u)dZ_{u}^{j}\Big{)}ds\Big{)},$
(4.2)

where $(Y_{i})_{i\in\mathbb{V}}$ is a family of independent unit rate Poisson
processes.

Under Assumption 2.4, the time-change equations from this lemma read, with
independent unit rate Poisson processes $Y_{1},...,Y_{N}$,

$\displaystyle Z_{t}^{N,i}$
$\displaystyle=Y_{i}\Big{(}\int_{0}^{t}h\Big{(}\sum_{j=1}^{N}\int_{0}^{s}\theta_{N}U_{j}V_{ji}\varphi(s-u)dZ_{u}^{N,j}\Big{)}ds\Big{)}.$
(4.3) We rewrite this as $\displaystyle Z_{t}^{N,i}$
$\displaystyle=Y_{i}\Big{(}\int_{0}^{t}h(I_{s}^{N,i})ds\Big{)}$ (4.4) with
$\displaystyle I_{s}^{N,i}$
$\displaystyle=\int_{0}^{s}\varphi(s-u)dJ_{u}^{N,i}\quad\text{ and }\quad
J_{u}^{N,i}=\theta_{N}\sum_{j=1}^{N}U_{j}V_{ji}Z_{u}^{N,j}=\theta_{N}\sum_{j=1}^{N}U_{j}V_{ji}Y_{j}\Big{(}\int_{0}^{\cdot}h(I_{s}^{N,j})ds\Big{)}.$
(4.5)

To obtain convergence results, we introduce the compensated point processes

$\displaystyle X^{N,i}_{t}$
$\displaystyle:=\sum_{j=1}^{N}U_{j}V_{ji}\Big{(}Y_{j}\Big{(}\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}-\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}.$

In Theorem 1.1 we show convergence uniformly in the vertices of the graph,
hence we need a result on convergence of $\frac{1}{N}X^{N,i}$, uniform in
$i\leq N$.

######  Lemma 4.3.

It holds that

$\displaystyle\sup_{i\leq N}\sup_{0\leq s\leq
t}\frac{1}{N}\big{|}X^{N,i}_{s}\big{|}\to 0$ (4.6)

almost surely (and in $L^{2}$).

###### Proof.

The sixth centered moment of a Poisson random variable with parameter
$\lambda$ is of order $\lambda^{3}$. For each $i$, we may split $X^{N,i}$ up
into its exciting and inhibiting part,

$\displaystyle X^{N,i}_{t}=\sum_{j:U_{j}=1}V_{ji}$
$\displaystyle\Big{(}Y_{j}\Big{(}\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}-\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}$
$\displaystyle-\sum_{j:U_{j}=-1}V_{ji}\Big{(}Y_{j}\Big{(}\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}-\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}.$

This is the difference of two compensated point processes with intensity
bounded by $N||h||$, hence the sixth moment of $X^{N,i}_{t}$ is at most of
order $(tN||h||)^{3}$. We obtain, using Doob’s martingale inequality for the
martingale $X^{N,i}$,

$\displaystyle\mathbb{E}\Big{[}\sup_{i\leq N}\sup_{0\leq s\leq
t}\big{(}\frac{1}{N}X^{N,i}_{s}\big{)}^{6}\Big{]}$
$\displaystyle\leq\sum_{i=1}^{N}\mathbb{E}\Big{[}\sup_{0\leq s\leq
t}\big{(}\frac{1}{N}X^{N,i}_{s}\big{)}^{6}\Big{]}$
$\displaystyle\leq\Big{(}\frac{6}{5}\Big{)}^{6}\sum_{i=1}^{N}\mathbb{E}\Big{[}\big{(}\frac{1}{N}X^{N,i}_{t}\big{)}^{6}\Big{]}=O(N^{-2}),$

which is summable. Using Borel-Cantelli, almost-sure convergence follows. ∎

In Theorem 1.3, we scale up by $\sqrt{N}$ and investigate on the correlation
structure at different vertices, hence we need a result on joint convergence
of $\big{(}\frac{1}{\sqrt{N}}X^{N,i}\big{)}_{i=1,...,n}$. Therefore we
introduce the processes

$\displaystyle X^{N,0}_{t}$
$\displaystyle=\sum_{j=1}^{N}\Big{(}Y_{j}\Big{(}\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}-\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)},$
$\displaystyle X^{N,U}_{t}$
$\displaystyle=\sum_{j=1}^{N}U_{j}\Big{(}Y_{j}\Big{(}\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}-\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)},$
$\displaystyle\widetilde{X}^{N,i}_{t}$
$\displaystyle=\sum_{j=1}^{N}U_{j}(V_{ji}-q)\Big{(}Y_{j}\Big{(}\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}-\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)},$
and split $\displaystyle X^{N,i}_{t}$
$\displaystyle=\widetilde{X}^{N,i}_{t}+qX^{N,U}_{t},\text{ for }i=1,2,...$

Note that only the first summand in the latter equation depends on the vertex
$i$.

######  Lemma 4.4.

Assume $\sup_{i\leq N}\sup_{s\leq t}|I^{N,i}_{s}-I_{s}|\to 0$ almost surely.
Then

$\displaystyle\frac{1}{\sqrt{N}}X^{N}=\frac{1}{\sqrt{N}}\big{(}X^{N,U},X^{N,0},\widetilde{X}^{N,1},...,X^{N,n}\big{)}\xRightarrow{N\to\infty}\big{(}M^{U},M^{0},\widetilde{M}^{1},...,\widetilde{M}^{n}\big{)}=:M,$

where $M^{U},M^{0},\widetilde{M}^{1},...,\widetilde{M}^{n}$ are local
martingales with covariance structure given by

$\displaystyle d\langle\widetilde{M}^{k},\widetilde{M}^{l}\rangle_{t}$
$\displaystyle=1_{k=l}q(1-q)h(I_{t})dt,\qquad$ $\displaystyle d\langle
M^{U},M^{U}\rangle_{t}=d\langle M^{0},M^{0}\rangle_{t}=h(I_{t})dt,$ (4.7)
$\displaystyle d\langle\widetilde{M}^{k},M^{U}\rangle_{t}$
$\displaystyle=d\langle\widetilde{M}^{k},M^{0}\rangle_{t}=0,$ $\displaystyle
d\langle M^{U},M^{0}\rangle_{t}=(2p-1)h(I_{t})dt.$

for any $k,l\neq 0$. We can extend the probability space such that the
convergence is almost surely and in $L^{2}$, uniformly on compact time
intervals.

We further extend the probability space by Brownian motions
$B^{U},B^{0},\widetilde{B}^{i}$, such that

$M^{U}_{t}=\int_{0}^{t}\sqrt{h(I_{s})}dB^{U}_{s},\quad
M^{0}_{t}=\int_{0}^{t}\sqrt{h(I_{s})}dB^{0}_{s},\quad\widetilde{M}^{k}_{t}=\int_{0}^{t}\sqrt{q(1-q)h(I_{s})}d\widetilde{B}^{k}_{s}.$

Necessarily, $\langle\widetilde{B}^{k},\widetilde{B}^{l}\rangle_{t}=1_{k=l}t$,
$\langle B^{U},B^{0}\rangle_{t}=(2p-1)t$ and
$\langle\widetilde{B}^{k},B^{U}\rangle_{t}=\langle\widetilde{B}^{k},B^{0}\rangle_{t}=0$
for $k,l=1,...,n$. Finally observe that, on this probability space,

$\displaystyle\frac{1}{\sqrt{N}}X^{N,k}\to\widetilde{M}^{k}+qM^{U}=q\int_{0}^{\cdot}\sqrt{h(I_{s})}B^{U}_{s}+\sqrt{q(1-q)}\int_{0}^{\cdot}\sqrt{h(I_{s})}\widetilde{B}^{k}_{s},$
(4.8)

almost surely and in $L^{2}$ by the continuous mapping theorem.

###### Proof of Lemma 4.4.

The result is a martingale central limit theorem, hence we need to identify
the covariance structure. By [16, Theorem I.4.52],

$\displaystyle\big{[}\frac{1}{\sqrt{N}}\widetilde{X}^{N,k},\frac{1}{\sqrt{N}}\widetilde{X}^{N,l}\big{]}_{t}$
$\displaystyle=\sum_{s\leq t}\sum_{j=1}^{N}\Delta
Z^{N,j}_{s}\frac{1}{\sqrt{N}}U_{j}(V_{jk}-q)\frac{1}{\sqrt{N}}U_{j}(V_{jl}-q)$
$\displaystyle=\sum_{j=1}^{N}Z^{N,j}_{t}\frac{1}{\sqrt{N}}U_{j}(V_{jk}-q)\frac{1}{\sqrt{N}}U_{j}(V_{jl}-q).$

The compensator of $Z^{N,j}$ is given by $\int_{0}^{\cdot}h(I^{N,j}_{s})ds$,
whence, by [16, Proposition I.4.50 b)],

$\displaystyle\big{\langle}\frac{1}{\sqrt{N}}\widetilde{X}^{N,k},\frac{1}{\sqrt{N}}\widetilde{X}^{N,l}\big{\rangle}_{t}$
$\displaystyle=\sum_{j=1}^{N}\frac{1}{\sqrt{N}}U_{j}(V_{jk}-q)\frac{1}{\sqrt{N}}U_{j}(V_{jl}-q)\int_{0}^{t}h(I^{N,j}_{s})ds$
$\displaystyle\xrightarrow{N\to\infty}1_{k=l}q(1-q)\int_{0}^{t}h(I_{s})ds.$

Analogously we obtain

$\displaystyle\langle\frac{1}{\sqrt{N}}\widetilde{X}^{N,U},\frac{1}{\sqrt{N}}\widetilde{X}^{N,U}\rangle_{t}$
$\displaystyle=\sum_{j=1}^{N}\frac{1}{\sqrt{N}}U_{j}\frac{1}{\sqrt{N}}U_{j}\int_{0}^{t}h(I^{N,j}_{s})ds\to\int_{0}^{t}h(I_{s})ds$
$\displaystyle\langle\frac{1}{\sqrt{N}}\widetilde{X}^{N,0},\frac{1}{\sqrt{N}}\widetilde{X}^{N,U}\rangle_{t}$
$\displaystyle=\sum_{j=1}^{N}\frac{1}{\sqrt{N}}\frac{1}{\sqrt{N}}U_{j}\int_{0}^{t}h(I^{N,j}_{s})ds\to(2p-1)\int_{0}^{t}h(I_{s})ds$
$\displaystyle\langle\frac{1}{\sqrt{N}}\widetilde{X}^{N,0},\frac{1}{\sqrt{N}}\widetilde{X}^{N,0}\rangle_{t}$
$\displaystyle=\sum_{j=1}^{N}\frac{1}{\sqrt{N}}\frac{1}{\sqrt{N}}\int_{0}^{t}h(I^{N,j}_{s})ds\to\int_{0}^{t}h(I_{s})ds$
$\displaystyle\langle\frac{1}{\sqrt{N}}\widetilde{X}^{N,k},\frac{1}{\sqrt{N}}\widetilde{X}^{N,0}\rangle_{t}$
$\displaystyle=\sum_{j=1}^{N}\frac{1}{\sqrt{N}}U_{j}(V_{jk}-q)\frac{1}{\sqrt{N}}\int_{0}^{t}h(I^{N,j}_{s})ds\to
0$
$\displaystyle\langle\frac{1}{\sqrt{N}}\widetilde{X}^{N,k},\frac{1}{\sqrt{N}}\widetilde{X}^{N,U}\rangle_{t}$
$\displaystyle=\sum_{j=1}^{N}\frac{1}{\sqrt{N}}U_{j}(V_{jk}-q)\frac{1}{\sqrt{N}}U_{j}\int_{0}^{t}h(I^{N,j}_{s})ds\to
0.$

The desired convergence in distribution follows from [16, Theorem VIII.3.8,
b)(ii)$\to$(i)]: $[\sup-\beta_{5}]$ holds as the first characteristic is 0,
and $[\hat{\delta}_{5}-\mathbb{R}]$ is obvious as the jumps of
$\widetilde{X}^{N,k},X^{N,U},X^{N,0}$ have size $1/{\sqrt{N}}$. Finally,
condition $[\gamma_{5}-\mathbb{R}]$ is precisely the convergence of the
predictable quadratic covariation above.
By Skorohod’s Theorem, we can extend our probability space such that this
convergence is almost surely with respect to Skorohod distance, and by
continuity of the limit it is equivalent to local uniform convergence. For
$L^{2}$ convergence, observe that $X^{N,U},X^{N,0},\widetilde{X}^{N,k}$ are
compensated point processes with intensity bounded by $||h||$. Let $Y$ be a
unit rate Poisson process, then uniform integrability follows from the
computation

$\displaystyle\mathbb{E}\Big{[}\sup_{0\leq s\leq
t||h||}\Big{(}\frac{1}{\sqrt{N}}(Y(Ns)-Ns)-B_{s}\Big{)}^{4}\Big{]}^{1/4}$
$\displaystyle\leq
C\mathbb{E}\Big{[}\Big{(}\frac{1}{\sqrt{N}}(Y(Nt)-Nt)-B_{t}\Big{)}^{4}\Big{]}^{1/4}$
$\displaystyle\leq
C\Big{(}\mathbb{E}\Big{[}\Big{(}\frac{1}{\sqrt{N}}(Y^{\pm}(Nt)-Nt)\Big{)}^{4}\Big{]}^{1/4}+\mathbb{[}(B_{t})^{4}]^{1/4}\Big{)}$
$\displaystyle=C((3t^{2}+t/N)^{1/4}+(3t^{2})^{1/4}),$

where we used Doob’s and Minkovski’s inequalities. ∎

###  4.3 Results on the synaptic weights

In this section we collect results on different types of mean values of
$U_{j},V_{ji}$. We start with laws of large numbers, which we need to derive
mean-field limits.

######  Lemma 4.5.

It holds that

$\displaystyle\sup_{j\leq N}|\frac{1}{N}\sum_{i=1}^{N}V_{ji}-q|$
$\displaystyle\xrightarrow{N\to\infty}0,$ (4.9) $\displaystyle\sup_{j\leq
N}|\frac{1}{N}\sum_{i=1}^{N}U_{i}V_{ji}-(2p-1)q|$
$\displaystyle\xrightarrow{N\to\infty}0,$ (4.10) $\displaystyle\sup_{j\leq
N}|\frac{1}{N}\sum_{i=1}^{N}U_{i}V_{ji}V_{ik}-(2p-1)q^{2}|$
$\displaystyle\xrightarrow{N\to\infty}0,$ (4.11)

almost surely.

###### Proof.

For (4.9), observe that the sixth central moment of a binomial distribution
with parameters $N$ and $p$ is of order $N^{3}$, whence

$\displaystyle\mathbf{P}\Big{(}\sup_{j\leq
N}|\frac{1}{N}\sum_{i=1}^{N}V_{ji}-q|>\varepsilon\Big{)}$
$\displaystyle\leq\frac{1}{\varepsilon^{6}}\sum_{j\leq
N}\mathbf{E}\Big{(}\frac{1}{N}\sum_{i=1}^{N}V_{ji}-q\Big{)}^{6}$ (4.12)
$\displaystyle=\frac{1}{\varepsilon^{6}N^{5}}E\Big{(}\sum_{i=1}^{N}V_{1i}-q\Big{)}^{6}=O(N^{-2}),$

is summable. Uniform convergence follows from the Borel-Cantelli Lemma.
Similarly in (4.10) and (4.11). ∎

In order to derive results on the fluctuation around the mean-field limit, we
need the following central limit theorems. First define
$W^{N}=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}(U_{j}-(2p-1))$ and
$\widetilde{W}^{N,i}=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}(V_{ji}-q)$, such
that

$\displaystyle
W^{N,i}:=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}(U_{j}V_{ji}-(2p-1)q)=qW^{N}+\widetilde{W}^{N,i}.$

######  Lemma 4.6.

For any $n\in\mathbb{N}$ it holds that

$\displaystyle\big{(}W^{N},\widetilde{W}^{N,1},...,\widetilde{W}^{N,n}\big{)}\xRightarrow{N\to\infty}\big{(}W,\sqrt{q(1-q)}\widetilde{W}^{1},...,\sqrt{q(1-q)}\widetilde{W}^{n}\big{)},$
(4.13)

where $W\sim N(0,4p(1-p))$, $\widetilde{W}^{i}\sim N(0,1)$ and
$W,\widetilde{W}^{1},\widetilde{W}^{2},...$ are independent. Furthermore, for
any $i\in\\{1,...,n\\}$,

$\displaystyle
W^{N,i}\xRightarrow{N\to\infty}qW+\sqrt{q(1-q)}\widetilde{W}^{i},$ (4.14)

as well as

$\displaystyle\overline{W}^{N}$
$\displaystyle:=\frac{1}{N}\sum_{i=1}^{N}W^{N,i}=qW^{N}+\frac{1}{N}\sum_{i=1}^{N}\widetilde{W}^{N,i}\xRightarrow{N\to\infty}qW,$
(4.15) $\displaystyle\overline{W}^{N,U}$
$\displaystyle:=\frac{1}{N}\sum_{i=1}^{N}U_{i}W^{N,i}=qW^{N}\frac{1}{N}\sum_{i=1}^{N}U_{i}+\frac{1}{N}\sum_{i=1}^{N}U_{i}\widetilde{W}^{N,i}\xRightarrow{N\to\infty}(2p-1)qW,$
(4.16) $\displaystyle\overline{W}^{N,UVk}$
$\displaystyle:=\frac{1}{N}\sum_{i=1}^{N}U_{i}V_{ik}W^{N,i}=qW^{N}\frac{1}{N}\sum_{i=1}^{N}U_{i}V_{ik}+\frac{1}{N}\sum_{i=1}^{N}U_{i}V_{ik}\widetilde{W}^{N,i}\xRightarrow{N\to\infty}(2p-1)qW,$
(4.17)

for any $k\in\mathbb{N}$.

###### Proof.

First, (4.13) follows from the central limit theorem, as

$\displaystyle\mathbf{E}[(U_{j}-(2p-1))^{2}]$ $\displaystyle=4p(1-p),$
$\displaystyle\mathbf{E}[U_{j}^{2}(V_{ji}-q)^{2}]=\mathbf{E}[(V_{ji}-q)^{2}]$
$\displaystyle=q(1-q),$

and the summands of $W^{N},\widetilde{W}^{N,1},...,\widetilde{W}^{N,n}$,
respectively, are uncorrelated. Then (4.14) follows by definition of $W^{N}$
and the continuous mapping theorem. For ,Next observe that, for any
$i,j,k,l,m\in\\{1,...,N\\}$, $k\neq l$,

$\displaystyle\mathbf{E}[U_{i}(V_{ik}-q)U_{j}(V_{jl}-q)]$ $\displaystyle=0,$
and hence $\displaystyle\mathbf{E}[\widetilde{W}^{N,k}\widetilde{W}^{N,l}]$
$\displaystyle=0,$
$\displaystyle\mathbf{E}[U_{k}\widetilde{W}^{N,k}U_{l}\widetilde{W}^{N,l}]$
$\displaystyle=0,$
$\displaystyle\mathbf{E}[U_{k}V_{km}\widetilde{W}^{N,k}U_{l}V_{lm}\widetilde{W}^{N,l}]$
$\displaystyle=0.$

Consequently the second summand in (4.15), (4.16), (4.17), respectively,
converges to $0$ in probability by the weak law of large numbers. The
convergence of the first summands follows from convergence of $W^{N}$ and
lemma 4.5 ∎

Assume from now on that we work on a probability space where $W^{N}\to W$
almost surely. Then the convergence in (4.15), (4.16), (4.17) is in
probability, and we obtain the following

######  Lemma 4.7.

Assume $W^{N}\to W$ almost surely. Then

$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\big{(}W^{N,i}\big{)}^{2}\to_{p}q^{2}W^{2}+q(1-q),$
(4.18)

where $\to_{p}$ denotes convergence in probability.

###### Proof.

We decompose

$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\big{(}W^{N,i}\big{)}^{2}$
$\displaystyle=q^{2}\underbrace{\big{(}W^{N}\big{)}^{2}}_{\to_{as}W^{2}}+q\underbrace{W^{N}}_{\to_{as}W}\underbrace{\frac{1}{N}\sum_{i=1}^{N}\widetilde{W}^{N,i}}_{\to_{p}0}+\underbrace{\frac{1}{N}\sum_{i=1}^{N}\big{(}\widetilde{W}^{N,i}\big{)}^{2}}_{\to_{p}q(1-q)}\to_{p}q^{2}W^{2}+q(1-q).$
(4.19)

The convergence $\frac{1}{N}\sum_{i=1}^{N}\widetilde{W}^{N,i}\to 0$ is in
probability, as $(\widetilde{W}^{N,i})_{i}$ are uncorrelated. For the
convergence in the last summand, observe that
$\big{(}\widetilde{W}^{N,i}\big{)}^{2}$ are independent given $(U_{j})_{j}$.
Denote by $\mathbf{E}_{U}$ conditional expectation with respect to
$(U_{j})_{j}$. As
$\mathbf{E}_{U}\big{[}\big{(}\widetilde{W}^{N,i}\big{)}^{2}\big{]}=q(1-q)$, it
holds that

$\displaystyle\mathbf{E}_{U}\Big{[}\frac{1}{N}\sum_{i=1}^{N}\big{(}\widetilde{W}^{N,i}\big{)}^{2}-q(1-q)\Big{]}\to
0,$

almost surely. It follows that

$\displaystyle\mathbf{E}_{U}\Big{[}\Big{(}\frac{1}{N}\sum_{i=1}^{N}\big{(}\widetilde{W}^{N,i}\big{)}^{2}-q(1-q)\Big{)}\wedge
1\Big{]}\to 0,$

and, by dominated convergence, the desired convergence in probability,

$\mathbf{E}\Big{[}\Big{(}\frac{1}{N}\sum_{i=1}^{N}\big{(}\widetilde{W}^{N,i}\big{)}^{2}-q(1-q)\Big{)}\wedge
1\Big{]}\to 0.$

∎

###  4.4 Proof of Theorem 1 and Corollary 3.4

###### Proof of Theorem 1.

Proof of 1.:
Define

$\displaystyle J_{t}=(2p-1)q\int_{0}^{t}h(I_{s})ds,\text{ such that
}I_{t}=\int_{0}^{t}\varphi(t-s)dJ_{s}.$ (4.20)

For $\theta_{N}=\tfrac{1}{N}$ we get that,

$\displaystyle J_{t}^{N,i}-J_{t}$ $\displaystyle=A^{N,i}+C^{N,i}+D^{N,i},$
where $\displaystyle A^{N,i}_{t}$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}U_{j}V_{ji}\Big{(}Y_{j}\Big{(}\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}-\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}$
$\displaystyle C^{N,i}_{t}$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}U_{j}V_{ji}\int_{0}^{t}h(I_{s}^{N,j})-h(I_{s})ds$
$\displaystyle D^{N,i}_{t}$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}\big{(}U_{j}V_{ji}-(2p-1)q\big{)}\int_{0}^{t}h(I_{s})ds$

We first show that $\sup_{s\leq
t}\frac{1}{N}\sum_{j=1}^{N}(J_{s}^{N,i}-J_{s})^{2}\to 0$ almost surely.
Observe that

$\displaystyle\sup_{s\leq
t}\frac{1}{N}\sum_{i=1}^{N}\big{(}J_{s}^{N,i}-J_{s}\big{)}^{2}$
$\displaystyle\leq\frac{3}{N}\sum_{i=1}^{N}\Big{(}\sup_{s\leq
t}(A^{N,i}_{s})^{2}+\sup_{s\leq t}(C^{N,i}_{s})^{2}+\sup_{s\leq
t}(D^{N,i}_{s})^{2}\Big{)}$ (4.21)

By Lemma 4.3, $\sup_{s\leq t}(A^{N,i}_{s})^{2}\to 0$ uniforlmly in $i\leq N$.
By boundedness of $h$ it follows from (4.10) that $\sup_{i\leq N}\sup_{s\leq
t}D^{N,i}_{s}\to 0$ almost surely. For $C^{N,i}$ we obtain, using Jensen’s
inequality twice and $U_{j}^{2}=1$, $V_{ji}^{2}\leq 1$,

$\displaystyle\sup_{s\leq
t}\frac{1}{N}\sum_{i=1}^{N}\Big{(}C^{N,i}_{s}\Big{)}^{2}$
$\displaystyle\leq\sup_{s\leq
t}\frac{1}{N}\sum_{i=1}^{N}\frac{1}{N}\sum_{j=1}^{N}\Big{(}\int_{0}^{t}h(I_{s}^{N,j})-h(I_{s})ds\Big{)}^{2}$
(4.22) $\displaystyle\leq
t\frac{1}{N}\sum_{j=1}^{N}\int_{0}^{t}\big{(}h(I_{s}^{N,j})-h(I_{s})\big{)}^{2}ds$
$\displaystyle\leq
th_{Lip}^{2}\frac{1}{N}\sum_{j=1}^{N}\int_{0}^{t}\big{(}I_{s}^{N,i}-I_{s}\big{)}^{2}ds$
$\displaystyle\leq Cth_{Lip}^{2}\int_{0}^{t}\sup_{u\leq
s}\frac{1}{N}\sum_{j=1}^{N}(J_{u}^{N,i}-J_{u})^{2}ds,$

where we have used Lemma 4.1 in the last step. Combining the results on
$A^{N,i},C^{N,i}$ and $D^{N,i}$ in (4.21) we obtain

$\displaystyle\sup_{s\leq
t}\frac{1}{N}\sum_{j=1}^{N}\big{(}J_{s}^{N,i}-J_{s}\big{)}^{2}\leq
3Cth_{lip}^{2}\int_{0}^{t}\sup_{u\leq
s}\frac{1}{N}\sum_{j=1}^{N}\big{(}J_{u}^{N,i}-J_{u}\big{)}^{2}ds+o(1),$

hence $\sup_{s\leq t}\frac{1}{N}\sum_{i=1}^{N}(J_{s}^{N,i}-J_{s})^{2}\to 0$ by
Gronwall’s inequality. Now fix $i$ and repeat the estimation in (4.22) to
obtain

$\displaystyle\sup_{s\leq t}\Big{(}C^{N,i}_{s}\Big{)}^{2}$
$\displaystyle\leq\sup_{s\leq
t}\frac{1}{N}\sum_{j=1}^{N}\Big{(}\int_{0}^{s}h(I_{u}^{N,j})-h(I_{u})ds\Big{)}^{2}$
$\displaystyle\leq Cth_{Lip}^{2}\int_{0}^{t}\sup_{u\leq
s}\frac{1}{N}\sum_{j=1}^{N}(J_{u}^{N,j}-J_{u})^{2}ds\to 0$

almost surely. As the right hand side does not depend on $i$, this convergence
is uniform in $i\leq N$. As we have already shown $\sup_{i\leq N}\sup_{s\leq
t}(A^{N,i}_{s})^{2}\to 0$ as well as $\sup_{i\leq N}\sup_{s\leq
t}(D^{N,i}_{s})^{2}\to 0$, we obtain

$\displaystyle\sup_{i\leq N}\sup_{s\leq
t}\big{(}J_{s}^{N,i}-J_{s}\big{)}^{2}\leq 4\Big{(}\sup_{i\leq N}\sup_{s\leq
t}(A^{N,i}_{s})^{2}+\sup_{i\leq N}\sup_{s\leq t}(C^{N,i}_{s})^{2}+\sup_{i\leq
N}\sup_{s\leq t}(D^{N,i}_{s})^{2}\Big{)}\to 0$

almost surely. By Lemma 4.1, we can conclude that $\sup_{i\leq N}\sup_{s\leq
t}\big{(}I^{N,i}_{s}-I_{s}\big{)}^{2}\to 0$ almost surely.

Proof of 2.:
Define $\bar{Z}^{i}_{t}=Y_{i}\big{(}\int_{0}^{t}h(I_{s})ds\big{)}$, where
$Y_{1},...,Y_{N}$ are independent Poisson process as in (4.4). Fix
$\omega\in\Omega$, such that

$\sup_{i\leq N}\sup_{0\leq s\leq t}|I_{s}^{N,i}(\omega)-I_{s}|\rightarrow 0.$

As $h$ is Lipschitz,
$\int_{0}^{t}h(I_{s}^{N,i}(\omega))ds\rightarrow\int_{0}^{t}h(I_{s})ds$, hence
for any point of continuity of $t\mapsto\bar{Z}^{i}_{t}(\omega)$ we can
conclude $Z^{i}_{t}(\omega)\rightarrow\bar{Z}^{i}_{t}(\omega)$. As the points
of continuity are dense in $[0,\infty)$, convergence in Skorohod-distance
follows from [16, Theorem 2.15 c)(ii)].

Proof of 3.:
First, strong existence and uniqueness of $\overline{K}$ follows from [1]. For
the convergence result, we work on a probability space where the convergence
in Lemma 4.4 and in (4.13) is in probability. We proceed in three steps:

Step 3.1:
$\frac{1}{N}\sum_{i=1}^{N}U_{i}\sqrt{N}(I^{N,i}-I)\to(2p-1)\overline{K}$
Use $B^{U}$ from Lemma 4.4 in the definition of $\overline{K}$ and define
$\overline{K}^{U}:=(2p-1)\overline{K}$. Then
$\overline{K}^{U}=\int_{0}^{t}\varphi(t-s)d\overline{G}^{U}_{s}$ with

$\displaystyle\overline{G}^{U}=\int_{0}^{t}(2p-1)qWh(I_{s})+(2p-1)qh^{\prime}(I_{s})\overline{K}^{U}_{s}ds+(2p-1)q\int_{0}^{t}\sqrt{h(I_{s})}dB^{U}_{s}.$

We show $\frac{1}{N}\sum_{i=1}^{N}U_{i}\sqrt{N}(I^{N,i}-I)\to\overline{K}^{U}$
in probability, uniformly on compact time intervals. Therefore write,

$\displaystyle\frac{1}{N}\sum_{i=1}^{N}U_{i}\sqrt{N}(J^{N,i}_{t}-J_{t})-\overline{G}^{U}_{t}=A^{N}_{t}+C^{N}_{t}+D^{N}_{t}+E^{N}_{t}+F^{N}_{t}$
with $\displaystyle A^{N}_{t}$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}U_{i}\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}V_{ji}\Big{(}Y_{j}\Big{(}\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}-\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}-(2p-1)q\int_{0}^{t}\sqrt{h(I_{s})}dB^{U}_{s},$
$\displaystyle C_{t}^{N}$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}U_{i}\frac{1}{N}\sum_{j=1}^{N}U_{j}V_{ji}\int_{0}^{t}\sqrt{N}\big{(}h(I_{s}^{N,j})-h({I}_{s})-(I_{s}^{N,j}-{I}_{s})h^{\prime}(I_{s})\big{)}ds,$
$\displaystyle D_{t}^{N}$
$\displaystyle=\Big{(}\frac{1}{N}\sum_{i=1}^{N}U_{i}\frac{1}{\sqrt{N}}\sum_{j=1}^{N}(U_{j}V_{ji}-(2p-1)q)-(2p-1)qW\Big{)}\int_{0}^{t}h(I_{s})ds,$
$\displaystyle E_{t}^{N}$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}U_{j}\Big{(}\frac{1}{N}\sum_{i=1}^{N}U_{i}V_{ji}-(2p-1)q\Big{)}\int_{0}^{t}\sqrt{N}(I_{s}^{N,j}-{I}_{s})h^{\prime}(I_{s})ds,$
$\displaystyle F_{t}^{N}$
$\displaystyle=(2p-1)q\int_{0}^{t}\Big{(}\frac{1}{N}\sum_{j=1}^{N}U_{j}\sqrt{N}(I_{s}^{N,j}-{I}_{s})-\overline{K}^{U}_{s}\Big{)}h^{\prime}(I_{s})ds$

Denote by $\mathbf{E}_{UV}$ conditional expectation with respect to the
configuration of the graph $(U_{j})_{j},(V_{ji})_{ji}$. In the following, we
show

$\mathbf{E}_{UV}\big{[}\sup_{0\leq s\leq
t}\big{(}A^{N}_{s}\big{)}^{2}\big{]},\mathbf{E}_{UV}\big{[}\sup_{0\leq s\leq
t}\big{(}C^{N}_{s}\big{)}^{2}\big{]},\mathbf{E}_{UV}\big{[}\sup_{0\leq s\leq
t}\big{(}C^{N}_{s}\big{)}^{2}\big{]},\mathbf{E}_{UV}\big{[}\sup_{0\leq s\leq
t}\big{(}E^{N}_{s}\big{)}^{2}\big{]}\to 0$

in probability, show

$\mathbf{E}_{UV}\big{[}\sup_{0\leq s\leq
t}\big{(}F^{N}_{s}\big{)}^{2}\big{]}\leq
C\int_{0}^{t}\mathbf{E}_{UV}\big{[}\sup_{0\leq u\leq
s}\big{(}\frac{1}{N}\sum_{i=1}^{N}U_{i}\sqrt{N}(J^{N,i}_{u}-J_{u})-\overline{G}^{U}_{u}\big{)}^{2}\big{]}ds$

and deduce convergence using Gronwall’s inequality. The constant $C$ depends
on $h,\varphi,t,p,q$ and may change from line to line.
First rewrite $A^{N}$,

$\displaystyle A^{N}_{s}$
$\displaystyle=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}\Big{[}\frac{1}{N}\sum_{i=1}^{N}U_{i}V_{ji}-(2p-1)q\Big{]}\Big{(}Y_{j}\Big{(}\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}-\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}$
$\displaystyle\qquad+(2p-1)q\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}\Big{(}Y_{j}\Big{(}\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}-\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}-\int_{0}^{t}\sqrt{h(I_{s})}dB^{U}_{s}.$

The predictable quadratic variation of the first line is given by

$\frac{1}{N}\sum_{j=1}^{N}\big{[}\frac{1}{N}\sum_{i=1}^{N}U_{i}V_{ji}-(2p-1)q\big{]}^{2}\int_{0}^{t}h(I_{s}^{N,j})ds$

which can be seen by Lemma 4.51 and Proposition 4.50 b) in [16], and converges
to $0$ almost surely by Theorem 1.1 and (4.10). The second line converges to
$0$ by Lemma 4.4. We obtain

$\displaystyle\mathbf{E}$ ${}_{UV}\Big{[}\sup_{0\leq s\leq
t}\big{(}A^{N}_{s}\big{)}^{2}\Big{]}$ $\displaystyle\leq
C\,\mathbf{E}_{UV}\Big{[}\Big{(}\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}\Big{[}\frac{1}{N}\sum_{i=1}^{N}U_{i}V_{ji}-(2p-1)q\Big{]}\Big{(}Y_{j}\Big{(}\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}-\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}\Big{)}^{2}\Big{]}$
$\displaystyle\quad+C\,\mathbf{E}_{UV}\Big{[}\Big{(}\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}\Big{(}Y_{j}\Big{(}\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}-\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}-\int_{0}^{t}\sqrt{h(I_{s})}dB^{U}_{s}\Big{)}^{2}\Big{]}\to
0$

almost surely.
For $C^{N}$, write $R_{1}h$ for the first order remainder in Taylor’s
expansion of the function $h$. We use $U_{j}^{2}=1,V_{ji}^{2}\leq 1$, Jensen
inequality, and some $\xi^{N,j}_{s}$ between $I^{N,j}_{s}$ and $I_{s}$

$\displaystyle\mathbf{E}_{UV}\Big{[}\sup_{0\leq s\leq
t}\big{(}C_{s}^{N}\big{)}^{2}\Big{]}$
$\displaystyle\leq\frac{1}{N}\sum_{j=1}^{N}\int_{0}^{t}\mathbf{E}_{UV}\Big{[}\big{(}\sqrt{N}R_{1}h(I^{N,j}_{s},I_{s})\big{)}^{2}ds\Big{]}$
$\displaystyle\leq
t\frac{1}{N}\sum_{j=1}^{N}\int_{0}^{t}N\mathbf{E}_{UV}\Big{[}(h^{\prime}(\xi^{N,j}_{s})-h^{\prime}(I_{s}))^{2}(I^{N,j}_{s}-I_{s})^{2}\Big{]}ds$
$\displaystyle\leq
t(h^{\prime}_{Lip})^{2}\int_{0}^{t}\sum_{j=1}^{N}\mathbf{E}_{UV}\Big{[}(I^{N,j}_{s}-I_{s})^{4}\Big{]}ds.$

We obtain convergence to zero almost surely of the last term similarly to the
Proof of Theorem 1.1: Recall the decomposition of
$J^{N,i}-J=A^{N,i}+C^{N,i}+D^{N,i}$ from (4.20). As the fourth centered moment
of a Poisson distribution with parameter $\lambda$ is of order $\lambda^{2}$
and $h$ is bounded, we obtain
$\sum_{j=1}^{N}\mathbf{E}_{UV}\big{[}(A^{N,i})^{4}\big{]}=O(N^{-1})$ similarly
to the proof of Lemma 4.3. For $D^{N,i}$ rewrite

$\displaystyle\sum_{i=1}^{N}\Big{(}\frac{1}{N}\sum_{j=1}^{N}\big{(}U_{j}V_{ji}-(2p-1)q\big{)}\Big{)}^{4}\leq\sup_{i\leq
N}\Big{(}\frac{1}{N}\sum_{j=1}^{N}\big{(}U_{j}V_{ji}-(2p-1)q\big{)}\Big{)}^{2}\frac{1}{N}\sum_{j=1}^{N}\big{(}W^{N,i}\big{)}^{2}\to
0$

in probability, by uniform convergence in (4.10) and (4.18). We obtain, using
$U_{j}^{4},V_{ji}^{4}\leq 1$, Jensen’s inequality and Lemma (4.1),

$\displaystyle\sum_{i=1}^{N}\mathbf{E}_{UV}\Big{[}\sup_{s\leq
t}(J^{N,i}_{s}-J_{s})^{4}\Big{]}$
$\displaystyle\leq\sum_{i=1}^{N}\mathbf{E}_{UV}\Big{[}\sup_{s\leq
t}\Big{(}\frac{1}{N}\sum_{j=1}^{N}U_{j}V_{ji}\int_{0}^{s}h(I_{u}^{N,j})-h(I_{u})du\Big{)}^{4}\Big{]}+o_{p}(1)$
$\displaystyle\leq
C\int_{0}^{t}\sum_{j=1}^{N}\mathbf{E}_{UV}\Big{[}\sup_{u\leq
s}(J^{N,j}_{u}-J_{u})^{4}\Big{]}ds+o_{p}(1).$

Then $\sum_{j=1}^{N}\mathbf{E}_{UV}\Big{[}\sup_{s\leq
t}(I^{N,j}_{s}-I_{s})^{4}\Big{]}\to 0$ in probability follows from Gronwall’s
inequality and again Lemma (4.1).
The convergence $\sup_{0\leq s\leq t}\big{(}D^{N}\big{)}^{2}\to 0$ in
probability follows from boundedness of $h$ and (4.16). In order to show
convergence of $E_{s}^{N}$, we need an a-priori bound on
$\frac{1}{N}\sum_{j=1}^{N}\mathbf{E}_{UV}\big{[}\big{(}\sqrt{N}(I_{s}^{N,j}-{I}_{s})\big{)}^{2}\big{]}$,
see (4.24) below. Then, by the uniform convergence in (4.10),

$\displaystyle\mathbf{E}_{UV}\Big{[}\sup_{0\leq s\leq
t}\big{(}E_{s}^{N}\big{)}^{2}\Big{]}$ $\displaystyle\leq C\sup_{j\leq
N}\Big{[}\frac{1}{N}\sum_{i=1}^{N}\big{(}U_{i}V_{ji}-(2p-1)q\big{)}\Big{]}^{2}\int_{0}^{t}\frac{1}{N}\sum_{j=1}^{N}\mathbf{E}_{UV}\Big{[}\big{(}\sqrt{N}(I_{s}^{N,j}-{I}_{s})\big{)}^{2}\Big{]}ds$
(4.23)

converges to 0 in probability. By Jensen’s inequality and Lemma 4.1,

$\displaystyle\mathbf{E}_{UV}\Big{[}\sup_{0\leq s\leq
t}\big{(}F_{s}^{N}\big{)}^{2}\Big{]}$ $\displaystyle\leq
C\int_{0}^{t}\mathbf{E}_{UV}\Big{[}\sup_{0\leq u\leq
s}\Big{(}\frac{1}{N}\sum_{j=1}^{N}U_{j}\sqrt{N}(I_{u}^{N,j}-{I}_{u})-\overline{K}^{U}_{u}\Big{)}^{2}\Big{]}ds$
$\displaystyle\leq C\int_{0}^{t}\mathbf{E}_{UV}\Big{[}\sup_{0\leq u\leq
s}\Big{(}\frac{1}{N}\sum_{j=1}^{N}U_{j}\sqrt{N}(J_{u}^{N,j}-{J}_{u})-\overline{G}^{U}_{u}\Big{)}^{2}\Big{]}ds.$

Hence we obtain

$\displaystyle\mathbf{E}_{UV}\Big{[}\sup_{0\leq s\leq t}$
$\displaystyle\Big{(}\frac{1}{N}\sum_{i=1}^{N}U_{i}\sqrt{N}(J^{N,i}_{s}-J_{s})-\overline{G}^{U}_{s}\Big{)}^{2}\Big{]}$
$\displaystyle\leq o_{p}(1)+C\int_{0}^{t}\mathbf{E}_{UV}\Big{[}\sup_{0\leq
u\leq
s}\Big{(}\frac{1}{N}\sum_{i=1}^{N}U_{i}\sqrt{N}(J^{N,i}_{u}-J_{u})-\overline{G}^{U}_{u}\Big{)}^{2}\Big{]}ds,$

and by Gronwalls inequality

$\displaystyle\mathbf{E}_{UV}\Big{[}1\wedge\sup_{0\leq s\leq t}$
$\displaystyle\Big{(}\frac{1}{N}\sum_{i=1}^{N}U_{i}\sqrt{N}(J^{N,i}_{s}-J_{s})-\overline{G}^{U}_{s}\Big{)}^{2}\Big{]}\to
0$

in probability. By bounded convergence, we have $\sup_{0\leq s\leq
t}\Big{(}\frac{1}{N}\sum_{i=1}^{N}U_{i}\sqrt{N}(J^{N,i}_{s}-J_{s})-\overline{G}^{U}_{s}\Big{)}^{2}\to
0$ in probability.
It remains to prove the a-priori bound on the mean fluctuations we used for
convergence of $E^{N}$ in (4.23). First write

$\displaystyle\sqrt{N}(J^{N,i}_{s}-J_{s})$
$\displaystyle=\mathcal{A}^{N,i}_{t}+\mathcal{C}^{N,i}_{t}+\mathcal{D}^{N,i}_{t}+\mathcal{E}^{N,i}_{t}$
with $\displaystyle\mathcal{A}^{N,i}_{t}$
$\displaystyle=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}V_{ji}\Big{(}Y_{j}\Big{(}\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}-\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}$
$\displaystyle\mathcal{C}_{t}^{N,i}$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}U_{j}V_{ji}\int_{0}^{t}\sqrt{N}\big{(}h(I_{s}^{N,j})-h({I}_{s})-(I_{s}^{N,j}-{I}_{s})h^{\prime}(I_{s})\big{)}ds,$
$\displaystyle\mathcal{D}_{t}^{N,i}$
$\displaystyle=\Big{(}\frac{1}{\sqrt{N}}\sum_{j=1}^{N}(U_{j}V_{ji}-(2p-1)q)\Big{)}\int_{0}^{t}h(I_{s})ds$
$\displaystyle\mathcal{E}_{t}^{N,i}$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}U_{j}V_{ji}\int_{0}^{t}\Big{(}\sqrt{N}(I_{s}^{N,j}-{I}_{s})\Big{)}h^{\prime}(I_{s})ds.$

By Lemma 4.51 and Proposition 4.50 b) in [16], the predictable quadratic
variation of $\mathcal{A}^{N,i}$ is given by
$\frac{1}{N}\sum_{i=1}^{N}V_{ji}\int_{0}^{t}\mathbf{E}_{UV}\Big{[}h(I_{s}^{N,j})\Big{]}ds$.
By Doob’s inequality, the uniform convergence from Theorem 1.1 and boundedness
of $h$,

$\displaystyle\mathbf{E}_{UV}\Big{[}\sup_{0\leq s\leq
t}\big{(}\mathcal{A}^{N,i}_{s}\big{)}^{2}\Big{]}\leq
C\mathbf{E}_{UV}\Big{[}\langle\mathcal{A}^{N,i}\rangle_{t}\Big{]}=\frac{1}{N}\sum_{i=1}^{N}V_{ji}\int_{0}^{t}\mathbf{E}_{UV}\Big{[}h(I_{s}^{N,j})\Big{]}ds\to
q\int_{0}^{t}h(I_{s})ds.$

Similarly to the arguments above for $C^{N}$, we can show $\sup_{i\leq
N}\mathbf{E}_{UV}\Big{[}\sup_{0\leq s\leq
t}\big{(}\mathcal{C}^{N,i}_{s}\big{)}^{2}\Big{]}\to 0$. Now observe (recall
$W^{N,j}$ from section 4.3)

$\displaystyle\mathbf{E}_{UV}\Big{[}\frac{1}{N}\sum_{j=1}^{N}\sup_{0\leq s\leq
t}\big{(}\mathcal{D}^{N,i}_{s}\big{)}^{2}\Big{]}$ $\displaystyle\leq
C\frac{1}{N}\sum_{j=1}^{N}\big{(}W^{N,j}\big{)}^{2}$ as well as
$\displaystyle\mathbf{E}_{UV}\Big{[}\sup_{0\leq s\leq
t}\big{(}\mathcal{E}^{N,i}_{s}\big{)}^{2}\Big{]}$ $\displaystyle\leq
C\int_{0}^{t}\mathbf{E}_{UV}\Big{[}\frac{1}{N}\sum_{j=1}^{N}\big{(}\sqrt{N}(I_{s}^{N,j}-{I}_{s})\big{)}^{2}\Big{]}ds$
whence
$\displaystyle\mathbf{E}_{UV}\Big{[}\frac{1}{N}\sum_{j=1}^{N}\sup_{0\leq s\leq
t}\big{(}\mathcal{E}^{N,i}_{s}\big{)}^{2}\Big{]}$ $\displaystyle\leq
C\int_{0}^{t}\mathbf{E}_{UV}\Big{[}\frac{1}{N}\sum_{j=1}^{N}\big{(}\sqrt{N}(I_{s}^{N,j}-{I}_{s})\big{)}^{2}\Big{]}ds$

Hence by Gronwall’s inequality

$\displaystyle\frac{1}{N}\sum_{j=1}^{N}\mathbf{E}_{UV}\Big{[}\sup_{0\leq s\leq
t}\big{(}\sqrt{N}(I_{s}^{N,j}-{I}_{s})\big{)}^{2}\Big{]}\leq
C\frac{1}{N}\sum_{j=1}^{N}\big{(}W^{N,j}\big{)}^{2}+Cq\int_{0}^{t}h(I_{s})ds+o_{p}(1),$
(4.24)

As $\frac{1}{N}\sum_{j=1}^{N}\big{(}W^{N,j}\big{)}^{2}$ converges in
probability by (4.18), this suffices to show convergence in (4.23).

Step 3.2: $\frac{1}{N}\sum_{i=1}^{N}\sqrt{N}(I^{N,i}-I)\to\overline{K}$ and
$\frac{1}{N}\sum_{i=1}^{N}U_{i}V_{ik}\sqrt{N}(I^{N,i}-I)\to(2p-1)q\overline{K}$
From the uniform convergence
$\frac{1}{N}\sum_{i=1}^{N}U_{i}\sqrt{N}(I^{N,i}-I)\to(2p-1)\overline{K}$ in
probability we can easily deduce convergence of

$\frac{1}{N}\sum_{i=1}^{N}\sqrt{N}(I^{N,i}-I)\to\overline{K}\quad\text{ as
well as
}\quad\frac{1}{N}\sum_{i=1}^{N}U_{i}V_{ik}\sqrt{N}(I^{N,i}-I)\to(2p-1)q\overline{K}$

as follows: We can split
$\frac{1}{N}\sum_{i=1}^{N}\sqrt{N}(I^{N,i}-I)\to\overline{K}$ in summands
similar to $A^{N},...,F^{N}$ from Step 3.1. Convergence of
$A^{N},C^{N},D^{N},E^{N}$ follows analogously, we simply use (4.9) instead of
(4.10) for $A^{N},E^{N}$ and (4.15) instead of (4.16) for $D^{N}$. Convergence
of $F^{N}$ follows from the uniform convergence of

$\frac{1}{N}\sum_{i=1}^{N}U_{i}\sqrt{N}(I^{N,i}-I)\to(2p-1)\overline{K}$

from Step 3.1. For $\frac{1}{N}\sum_{i=1}^{N}U_{i}V_{ik}\sqrt{N}(I^{N,i}-I)$,
use (4.11) instead of (4.10) and (4.17) instead of (4.16).

Step 3.3: $\sqrt{N}(I^{N,i}-I)\to{K^{i}}$
Use $\widetilde{B}^{i}$ from Lemma 4.4 in the definition of $G^{i}$ and recall
$\widetilde{W}^{i}$ from 4.3. As always, rewrite

$\displaystyle\sqrt{N}(J^{N,i}-J)-G^{i}$
$\displaystyle=A^{N,i}_{t}+C^{N,i}_{t}+D^{N,i}_{t}+E^{N,i}_{t}$ with
$\displaystyle A^{N,i}_{t}$
$\displaystyle=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}U_{j}V_{ji}\Big{(}Y_{j}\Big{(}\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}-\int_{0}^{t}h(I_{s}^{N,j})ds\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad-q\int_{0}^{t}\sqrt{h(I_{s})}B^{U}_{s}+\sqrt{q(1-q)}\int_{0}^{t}\sqrt{h(I_{s})}\widetilde{B}^{k}_{s}$
$\displaystyle C_{t}^{N,i}$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}U_{j}V_{ji}\int_{0}^{t}\sqrt{N}\big{(}h(I_{s}^{N,j})-h({I}_{s})-(I_{s}^{N,j}-{I}_{s})h^{\prime}(I_{s})\big{)}ds,$
$\displaystyle D_{t}^{N,i}$
$\displaystyle=\Big{(}\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\big{(}U_{j}V_{ji}-(2p-1)q\big{)}-\big{(}qW+\sqrt{q(1-q)}\widetilde{W}^{i}\big{)}\Big{)}\int_{0}^{t}h(I_{s})ds$
$\displaystyle E_{t}^{N,i}$
$\displaystyle=\int_{0}^{t}\Big{(}\Big{(}\frac{1}{N}\sum_{j=1}^{N}U_{j}V_{ji}\sqrt{N}(I_{s}^{N,j}-{I}_{s})\Big{)}-(2p-1)q\overline{K}\Big{)}h^{\prime}(I_{s})ds.$

From step 3.2 we know $\sup_{0\leq s\leq t}\big{(}E_{s}^{N,i}\big{)}^{2}\to 0$
in probability. The convergence $\sup_{0\leq s\leq
t}\big{(}A_{s}^{N,i}\big{)}^{2}\to 0$ is exactly (4.8). Convergence of
$\big{(}C^{N,i}\big{)}^{2},\big{(}D^{N,i}\big{)}^{2}\to 0$ uniformly on
compact time intervals in probability can be shown as in step 3.1, just use
(4.14) instead of (4.16) for $D^{N,i}$. We conclude $\sup_{0\leq s\leq
t}\big{(}\sqrt{N}(J_{s}^{N,i}-J_{s})\to{G_{s}^{i}}\big{)}^{2}\to 0$ and, by
Lemma 4.1, $\sup_{0\leq s\leq
t}\big{(}\sqrt{N}(I_{s}^{N,i}-I_{s})\to{K_{s}^{i}}\big{)}^{2}\to 0$ in
probability. ∎

###### Proof of Corollary 3.4.

First, (3.4) follows from the convergence of $\frac{1}{\sqrt{N}}X^{N,0}$ in
Lemma 4.4. For (3.5), by the law of large numbers,

$\frac{1}{N}\sum_{j=1}^{N}\bar{Z}^{j}_{T}\xrightarrow{N\to\infty}\int_{0}^{T}h(I_{t})dt,\quad\text{as
well
as}\quad\frac{1}{N}\sum_{i=1}^{N}h(I^{N,i})-h(I)\xrightarrow{N\to\infty}0$

almost surely, uniformly on compact time intervals by Theorem 1.1. Therefore

$\displaystyle\frac{1}{N}\sum_{j=1}^{N}\Big{(}Z^{N,j}-\bar{Z}^{j}\Big{)}$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}\Big{(}Z^{N,j}-\int_{0}^{.}h(I_{s}^{N,i})ds\Big{)}$
$\displaystyle\qquad-\frac{1}{N}\sum_{j=1}^{N}\Big{(}\bar{Z}^{j}-\int_{0}^{.}h(I_{s})ds\Big{)}+\int_{0}^{.}\frac{1}{N}\sum_{i=1}^{N}h(I^{N,i})-h(I)ds\xrightarrow{N\to\infty}0,$

almost surely. Next, (3.6) follows from the reformulation (4.4), Lemma 4.4 and
Theorem 1.1. For (3.7), with $K^{i}$ as in Theorem 1.3,

$\displaystyle\sqrt{N}(h(I_{t}^{N,i})-h(I_{t}))=\sqrt{N}h^{\prime}(I_{t})(I_{t}^{N,i}-I_{t})+o(1)=h^{\prime}(I_{t})K^{i}_{t}+o(1).$

Last, for (3.8), recall $B,B^{0}$ from section 4.2. Use $B$ in the definition
of $\overline{K}$, then $\mathbf{E}[B_{t}B^{0}_{t}]=(2p-1)qt,t\geq 0$ and

$\displaystyle\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\big{(}h(I^{N,j})-h(I)\big{)}=h^{\prime}(I_{s})\frac{1}{N}\sum_{j=1}^{N}\sqrt{N}\big{(}I^{N,j}-I\big{)}+o(1)\xRightarrow{N\to\infty}h^{\prime}(I_{s})\overline{K}_{s},$

by Theorem 1.3, whence

$\displaystyle\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\Big{(}Z^{N,j}-\int_{0}^{t}h(I_{s})ds\Big{)}$
$\displaystyle=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\Big{(}Z^{N,j}-\int_{0}^{t}h(I^{N,j}_{s})ds\Big{)}+\int_{0}^{t}\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\big{(}h(I^{N,j})-h(I)\big{)}ds$
$\displaystyle\xRightarrow{N\to\infty}\int_{0}^{t}\sqrt{h(I_{s})}dB^{0}_{s}+\int_{0}^{t}h^{\prime}(I_{s})\overline{K}_{s}ds$

by (3.6). ∎

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# Vector Quantized Semantic Communication System

Qifan Fu, Huiqiang Xie, Zhijin Qin, Gregory Slabaugh, and Xiaoming Tao This
work was supported in part by the National Natural Science Foundation of China
(NSFC Nos 62293484 and 61925105) and in part by Tsinghua University-China
Mobile Communications Group Company Ltd. Joint Institute. (Corresponding
author: Zhijin Qin.)Qifan Fu, Huiqiang Xie and Gregory Slabaugh are with the
School of Electronic Engineering and Computer Science, Queen Mary University
of London, London E1 4NS, U.K. (e-mail: q.fu, h.xie,
g.slabaugh@qmul.ac.uk).Zhijin Qin and Xiaoming Tao are with the Department of
Electronic Engineering, Tsinghua University, Beijing 100084, China (e-mail:
qinzhijin, taoxm@tsinghua.edu.cn).

###### Abstract

Although analog semantic communication systems have received considerable
attention in the literature, there is less work on digital semantic
communication systems. In this paper, we develop a deep learning (DL)-enabled
vector quantized (VQ) semantic communication system for image transmission,
named VQ-DeepSC. Specifically, we propose a convolutional neural network
(CNN)-based transceiver to extract multi-scale semantic features of images and
introduce multi-scale semantic embedding spaces to perform semantic feature
quantization, rendering the data compatible with digital communication
systems. Furthermore, we employ adversarial training to improve the quality of
received images by introducing a PatchGAN discriminator. Experimental results
demonstrate that the proposed VQ-DeepSC is more robustness than BPG in digital
communication systems and has comparable MS-SSIM performance to the DeepJSCC
method.

###### Index Terms:

Deep learning, semantic communication, vector quantization, image
transmission.

##  I Introduction

As we are moving towards an era of “Connected Intelligence”[1], intelligent
communication systems have attracted extensive attention. Semantic
communication is a new intelligent communication paradigm that has shown many
advantages, particularly for the scenarios that involve with large data
volume, such as Internet of Things systems and self-driving cars. Compared to
the traditional communication system transmitting all the data regardless of
its content, semantic communication[2] extracts and transmits the most
valuable task-oriented semantic features of the source data and thereby better
utilizes communication resources. Existing semantic communication systems can
be divided into two categories: data reconstruction and task execution.

For data reconstruction systems, the transmitter must extract and send the
global semantic features of the source data to the receiver, so that the
receiver can recover the large and small scale features of the source data.
Bourtsoulatze et al. proposed a deep learning (DL) enabled joint source-
channel coding (JSCC) system for image transmission, named Deep-JSCC, in which
the source-channel encoder and decoder are jointly designed[3]. Then, Xie et
al. developed a powerful transformer-based semantic communication system for
text transmission, named DeepSC, which is robust to channel variations and
applicable to many scenarios[4]. Besides, Weng et al. [5] designed a speech
semantic reconstruction system that leverages an attention mechanism.

To make semantic communication compatible with digital communication systems,
semantic features must be converted to bits for transmission. Vector
quantization[6] can map features into indices, which can then be converted to
bits and transmitted by digital communication methods. Hu et al. [7] designed
a robust vector quantized semantic communication system to mitigate semantic
noise, which outperforms traditional methods on a classification task. Nemati
et al. [8] designed a vector quantized DL-based JSCC system and showed its
robustness against noisy wireless channels. Although existing vector
quantization-based semantic communication systems show good performance for
many downstream tasks, their application to data reconstruction remain
limited.

In this paper, we propose a vector quantization-based semantic communication
system, named VQ-DeepSC, which uses a semantic embedding space for wireless
transmission of images and is compatible with digital communication systems.
By sharing semantic embedding spaces between the transmitter and receiver, VQ-
DeepSC can perform image transmission with low transmission overheads. The
main contributions of this paper can be summarized as follows:

  * •

VQ-DeepSC excels at feature extraction and compression to reduce transmitted
bits. Specifically, VQ-DeepSC extracts multi-scale semantic features of an
image with a U-Net structure [9], and compresses these features into
transmitted bit stream at transmitter. Then, the image can be reconstructed
with multi-scale features at receiver.

  * •

In the model training process, we introduce a PatchGAN discriminator[10] for
adversarial training and a generative adversarial network (GAN)-based loss to
improve the quality of image generation.

Following notations are used in this paper: $\mathbb{R}^{n\times m}$,
$\mathbb{Z}^{n\times m}$, and $\mathbb{C}^{n\times m}$ represent sets of real,
positive integer, and complex matrices of size $n\times m$, respectively.
Boldface lower-case and upper-case letters denote vectors and matrices,
respectively. $\odot$ and $\oslash$ are the element-wise multiplication and
division, respectively. ${\left(\cdot\right)^{\rm{H}}}$ denotes the Hermitian.
${\cal C}{\cal N}\left({\mu,{\sigma^{2}}}\right)$ means the circularly-
symmetric complex Gaussian distribution with mean $\mu$ and covariance
$\sigma^{2}$.

##  II System Model

In this section, we introduce the vector quantized semantic communication
system with semantic embedding space. As shown in Fig. 1, we consider a DL-
enabled end-to-end semantic communication system with a stochastic physical
channel, where the transmitter and receiver share a pre-trained semantic
embedding space that contains semantic vectors as prior information.

###  II-A Transmitter

As shown in Fig. 1, the transmitter consists of three parts: semantic encoder,
vector quantization with semantic embedding space, and channel encoder. We
denote the input image as $\bf I$. The semantic feature tensor $\bf F$ is
extracted first as

$\displaystyle{\bf{F}}=T\left({\bf I};{\bm{\alpha}}\right),$ (1)

where $T\left(\cdot;{\bm{\alpha}}\right)$ is the semantic encoder with
learnable parameters ${\bm{\alpha}}$, and
${\mathbf{F}}=\left[{\mathbf{f}}_{1},{\mathbf{f}}_{2},\cdots,{\mathbf{f}}_{M}\right]\in{\mathbb{R}}^{M\times
K}$. Specifically, a local semantic embedding space shared by transceivers is
denoted as ${{\bf
E}=\left[{\mathbf{e}}_{1},{\mathbf{e}}_{2},\cdots,{\mathbf{e}}_{N}\right]\in\mathbb{R}^{N\times
K}}$, where $N$ represents the number of vectors, and $K$ represents the
dimension of each vector. In other words, a semantic embedding space contains
$N$ semantic embedding vectors with $K$ dimensions. Then with vector
quantization[6], the semantic features can be mapped to a set of indices in
the embedding space and the transmitter only needs to send these indices
rather than the semantic features to the receiver, which performs the nearest
neighbor reconstruction:

$\displaystyle{n}=\mathop{\arg\min}\limits_{n}{\left\|{{\mathbf{f}}_{m}-{{{\mathbf{e}}_{n}}}}\right\|_{2}},$
(2)

where $m\in\left[{1,M}\right],{n}\in\left[{1,N}\right]$. With the (2), the
semantic features can be mapped into the sequence of indices,
${\mathbf{s}}\in{\mathbb{Z}}^{M\times 1}$. Then, after conventional channel
coding and modulation, ${\mathbf{s}}$ is processed as a transmitted complex
signal $\bm{x}\in\mathbb{C}^{B\times 1}$, where $B$ is the length of the
transmitted signal.

###  II-B Receiver

The receiver also consists of three parts: channel decoder, de-quantization
with semantic embedding space, and semantic decoder. The received signal ${\bf
y}\in\mathbb{C}^{B\times 1}$ can be expressed as

$\displaystyle{\mathbf{y}}={\mathbf{h}}\odot{\mathbf{x}}+{\mathbf{w}},$ (3)

where $\mathbf{h}$ represents the coefficients of a linear channel between the
transmitter and receiver. For the Rayleigh fading channel, the channel
coefficient follows ${\cal C}{\cal N}(0,1)$; for the Rician fading channel, it
follows ${\cal C}{\cal N}\left({\mu,{\sigma^{2}}}\right)$ with
$\mu=\sqrt{{r\mathord{\left/{\vphantom{r{\left({r+1}\right)}}}\right.\kern-1.2pt}{\left({r+1}\right)}}}$
and
$\sigma=\sqrt{{1\mathord{\left/{\vphantom{1{\left({r+1}\right)}}}\right.\kern-1.2pt}{\left({r+1}\right)}}}$,
where $r$ is the Rician coefficient. ${\mathbf{w}}$ denotes the additive white
Gaussian noise (AWGN) whose elements are i.i.d with zero mean and variance
$\sigma^{2}_{n}$.

Assuming that the receiver has perfect channel state information (CSI), the
recovered signals are

$\displaystyle\hat{\mathbf{x}}={{\mathbf{h}}}^{\tt
H}\odot{\mathbf{y}}\oslash\left({{\mathbf{h}}}\odot{\hat{\mathbf{h}}^{\tt
H}}\right).$ (4)

Next, the channel decoder recovers the indices $\hat{\mathbf{s}}$ from
${\bm{\hat{x}}}$. Then, by selecting the corresponding embedding vectors in
the semantic embedding space with $\hat{\mathbf{s}}$, the receiver can
reconstruct the semantic feature tensor $\hat{\bf F}$. Finally, the semantic
decoder generates the reconstructed output ${\bf{\hat{I}}}=R(\hat{\bf
F};{{\bm{\gamma}}})$, where $R\left(\cdot;{{\bm{\gamma}}}\right)$ is the
semantic decoder with learnable parameters ${{\bm{\gamma}}}$.

Figure 1: Vector quantized semantic communication system with multi-scale
semantic features.

##  III The Proposed VQ-DeepSC

As shown in Fig. 2, this section first introduces the framework of VQ-DeepSC,
which consists of the transmitter, receiver, and discriminator. Then, the loss
function and training of the proposed model are discussed.

Figure 2: The multi-scale architecture of VQ-DeepSC represents the embedding
space at different scales.

###  III-A Model Description

####  III-A1 Transmitter

The transmitter consists of a semantic encoder, vector quantization, and
channel encoder. In semantic encoder, a convolution layer with a PReLU
activation function and $L$ down-sample blocks are connected sequentially.

The down-sample block shown in Fig. 2 is successively connected by a residual
block, a strided convolutional down-sample layer, and another residual block.
After being processed by batch normalization and PReLU activation function, it
ends with a convolutional layer as the last layer. The convolutional down-
sample layer is a strided convolutional layer, which can reduce the height and
width of the feature tensor. The output feature tensor of each down-sample
block enters the vector quantization part.

The residual block consists of two convolution layers with a batch
normalization and a PReLU activation function between them, and a skip
connection between the input of the first convolutional layer and the output
of the second convolution layer. Residual blocks can be used to train deeper
neural networks to achieve good performance[11].

####  III-A2 Receiver

The receiver consists a channel decoder, de-quantization, and semantic
decoder. In the semantic decoder, a convolutional layer with a PReLU
activation function is used as the input layer, followed by $L$ up-sample
blocks and ending with a transposed convolution layer.

The up-sample block consists of a sequence connection of a residual block, an
up-sample layer, and a convolutional layer, followed by batch normalization
and PReLU activation function. The up-sample layer and the convolutional down-
sample layer achieve the opposite effect. In the above layers, only the up-
sample layers and the down-sample layers will change the height and width of
feature tensor.

For the pipeline of whole system, the fusion of multi-scale features at the
receiver is beneficial to restore detailed features and preserve the
structural information of image. Given the input image, $L$ different scales
feature tensors ${{\bf F}^{l}}$ are extracted by the different down-sample
blocks of the semantic encoder,

$\displaystyle{{\bf
F}^{l}}=T_{\rm{down}}^{l}\left({{{\mathbf{F}}^{l-1}}};{\bm{\alpha}^{l}}\right),l=1,2,\cdots,L,$
(5)

where $T_{\rm{down}}^{l}\left(;{\bm{\alpha}^{l}}\right)$ represents the $l$-th
down-sample blocks in the semantic encoder with learnable parameters
${\bm{\alpha}^{l}}$.

By (2), ${{{\bf F}}^{l}}$ can be further compressed into ${\mathbf{s}}^{l}$ by
embedding space ${\mathbf{E}}^{l}$. Then, feature indices are converted into
bit streams, which are transmitted after channel coding and modulation.

After transmitting over the physical channel, the received symbols reach the
receiver. The receiver inverts the operations performed by the transmitter.
The demodulation maps these symbols to bits and the channel decoder decodes
the received bits into recovered indices $\hat{\mathbf{s}}^{l}$. Next in the
de-quantization part, the $\hat{\mathbf{F}}^{l}$ is recovered by selecting the
corresponding embedding vectors from the embedding space with the
$\hat{\mathbf{s}}^{l}$. Then, the reconstructed feature tensors of different
scales ${\hat{\bf F}^{l}}$ are fed into the corresponding layers of the
semantic decoder.

In the semantic decoder, feature fusion and reconstruction are carried out
through $L$ up-sample blocks. With the recovered $\hat{\mathbf{F}}^{l}$, the
up-sample block can recover the multi-scale semantic features by

$\hat{\mathbf{F}}^{l}_{\text{c}oncat}={\text{concat}}\left(R^{l}_{up}\left(\hat{\mathbf{F}}^{l-1}_{\text{c}oncat}\right),\hat{\mathbf{F}}^{l-1}\right),$
(6)

where
${\hat{{\mathbf{F}}}^{l-1}_{{\text{c}oncat}}}={{\text{c}oncat}}(R^{l}_{{\rm{up}}}({{{\hat{{\bf
F}}}^{l};{\bm{\gamma}^{l}}}}),{\hat{{\bf F}}^{l-1}})$, where
${{\text{c}oncat}}\left(\cdot\right)$ represents the concatenation operation.
At the end of the receiver, after all up-sample blocks, the image is
reconstructed by a transposed convolution layer.

(a) AWGN channels.

(b) Rician channels.

Figure 3: MS-SSIM performance comparisons with a similar number of transmitted
symbols over different channels.

###  III-B Loss Function and Training Process

To further improve the quality of the generated image, we introduce a PatchGAN
discriminator[10] with binary cross entropy (BCE) loss for adversarial
training. The PatchGAN discriminator can make judgements of the patches of the
generated image, which makes the model pay more attention to image details.
For the input image $\bf I$ that follows distribution ${P_{\bf I}}$, $\hat{\bf
I}$ is the recovered image generated by VQ-DeepSC. Then, the loss function of
adversarial training is as follows:

$\displaystyle{{\cal L}_{{\rm{GAN}}}}=\log
D\left({\mathbf{I}};{\mu}\right)+\log({1-D({\hat{\bf I}};{\mu})}),$ (7)

where $D\left(\cdot;{\bm{\mu}}\right)$ is the discriminator with learnable
parameters $\bm{\mu}$. The introduction of the GAN loss not only enables VQ-
DeepSC to generate realistic images, but also enables the model to use
unlabeled data for unsupervised training.

The back-propagation through (2) is a non-differentiable due to the
quantization operation. To solve this problem, we use a straight-through
gradient estimator approach [12] to copy the gradients from the receiver to
the transmitter. In this way, the entire model can be trained in an end-to-end
fashion using the following loss function:

$\displaystyle{{{\cal L}_{{\rm{VQ-DeepSC}}}}=}$
$\displaystyle{\mathbb{E}}\left|{{\bf I}-\hat{\bf I}}\right|+$ (8)
$\displaystyle{\mathbb{E}}{\left({\left\|{sg\left[{\mathbf{f}}^{l}_{m}\right]-{{{{\mathbf{e}}^{l}_{n}}}}}\right\|_{2}^{2}+\beta_{c}\left\|{sg\left[{{{{\mathbf{e}}^{l}_{n}}}}\right]-{\mathbf{f}}^{l}_{m}}\right\|_{2}^{2}}\right)},$

where the first term is the mean absolute error (MAE) loss of the original
input image ${\bf I}$ and generated image $\hat{\bf I}$, ${{\bf e}_{n}^{l}}$
is the $n$-th semantic embedding vector from the $l$-th embedding space,
${\mathbf{f}}^{l}_{m}$ is the $m$-th feature vector from the $l$-th feature
tensor,
$\left\|{sg\left[{\mathbf{f}}^{l}_{m}\right]-{{\mathbf{e}}^{l}_{n}}}\right\|_{2}^{2}$
is the embedding loss of $l$-th layer feature vector and the corresponding
embedding space, which moves the embedding vector toward the feature vector in
training, $sg\left[\cdot\right]$ represents the stop-gradient operation,
$\left\|{sg\left[{{{\mathbf{e}}^{l}_{n}}}\right]-{\mathbf{f}}^{l}_{m}}\right\|_{2}^{2}$
is the so-called “commitment loss” [6] of the $l$-th layer feature vector and
the $l$-th scale embedding space, which is used to avoid instabilities
resulting from large changes in the output of down-sample block, and
$\beta_{c}$ is the commitment loss scalar.

The final objective of training the entire VQ-DeepSC model to the optimize
${{\cal
G}^{*}}=\left\\{{{{\bm{\alpha}}^{*}},{{\bm{\gamma}}^{*}},{{\bm{\beta}}^{*}}}\right\\}$
as:

$\displaystyle{{\cal
G}^{*}}=\arg\mathop{\min}\limits_{{\bm{\alpha}},{\bm{\gamma}},{\bm{\beta}}}\mathop{\max}\limits_{\bm{\mu}}{\mathbb{E}_{{\bf
I}\sim p\left({\bf I}\right)}}[\lambda{{\cal L}_{{\rm{VQ-DeepSC}}}}+{{\cal
L}_{{\rm{GAN}}}}],$ (9)

where ${\bm{\beta}}$ represents the learnable parameters of the embedding
spaces, $\lambda$ is a weight to balance ${{\cal L}_{{\rm{VQ-DeepSC}}}}$ and
${{\cal L}_{{\rm{GAN}}}}$. As [10] shows, the proper combination of the least
absolute deviation (LAD) loss and GAN loss can not only improve the clarity of
the generated images, but also ensure that the generated images are more
realistic.

##  IV Discussions and Simulations

This section evaluates the proposed VQ-DeepSC over different channels with
perfect channel state information (CSI).

The Cars196 dataset[13] is used for training and the size of training input
image is resized to $256\times 256$. We save the model with the best
performance in training for the final test, which is carried out on the Kodak
dataset111http://r0k.us/graphics/kodak/ and the testing input image size is
768×512. Our models were trained in noiseless conditions with the hyper-
parameters $\beta_{c}=0.25$ and $\lambda=0.1$. Each convolutional layer of
semantic encoder and decoder has a $3\times 3$ kernel. Besides, the PatchGAN
discriminator[10] begins with a convolutional layer and activation function
followed by $3$ sets of convolution layers, batch normalization and activation
function, and ends with another convolution layer as output. The size of the
first three convolution kernels of the discriminator is $4\times 4$ and the
stride is two, respectively. The size of the last two convolution kernels is
$4\times 4$ and the stride is one. The Adam optimizer with learning rate
$1.75\times{10^{-4}}$ and StepLR is used for VQ-DeepSC training, with betas of
0.5 and 0.999, batch size of 24, and 400 epochs. The training setting of the
discriminator is the Adam optimizer with learning rate $1\times{10^{-5}}$ and
LambdaLR, betas of 0.5 and 0.999. The vector dimensions in the four embedding
spaces are $K_{1}=128$, $K_{2}=256$, $K_{3}=512$ and $K_{4}=1024$.

In particular, we test the performance of the following methods and evaluated
results using the MS-SSIM [14]

  * •

UNet-DeepJSCC: The remaining 4-layer VQ-DeepSC structures without vector
quantization, which transmits images in the form of direct features
transmition and compress the transmitted features by $1\times 1$ convolution.
The training channel condations of UNet-DeepJSCC models are SNR = 9 dB for
both AWGN and Rician channels.

  * •

BPG: The Better Portable Graphics (BPG) [15] codec.

  * •

The VQ-DeepSC1: uses three embedding spaces with vector number $N_{2}=8$,
$N_{3}=4$, $N_{4}=2$.

  * •

The VQ-DeepSC2: uses three embedding spaces with vector number $N_{2}=64$,
$N_{3}=16$, $N_{4}=4$.

  * •

The VQ-DeepSC3: uses four embedding spaces with vector number $N_{1}=2$,
$N_{2}=64$, $N_{3}=4$, $N_{4}=4$.

  * •

The VQ-DeepSC4: uses four embedding spaces with vector number $N_{1}=8$,
$N_{2}=4$, $N_{3}=2$, $N_{4}=2$.

Fig. 3 illustrates the MS-SSIM performance comparison for various methods with
the similar number of transmitted symbols over AWGN and Rician channels, in
which the adaptive modulation and coding (AMC) is employed. Specifically, the
LDPC blocklength is 64800 bits. In the case of AWGN channels, at the low SNR
regimes, the VQ-DeepSC can achieve cliff point about 1 dB lower in SNR than
BPG when using the better AMC mode (LDPC with 1/2 rate and BPSK). As the AMC
mode changes, the VQ-DeepSC achieves the cliff point at lower SNR regimes than
the BPG, which means that the proposed method has better robustness to the
channel variations. In addition, VQ-DeepSC can achieve higher MS-SSIM than
UNet-DeepJSCC in certain SNR regimes by adjusting coding rates and modulation.
For fading channels, the VQ-DeepSC outperforms BPG in terms of MS-SSIM at all
SNR regimes and achieve the similar MS-SSIM (above 0.8) as the DeepJSCC when
SNR is larger than 5dB. In general, the VQ-DeepSC is more robustness than BPG
and has the similar MS-SSIM as the DeepJSCC method.

Figure 4: MS-SSIM performance comparisons with a similar compression ratio of
source over AWGN channels.

Fig. 4 shows the MS-SSIM comparison for VQ-DeepSC3 and BPG method with a
similar compression ratio of source over AWGN channels. Compared to the BPG,
VQ-DeepSC can employ higher modulation order to achieve similar cliff point
with higher transmission rate. In Fig. 4, with the same channel coding rate,
VQ-DeepSC with 32-QAM has higher MS-SSIM than the BPG with 16-QAM at low SNR
regimes, and achieves the same cliff point at SNR=20dB, which shows the high
achievable transmission rate and robustness of VQ-DeepSC.

Figure 5: MS-SSIM comparisons with a similar compression ratio and different
LDPC blocklength over AWGN channels.

Fig. 5 shows the impact of LDPC blocklength on MS-SSIM of VQ-DeepSC3 and BPG
with a similar compression ratio of source. Specifically, the blocklength of
LDPC1, LDPC2, and LDPC3 are 20 bits, 648 bits, and 64800 bits, respectively.
BPG with the longer LDPC blocklength outperforms BPG with shorter LDPC
blocklength, in which the gain can approximately be 8dB. Obviously, the
blocklength of LDPC has less impact on VQ-DeepSC3 than on BPG, which
demonstrates the robustness of VQ-DeepSC3 to digital communication systems.

## V. Conclusions

In this paper, we have proposed a framework of vector quantized semantic
communication system and introduced a GAN-based implementation for image
transmission. Our approach, named VQ-DeepSC, contains different scales of
semantic embedding space and realises different scales of semantic features
fusion at receiver with low transmission overheads. Specifically, by
leveraging vector quantization, the semantic feature vectors were compressed
into a set of semantic feature indices in the corresponding embedding space,
which also made the model suitable for digital communication systems. In
particular, VQ-DeepSC was found to outperform BPG and be comparable to
DeepJSCC as measured with MS-SSIM. Hence, the proposed VQ-DeepSC is a
promising candidate for image semantic communication systems.

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