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[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 91.10, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 91.56, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 92.35, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 92.93, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 96.31, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 98.05, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 98.10, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 98.16, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (101.26, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (102.25, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (103.08, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (103.68, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (104.31, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (108.58, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (109.47, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (110.03, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (111.33, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (112.40, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (115.78, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (116.46, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (116.90, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (117.38, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (119.02, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (120.26, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (120.84, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (123.20, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (124.49, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (126.75, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (127.81, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (129.97, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (130.19, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (130.86, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (132.07, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (133.61, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 32.50, 99.62) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 53.28, 99.62) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 58.48, 99.62) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 66.45, 99.62) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 87.19, 99.62) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (106.32, 99.62) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (114.50, 99.62) circle (  0.57);

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 17.48, 30.94) $N^0$;

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 17.48, 44.07) $N^1$;

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 17.48, 57.20) $N^2$;

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 17.48, 70.33) $N^3$;

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 17.48, 83.46) $N^4$;

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 17.48, 96.59) $N^5$;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 27.18, 19.68) 0.0;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 53.86, 19.68) 2.5;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 80.54, 19.68) 5.0;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at (107.23, 19.68) 7.5;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at (133.91, 19.68) 10.0;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  1.10] at ( 80.73,  7.64) Time;

(B0) at (0, 3)  $N^0$;
(B1) at (1, 2)  $N^1$;
(B2) at (1, 1)  $N^2$;
(B3) at (0, 0)  $N^3$;
(B4) at (-1, 1) $N^4$;
(B5) at (-1, 2) $N^5$;

(C0) at (3, 3)$N^0$;
(C1) at (4, 2)$N^1$;
(C2) at (4, 1)$N^2$;
(C3) at (3, 0)$N^3$;
(C4) at (2, 1) $N^4$;
(C5) at (2, 2) $N^5$;

at (0, -0.5) First;
at (3, -0.5) Second;
[->] (B0) edge[bend right=5] (B1);
[->] (B0) edge[bend right=5] (B3);
[->] (B0) edge[bend right=5] (B4);
[->] (B0) edge[bend right=5] (B5);
[->] (B1) edge[bend right=5] (B0);
[->] (B1) edge[bend right=5] (B2);
[->] (B1) edge[bend right=5] (B3);
[->] (B1) edge[bend right=5] (B4);
[->] (B1) edge[bend right=5] (B5);
[->] (B2) edge[bend right=5] (B0);
[->] (B2) edge[bend right=5] (B4);
[->] (B2) edge[bend right=5] (B5);
[->] (B3) edge[bend right=5] (B1);
[->] (B3) edge[bend right=5] (B2);
[->] (B3) edge[bend right=5] (B4);
[->] (B3) edge[bend right=5] (B5);
[->] (B4) edge[bend right=5] (B0);
[->] (B4) edge[bend right=5] (B1);
[->] (B4) edge[bend right=5] (B3);
[->] (B5) edge[bend right=5] (B1);
[->] (B5) edge[bend right=5] (B3);

[->] (C1) edge[bend right=5] (C3);
[->] (C2) edge[bend right=5] (C0);
[->] (C2) edge[bend right=5] (C3);
[->] (C2) edge[bend right=5] (C5);
[->] (C3) edge[bend right=5] (C4);

(left) The first repetition of the experiment. Each point corresponds to one neuron firing.
(right) Output of the CA algorithm on the first repetition, when the test of local independence either uses a first-order test (`First') or a second-order test (`Second'). <Ref> in the appendix show the similar plots and graphs for the repetitions two through five.

Consistency of estimated graphs from the $5$ repetitions of the stimulus experiment.

Baseline     First        Second
Edges consistent in all 5 repetitions         $6.25\%$     $23.3\%$     $\mathbf{26.7\%}$

Edges consistent in a least 4 repetitions     $37.5\%$     $40.0\%$     $\mathbf{56.7\%}$

Number of edges present                       $50.0\%$     $62.7\%$     $30.7\%$

§ DISCUSSION

In this paper, we formulated a framework for testing local independence in point processes. We introduced a test of local independence that fits intensities using basis expansions and tests the local independence hypothesis $j \not\rightarrow k \mid C$ by testing significance of contributions of the process $N^j$ to the intensity $\lambda^{k, C\cup\{j\}}$.

We addressed the issue of marginalization: Even if the full data generating mechanism is a known and simple model class, such as Hawkes processes, a partially observed system with some event types unobserved cannot necessarily be modelled as a Hawkes process. This issue is native to (conditional) local independence testing, since the local independence $j \not\rightarrow k\mid C$ relates to the marginal distribution of $N^{\{j,k\}\cup C}$.
To overcome this misspecification, we proved that, when facing marginalized variables, the intensity can be arbitrarily well approximated by expansions in terms of iterated integrals, and we have verified that including higher-order interactions leads to an improved level of the test of $j \not\rightarrow k \mid C$.

The availability of an empirical local independence test is quintessential to constraint-based causal structure learning algorithms for point processes, and we have validated in simulation studies that using our proposed test, one can from data obtain good estimates of the underlying graph.
We applied our approach to a real-world data set on neuron spiking in turtles, and found that including higher-order interactions resulted in sparser, more informative estimated networks.

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§ APPENDIX

§.§ Proof of Theorem <ref>

In this appendix, we prove <ref>. The proof first shows the result for a univariate process ($|C| = 1$), and then argues that result can easily be extended to the multivariate setting.

We stress that the motivation for the theorem is to show convergence of the representation. In practice, many other kernel functions than those appearing in the proof, could also be used to describe the system, and so our interest lies very little in the concrete functional forms used.

Let $\tau_1, \tau_2, \ldots$ be the jumps of $N$ starting at $0$ and moving backwards in time. That is $\ldots < \tau_2 < \tau_1 < 0$.

For $s < 0$, let $\mathcal{F}_s$ denote the $\sigma$-algebra generated by events in $[s, 0)$. That is
$$\mathcal{F}_s = \sigma(\tau_1 \lor s, \tau_2 \lor s, \ldots),$$

where $\tau\lor s = \max(\tau,s)$.
Define also $\mathcal{F}_{0-} = \sigma(\cup_{s < 0} \mathcal{F}_s)$.

$\mathcal{F}_0$ equals $\mathcal{F}_{-\infty}:= \sigma(\tau_1, \tau_2, \ldots)$.

For all $i$, $\tau_i \lor s$ is $\sigma(\tau_i)$-measurable, and in particular, $\mathcal{F}_{-\infty}$-measurable. Therefore $\mathcal{F}_{s} = \sigma(\tau_1 \lor s, \ldots) \subseteq \mathcal{F}_{-\infty}$ and so $\mathcal{F}_{0-} = \sigma(\cup_{s<0} \mathcal{F}_s) \subseteq \mathcal{F}_{-\infty}$.

Reversely, $\tau_n$ is $\mathcal{F}_0$-measurable for each $n$. $\mathcal{F}_{-\infty}$ is the smallest $\sigma$-algebra making all $\tau_n$'s measurable, so $\mathcal{F}_{-\infty} \subseteq \mathcal{F}_0$ will follow. To see that $\tau_n$ is $\mathcal{F}_0$-measureable, consider any $n$. $(\tau_n \lor s) \rightarrow \tau_n$ for $s \rightarrow -\infty$ (potentially with $\tau_n = -\infty)$, and so since $(\tau_n \lor s)$ is $\mathcal{F}_0$-measurable for each $s$, $\tau_n$ is $\mathcal{F}_0$-measurable.

The union of function spaces $\cup_{s < 0} \mathcal{L}^1(\mathcal{F}_s)$ is dense in $\mathcal{L}^1(\mathcal{F}_{0-})$.

Take any $\lambda \in \mathcal{L}^1(\mathcal{F}_{0-})$. By the tower property, $\lambda_s := E[\lambda \mid \mathcal{F}_s] \in \mathcal{L}^1(\mathcal{F}_s)$ and further from the martingale convergence theorem, $(\lambda_s)_{s < 0}$ is a martingale (in $-s$) and $E[\lambda \mid \mathcal{F}_s]$ converges in $\mathcal{L}^1$ to $E[\lambda \mid \mathcal{F}_{0-}] = \lambda$ as $s \to -\infty$.

Because each $\lambda_s \in \mathcal{L}^1(\mathcal{F}_s) \subseteq \cup_{s<0} \mathcal{L}^1(\mathcal{F}_s)$, it follows that $\cup_s \mathcal{L}^1(\mathcal{F}_s)$ is dense in $\mathcal{L}^1(\mathcal{F}_{0-})$.

We now show that for any $\lambda \in \mathcal{L}^1(\mathcal{F}_s)$ and for each $M \in \mathbb{N}$ that $\lambda 1_{N([s, 0)) = M}$ can be written as a sum of integrals of deterministic functions. These integrands will play a role similar to Volterra kernels, but only given the count $N([s, 0))$. We then sum over these terms, to obtain a general representation of $\lambda$.

It is well known that if $Y \in \mathcal{L}^1(\sigma(X_1, X_2, \ldots))$ for some random variables $X_1, \ldots$, then there exists a measurable map $f$ such that $Y = f(X_1, X_2, \ldots)$. In the case of event times truncated at $s$, $\tau_n \lor s$, this corresponds to that if $\lambda \in \mathcal{L}^1(\mathcal{F}_s)$ there exists a function $f$ such that
$$\lambda = f(\tau_1 \lor s, \tau_2 \lor s, \ldots)$$

To obtain an integral representation of $\lambda$, we can utilize this function.
Define $f_s^n(t_1, \ldots, t_n) = f(t_1, \ldots, t_n, s, s, \ldots)$ as the evaluation of $f$ in $(t_1, \ldots, t_n)$ and then the $s$ in all other entries of the function. We will write $f^n$ if $s$ is clear from the context or even $f(t_1, \ldots, t_n)$.

As a motivation for the below proof, suppose that we knew that exactly one event occurred in the interval $A := [s,0)$, i.e. $\tau_1 \in A, \tau_n \notin A$ for $n\geq 2$. Then one could write:

\begin{align*}
\lambda &= f(\tau_1\lor s, \tau_2\lor s,\ldots) = f(\tau_1, s, s, \ldots) \\
&= \int_s^{0-} f(t, s, s, \ldots) N(\dd t) = \int_s^{0-} f^1 N(\dd t)
\end{align*}

This however depends heavily on the assumption that $N(A) = 1$. If instead the interval contained $m$ events, then $\int_s^{0-} f^1(t) N(\dd t) = f^1(\tau_1) + \ldots + f^1(\tau_m)$ which is not equal to $\lambda$ (because in this case $\lambda = f^m(\tau_1, \ldots, \tau_m)$).

The following proposition devices a procedure, such that one can obtain $f(\tau_1)$ exactly if $N(A) = 1$ and else $0$, using only integrals of deterministic functions. For a function $h(t_1, \ldots, t_n)$, we use the shorthand notation

\begin{align*}
\int_A h \dd N(t^n) := \int_A\cdots\int_A h(t_1, \ldots, t_n) \dd N(t_1) \cdots \dd N(t_n).
\end{align*}

Assume $N$ is a simple, non-exploding point process. Let $\lambda \in \mathcal{L}^1(\mathcal{F}_s)$ and $A = [s, 0)$.

\begin{align}\label{rep:eq_compensation_scheme_t1}
\sum_{n=1}^L \beta_n \int_A f(t_1) 1_{D_n} \dd N(t^n) \stackrel{\textrm{a.s.}}{\longrightarrow} \lambda 1_{N(A) = 1}
\end{align}

for $L \longrightarrow \infty$ where $\beta_n = \frac{(-1)^{n-1}}{(n-1)!}, n \geq 1$, and:

\begin{align}
D_n = \{(t_1, \ldots, t_n) \in [-s, 0)^n\,\mid \, t_i \neq t_j \text{ for }i \neq j\}\notag
\end{align}

Observe that while we integrate over sequences $(t_1, \ldots, t_n)$, we evaluate only the function $f^1(t_1)$ in $t_1$. The indicator function $1_{D_n}$ still is evaluated in $(t_1, \ldots, t_n)$.
For this reason

\begin{align*}
&\int_A f(t_1) 1_{D_n}(t_1, \ldots, t_n) \dd N(t^n) \\
&= \left[\int_A f(t_1) \dd N(t_1)\right]\begin{pmatrix}N(A) - 1 \\ n - 1\end{pmatrix}(n-1)!
\end{align*}

This follows because for each event time $\tau \in A$, there are exactly $\begin{pmatrix}N(A) - 1 \\ n - 1\end{pmatrix}(n-1)!$ tuples 	$(\tau, t_2, \ldots, t_n)$ where $\tau$ is the first element and no elements are identical.

It then follows that:

\begin{align*}
&\sum_{n=1}^{N(A)} \beta_n \int_A f(t_1) 1_{D_n} \dd N(t^n) \\
&= \left[\int_A f(t) \dd N(t)\right] \sum_{n=1}^{N(A)} (-1)^{n-1} \begin{pmatrix}N(A)-1\\ n-1\end{pmatrix} \\
&= \begin{cases}
f(\tau_1) & N(A) = 1 \\
0 & \textrm{else}
\end{cases} = \lambda 1_{N(A) = 1}
\end{align*}

This last step is utilizes that for $M = 1$,
$\sum_{n=1}^M (-1)^{n-1} \binom{M-1}{n-1} = 1$, and for $M > 1$, the binomial formula implies that

\begin{align*}
0 &= \left(1 + (-1)\right)^{M-1} \\
&=\sum_{n=0}^{M-1} (-1)^n \begin{pmatrix}M-1 \\ n\end{pmatrix} \\
&=\sum_{n=1}^{M} (-1)^{n-1} \begin{pmatrix}M-1 \\ n-1\end{pmatrix}.
\end{align*}

Since the integrand $1_{D_n}$ is $0$ for $n \geq N(A)$, and $P\left(N(A) < \infty\right) = 1$, it follows that

\sum_{n=1}^{L} \beta_n \int_A f(t_1) 1_{D_n} \dd N(t^n) \stackrel{\textrm{a.s.}}{\longrightarrow} \lambda 1_{N(A) = 1} \textrm{ for } L \rightarrow \infty

This extends to the following corollary:

Let $\lambda\in \mathcal{L}^1(\mathcal{F}_s)$. For $M \in \mathbb{N}$, one has:

\begin{align}
\sum_{n=M}^L \beta_n^M \int_A f(t_1, \ldots, t_M) 1_{D_n} 1_{O_M} \dd N(t^{n}) \stackrel{\textrm{a.s.}}{\longrightarrow} \lambda 1_{N(A) = M} \notag
\end{align}

with $\beta_n^M = \frac{(-1)^{n-M}}{(n-M)!}$ for $n \geq M$ and

\begin{align}
O_M = \{(t_1, \ldots, t_M) \in [-s, 0)^n\,\mid \, t_1 < t_2 \ldots < t_M \}\notag
\end{align}

The case $M = 1$ is covered in <ref>. For $M \geq 2$, the result essentially is the same, with the additional requirement that the first $M$ jumps should be ordered, which is handled by $1_{O_M}$.

Apart from this, combinatorics of how many tuples $(t_1, \ldots, t_n)$ with $t_1 < \ldots < t_M$ ordered (as fixed by $O_M$) and all $t$'s distinct (by $D_n$) remains the same, in particular

\begin{align*}
&\int_A f(t_1, \ldots, t_M) 1_{D_n} 1_{O_M} \dd N(t^n)\\
&= \left[\int_A f(t_1, \ldots, t_M) 1_{O_M} \dd N(t^M)\right]\begin{pmatrix}N(A) - M \\ n - M\end{pmatrix}(n-M)!
\end{align*}

Consequently, the proof from <ref> also applies in the case of $1_{N(A) = M}$.

Extending further on <ref> and Corollary <ref>, we may include the base-rate $\lambda 1_{N(A) = 0}$. Let $h^0$ be the value of $\lambda$ on the set $\{N(A) = 0\}$ (that is $h^0 = f(s, s, \ldots)$). Now $\sum_{n=1}^L \int_A \left[f(t_1) - h^0 \right] 1_{D_n} \dd N(t^{n})$ will return the additional to base-rate intensity $f(\tau_1) - h^0$ if $N(A) = 1$ and $0$ else.
We combine the above:

Assume $N$ is a non-exploding point process, and assume $\lambda \in \mathcal{L}^1(\mathcal{F}_s)$. Then

\begin{equation*}
\sum_{M=1}^L \sum_{n=M}^L \beta_n^M \int_A f(t_1, \ldots, t_M) 1_{D_n} 1_{O_M} \dd N(t^n)\stackrel{\textrm{a.s.}}{\longrightarrow}\lambda,
\end{equation*}

for $L \longrightarrow \infty$.

As above, the almost sure convergence follows simply by decomposing $\lambda = \lambda 1_{N(A) = 0} + \sum_{M \in \mathbb{N}} \lambda 1_{N(A) = M}$, and again observing that since $P(N(A) < \infty)$, for every $\omega$, the left hand side will arrive at the true value for some finite $L$.

Finally we are able to prove the main result.

Observe that each function $\beta_n^M f 1_{D_n} 1_{O_M} \in E_n$, and
$$\sum_{M=1}^L \sum_{n=M}^L \beta_n^M \int_A f(t_1, \ldots, t_M) 1_{D_n} 1_{O_M} \dd N(t^n)$$
is in $\bigoplus_{n=0}^L W_n$.
Be reminded that by <ref>, $\cup_{s < 0}\mathcal{L}^1(\mathcal{F}_s)$ is ($\mathcal{L}^1$-)dense in $\mathcal{L}^1(\mathcal{F}_{0-})$, and for every element $\lambda$ of $\cup_{s< 0}\mathcal{L}^1(\mathcal{F}_s)$ there exists a sequence in $\bigoplus_{n \in \mathbb{N}} W_n$ converging almost surely to $\lambda$. Consequently, as both $\mathcal{L}^1$ and almost sure convergence implies convergence in probability, for any $\lambda \in \mathcal{L}^1(\mathcal{F}_{0-})$ there exist a sequence in $\bigoplus_{n\in\mathbb{N}} W_n$ converging to $\lambda$ in probability.

Consider now any $\lambda \in \mathcal{L}(\mathcal{F}_{0-}^C)$ with $|\lambda| < \infty$ a.s. Trivially $\lambda_k := 1_{|\lambda| < K} \lambda$ converges in probability to $\lambda$ for $k\to\infty$. Further each $\lambda_k \in \L^1(\mathcal{F}_{0-})$, and hence there exists a sequence there exists a sequence in $\bigoplus_{n \in \mathbb{N}} W_n$ converging almost surely to $\lambda$, completing the proof in the case without marks.

[>=latex,font=,every node/.style=inner sep = 3pt]

[draw, circle] (A) at (4, 0) $j$;
[draw, circle] (C) at (5,0) $c$;
[draw, circle] (B) at (6,0) $k$;

[->] (C) edge node[above] $0.4$ (A);
[->] (C) edge node[above] $0.4$ (B);

[draw, circle] (A) at (8, 0) $j$;
[draw, circle] (C1) at (9,0) $c$;
[draw] (C2) at (10,0) $h$;
[draw, circle] (B) at (11,0) $k$;

[->] (A) edge node[above] $0.9$ (C1);
[->] (C1) edge node[above] $-0.6$ (C2);
[->] (C2) edge node[above] $0.4$ (B);

[draw, circle] (A) at (13, 0) $j$;
[draw, circle] (C1) at (14,0) $c$;
[draw] (C2) at (15,0.5) $h$;
[draw, circle] (B) at (16,0) $k$;

[->] (C1) edge node[above] $0.4$ (A);
[->] (C2) edge node[above] $0.4$ (C1);
[->] (C2) edge node[above] $0.4$ (B);
[->] (C1) edge[bend right=40] node[above] $-0.4$ (B);

at (3.2, 0) $L_1$:;
at (7.2, 0) $L_2$:;
at (12.2, 0) $L_3$:;

[draw, circle] (A) at (4, -2) $j$;
[draw, circle] (B) at (6, -2) $k$;

[->] (A) edge node[above] $-0.6$ (B);

[draw, circle] (A) at (8, -2) $j$;
[draw] (C2) at (9.5,-2) $h$;
[draw, circle] (B) at (11, -2) $k$;

[->] (A) edge node[above] $0.4$ (C2);
[->] (C2) edge node[above] $-0.4$ (B);

[draw, circle] (A) at (13, -2) $j$;
[draw, circle] (C1) at (14.5, -2.5) $c$;
[draw] (C2) at (14.5, -1.5) $h$;
[draw, circle] (B) at (16, -2) $k$;

[->] (A) edge node[below] $0.3$ (C1);
[->] (A) edge node[above] $0.3$ (C2);
[->] (C1) edge node[below] $0.2$ (B);
[->] (C2) edge node[above] $-0.25$ (B);

at (3.2, -2) $P_1$:;
at (7.2, -2) $P_2$:;
at (12.2, -2) $P_3$:;

Simulation parameters for the experiment in <ref>

The above framework is readily extended to marked point processes. Remember that with $V = \{1, \ldots, d\}$ and $C \subseteq V$, one has for any Borel measurable set $A$ that:

\begin{align}
N(A\times C) = \sum_{v\in C}N(A\times\{v\}) = \sum_{v\in C} N^{v}(A)\notag
\end{align}

When integrating, this factorizes:

\begin{align*}
&\int_{A\times C} f(x, v) N(\dd x, \dd v) \\
&=  \int_{A} f(x, v) \sum_{v \in C} N^{v}(\dd x)\\
&= \sum_{v\in C}\int_A f^{v}(x) N^{v}(\dd x)
\end{align*}

where we let $f^{v}(x) := f(x, v)$. Similarly in higher dimensions:

\begin{align*}
&\int_{A\times C} f(x_1, v_1, \ldots, x_n, v_n) N(\dd x^n \times \dd v^n)\\
&= \sum_{|\alpha|= n}\int_{A\times C}  f^\alpha(x_1, \ldots, x_n) \underbrace{N^{\alpha_1}(x^1)\cdots N^{\alpha_n}(x^n)}_{=: N^\alpha(\dd x^n)}
\end{align*}

where $f^\alpha(x_1, \ldots, x_n) = f(x_1, \alpha_1, \ldots x_n, \alpha_n)$ and $\alpha \in V^n$ is some tuple of length $n$.

Thus, the combinatorics of the one-dimensional case apply also in the marked setting and the result thus directly transfers to the multi-dimensional case:
In the marked setting, the generated $\sigma$-field becomes $\mathcal{F}_s = \sigma((\tau_1 \lor s, v_1 1_{\tau_1 > s}), \ldots )$. A multivariate version of <ref> follows because $(\tau_1\lor s, v_11_{\tau_1>s}) \rightarrow (\tau_1, v_11_{\tau_1 > -\infty})$[Which is the desired limit, with the convention  that $v_n = 0$ if $\tau_n = -\infty$.], and so denseness of $\cup_s \mathcal{L}^1(\mathcal{F}_s)$ also follows in the marked case.
Thus the function $f$ could have been written:

\begin{align}
\lambda = f\left((\tau_1 \lor s, v_1 1_{\tau_1 > s}), \ldots\right)\notag
\end{align}

In <ref>, one could have proceeded in exactly the same way, but using integrals $\int_{A\times V} f(t_1, v_1) N(\dd t_1 \times v_1)$ instead.

Therefore also <ref> generalizes such that any $\lambda \in \mathcal{L}^1(\mathcal{F}_s)$ can be approximated by an almost surely converging sequence, and combined with the denseness result, the result extends to the multivariate case.

§.§ Simulation details

In this section, we provide simulation details for the experiments in <ref>.

§.§.§ Details from

Recall that from each structure, we sampled point processes with kernels $g^{i_1i_2}(s) = \alpha_{i_1i_2}\beta_{i_1i_2}e^{-\beta_{i_1i_2} s}$ if $(i_1, i_2) \in \mathcal{G}$ and otherwise $g^{i_1i_2}(s)=0$. We simulated data using the link-function $\eta(x) = 1_{x\geq 1}\cdot x + 1_{x<1}\cdot (\log(x)+1)$.

For all structures and edges, the decay parameter $\beta_{i_1i_2}$ is $0.8$, the baseline intensity is $\beta_0 = 0.25$ and the rate parameter on self-edges is $\alpha_{i_1i_1} = 0.4$.
The remaining rate parameters $\alpha_{i_1i_2}$ are given in <ref>.

§.§.§ Details from

All graphs are sampled randomly with all self-edges present and all other edges sampled with a probability of an edge occuring at $p=0.2$.
Given the graph, Hawkes processes are sampled with kernels $g^{i_1i_2}(s) = \alpha_{i_1i_2}\beta_{i_1i_2}e^{-\beta_{i_1i_2} s}$ if $(i_1, i_2) \in \mathcal{G}$ and otherwise $g^{i_1i_2}(s)=0$, and again using the link function $\eta(x) = 1_{x\geq 1}\cdot x + 1_{x<1}\cdot (\log(x)+1)$.
The decay parameter is $\beta^{i_1i_2} = 0.8$, the baseline intensities $\beta_0^{i_1} = 0.25$, and for self-edges the rate parameter is $\alpha_{i_1i_1}=0.3$. The rate parameters between two different nodes is $s\cdot 0.4$ where $P(s=1) = P(s=-1) = 1/2$.

§.§ Estimated graphs for remaining 4 experiments

<Ref> in <ref> we displayed data and resulting estimated graphs from the first repetition in an experiment that was repeated $5$ times.
This section contains plots similar to <ref>, but for the other $4$ repetitions. These are displayed in <ref>.

[draw=drawColor,line width= 0.6pt,line join=round] ( 22.43, 33.97) –
(139.04, 33.97);

[draw=drawColor,line width= 0.6pt,line join=round] ( 22.43, 47.10) –
(139.04, 47.10);

[draw=drawColor,line width= 0.6pt,line join=round] ( 22.43, 60.23) –
(139.04, 60.23);

[draw=drawColor,line width= 0.6pt,line join=round] ( 22.43, 73.36) –
(139.04, 73.36);

[draw=drawColor,line width= 0.6pt,line join=round] ( 22.43, 86.49) –
(139.04, 86.49);

[draw=drawColor,line width= 0.6pt,line join=round] ( 22.43, 99.62) –
(139.04, 99.62);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 28.91, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 29.89, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 31.90, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 33.19, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 34.04, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 35.50, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 36.87, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 39.53, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 42.02, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 43.67, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 44.34, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 52.82, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 54.57, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 54.76, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 56.37, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 56.56, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 56.80, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 58.60, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 60.80, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 61.11, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 63.91, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 64.23, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 64.69, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 67.59, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 68.12, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 69.78, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 70.53, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 70.60, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 73.05, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 74.72, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 75.18, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 75.42, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 76.59, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 78.25, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 78.59, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 80.11, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 81.41, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 81.76, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 82.08, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 86.08, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 88.59, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 89.20, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 91.36, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 92.72, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 94.83, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 96.69, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 96.77, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 97.66, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 97.68, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 98.69, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 98.75, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 98.96, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 99.38, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (100.25, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (100.44, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (100.87, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (101.21, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (101.94, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (102.69, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (102.87, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (103.68, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (104.55, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (104.68, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (104.70, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (105.78, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (106.97, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (107.59, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (108.96, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (110.11, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (110.48, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (111.68, 33.97) circle (  0.57);

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[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (126.77, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (128.61, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (129.39, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (130.92, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (133.30, 86.49) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 32.81, 99.62) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 34.41, 99.62) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 38.98, 99.62) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 41.47, 99.62) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 89.13, 99.62) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (106.80, 99.62) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (108.14, 99.62) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (110.26, 99.62) circle (  0.57);

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 17.48, 30.94) $N^0$;

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 17.48, 44.07) $N^1$;

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 17.48, 57.20) $N^2$;

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 17.48, 70.33) $N^3$;

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 17.48, 83.46) $N^4$;

[text=drawColor,anchor=base east,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 17.48, 96.59) $N^5$;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 26.18, 19.68) 0.0;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 53.11, 19.68) 2.5;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at ( 80.04, 19.68) 5.0;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at (106.97, 19.68) 7.5;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  0.88] at (133.90, 19.68) 10.0;

[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  1.10] at ( 80.73,  7.64) Time;

(B0) at (0, 3)  $N^0$;
(B1) at (1, 2)  $N^1$;
(B2) at (1, 1)  $N^2$;
(B3) at (0, 0)  $N^3$;
(B4) at (-1, 1) $N^4$;
(B5) at (-1, 2) $N^5$;

(C0) at (3, 3)$N^0$;
(C1) at (4, 2)$N^1$;
(C2) at (4, 1)$N^2$;
(C3) at (3, 0)$N^3$;
(C4) at (2, 1) $N^4$;
(C5) at (2, 2) $N^5$;

at (0, -0.5) First;
at (3, -0.5) Second;
[->] (B0) edge[bend right=5] (B1);
[->] (B0) edge[bend right=5] (B3);
[->] (B1) edge[bend right=5] (B2);
[->] (B1) edge[bend right=5] (B3);
[->] (B2) edge[bend right=5] (B0);
[->] (B2) edge[bend right=5] (B1);
[->] (B2) edge[bend right=5] (B3);
[->] (B2) edge[bend right=5] (B4);
[->] (B2) edge[bend right=5] (B5);
[->] (B3) edge[bend right=5] (B0);
[->] (B3) edge[bend right=5] (B2);
[->] (B3) edge[bend right=5] (B4);
[->] (B4) edge[bend right=5] (B0);
[->] (B4) edge[bend right=5] (B1);
[->] (B4) edge[bend right=5] (B2);
[->] (B4) edge[bend right=5] (B3);
[->] (B5) edge[bend right=5] (B1);
[->] (B5) edge[bend right=5] (B3);
[->] (B5) edge[bend right=5] (B4);

[->] (C1) edge[bend right=5] (C3);
[->] (C2) edge[bend right=5] (C5);
[->] (C4) edge[bend right=5] (C0);
[->] (C4) edge[bend right=5] (C2);
[->] (C4) edge[bend right=5] (C3);

Data and estimated graphs from repetition 2

[draw=drawColor,line width= 0.6pt,line join=round] ( 22.43, 33.97) –
(139.04, 33.97);

[draw=drawColor,line width= 0.6pt,line join=round] ( 22.43, 47.10) –
(139.04, 47.10);

[draw=drawColor,line width= 0.6pt,line join=round] ( 22.43, 60.23) –
(139.04, 60.23);

[draw=drawColor,line width= 0.6pt,line join=round] ( 22.43, 73.36) –
(139.04, 73.36);

[draw=drawColor,line width= 0.6pt,line join=round] ( 22.43, 86.49) –
(139.04, 86.49);

[draw=drawColor,line width= 0.6pt,line join=round] ( 22.43, 99.62) –
(139.04, 99.62);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 29.42, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 35.11, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 35.44, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 37.16, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 37.93, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 38.51, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 41.14, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 41.49, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 42.33, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 44.11, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 45.71, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 46.06, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 46.78, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 47.48, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 49.41, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 50.01, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 50.94, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 52.01, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 52.03, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 52.26, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 53.49, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 56.49, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 58.22, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 58.48, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 58.61, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 61.42, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 63.24, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 64.12, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 64.44, 33.97) circle (  0.57);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 65.06, 33.97) circle (  0.57);