$X^{H,\xi_{W}}_{U_{m}}[\tau_{m})=X^{H_{\ell},\xi_{H_{\ell}}}_{U_{m}}[\tau_{m})\quad\Rightarrow\quad
X^{H,\xi_{W}}_{U_{m-1}}[\tau_{m-1})=X^{H_{\ell},\xi_{H_{\ell}}}_{U_{m-1}}[\tau_{m-1})\text{
a.s. on }\\{U_{m}\subseteq V_{H_{\ell}}\\}.$ (5.23)

Indeed, $U_{m-1}\subseteq U_{m}$ implies $U_{m-1}\subseteq V_{H_{\ell}}$ on
the event $\\{U_{m}\subseteq V_{H_{\ell}}\\}$, and so (5.23) and (5.22), with
$k=m-1$, shows that (5.22) holds for $k=m$. Claim 3B then follows by
induction.

To establish (5.23), assume $U_{m}\subseteq V_{H_{\ell}}$ and
$X^{H,\xi_{W}}_{U_{m}}[\tau_{m})=X^{H_{\ell},\xi_{H_{\ell}}}_{U_{m}}[\tau_{m}).$
Since $U_{m-1}\subset U_{m}$, this implies
$X^{H,\xi_{W}}_{U_{m-1}}[\tau_{m})=X^{H_{\ell},\xi_{H_{\ell}}}_{U_{m-1}}[\tau_{m})$.
Moreover, note from (5.19) that $\tau_{m}$ is the largest time prior to
$\tau_{m-1}$ that there is an event for any of the Poisson processes in
$U_{m-1}$ and hence, $\sum_{v\in
U_{m-1}}\mathbf{N}^{G,T}_{v}(\tau_{m},\tau_{m-1})=0$. The form of the SDE
(3.3) then implies that both $X^{H,\xi_{W}}_{U_{m-1}}$ and
$X^{H_{\ell},\xi_{H_{\ell}}}_{U_{m-1}}$ are constant on
$(\tau_{m},\tau_{m-1})$. Thus, to establish (5.23), it suffices to show that
$X^{H,\xi_{W}}_{U_{m-1}}(\tau_{m})=X^{H_{\ell},\xi_{H_{\ell}}}_{U_{m-1}}(\tau_{m})$
a.s.. Now, by (5.19)-(5.20), $v_{m}$ is the only vertex in $U_{m-1}$ such that
$\mathbf{N}^{G,T}_{v_{m}}(\\{\tau_{m}\\})=1$. Thus,

$X^{H,\xi_{W}}_{U_{m-1}\setminus\\{v_{m}\\}}(\tau_{m})=X^{H,\xi_{W}}_{U_{m-1}\setminus\\{v_{m}\\}}(\tau_{m}-)=X^{H_{\ell},\xi_{H_{\ell}}}_{U_{m-1}\setminus\\{v_{m}\\}}(\tau_{m}-)=X^{H_{\ell},\xi_{H_{\ell}}}_{U_{m-1}\setminus\\{v_{m}\\}}(\tau_{m}),$
(5.24)

and so it only remains to show that
$X^{H,\xi_{W}}_{v_{m}}(\tau_{m})=X^{H_{\ell},\xi_{H_{\ell}}}_{v_{m}}(\tau_{m})$
a.s.. Since for $m\in{\mathbb{N}}$, $\text{cl}_{v_{m}}(G)\subset U_{m}$ by
(5.21), and $U_{m}\subseteq V_{H_{\ell}}\subseteq W$ by assumption,
$\text{cl}_{v_{m}}(H_{\ell})=\text{cl}_{v_{m}}(G)=:\text{cl}_{v_{m}}$. Then
the assumption
$X^{H,\xi_{W}}_{U_{m}}[\tau_{m})=X^{H_{\ell},\xi_{H_{\ell}}}_{U_{m}}[\tau_{m})$
implies
$X^{H,\xi_{W}}_{\text{cl}_{v_{m}}}[\tau_{m})=X^{H_{\ell},\xi_{H_{\ell}}}_{\text{cl}_{v_{m}}}[\tau_{m})$
a.s. The locality and predictability (see Definition 3.1) of the jump rates
stated in the Standing Assumption then imply that for $j\in{\mathcal{J}}$,
denoting by $(\overline{\mathbf{\kappa}},\mathbf{\kappa})$ the marks of
$G[\text{cl}_{v_{m}}]$ we have

$r^{H,v_{m}}_{j}(s,X^{H,\xi_{W}})=\bar{r}^{H[\text{cl}_{v_{m}}]}_{j}(s,X^{H,\xi_{W}}_{\text{cl}_{v_{m}}},\overline{\mathbf{\kappa}},\mathbf{\kappa})=r^{H_{\ell},v_{m}}_{j}(s,X^{H_{\ell},\xi_{H_{\ell}}}),\quad
s\in[0,\tau_{m}].$

Due to the form of the SDE (3.3), this shows
$X^{H,\xi_{W}}_{v_{m}}(\tau_{m})=X^{H_{\ell},\xi_{H_{\ell}}}_{v_{m}}(\tau_{m})$,
as desired. ∎

We now turn to the proof of the second assertion of the proposition. Let
$\\{G_{n}\\}$ be a sequence of deterministic finitely dissociable
$[\overline{{\mathcal{K}}},{\mathcal{K}}]$-graphs, and for each
$n\in{\mathbb{N}}$, let $\mathbf{N}^{G_{n}}$ be a driving noise compatible
with $G_{n}$. Then define $\mathcal{A}^{n}_{t_{1},t_{2}}$ analogously to
$\mathcal{A}_{t_{1},t_{2}}$, but with $G$ and $\mathbf{N}^{G}$ replaced by
$G_{n}$ and $\mathbf{N}^{G_{n}}$, respectively. By the first assertion of the
proposition just established above, for each $n$, $\mathcal{A}^{n}_{T}$ is a
localizing map of the SDE (3.3) on $(G_{n},\mathbf{N}^{G_{n}})$. It follows
that
$\mathcal{S}_{T}(\cdot;G_{n},\mathbf{N}^{G_{n}}):=\text{cl}_{\mathcal{A}^{n}_{T}(\cdot)}(G_{n})$
is likewise a localizing map of the SDE (3.3) on $(G_{n},\mathbf{N}^{G_{n}})$.
Note that $\mathcal{S}_{T}(\cdot;G_{n},\mathbf{N}^{G_{n}})$ satisfies (5.1) as
a consequence of the fact that $\\{\mathcal{A}^{n}_{T}(U)\subset
U^{\prime}\\}$ lies in ${\mathcal{F}}_{T}^{\mathbf{N}^{G_{n}}}$ for every
$U,U^{\prime}\in\Lambda_{G_{n}}$. With this choice of localizing maps
$\mathcal{S}_{T}(\cdot;G_{n},\mathbf{N}^{G_{n}})$, clearly Property 1 of
Definition 5.4 holds.

It only remains to show that Property 2 of Definition 5.4 is also satisfied.
Fix $n,n^{\prime},\ell\in{\mathbb{N}}$ such that there exists an isomorphism
$\varphi\in I(B_{\ell}([G_{n^{\prime},*}]),B_{\ell}([G_{n,*}]))$. Define the
${\mathcal{F}}_{0}$-measurable event

${\mathcal{I}}_{\varphi}:=\\{\varphi\in
I(B_{\ell}([G_{n^{\prime},*}],\mathbf{N}^{G_{n^{\prime}}}),B_{\ell}([G_{n,*}],\mathbf{N}^{G_{n}}))\\}.$

and for notational conciseness, set $W:=B_{\ell-1}(G_{n})$ and
$W^{\prime}:=B_{\ell-1}(G_{n^{\prime}})$. Let $U_{n}\subset V_{G_{n}}$ and
$U_{n^{\prime}}\subset V_{G_{n^{\prime}}}$ be any pair of sets such that
$\varphi(U_{n^{\prime}})=U_{n}$. Then it suffices to prove that on the event
${\mathcal{I}}_{\varphi}\cap\\{W\supseteq\mathcal{A}^{n}_{T}(U_{n})\\}={\mathcal{I}}_{\varphi}\cap\\{B_{\ell}(G_{n})\supseteq\mathcal{S}_{T}(U_{n};G_{n},\mathbf{N}^{G_{n}})\\}$,
$\mathcal{A}^{n}_{T}(U_{n})=\varphi(\mathcal{A}^{n^{\prime}}_{T}(U_{n^{\prime}}))$
a.s..

Let $\Gamma=(v=u_{0},u_{1},\dots,u_{m}\in U_{n})$ be a
$(G_{n},\mathbf{N}^{G_{n},T})$-causal chain up to time $T$ which we will refer
to as a $G_{n}$-causal chain for now. Let $(0=t_{0}<t_{1}<\cdots<t_{n}\leq T)$
be the corresponding times associated with $\Gamma$. By definition of
$\mathcal{A}^{n}_{T}(U_{n})$, $\Gamma\subseteq W$ on the event
${\mathcal{I}}_{\varphi}\cap\\{W\supseteq\mathcal{A}^{n}_{T}(U_{n})\\}$.
Moreover, $\varphi^{-1}|_{W}$ is an isomorphism in
$I(([G_{n,*}[W]],\mathbf{N}^{G_{n}}_{W}),([G_{n^{\prime},*}[W^{\prime}]],\mathbf{N}^{G_{n^{\prime}}}_{W^{\prime}}))$,
so $\varphi^{-1}(\Gamma)$ is a path in $G_{n^{\prime}}$. Furthermore, for
every $k\in\\{1,\dots,m\\}$, $\mathbf{N}^{G_{n}}_{u_{k}}(\\{t_{k}\\})=1$, so
$\mathbf{N}^{G_{n^{\prime}}}_{\varphi^{-1}(u_{k})}(\\{t_{k}\\})=\mathbf{N}^{G_{n}}_{u_{k}}(\\{t_{k}\\})=1$.
It follows that $\varphi^{-1}(\Gamma)$ is also a
$(G_{n^{\prime}},\mathbf{N}^{G_{n^{\prime}}})$-causal chain which we will
refer to as a $G_{n^{\prime}}$-causal chain.

We not claim that for any $G_{n^{\prime}}$-causal chain $\Gamma^{\prime}$,
$\varphi(\Gamma^{\prime})$ is a $G_{n}$-causal chain. To see why this claim
holds, let
$\Gamma^{\prime}=(v^{\prime}=u^{\prime}_{0},u^{\prime}_{1},\dots,u^{\prime}_{m}\in
U_{n^{\prime}})$. By an identical argument to what we applied in the previous
paragraph, if $\Gamma^{\prime}\subseteq W^{\prime}$, then
$\varphi(\Gamma^{\prime})$ is a $G_{n}$-causal chain. We now argue by
contradiction to justify that this is the only possible case. Indeed, suppose
there exists $k\in\\{0,1,\dots,m\\}$ such that $u^{\prime}_{k}\notin
W^{\prime}$. Note that $u^{\prime}_{m}\in U_{n^{\prime}}\subseteq W^{\prime}$,
so $k<m$. Choose the maximal such $k$ so that
$(u^{\prime}_{k+1},\dots,u^{\prime}_{m})\subseteq W^{\prime}$. Then because
subpaths of causal chains are causal chains,
$\overline{\Gamma}^{\prime}:=(u^{\prime}_{k},\dots,u^{\prime}_{m})$ is also a
$G_{n^{\prime}}$-causal chain. Furthermore,
$u^{\prime}_{k}\in{\mathcal{N}}_{W^{\prime}}(G_{n^{\prime}})$, so on the event
${\mathcal{I}}_{\varphi}$, $\varphi(\overline{\Gamma}^{\prime})$ is a path and
a causal chain in $G_{n}$. Moreover,
$\varphi(\overline{\Gamma}^{\prime})\nsubseteq W$ which contradicts our
assumption that $\mathcal{A}_{T}^{n}(U_{n})\subseteq W$. This proves the
claim.

Because $\varphi$ a.s. induces a bijection between causal chains in $G_{n}$
and $G_{n^{\prime}}$ that end in $U_{n}$ and $U_{n^{\prime}}$ respectively, it
follows that
$\mathcal{A}_{T}^{n}(U_{n})=\varphi(\mathcal{A}_{T}^{n^{\prime}}(U_{n^{\prime}}))$
on the event
${\mathcal{I}}_{\varphi}\cap\\{\mathcal{A}_{T}^{n}(U_{n})\subseteq W\\}$.
Equivalently,
$\mathcal{S}_{T}(U_{n};G_{n},\mathbf{N}^{G_{n}})=\varphi(\mathcal{S}_{T}(U_{n^{\prime}};G_{n^{\prime}},\mathbf{N}^{G_{n^{\prime}}}))$
on the event $\\{\mathcal{S}_{T}(U_{n};G_{n},\mathbf{N}^{G_{n}})\subseteq
B_{\ell}(G_{n})\\}\cap{\mathcal{I}}_{\varphi}$. It follows that Property 2 of
Definition 5.4 also holds for the sequence $\\{G_{n}\\}_{n\in{\mathbb{N}}}$.

This proves the second assertion and hence, concludes the proof of the
proposition. ∎

##  6 Local Convergence of the IPS

The main goal of this section is to prove the local convergence result of
Theorem 4.3. When the initial data have finite, deterministic and equal
unmarked representatives $[G_{n,*}]=[G_{*}]$, $n\in{\mathbb{N}}$,
$|V_{G}|<\infty$, and the corresponding SDEs in (3.3) are driven by the same
Poisson processes for different $n$, Theorem 4.3 is essentially a consequence
of the continuity condition in Assumption 2, which implies pathwise continuity
of the dynamics of the SDE (3.3) with respect to the initial data. For more
general finite initial data, the proof entails carefully constructed
couplings, and when dealing with infinite graphs, it will also involve
applications of consistent spatial localization to reduce the analysis to the
finite graph case. While the localizing maps associated with consistent
spatial localization (see Definition 5.4) only make sense on graphs, rather
than their equivalence classes, local convergence results are defined in terms
of equivalence classes. To bridge this gap, we need to carefully select
suitable representatives of equivalence classes, as well as establish
correspondences between statements about convergence of equivalence classes
and statements about representative graphs.

Section 6.1 defines canonical representatives, with a view to constructing
suitable couplings. These are then used in Section 6.2 to establish a more
general almost sure local convergence result in Proposition 6.11). The latter
result is used in Section 6.3 to prove Theorem 4.3, and also in Section 7 in
the proof of the hydrodynamic limit.

###  6.1 Canonical Representative Graphs and Consistent Extensions

Let $\overline{{\mathcal{Z}}},{\mathcal{Z}}$ be Polish spaces. We begin by
defining a space of $[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$-graphs:

$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]=\\{[\overline{{\mathcal{Z}}},{\mathcal{Z}}]\text{-graphs
}G:=(V,E,\o,\overline{\vartheta},\vartheta)\text{ s.t.
}V\subseteq{\mathbb{N}}\\}.$ (6.1)

It follows from Lemmas B.5, B.6 and B.7 that
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ can be
equipped with a Polish topology that is compatible with the topology of
${\mathscr{G}}_{*}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$, and that for any
${\mathscr{G}}_{*}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$-random element
$\langle G\rangle$, there exists a $\sigma(\langle G\rangle)$-measurable
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$-random
element $G$ such that $G\in\langle G\rangle$ almost surely. In the latter
case, $G$ is referred to as a
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}]$-random
representative of $\langle G\rangle$, and thus,
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ can be
viewed as a canonical space of measurable representative graphs compatible
with the local topology. We begin with the definition of a representative
convergent sequence.

######  Definition 6.1 (Representative convergent sequences).

Let $\langle(G_{n},\xi^{n})\rangle$ be a random sequence converging a.s. to
$\langle(G,\xi)\rangle$ in
${\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$
on some complete probability space. Then a _representative convergent
sequence_ (henceforth abbreviated to _rep-con sequence_) of
$(\\{\langle(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}},\langle(G,\xi)\rangle)$
is a
$\sigma(\\{\langle(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}},\langle(G,\xi)\rangle)$-measurable
tuple
$\left(\\{(G_{n},\xi^{n}),M_{n}\\}_{n\in{\mathbb{N}}},(G,\xi),\\{\varphi_{n,m}\\}_{n\in{\mathbb{N}},m\leq
M_{n}}\right)$, defined on the same probability space, that satisfies the
following properties:

  1. 1.

for each $n\in{\mathbb{N}}$,
$(G_{n},\xi^{n})=(V_{n},E_{n},\o_{n},\overline{\mathbf{\kappa}}^{n},\mathbf{\kappa}^{n},\xi^{n})$
is a
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$-random
representative of $\langle(G_{n},\xi^{n})\rangle$;

  2. 2.

$(G,\xi)=(V,E,\o,\overline{\mathbf{\kappa}},\mathbf{\kappa},\xi)$ is a
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$-random
representative of $\langle(G,\xi)\rangle$;

  3. 3.

$\lim_{n\to\infty}M_{n}=\infty$ a.s. and $B_{m}([G_{*}],\xi)\cong
B_{m}([G_{n,*}],\xi^{n})$ on the event $\\{m\leq M_{n}\\}$;

  4. 4.

for each $n\in{\mathbb{N}},m\leq m^{\prime}\leq M_{n}$, $\varphi_{n,m}\in
I(B_{m}([G_{*}],\xi),B_{m}([G_{n,*}],\xi^{n}))$ and
$\varphi_{n,m^{\prime}}|_{B_{m}(G)}=\varphi_{n,m}$;

  5. 5.

for every $n^{\prime},m\in{\mathbb{N}}$ such that $m\leq M_{n^{\prime}}$ and
$e\in E_{B_{m}(G)}$,
$\lim_{n\to\infty,n>n^{\prime}}\overline{\mathbf{\kappa}}^{n}_{\varphi_{n,m}(e)}=\overline{\mathbf{\kappa}}_{e}$;

  6. 6.

for every $n^{\prime},m\in{\mathbb{N}}$ such that $m\leq M_{n^{\prime}}$ and
$v\in B_{m}(G)$,
$\lim_{n\to\infty,n>n^{\prime}}\mathbf{\kappa}^{n}_{\varphi_{n,m}(v)}=\mathbf{\kappa}_{v}$.

The next lemma guarantees the existence of rep-con sequences. A constructive
proof of the lemma, which leverages the existence of suitably measurable
representative graph sequences, isomorphisms and driving maps is deferred to
in Appendix B.2.

######  Lemma 6.2.

Fix a complete probability space
$(\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}})$ that
supports the random sequence $\langle(G_{n},\xi^{n})\rangle$ converging a.s.
to $\langle(G,\xi)\rangle$ in
${\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$.
Then there exists an $\widehat{{\mathcal{F}}}$-measurable sequence of
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$-random
elements
$\\{(\overline{G}_{n},\overline{\xi}^{n})\\}_{n\in{\mathbb{N}}},(\overline{G},\overline{\xi})$
satisfying Properties 1 and 2 of Definition 6.1. In addition, given any
sequence $\\{(G_{n},\xi^{n})\\}_{n\in{\mathbb{N}}},(G,\xi)$ satisfying
Properties 1 and 2 of Definition 6.1, the probability space
$(\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}})$ also
supports a rep-con sequence
$\left(\\{(G_{n},\xi^{n}),M_{n}\\},(G,\xi),\\{\varphi_{n,m}\\}\right)$.

To extend the notion of rep-con sequences from initial data consisting of
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$-random
elements to the corresponding IPS characterized by
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{D}}]$-random
elements, we will need a common probability space on which we can define both
the driving noise and a rep-con sequence.

######  Definition 6.3 (Consistent representative convergent extensions).

Given a complete probability space
$(\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}})$ that
supports a sequence $\langle(G_{n},\xi^{n})\rangle$ converging a.s. to
$\langle(G,\xi)\rangle$ in
${\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$,
a _consistent representative convergent extension_ (henceforth abbreviated to
consistent rep-con extension) of
$((\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}}),\\{\langle(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}},\langle(G,\xi)\rangle)$
is a 4-tuple
$((\Omega,{\mathcal{F}},{\mathbb{F}},{\mathbb{P}}),\\{(G_{n},\xi^{n},\mathbf{N}^{G_{n}}),M_{n}\\}_{n\in{\mathbb{N}}}$,
$(G,\xi,\mathbf{N}^{G}),\\{\varphi_{n,m}\\}_{n\in{\mathbb{N}},m\leq M_{n}})$
such that

  1. 1.

$(\\{(G_{n},\xi^{n}),M_{n}\\},(G,\xi),\\{\varphi_{n,m}\\})$ is a rep-con
sequence of
$(\\{\langle(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}},\langle(G,\xi)\rangle)$;

  2. 2.

$(\Omega,{\mathcal{F}},{\mathbb{F}},{\mathbb{P}})$ is a complete extension of
the probability space
$(\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}})$ such that
${\mathbb{F}}$ satisfies the usual conditions and
$(G_{n},\mathbf{N}^{G_{n}})$, $n\in{\mathbb{N}},$ and $(G,\mathbf{N}^{G})$,
respectively, are ${\mathbb{F}}$-driving noises that are compatible with
$G_{n}$, $n\in{\mathbb{N}},$ and $G$, as defined in Definition 3.3;

  3. 3.

for each $n\in{\mathbb{N}}$ and $v\in B_{M_{n}}(G)$,
$\mathbf{N}^{G_{n}}_{\varphi_{n,M_{n}}(v)}=\mathbf{N}^{G}_{v}$ a.s..

The construction of consistent rep-con extensions is facilitated by the use of
so-called _driving maps_ defined below. Let ${\mathcal{M}}$ be the space of
maps from subspaces $W\subseteq{\mathbb{N}}$ to ${\mathbb{N}}$, which can be
equipped with a Polish topology by Remark B.8.

######  Definition 6.4.

Given a
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$-random
element $(G,\xi):=(V,E,\o,\overline{\mathbf{\kappa}},\mathbf{\kappa},\xi)$, a
_driving map_ is a random injective map $\psi:V\to{\mathbb{N}}$, that is,
$\psi$ is a $\sigma(G,\xi)$-measurable random element taking values in
${\mathcal{M}}$. Suppose $(\Omega,{\mathcal{F}},{\mathbb{F}},{\mathbb{P}})$ is
a filtered probability space supporting a collection of i.i.d.
${\mathbb{F}}$-Poisson processes $\\{\mathbf{N}_{k}\\}_{k\in{\mathbb{N}}}$ on
${\mathbb{R}}_{+}^{2}\times{\mathcal{J}}$ with intensity ${\rm
Leb}^{2}\otimes\varsigma$. Suppose also that ${\mathbb{F}}$ satisfies the
usual conditions and $(G,\xi)$ is ${\mathcal{F}}_{0}$-measurable. Then the
$[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{M}}_{{\mathbb{N}}}({\mathbb{R}}_{+}^{2}\times{\mathcal{J}})]$-random
graph $(G,\mathbf{N}^{G})$ defined by
$\mathbf{N}^{G}_{v}=\mathbf{N}_{\psi(v)}$ is said to be an
${\mathbb{F}}$-driving noise generated by $\psi$.

######  Remark 6.5.

We now justify our reference to $(G,\mathbf{N}^{G})$ as a
${\mathbb{F}}$-driving noise in Definition 6.4. It is easy to see that on any
complete, filtered probability space
$(\Omega,{\mathcal{F}},{\mathbb{F}},{\mathbb{P}})$ supporting the
${\mathcal{F}}_{0}$-measurable initial data $(G,\xi)$ and
${\mathbb{F}}$-Poisson processes $\\{\mathbf{N}_{k}\\}_{k\in{\mathbb{N}}}$,
the $\sigma(G,\xi)$-measurability (and thus ${\mathcal{F}}_{0}$-measurability)
of driving maps ensures that for any ${\mathcal{F}}_{0}$-measurable $v\in
V_{G}$, $\mathbf{N}^{G}_{v}$ is ${\mathbb{F}}$-adapted and therefore supported
on $(\Omega,{\mathcal{F}},{\mathbb{F}},{\mathbb{P}})$, and Condition 3 of
Definition 3.3 is satisfied. Furthermore, conditioned on ${\mathcal{F}}_{0}$,
$\mathbf{N}^{G}$ is a collection of i.i.d. Poisson processes indexed by
$V_{G}$. Thus, Condition 1 of Definition 3.3 is satisfied. Lastly, because
$\\{\mathbf{N}_{k}\\}_{k\in{\mathbb{N}}}$ are ${\mathbb{F}}$-Poisson
processes, for any $t>0$ and
$A\in{\mathcal{B}}((t,\infty)\times{\mathbb{R}}_{+}\times{\mathcal{J}})$,
$(G,\mathbf{N}^{G}(A))$ is conditionally independent of ${\mathcal{F}}_{t}$
given ${\mathcal{F}}_{0}$ so Condition 2 of Definition 3.3 is satisfied.
Assuming ${\mathbb{F}}$ satisfies the usual conditions, all three conditions
of Definition 3.3 are satisfied.

The following lemma shows that there always exists a consistent rep-con
extension of any convergent sequence of random elements in
${\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$.
We prove this at the end of Appendix B.

######  Lemma 6.6.

Let $\\{\langle(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}}$ be a random
sequence on $(\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}})$
that converges a.s. to $\langle(G,\xi)\rangle$ in
${\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}]$. Then there exists
a consistent rep-con extension
$((\Omega,{\mathcal{F}},{\mathbb{F}},{\mathbb{P}}),\\{(G_{n},\xi^{n},\mathbf{N}^{G_{n}}),M_{n}\\}_{n\in{\mathbb{N}}},(G,\xi,\mathbf{N}^{G}),\\{\varphi_{n,m}\\}_{n\in{\mathbb{N}},m\leq
M_{n}})$ of
$((\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}}),\\{\langle(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}},\langle(G,\xi)\rangle)$
such that $(\Omega,{\mathcal{F}},{\mathbb{F}},{\mathbb{P}})$ supports a
collection of i.i.d. ${\mathbb{F}}$-Poisson processes
$\mathbf{N}:=\\{\mathbf{N}_{k}\\}_{k\in{\mathbb{N}}}$ and driving maps
$\psi_{n},$ $n\in{\mathbb{N}}$, and $\psi$ that generate the respective
${\mathbb{F}}$-driving maps $\mathbf{N}^{G_{n}}$, $n\in{\mathbb{N}}$, and
$\mathbf{N}^{G}$.

Consistent rep-con extensions are useful because, as shown in Proposition 6.11
of the next section, under the conditions of Theorem 4.3, a sequence of IPS
will converge almost surely if it is generated by a consistent rep-con
extension in the following sense.

######  Definition 6.7.

A sequence of IPS
$\\{(G_{n},X^{G_{n},\xi^{n}})\\}_{n\in{\mathbb{N}}},(G,X^{G,\xi})$ is said to
be generated by a consistent rep-con extension
$((\Omega,{\mathcal{F}},{\mathbb{F}},{\mathbb{P}}),\\{(G_{n},\xi^{n},\mathbf{N}^{G_{n}}),M_{n}\\}_{n\in{\mathbb{N}}},(G,\xi,\mathbf{N}^{G}),\\{\varphi_{n,m}\\}_{n\in{\mathbb{N}},m\leq
M_{n}})$ if for each $n\in{\mathbb{N}}$, $(G,X^{G_{n},\xi^{n}})$ is a
$\mathbf{N}^{G_{n}}$-strong solution to (3.3) for $(G_{n},\xi^{n})$ and,
likewise, $(G,X^{G,\xi})$ is a $\mathbf{N}^{G}$-strong solution to (3.3) for
$(G,\xi)$.

Not every sequence of IPS can be generated by a consistent rep-con extension
of its initial data. For example, this would fail if
$\\{\langle(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}},\langle(G,\xi)\rangle$
is a deterministic sequence of isomorphism classes such that
$\langle(G_{n},\xi^{n})\rangle\to\langle(G,\xi)\rangle$ in
${\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$,
and $\\{(G,X^{G_{n},\xi^{n}})\\}_{n\in{\mathbb{N}}},(G,X^{G,\xi})$ are
mutually independent. However, as noted below, consistent rep-con extensions
are preserved under conditioning on ${\mathcal{F}}_{0}$.

######  Remark 6.8.

If $\\{(G_{n},X^{G_{n},\xi^{n}})\\}_{n\in{\mathbb{N}}},(G,X^{G,\xi})$, are IPS
generated by a consistent rep-con extension of
$((\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}}),\\{\langle(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}},\langle(G,\xi)\rangle)$,
then for almost every $\omega\in\Omega$,
$\mathcal{L}(\\{(G_{n},X^{G_{n},\xi^{n}})\\}_{n\in{\mathbb{N}}},(G,X^{G,\xi})|{\mathcal{F}}_{0})(\omega)$
describes the law of a sequence of IPS generated by a consistent rep-con
extension of a tuple that contains the terms
$\\{\langle(G_{n},\xi^{n})\rangle(\omega)\\}_{n\in{\mathbb{N}}}$ and
$\langle(G,\xi)\rangle(\omega)$. A rigorous justification of this would follow
along the same lines as Lemma 3.8 and is thus omitted.

###  6.2 Proof of Local Convergence in the almost sure setting

We start by stating a “finite convergence” condition on IPS defined on finite
truncations of graphs (equipped with the associated marks).

######  Assumption 2′.

Given any consistent rep-con extension
$((\Omega,{\mathcal{F}},{\mathbb{F}},{\mathbb{P}}),\\{(G_{n},\xi^{n},\mathbf{N}^{G_{n}}),M_{n}\\}_{n\in{\mathbb{N}}},\\\
(G,\xi,\mathbf{N}^{G}),\\{\varphi_{n,m}\\}_{n\in{\mathbb{N}},m\leq M_{n}})$ of
$((\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}}),\\{\langle(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}},\langle(G,\xi)\rangle)$,
$T\in{\mathbb{R}}_{+}$ and $m\in{\mathbb{N}}$, there exists an a.s. finite,
${\mathcal{F}}_{T}$-measurable random variable
$\overline{N}_{m}:=\overline{N}_{m,T}$ such that for every $n\in{\mathbb{N}}$,

$\left(B_{m}([G_{n,*}]),X^{m,n}[T]\right)\cong\left(B_{m}([G_{*}]),X^{m,\infty}[T]\right)\text{
a.s. on the event }\\{n\geq\overline{N}_{m}\\}\cap\\{m\leq M_{n}\\},$ (6.2)

where $X^{m,n}$ and $X^{m,\infty}$ are the respective
$\mathbf{N}^{G_{n}}_{B_{m}(G_{n})}$\- and $\mathbf{N}^{G}_{B_{m}(G)}$-strong
solutions to (3.3) for the initial data $B_{m}(G_{n},\xi^{n})$ and
$B_{m}(G,\xi)$.

######  Remark 6.9.

Given Assumption 1′, as a consequence of Remark 6.8, if Assumption 2′ holds
for a tuple
$((\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}}),\\{\langle(G_{n},\xi^{n})\rangle\\},\langle(G,\xi)\rangle)$
then it holds for a.s. every realization of the isomorphism classes
$\\{\langle(G_{n},\xi^{n})\rangle\\},\langle(G,\xi)\rangle$.

The following lemma shows that the finite convergence property of Assumption
2′ holds under our basic Assumptions 1 and 2, together with consistent spatial
localization.

######  Lemma 6.10.

Suppose the family of jump rate functions $\bf{r}$ satisfies Assumption 1, and
$\langle(G,\xi)\rangle$ satisfies Assumption 2. If the sequence $\\{\langle
G_{n}\rangle\\}_{n\in{\mathbb{N}}},\langle G\rangle$ defined on
$(\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}})$ consistently
spatially localizes the SDE (3.3), then the tuple
$((\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}}),\\{\langle(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}},\langle(G,\xi)\rangle)$
satisfies Assumption 2′.

The weaker Assumptions 1′ and 2′ have been introduced in place of Assumptions
1 or Assumption 2 with a view to future extensions of our results to more
general IPS for which the latter assumptions may not hold but the former
assumptions and consistent spatial localization can nevertheless be
established. Indeed, as we now show, the main convergence result of this
section holds under these weaker assumptions. We defer the proof of Lemma 6.10
to after the proof of this main result.

######  Proposition 6.11.

Suppose that Assumption 1′ holds, the tuple
$((\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}}),\\{\langle(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}}$,
$\langle(G,\xi)\rangle)$ satisfies Assumption 2′ and $\\{\langle
G_{n}\rangle\\}_{n\in{\mathbb{N}}},\langle G\rangle$ a.s. consistently
spatially localizes the SDE (3.3). Also, let
$\\{(G_{n},X^{G_{n},\xi^{n}})\\}_{n\in{\mathbb{N}}},(G,X^{G,\xi})$ be a
collection of IPS generated by a consistent rep-con extension of the tuple. If
$\langle(G_{n},\xi^{n})\rangle\to\langle(G,\xi)\rangle$ a.s. in
${\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$,
then $\langle(G_{n},X^{G_{n},\xi^{n}})\rangle\to\langle(G,X^{G,\xi})\rangle$
a.s. in
${\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{D}}]$.

###### Proof.

Let ${\mathbb{N}}_{\infty}:={\mathbb{N}}\cup\\{\infty\\}$ and let
$\langle(G_{\infty},\xi^{\infty})\rangle:=\langle(G,\xi)\rangle$, and
$(G_{\infty},X^{G_{\infty},\xi^{\infty}}):=(G,X^{G,\xi})$ respectively. We
first prove the proposition under the additional assumption that
$\\{(G_{n},\xi^{n})\\}_{n\in{\mathbb{N}}_{\infty}}$ are deterministic. In this
case, let
$((\Omega,{\mathcal{F}},{\mathbb{F}},{\mathbb{P}}),\\{(G_{n},\xi^{n},\mathbf{N}^{G_{n}}),M_{n}\\}_{n\in{\mathbb{N}}},(G,\xi,\mathbf{N}^{G}),\\{\varphi_{n,m}\\}_{n\in{\mathbb{N}},m\leq
M_{n}})$ be a consistent rep-con extension of
$((\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}}),\\{\langle(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}_{\infty}})$,
where $(G_{\infty},\xi^{\infty}):=(G,\xi)$. In addition, let
$\mathbf{N}^{G_{\infty}}:=\mathbf{N}^{G}$. Due to Assumption 1′ and the
assumption of of consistent spatial localization, there exists a consistent
sequence of localizing maps
$\\{\mathcal{S}_{T}(\cdot;G_{n},\mathbf{N}^{G_{n}})\\}_{n\in{\mathbb{N}}_{\infty}}$
for the SDE (3.3) on
$\\{(G_{n},\mathbf{N}^{G_{n}})\\}_{n\in{\mathbb{N}}_{\infty}}$, and by
Proposition 5.7, for each $n\in{\mathbb{N}}_{\infty}$, the SDE (3.3) is
strongly well-posed for $(G_{n},\xi^{n})$. Hence, the
$\mathbf{N}^{G_{n}}$-strong solution $X^{G_{n},\xi^{n}}$ to (3.3) for the
initial data $(G_{n},\xi^{n})$ is well defined.

Fix $n^{\prime},m,M\in{\mathbb{N}}$ such that $m\leq M\leq M_{n^{\prime}}$. By
Property 3 of Definition 6.3, for every $n\in{\mathbb{N}}$, $m\leq M_{n}$ and
$v\in B_{m}(G)$, $\mathbf{N}^{G_{n}}_{\varphi_{n,m}(v)}=\mathbf{N}^{G}_{v}$.
Thus, by an application of Property 2 of the consistent spatial localization
property in Definition 5.4, with $\ell=M$, $n=\infty$,
$n^{\prime}=n^{\prime}$, $\varphi=\varphi_{n^{\prime},M}^{-1}$ and
$U=B_{m}(G)$, and noting that then ${\mathcal{I}}_{\varphi}=\\{M\leq
M_{n^{\prime}}\\}=\Omega$, we see that

$\mathcal{S}_{T}(U;G,\mathbf{N}^{G})=\varphi(\mathcal{S}_{T}(\varphi^{-1}(U);G_{n^{\prime}},\mathbf{N}^{G_{n^{\prime}}}))\Rightarrow\mathcal{S}_{T}(\varphi_{n^{\prime},M}(U);G_{n^{\prime}},\mathbf{N}^{G_{n^{\prime}}})=\varphi_{n^{\prime},M}(\mathcal{S}_{T}(U;G,\mathbf{N}^{G}))$

a.s. on the event $\\{\mathcal{S}_{T}(U;G,\mathbf{N}^{G})\subseteq
B_{M}(G)\\}$. Because $\mathcal{S}_{T}(U;G,\mathbf{N}^{G})$ is a.s. finite,
there must exist an ${\mathcal{F}}_{T}$-measurable a.s. finite random variable
$\overline{M}_{m}$ such that $\mathcal{S}_{T}(U;G,\mathbf{N}^{G})\subseteq
B_{\overline{M}_{m}}(G)$ a.s.. Furthermore, by (5.3) this implies that

$\displaystyle X^{G,\xi}_{U}[T]$
$\displaystyle=X^{B_{M}(G),\xi_{B_{M}(G)}}_{U}[T]\quad\text{and}\quad
X^{G_{n^{\prime}},\xi^{n^{\prime}}}_{\varphi_{n^{\prime},M}(U)}[T]=X^{B_{M}(G_{n^{\prime}}),\xi^{n^{\prime}}_{B_{M}(G_{n^{\prime}})}}_{\varphi_{n^{\prime},M}(U)}[T]\text{
a.s.\ on }\\{M\geq\overline{M}_{m}\\}.$

By (6.2) of Assumption 2′, there exists an a.s. finite,
${\mathcal{F}}_{T}$-measurable random variable $N_{M}$ such that

$X^{B_{M}(G),\xi_{B_{M}(G)}}_{U}[T]=X^{B_{M}(G_{n^{\prime}}),\xi^{n^{\prime}}_{B_{M}(G_{n^{\prime}})}}_{\varphi_{n^{\prime},M}(U)}[T]\text{
a.s. on the event }\\{n^{\prime}\geq N_{M}\\}.$

The last two displays together show that a.s. on the event
$\\{M\geq\overline{M}_{m}\\}\cap\\{n^{\prime}\geq N_{M}\\}$,

$\displaystyle X^{G,\xi}_{U}[T]$
$\displaystyle=X^{B_{M}(G),\xi_{B_{M}(G)}}_{U}[T]=X^{B_{M}(G_{n^{\prime}}),\xi^{n^{\prime}}_{B_{M}(G_{n^{\prime}})}}_{\varphi_{n^{\prime},M}(U)}[T]=X^{G_{n^{\prime}},\xi^{n^{\prime}}}_{\varphi_{n^{\prime},M}(U)}[T].$

Applying the last display for each $M$ satisfying $m\leq M\leq M_{n^{\prime}}$
and noting by Property 1 of Definition 5.1 that $\overline{M}_{m}\geq m$, it
follows that

$X^{G,\xi}_{U}[T]=X^{G_{n^{\prime}},\xi^{n^{\prime}}}_{\varphi_{n^{\prime},\overline{M}_{m}}(U)}[T]\text{
a.s. on the event }\\{\overline{M}_{m}\leq
M_{n^{\prime}}\\}\cap\\{n^{\prime}\geq N_{\overline{M}_{m}}\\}.$

Sending $n^{\prime}\to\infty$ and noting that then
$M_{n^{\prime}}\rightarrow\infty$ by Property 3 of Definition 6.1, and
$\overline{M}_{m}$ is a.s. finite it follows that

$\lim_{n^{\prime}\to\infty}X^{G_{n^{\prime}},\xi^{n^{\prime}}}_{\varphi_{n^{\prime},\overline{M}_{m}}(U)}[T]=X^{G,\xi}_{U}[T]\text{
a.s. }$

This concludes the proof for deterministic sequences.

The random case can be obtained by conditioning on the initial data. More
precisely, suppose $\\{\langle(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}}$,
$\langle(G,\xi)\rangle$ is random and let
$\\{(G_{n},X^{G_{n},\xi^{n}})\\}_{n\in{\mathbb{N}}}$, $(G,X^{G,\xi})$ be the
sequence of IPS generated by a consistent rep-con extension of the tuple
$((\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}}),\\{\langle(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}},\langle(G,\xi)\rangle)$.
Then by Remark 6.8, for almost every $\omega\in\Omega$,
$\mathcal{L}(\\{(G_{n},X^{G_{n},\xi^{n}})\\},(G,X^{G,\xi})|{\mathcal{F}}_{0})(\omega)$
describes the law of a sequence of IPS generated by a consistent rep-con
extension of a tuple that contains the terms
$\\{\langle(G_{n},\xi^{n})\rangle(\omega)\\}_{n\in{\mathbb{N}}},\langle(G,\xi)\rangle(\omega)$.
Moreover, by Remark 6.9, Assumption 2′ holds a.s. conditioned on
${\mathcal{F}}_{0}$. Therefore, by the proof of the proposition for
deterministic initial data,
${\mathbb{P}}(\langle(G_{n},X^{G_{n},\xi^{n}})\rangle\to\langle(G,X^{G,\xi})\rangle|{\mathcal{F}}_{0})=1$
a.s.. Therefore,
$\langle(G_{n},X^{G_{n},\xi^{n}})\rangle\to\langle(G,X^{G,\xi})\rangle$ a.s.,
as desired. ∎

We finish the section with a proof of Lemma 6.10.

###### Proof of Lemma 6.10:.

Fix a consistent rep-con extension of the given tuple, write
$G=(V,E,\o,\overline{\mathbf{\kappa}},\mathbf{\kappa})$,
$G_{n}=(V_{n},E_{n},\o_{n},\overline{\mathbf{\kappa}}^{n},\mathbf{\kappa}^{n})$,
$n\in{\mathbb{N}}$, and set $M:=\max_{v\in V}d_{G}(v,\o)$. We first consider
the case when $\\{G_{n}\\}_{n\in{\mathbb{N}}}$ and $G$ are a.s. finite graphs
and additionally assume that $\max_{v\in V_{n}}d_{G_{n}}(v,\o_{n})\leq M$ for
all $n\in{\mathbb{N}}$, and show that for any $T\in{\mathbb{R}}_{+}$, there
exists an a.s. finite, ${\mathcal{F}}_{T}$-measurable random variable
$\overline{N}:=\overline{N}_{M,T}$ such that,

$\left([G_{n,*}],X^{G_{n},\xi^{n}}[T]\right)\cong\left([G_{*}],X^{G,\xi}[T]\right)\text{
a.s. on the event }\\{n\geq\overline{N}\\}\cap\\{M\leq M_{n}\\}.$ (6.3)

Note that on the event $A_{n}:=\\{M\leq M_{n}\\}$,
$([G_{n,*}],\xi^{n})\cong([G_{*}],\xi^{n})$. To show (6.3), fix
$T\in{\mathbb{R}}_{+}$ and for each $n\in{\mathbb{N}}$, let $\varphi_{n}:V\to
V_{n}$ be the ${\mathcal{F}}_{0}$-measurable map given by
$\varphi_{n}:=\varphi_{n,M}$ on the event $A_{n}$ (on $A_{n}^{c}$, we may
define $\varphi_{n}$ to be any measurable function with the appropriate domain
and range, for instance, the function that maps all vertices of $G$ to the
root of $G_{n}$). For each $n\in{\mathbb{N}}$ and $v\in V_{G}$, recall from
Property 3 of Definition 6.3 that
$\mathbf{N}^{G_{n}}_{\varphi_{n}(v)}=\mathbf{N}^{G}_{v}$ on the event $A_{n}$.
In terms of the family of constants $\\{C_{k,T}\\}_{k\in{\mathbb{N}},T>0}$
from Assumption 1, define

$\overline{{\mathcal{E}}}=\overline{{\mathcal{E}}}_{T}:=\left\\{(v,t,r,j)\in
V\times[0,T]\times(0,C_{|\text{cl}_{v}(G)|,T}]\times{\mathcal{J}}:\mathbf{N}^{G}_{v}\left(\\{(t,r,j)\\}\right)=1\right\\}.$

Since $|\overline{{\mathcal{E}}}|<\infty$ a.s., we can a.s. order the elements
$\\{(v_{k},\tau_{k},r_{k},j_{k})\\}_{k=1}^{|\overline{{\mathcal{E}}}|}$ of
$\overline{{\mathcal{E}}}$ such that $\\{\tau_{k}\\}_{k\in{\mathbb{N}}}$ is
strictly increasing. Note that $\\{\tau_{k}\\}_{k\in{\mathbb{N}}}$ is the
sequence of points in a time-homogeneous Poisson process, which implies that
each $\tau_{k}$ is an absolutely continuous (Gamma distributed) non-negative
random variable. Thus, for any Borel set $O\subset[0,T]$ such that ${\rm
Leb}(O)=0$, ${\mathbb{P}}(\tau_{k}\in O\text{ for any
}k)=\sum_{k=1}^{\infty}{\mathbb{P}}(\tau_{k}\in
O,k\leq|\overline{{\mathcal{E}}}|)=0$. Let
$R:=\inf\left\\{|r_{k}-r^{G,v_{k}}_{j_{k}}(\tau_{k},X^{G,\xi})|:k=1,\dots,|\overline{{\mathcal{E}}}|\right\\}$.
By the predictability of the jump rate function $r^{G,v_{k}}_{j_{k}}$, the
random variables $r_{k}$ and $r^{G,v_{k}}_{j_{k}}(\tau_{k},X^{G,\xi})$ are
independent and therefore $r_{k}\neq r^{G,v_{k}}_{j_{k}}(\tau_{k},X^{G,\xi})$
a.s. which implies that $R>0$ a.s.. On the event $A_{n}$, let

$R_{n}:=\sup\left\\{\left|r^{G,v_{k}}_{j_{k}}(\tau_{k},X^{G,\xi})-r^{G_{n},\varphi_{n}(v_{k})}_{j_{k}}(\tau_{k},X^{G,\xi}_{\varphi^{-1}_{n}(V_{n})})\right|:k=1,\dots,|\overline{{\mathcal{E}}}|\right\\},$

where
$X^{G,\xi}_{\varphi^{-1}_{n}(V_{n})}=(X^{G,\xi}_{\varphi^{-1}_{n}(v)})_{v\in
V_{n}}$. By Property 3 of Definition 6.1, $M_{n}$ a.s. diverges to infinity,
so $\\{A_{n}\\}_{n\in{\mathbb{N}}}$ is a sequence of events such that
${\mathbb{P}}(\cap_{n^{\prime}\geq n}A_{n^{\prime}})\to 1$ so that
$\lim_{n\to\infty}\mathbb{I}_{\left\\{A_{n}\right\\}}=1$ a.s.. By Properties 5
and 6 of Definition 6.1, it follows that for each $v\in V$ and $e\in E$,

$\lim_{n\to\infty}\mathbb{I}_{A_{n}}(\overline{\mathbf{\kappa}}^{n}_{\varphi_{n}(e)},\mathbf{\kappa}^{n}_{\varphi_{n}(v)})=(\overline{\mathbf{\kappa}}_{e},\mathbf{\kappa}_{v})\quad\text{a.s..}$
(6.4)

Let
$\overline{\mathbf{\kappa}}^{n}_{\varphi_{n}(E)}=(\overline{\mathbf{\kappa}}^{n}_{\varphi_{n}(e)})_{e\in
E}$ and
$\mathbf{\kappa}^{n}_{\varphi_{n}(V)}=(\mathbf{\kappa}^{n}_{\varphi_{n}(v)})_{v\in
V}$. Then using the fact that the local rate functions are class functions
(specifically, applying Remark 3.2 with
$G_{1}=(V,E,\o,\overline{\mathbf{\kappa}}^{n}_{\varphi_{n}(E)},\mathbf{\kappa}^{n}_{\varphi_{n}(V)})$,
$G_{2}=G_{n}$ and $\varphi=\varphi_{n}^{-1}$) in the first equality below, and
combining (6.4) with the absolute continuity of $\tau_{k}$ and the fact that
$\langle(G,\xi)\rangle$ satisfies Assumption 2, we have for every
$k\in{\mathbb{N}}$,

$\displaystyle\lim_{n\to\infty}r^{G_{n},\varphi_{n}(v_{k})}_{j_{k}}\left(\tau_{k},X^{G,\xi}_{\varphi_{n}^{-1}(V_{n})}\right)\mathbb{I}_{A_{n}}=\lim_{n\to\infty}r^{(V,E,\o,\overline{\mathbf{\kappa}}^{n}_{\varphi_{n}(E)},\mathbf{\kappa}^{n}_{\varphi_{n}(V)}),v_{k}}_{j_{k}}\left(\tau_{k},X^{G,\xi}\right)\mathbb{I}_{A_{n}}=r^{G,v_{k}}_{j_{k}}\left(\tau_{k},X^{G,\xi}\right)$

a.s. on the event $\\{k\leq|\overline{{\mathcal{E}}}|\\}$. Because
$|\overline{{\mathcal{E}}}|$ is a.s. finite, it follows that
$\lim_{n\to\infty}R_{n}=0$ a.s.. Moreover, the fact that $R>0$ a.s. implies
the existence of an a.s. finite random variable $\overline{N}$ such that,

$R_{n}<\frac{R}{2}\text{ on the event }\\{n\geq\overline{N}\\}\cap A_{n}.$
(6.5)

We now argue that $X^{G,\xi}[T]=X^{G_{n},\xi^{n}}[T]$ on the event
$\\{n\geq\overline{N}\\}\cap A_{n}$ by making use of the fact that for each
$n\in{\mathbb{N}}$ and $v\in V$, $X^{G,\xi}_{v}$ and
$X^{G_{n},\xi^{n}}_{\varphi_{n}(v)}$ are driven by the same Poisson processes
on the event $A_{n}$. Fix $n\in{\mathbb{N}}$ and note that by the SDE (3.3),
$X^{G,\xi}$ and $X^{G_{n},\xi^{n}}$ are both a.s. continuous on the random set
$\\{t\in[0,\infty)\setminus\\{\tau_{k}\\}_{k\in{\mathbb{N}}}\\}\cap A_{n}$.
Furthermore, at time $\tau_{k}$, the processes $X^{G,\xi}$ and
$X^{G_{n},\xi^{n}}$ may either remain constant or experience a jump of size
$j_{k}$ at the respective vertices $v_{k}$ and $\varphi_{n}(v_{k})$. It
follows from the SDE (3.3) that the processes will either simultaneously jump
or both fail to jump if and only if

$\text{sgn}\left(r_{k}-r^{G,v_{k}}_{j_{k}}(\tau_{k},X^{G,\xi})\right)=\text{sgn}\left(r_{k}-r^{G_{n},\varphi_{n}(v_{k})}_{j_{k}}(\tau_{k},X^{G_{n},\xi^{n}})\right),$

where $\text{sgn}:{\mathbb{R}}\to{\mathbb{R}}$ is the càdlàg function given by
$\text{sgn}(a)=\mathbb{I}_{\left\\{a\geq
0\right\\}}-\mathbb{I}_{\left\\{a<0\right\\}}$. Suppose that
$X^{G,\xi}[\tau_{k})=X^{G_{n},\xi^{n}}_{\varphi_{n}(V)}[\tau_{k})$. Then,
using the predictability of the jump rates (by the Standing Assumption and
Definition 3.1) in the first equality below, we have

sgn
$\displaystyle\left(r_{k}-r^{G_{n},\varphi_{n}(v_{k})}_{j_{k}}(\tau_{k},X^{G_{n},\xi^{n}})\right)$
$\displaystyle\hskip
24.0pt=\text{sgn}\left(r_{k}-r^{G_{n},\varphi_{n}(v_{k})}_{j_{k}}(\tau_{k},X^{G,\xi}_{\varphi^{-1}_{n}(V_{n})})\right)$
$\displaystyle\hskip
24.0pt=\text{sgn}\left(r_{k}-r^{G,v_{k}}_{j_{k}}(\tau_{k},X^{G,\xi})+r^{G,v_{k}}_{j_{k}}(\tau_{k},X^{G,\xi})-r^{G_{n},\varphi_{n}(v_{k})}_{j_{k}}(\tau_{k},X^{G,\xi}_{\varphi_{n}^{-1}(V_{n})})\right).$

However, the last line of the above display is a.s. equal to
$\text{sgn}\left(r_{k}-r^{G,v_{k}}_{j_{k}}(\tau_{k},X^{G,\xi})\right)$ on the
event $\\{n\geq\overline{N}\\}$ by (6.5). Thus,
$X^{G,\xi}[\tau_{k}]=X^{G_{n},\xi^{n}}_{\varphi_{n}(V)}[\tau_{k}]$ a.s. on the
event $\\{n\geq\overline{N}\\}\cap A_{n}$. It follows that
$X^{G,\xi}[\tau_{k+1})=X^{G_{n},\xi^{n}}_{\varphi_{n}(V)}[t_{k+1})$ a.s. if
$|\overline{{\mathcal{E}}}|>k$ and
$X^{G,\xi}_{V}[T]=X^{G_{n},\xi^{n}}_{\varphi_{n}(V)}[T]$ a.s. if
$|\overline{{\mathcal{E}}}|=k$. Applying induction, we see that
$X^{G,\xi}[T]=X^{G_{n},\xi^{n}}_{\varphi_{n}(V)}[T]$ a.s. on the event
$\\{n\geq\overline{N}\\}\cap A_{n}$, and so (6.3) follows.

To see why this implies Lemma 6.10 in the general case of possibly infinite
graphs, note that (6.3) implies that for any $m\in{\mathbb{N}}$ and any
sequence $\\{\langle G_{n}\rangle\\}_{n\in{\mathbb{N}}},\langle G\rangle$ we
may replace $(G_{n},\xi^{n}),n\in{\mathbb{N}}$ by
$B_{m}(G_{n},\xi^{n}),n\in{\mathbb{N}}$, $(G,\xi)$ by $B_{m}(G,\xi)$,
$\overline{N}_{m}$ by $\overline{N}$ and $M$ by $m$ in (6.3) to get (6.2).
This concludes the proof of the lemma. ∎

###  6.3 Proof of Theorem 4.3

We now show how Theorem 4.3 follows from Proposition 6.11. The proof uses a
simple argument involving the Skorokhod representation theorem and Proposition
5.19.

###### Proof of Theorem 4.3.

Set $\langle(G_{\infty},\xi^{\infty})\rangle:=\langle(G,\xi)\rangle$ and
${\mathbb{N}}_{\infty}:={\mathbb{N}}\cup\\{\infty\\}$. By Assumption 1 and the
a.s. finite dissociability of $\\{\langle
G_{n}\rangle\\}_{n\in{\mathbb{N}}_{\infty}}$, the conditions of Proposition
5.19 are satisfied a.s.. Hence, the collection $\\{\langle
G_{n}\rangle\\}_{n\in{\mathbb{N}}_{\infty}}$ a.s. consistently spatially
localizes the SDE (3.3). Assumption 1, Proposition 4.1 and Proposition 5.7
then imply that the SDE (3.3) is strongly well-posed for all initial data in
$\\{\langle(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}_{\infty}}$. Moreover,
since $\langle(G_{n},\xi^{n})\rangle\Rightarrow\langle(G,\xi)\rangle$ in
${\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$,
by the Skorokhod representation theorem there exists a (complete) probability
space $(\widetilde{\Omega},\widetilde{{\mathcal{F}}},\widetilde{\mathbb{P}})$
that supports random elements
$\langle(\widetilde{G}_{n},\widetilde{\xi}^{n})\rangle\overset{\text{(d)}}{=}\langle(G_{n},\xi^{n})\rangle$,
$n\in{\mathbb{N}}_{\infty},$ such that
$\langle(\widetilde{G}_{n},\widetilde{\xi}^{n})\rangle\to\langle(\widetilde{G},\widetilde{\xi})\rangle:=\langle(\widetilde{G}_{\infty},\widetilde{\xi}^{\infty})\rangle$
$\widetilde{\mathbb{P}}$-a.s..

By Lemma 6.6 there exists a consistent rep-con extension of
$((\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}}),\\{\langle(\widetilde{G}_{n},\widetilde{\xi}^{n})\rangle\\}_{n\in{\mathbb{N}}_{\infty}})$
given by
$((\Omega,{\mathcal{F}},{\mathbb{F}},{\mathbb{P}}),\\{(\widetilde{G}_{n},\widetilde{\xi}^{n},\mathbf{N}^{\widetilde{G}_{n}}),\widetilde{M}_{n}\\}_{n\in{\mathbb{N}}},(\widetilde{G},\widetilde{\xi},\mathbf{N}^{\widetilde{G}}),\\{\widetilde{\varphi}_{n,m}\\}_{n\in{\mathbb{N}},m\leq
M_{n}})$. For each $n\in{\mathbb{N}}_{\infty}$, let
$(G_{n},X^{\widetilde{G_{n}},\widetilde{\xi}^{n}})$ be the resulting
$\mathbf{N}^{\widetilde{G}_{n}}$-strong solution to (3.3) for
$(\widetilde{G}_{n},\widetilde{\xi}^{n})$, where
$(\widetilde{G}_{\infty},\widetilde{\xi}^{\infty})=(\widetilde{G},\widetilde{\xi})$
and let
$\langle(\widetilde{G}_{n},X^{\widetilde{G}_{n},\widetilde{\xi}^{n}})\rangle$
denote its isomorphism class. Such strong solutions are well defined by Lemma
3.10 (which establishes the existence of a pathwise unique strong solution for
the initial data $\langle(\widetilde{G}_{n},\widetilde{\xi}^{n})\rangle$,
$n\in{\mathbb{N}}_{\infty}$). Because
$\\{\widetilde{G}_{n}\\}_{n\in{\mathbb{N}}_{\infty}}$ spatially localizes
(3.3), by Assumption 1 and the fact that $\langle(G,\xi)\rangle$ (and
therefore $\langle(\widetilde{G},\widetilde{\xi})\rangle$) satisfies
Assumption 2, Lemma 6.10 implies that the tuple
$((\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}}),\\{\langle(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}},\langle(G,\xi)\rangle)$
satisfies Assumption 2′. Proposition 6.11 then implies

$\lim_{n\to\infty}\langle(\widetilde{G}_{n},X^{\widetilde{G}_{n},\widetilde{\xi}^{n}})\rangle=\langle(\widetilde{G}_{\infty},X^{\widetilde{G}_{\infty},\widetilde{\xi}^{\infty}})\rangle\quad\text{a.s.}$

By well-posedness of (3.3),
$\langle(\widetilde{G}_{n},X^{\widetilde{G}_{n},\widetilde{\xi}^{n}})\rangle\overset{\text{(d)}}{=}\langle(G_{n},X^{G_{n},\xi^{n}})\rangle$
for every $n\in{\mathbb{N}}_{\infty}$. Thus,
$\langle(G_{n},X^{G_{n},\xi^{n}})\rangle$
$\overset{\text{(d)}}{=}\langle(\widetilde{G}_{n},X^{\widetilde{G}_{n},\widetilde{\xi}^{n}})\rangle\Rightarrow\langle(\widetilde{G},X^{\widetilde{G},\widetilde{\xi}})\rangle\overset{\text{(d)}}{=}\langle(G,X^{G,\xi})\rangle$
in
${\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{D}}]$.
∎

##  7 Proof of Asymptotic Correlation Decay

This section is devoted to the proof of Theorem 4.5. Recall that for an
unrooted $[\overline{{\mathcal{K}}},{\mathcal{K}}]$-graph $G$ and a vertex
$v\in V_{G}$, $\mathscr{C}_{v}(G)$ is the connected component of $G$ equipped
with $v$ as its root. For the remainder of the section, we fix a sequence of
finite, (possibly disconnected) unrooted
$[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$-random graphs
$(G_{n},\xi^{n})$, $n\in{\mathbb{N}},$ and a
${\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$-random
element $\langle\mathscr{C}_{\o}(G,\xi)\rangle$ (henceforth, denoted just
$\langle(G,\xi)\rangle$), all defined on a common complete probability space
$(\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}})$. We
additionally assume that, by extending the probability space if necessary,
$(\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}})$ also
supports an i.i.d. pair of vertices $(o^{1}_{n},o^{2}_{n})$, each uniformly
distributed on $G_{n}$, for all $n\in{\mathbb{N}}$. The proof of Theorem 4.5
is comprised of two steps. The first step is to establish an asymptotic
independence property stated in Lemma 7.1 below.

######  Lemma 7.1.

Suppose $(G_{n},\xi^{n})$ converges in probability in the local weak sense to
$\langle(G,\xi)\rangle$. Then

$(\langle\mathscr{C}_{o_{n}^{1}}(G_{n},\xi^{n})\rangle,\langle\mathscr{C}_{o_{n}^{2}}(G_{n},\xi^{n})\rangle)\Rightarrow(\langle(G^{(1)},\xi^{(1)})\rangle,\langle(G^{(2)},\xi^{(2)})\rangle),$
(7.1)

where $\Rightarrow$ represents convergence in distribution in
$({\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}])^{2}$
and $\langle(G^{(i)},\xi^{(i)})\rangle,i=1,2,$ are two independent copies of
$\langle(G,\xi)\rangle$.

###### Proof.

Let
$f_{1},f_{2}:{\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]\to{\mathbb{R}}$
be bounded, continuous functions. Then by [LacRamWuLWC21, Lemma 2.12],

$\displaystyle\lim_{n\to\infty}\mathbb{E}\left[f_{1}(\langle\mathscr{C}_{o_{n}^{1}}(G_{n},\xi^{n})\rangle)f_{2}(\langle\mathscr{C}_{o_{n}^{2}}(G_{n},\xi^{n})\rangle)\right]$
$\displaystyle=\mathbb{E}\left[f_{1}(\langle(G,\xi)\rangle)\right]\mathbb{E}\left[f_{2}(\langle(G,\xi)\rangle)\right]$
$\displaystyle=\mathbb{E}\left[f_{1}(\langle(G^{(1)},\xi^{(1)})\rangle)f_{2}(\langle(G^{(2)},\xi^{(2)})\rangle)\right].$
(7.2)

Since
${\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$
is a metric space, the algebra of separable bounded functions on
$({\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}])^{2}$
strongly separates points and hence, is convergence determining [EthKur86,
Theorem 3.4.5(b)]. Thus, (7.2) implies (7.1). ∎

Invoking the Skorokhod representation theorem, the second and main step of the
proof assumes joint local convergence of the initial data
$\\{\langle\mathscr{C}_{o_{n}^{1}}(G_{n},\xi^{n})\rangle,\langle\mathscr{C}_{o_{n}^{2}}(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}}$
to the i.i.d. pair
$(\langle(G^{(1)},\xi^{(1)}\rangle,\langle(G^{(2)},\xi^{(2)})\rangle)$ and
proves convergence in probability of the corresponding pairs of strong
solutions
$\\{\langle\mathscr{C}_{o_{n}^{1}}(G_{n},X^{G_{n},\xi^{n}})\rangle,\langle\mathscr{C}_{o_{n}^{2}}(G_{n},X^{G_{n},\xi^{n}})\rangle\\}_{n\in{\mathbb{N}}}$
to
$(\langle(G^{(1)},X^{G^{(1)},\xi^{(1)}})\rangle,\langle(G^{(2)},X^{G^{(2)},\xi^{(2)}})\rangle)$.
The coupling proof proceeds as follows. For each $i=1,2$, we first construct a
sequence of driving noises $\\{(G,\mathbf{N}^{n,i})\\}_{n\in{\mathbb{N}}}$
such that the isomorphism class of the corresponding
$\\{\mathbf{N}^{n,i}\\}$-strong solution
$\\{\langle\mathscr{C}_{o_{n}^{i}}(G_{n},X^{n,i})\rangle\\}$ for the initial
data $\langle\mathscr{C}_{o_{n}^{i}}(G_{n},\xi^{n})\rangle$ converges a.s. as
$n\rightarrow\infty$ to $\langle(G^{(i)},X^{G^{(i)},\xi^{(i)}})\rangle$. From
this, we construct a single sequence of _common_ driving noises
$\\{(G_{n},\mathbf{N}^{G_{n}})\\}_{n\in{\mathbb{N}}}$ such that for each
$i=1,2,$ the corresponding
$\\{\mathbf{N}^{G_{n}}\\}_{n\in{\mathbb{N}}}$-strong solutions
$\mathscr{C}_{o_{n}^{i}}(G_{n},X^{G_{n},\xi^{n}}),$ $n\in{\mathbb{N}},$ and
$(G^{(i)},X^{G^{(i)},\xi^{(i)}})$ satisfy

$\langle\mathscr{C}_{o^{i}_{n}}(G_{n},X^{G_{n},\xi^{n}})\rangle\to\langle(G^{(i)},X^{G^{(i)},\xi^{(i)}})\rangle\text{
in probability.}$

By the independence of
$\\{\langle(G^{(i)},X^{G^{(i)},\xi^{(i)}})\rangle\\}_{i=1,2}$, this would
imply the desired correlation decay result.

######  Lemma 7.2.

Given $\\{(G_{n},\xi^{n})\\}_{n\in{\mathbb{N}}}$, suppose there exists a
countable set ${\mathcal{S}}$ such that for ${\mathbb{P}}$-a.s. every
$\omega\in\Omega$, $V_{G_{n}(\omega)}\subset{\mathcal{S}}$ for all
$n\in{\mathbb{N}}$. Suppose also that Assumption 1′ holds and that the
collection of isomorphism classes
$\\{\langle\mathscr{C}_{o^{i}_{n}}(G_{n})\rangle,\langle
G^{(i)}\rangle\\}_{n\in{\mathbb{N}},i=1,2}$ a.s. consistently spatially
localizes the SDE (3.3). In addition, assume that for each $i=1,2,$,

$\langle\mathscr{C}_{o_{n}^{i}}(G_{n},\xi^{n})\rangle\to\langle(G^{(i)},\xi^{(i)})\rangle$
(7.3)

and the tuple
$(\\{\langle\mathscr{C}_{o^{i}_{n}}(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}},\langle(G^{(i)},\xi^{(i)})\rangle)$
satisfies Assumption 2′. Then it is possible to define a filtered probability
space $(\Omega,{\mathcal{F}},{\mathbb{F}},{\mathbb{P}})$ supporting solutions
$(G_{n},X^{G_{n},\xi^{n}}),n\in{\mathbb{N}},$ and
$\langle(G^{(i)},X^{G^{(i)},\xi^{(i)}})\rangle$, $i=1,2,$ to the SDE (3.3) for
the respective initial data $(G_{n},\xi^{n}),n\in{\mathbb{N}},$ and
$\langle(G^{(i)},\xi^{(i)})\rangle$, $i=1,2,$ such that as $n\to\infty$,

$\langle\mathscr{C}_{o_{n}^{i}}(G_{n},X^{G_{n},\xi^{n}})\rangle\to\langle(G^{(i)},X^{G^{(i)},\xi^{(i)}})\rangle,\text{
}i=1,2,\text{ in probability.}$ (7.4)

###### Proof.

Fix any deterministic injection $\widehat{\psi}:{\mathcal{S}}\to{\mathbb{N}}$.
For each $n\in{\mathbb{N}}$, let $(\widetilde{G}_{n},\widetilde{\xi}^{n})$ be
the unique
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$-random
element such that $\widehat{\psi}|_{V_{G_{n}}}\in
I((G_{n},\xi^{n}),(\widetilde{G}_{n},\widetilde{\xi}^{n}))$. To be precise, if
for each $n\in{\mathbb{N}}$,
$(G_{n},\xi^{n})=(V_{n},E_{n},\o_{n},\overline{\mathbf{\kappa}}^{n},\mathbf{\kappa}^{n},\xi^{n})$,
then

$(\widetilde{G}_{n},\widetilde{\xi}^{n}):=(\widetilde{V}_{n},\widetilde{E}_{n},\widetilde{\o}_{n},\widetilde{\overline{\mathbf{\kappa}}}^{n},\widetilde{\mathbf{\kappa}}^{n},\widetilde{\xi}^{n})=(\widehat{\psi}(V_{n}),\widehat{\psi}(E_{n}),\widehat{\psi}(\o_{n}),(\overline{\mathbf{\kappa}}^{n}_{\widehat{\psi}(e)})_{e\in\widetilde{E}_{n}},(\mathbf{\kappa}^{n}_{\widehat{\psi}(v)},\xi^{n}_{\widehat{\psi}(v)})_{v\in\widetilde{V}_{n}}).$

Noting that all the remaining statements of the lemma depend only on
$\langle\mathscr{C}_{o^{i}_{n}}(G_{n},\xi^{n})\rangle=\langle\mathscr{C}_{\widehat{\psi}(o^{i}_{n})}(\widetilde{G}_{n},\widetilde{\xi}^{n})\rangle$,
we may assume without loss of generality that
${\mathcal{S}}\subseteq{\mathbb{N}}$ and therefore that
$\mathscr{C}_{o^{i}_{n}}(G_{n},\xi^{n})$ is a
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$-random
element for each $n\in{\mathbb{N}}$ and $i=1,2$.

Now, by Assumption 1′ and the spatial localization assumption, Proposition 5.7
implies that the SDE (3.3) is strongly well-posed for all initial data in the
collection of marked graph representatives
$\\{\mathscr{C}_{o^{i}_{n}}(G_{n},\xi^{n}),(G^{(i)},\xi^{(i)})\\}_{n\in{\mathbb{N}},i=1,2}$,
where for $i=1,2$ $(G^{(i)},\xi^{(i)})$ is a random representative of
$\langle(G^{(i)},\xi^{(i)})\rangle$ (whose existence is guaranteed by Lemma
B.7). Also, by assumption, the collection of
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$-random
elements
$\\{\mathscr{C}_{o^{i}_{n}}(G_{n},\xi^{n})\\}_{n\in{\mathbb{N}},i=1,2}$ is
$\widehat{{\mathcal{F}}}$-measurable. For each $i=1,2,$ Lemma 6.6 implies the
existence of an associated consistent rep-con extension
$((\Omega^{i},{\mathcal{F}}^{i},{\mathbb{F}}^{i},{\mathbb{P}}^{i}),\\{\mathscr{C}_{o^{i}_{n}}(G_{n},\xi^{n},\mathbf{N}^{n,i}),M_{n}^{(i)}\\}_{n\in{\mathbb{N}}},(G^{(i)},\xi^{(i)},\mathbf{N}^{G^{(i)}}),\\{\varphi^{(i)}_{n,m}\\}_{n\in{\mathbb{N}},m\leq
M_{n}})$ where
$(\Omega^{i},{\mathcal{F}}^{i},{\mathbb{F}}^{i},{\mathbb{P}}^{i})$ supports a
collection of i.i.d. ${\mathbb{F}}^{i}$-Poisson processes
$\overline{\mathbf{N}}^{i}:=\\{\overline{\mathbf{N}}^{i}_{k}\\}_{k\in{\mathbb{N}}}$
which together with the driving maps $\overline{\psi}^{n,i}$,
$n\in{\mathbb{N}}$ and $\overline{\psi}^{(i)}$ generate the respective
${\mathbb{F}}^{i}$-driving noises $\mathbf{N}^{n,i}$, $n\in{\mathbb{N}}$ and
$\mathbf{N}^{G^{(i)}}$. We may also identify
$(\Omega,{\mathcal{F}},{\mathbb{F}},{\mathbb{P}}):=(\Omega^{1},{\mathcal{F}}^{1},{\mathbb{F}}^{1},{\mathbb{P}}^{1})=(\Omega^{2},{\mathcal{F}}^{2},{\mathbb{F}}^{2},{\mathbb{P}}^{2})$
and assume that $\overline{\mathbf{N}}^{1}$ is independent of
$\overline{\mathbf{N}}^{2}$ so that both collections consist of
${\mathbb{F}}$-Poisson processes. On this space, define for each
$k\in{\mathbb{N}}$ and $i=1,2,$
$\mathbf{N}_{2k-2+i}:=\overline{\mathbf{N}}^{i}_{k}$. Then for each
$n\in{\mathbb{N}}$ and $i=1,2,$ $\mathbf{N}^{n,i}$ and $\mathbf{N}^{G^{(i)}}$
are generated from $\\{\mathbf{N}_{k}\\}_{k\in{\mathbb{N}}}$ by the driving
maps $\psi^{n,i}(\cdot):=2\overline{\psi}^{n,i}(\cdot)-2+i$ and
$\psi^{(i)}(\cdot):=2\overline{\psi}^{(i)}(\cdot)-2+i$, respectively.

For each $n$, let $L_{n}:=d_{G_{n}}(o_{n}^{1},o_{n}^{2})$ and let
$\overline{M}_{n}$ be the maximal ${\mathcal{F}}_{0}$-measurable random
variable such that $\overline{M}_{n}<\frac{L_{n}}{2}$ and
$\overline{M}_{n}\leq\min_{i=1,2}M_{n}^{(i)}$ a.s.. Define the mapping
$\psi_{n}:V_{n}\to{\mathbb{N}}$ by

$\psi^{n}(v):=\begin{cases}\psi^{n,1}(v)&\text{ on the event
}\\{d_{G_{n}}(v,o^{1}_{n})\leq\overline{M}_{n}\\},\\\ \psi^{n,2}(v)&\text{
otherwise.}\end{cases}$

Since for $i=1,2$, the driving maps $\psi^{n,i}$ have disjoint images and
$\overline{M}_{n}<\frac{1}{2}d_{G_{n}}(o_{n}^{1},o_{n}^{2})$, it follows that
$\psi^{n}$ is ${\mathcal{F}}_{0}$-measurable and injective and is therefore
also a driving map. For each $n\in{\mathbb{N}}$, let $\mathbf{N}^{G_{n}}$ be
the ${\mathbb{F}}$-driving noise generated by the driving map $\psi^{n}$. Then
$\mathbf{N}^{G_{n}}$ is compatible with $\mathscr{C}_{o^{i}_{n}}(G_{n}),$
$i=1,2$. Because $\overline{M}_{n}\leq M_{n}^{(i)}$, $i=1,2$, it follows that
for any $i=1,2,$ $n\in{\mathbb{N}}$ and $v\in
B_{m}(\mathscr{C}_{o^{i}_{n}}(G_{n}))$,

$\mathbf{N}^{G_{n}}_{v}=\mathbf{N}^{n,i}_{v}=\mathbf{N}^{G^{(i)}}_{(\varphi_{n,m}^{(i)})^{-1}(v)}\text{
on the event }\\{m\leq\overline{M}_{n}\\}.$ (7.5)

Let $\\{n_{k}\\}_{k\in{\mathbb{N}}}$ be any deterministic sequence such that
$L_{n_{k}}$ (and therefore $\overline{M}_{n_{k}}$) diverges to infinity a.s.
as $k\rightarrow\infty$. Such a sequence exists because $L_{n}\to\infty$ in
probability. Fix $i\in\\{1,2\\}$ and consider the tuple
$\mathbf{T}_{i}:=((\Omega,{\mathcal{F}},{\mathbb{F}},{\mathbb{P}}),\\{\mathscr{C}_{o^{i}_{n_{k}}}(G_{n_{k}},\xi^{n_{k}},\mathbf{N}^{G_{n_{k}}}),\overline{M}_{n_{k}}\\}_{k\in{\mathbb{N}}},$
$(G^{(i)},\xi^{(i)},\mathbf{N}^{G^{(i)}}),\\{\varphi^{(i)}_{n_{k},m}\\}_{k\in{\mathbb{N}},m\leq\overline{M}_{n_{k}}})$.
Using (7.5), the definition of $\overline{M}_{n}$ and directly checking the
three properties of Definition 6.3, it is easy to see that this tuple is a
consistent rep-con extension of
$\mathbf{R}_{i}:=((\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}}),\\{\langle\mathscr{C}_{o^{i}_{n_{k}}}(G_{n_{k}},\xi^{n_{k}})\rangle\\},\langle(G^{(i)},\xi^{(i)})\rangle)$.
For completeness the verification is provided below.

Property 1:

By construction,
$(\\{\mathscr{C}_{o^{i}_{n}}(G_{n},\xi^{n}),M^{(i)}_{n}\\}_{n\in{\mathbb{N}}},(G^{(i)},\xi^{(i)})$,$\\{\varphi^{(i)}_{n,m}\\}_{n\in{\mathbb{N}},m\leq
M_{n}})$ is a rep-con sequence of
$(\\{\langle\mathscr{C}_{o^{i}_{n}}(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}},\langle(G^{(i)},\xi^{(i)})\rangle)$.
Since $\overline{M}_{n}\leq M^{(i)}_{n}$ for every $n\in{\mathbb{N}}$, the
tuple
$\mathbf{\overline{R}}_{i}:=(\\{\mathscr{C}_{o^{i}_{n_{k}}}(G_{n_{k}},\xi^{n_{k}}),\overline{M}_{n_{k}}\\},(G^{(i)},\xi^{(i)}),\\{\varphi^{(i)}_{n_{k},m}\\}_{k\in{\mathbb{N}},m\leq\overline{M}_{n_{k}}})$
satisfies Properties 1,2 and 4-6 of Definition 6.1. Because
$\overline{M}_{n_{k}}\to\infty$ a.s. and $\overline{M}_{n_{k}}\leq
M^{(i)}_{n_{k}}$ for each $k\in{\mathbb{N}}$, it follows that Property 3 of
Definition 6.1 holds. Thus, $\mathbf{\overline{R}}_{i}$ is a rep-con sequence
of
$(\\{\langle\mathscr{C}_{o^{i}_{n_{k}}}(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}},\langle(G^{(i)},\xi^{(i)})\rangle)$.

Property 2:

This holds because $(\Omega,{\mathcal{F}},{\mathbb{F}},{\mathbb{P}})$ is part
of the consistent rep-con extension of $\mathbf{R}_{i}$ and
$\mathbf{N}^{G_{n}}$ is an ${\mathbb{F}}$-driving noise.

Property 3:

This holds by (7.5).

Given that $\mathbf{T}_{i}$ is a consistent rep-con extension of
$\mathbf{R}_{i}$, by Proposition 6.11,

$\langle\mathscr{C}_{o^{i}_{n_{k}}}(G_{n_{k}},X^{G_{n_{k}},\xi^{n_{k}}})\rangle\to\langle(G^{(i)},\xi^{(i)})\rangle\text{
a.s..}$

Lastly, note that for any deterministic subsequence
$\\{n_{k}\\}_{k\in{\mathbb{N}}}\subseteq{\mathbb{N}}$, there exists a further
deterministic subsequence $\\{n_{k_{\ell}}\\}_{\ell\in{\mathbb{N}}}$ such that
$\lim_{\ell\to\infty}L_{n_{k_{\ell}}}=\infty$ a.s.. By the above result, this
implies that

$\lim_{\ell\to\infty}\langle\mathscr{C}_{o^{i}_{n_{k_{\ell}}}}(G_{n_{k_{\ell}}},X^{G_{n_{k_{\ell}}},\xi^{n_{k_{\ell}}}})\rangle=\langle(G^{(i)},\xi^{(i)})\rangle\text{
a.s.,}$

which immediately implies (7.4) and completes the proof. ∎

######  Remark 7.3.

Note that the proof of Lemma 7.2 does not make direct use of the form of (3.3)
but only relies on spatial localization of the SDE, Assumptions 1′ and 2′ and
Proposition 6.11, all of which could potentially be verified for solutions
$X^{G,\xi}$ of more general Poisson-driven SDEs.

Applying Lemmas 7.1 and 7.2, we now prove Theorem 4.5.

###### Proof of Theorem 4.5:.

Lemma 7.1 implies that

$(\langle\mathscr{C}_{o_{n}^{1}}(G_{n},\xi^{n})\rangle,\langle\mathscr{C}_{o_{n}^{2}}(G_{n},\xi^{n})\rangle)\Rightarrow(\langle(G^{(1)},\xi^{(1)})\rangle,\langle(G^{(2)},\xi^{(2)})\rangle),$

where $\langle(G^{(i)},\xi^{(i)})\rangle,i=1,2,$ are i.i.d. copies of
$\langle(G,\xi)\rangle$. By the Skorokhod representation theorem and Lemma
B.7, there exist
$\\{\langle(\widetilde{G}^{(i)},\widetilde{\xi}^{(i)})\rangle\\}_{i=1,2}\overset{\text{(d)}}{=}\\{\langle(G^{(i)},\xi^{(i)})\rangle\\}_{i=1,2}$,
and finite,
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$-random
elements
$\\{(\widetilde{G}_{n},\widetilde{o}^{i}_{n},\widetilde{\xi}^{n})\\}_{n\in{\mathbb{N}},i=1,2}\overset{\text{(d)}}{=}\\{(G_{n},o^{i}_{n},\xi^{n})\\}_{n\in{\mathbb{N}},i=1,2}$
such that

$\langle\mathscr{C}_{\widetilde{o}^{i}_{n}}(\widetilde{G}_{n},\widetilde{\xi}^{n})\rangle\to\langle(\widetilde{G}^{(i)},\widetilde{\xi}^{(i)})\rangle\text{,
}i=1,2,\text{ a.s..}$

By assumption, $\widetilde{G}_{n}\overset{\text{(d)}}{=}G_{n}$ and
$\widetilde{G}^{(i)}\overset{\text{(d)}}{=}G$ are a.s. finitely dissociable
for $i=1,2,$ and $n\in{\mathbb{N}}$. So by Assumption 1 and Proposition 5.19,
$\\{(\widetilde{G}_{n},\widetilde{o}^{i}_{n},\widetilde{\xi}^{n}),(\widetilde{G}^{(i)},\widetilde{\xi}^{(i)})\\}_{n\in{\mathbb{N}},i=1,2}$
consistently spatially localizes the SDE (3.3). Assumptions 1 and 2 imply by
Lemma 6.10 that the tuple
$((\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}}),\\{\langle\mathscr{C}_{o^{i}_{n}}(\widetilde{G}_{n},\widetilde{\xi}^{n})\rangle\\}_{n\in{\mathbb{N}}},\langle(\widetilde{G}^{(i)},\widetilde{\xi}^{(i)})\rangle$
satisfies Assumption 2′ for each $i$. Thus, by Lemma 7.2, it is possible to
construct a collection of driving noises
$\\{\mathbf{N}^{\widetilde{G}_{n}},\mathbf{N}^{\widetilde{G}^{(i)}}\\}_{n\in{\mathbb{N}},i=1,2}$
such that for each $n\in{\mathbb{N}}$ and $i=1,2,$ the respective
$\mathbf{N}^{\widetilde{G}_{n}}$ and
$\mathbf{N}^{\widetilde{G}^{(i)}}$-solutions
$X^{\widetilde{G}_{n},\widetilde{\xi}_{n}}$ and
$X^{\widetilde{G}^{(i)},\widetilde{\xi}^{(i)}}$ satisfy

$\langle\mathscr{C}_{\widetilde{o}^{n}_{i}}(\widetilde{G}_{n},X^{\widetilde{G}_{n},\widetilde{\xi}^{n}})\rangle\to\langle(\widetilde{G}^{(i)},X^{\widetilde{G}^{(i)},\widetilde{\xi}^{(i)}})\rangle\text{
in probability as }n\to\infty.$

By the bounded convergence theorem, for any bounded, continuous function
$f:({\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{D}}])^{2}\to{\mathbb{R}}$,

$\lim_{n\to\infty}\mathbb{E}\left[f\left(\left(\langle\mathscr{C}_{\widetilde{o}^{n}_{i}}(\widetilde{G}_{n},X^{\widetilde{G}_{n},\widetilde{\xi}^{n}})\rangle\right)_{i=1,2}\right)\right]=\mathbb{E}\left[f\left(\left(\langle(\widetilde{G}^{(i)},X^{\widetilde{G}^{(i)},\widetilde{\xi}^{(i)}})\rangle\right)_{i=1,2}\right)\right].$
(7.6)

By well-posedness of the SDE (3.3) on each of the graphs
$(\widetilde{G}_{n},\widetilde{\xi}^{n}),n\in{\mathbb{N}},$ and
$(\widetilde{G},\widetilde{\xi})$, which holds by Theorem 4.2, it follows that
for every $n\in{\mathbb{N}}$,
$\left(\langle\mathscr{C}_{\widetilde{o}^{n}_{i}}(\widetilde{G}_{n},X^{\widetilde{G}_{n},\widetilde{\xi}^{n}})\rangle\right)_{i=1,2}\overset{\text{(d)}}{=}\left(\langle\mathscr{C}_{o_{n}^{i}}(G_{n},X^{G_{n},\xi^{n}})\rangle\right)_{i=1,2}$
and
$(\widetilde{G}^{(1)},X^{G^{(1)},\xi^{(1)}})\overset{\text{(d)}}{=}(\widetilde{G}^{(2)},X^{G^{(2)},\xi^{(2)}})\overset{\text{(d)}}{=}(G,X^{G,\xi})$.
Together with (7.6) and the independence of
$\langle(\widetilde{G}^{(1)},X^{\widetilde{G}^{(1)},\widetilde{\xi}^{(1)}})\rangle$
and
$\langle(\widetilde{G}^{(2)},X^{\widetilde{G}^{(2)},\widetilde{\xi}^{(2)}})\rangle$,
for any bounded, continuous functions
$f_{1},f_{2}:{\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{D}}]\to{\mathbb{R}}$,
it follows that

$\begin{array}[]{l}\lim_{n\to\infty}\mathbb{E}\left[f_{1}(\langle\mathscr{C}_{o_{n}^{1}}(G_{n},X^{G_{n},\xi^{n}})\rangle)f_{2}(\langle\mathscr{C}_{o_{n}^{2}}(G_{n},X^{G_{n},\xi^{n}})\rangle)\right]\\\
\qquad\qquad=\lim_{n\to\infty}\mathbb{E}\left[f_{1}(\langle\mathscr{C}_{\widetilde{o}^{n}_{1}}(\widetilde{G}_{n},X^{\widetilde{G}_{n},\widetilde{\xi}^{n}})\rangle)f_{2}(\langle\mathscr{C}_{\widetilde{o}^{n}_{2}}(\widetilde{G}_{n},X^{\widetilde{G}_{n},\widetilde{\xi}^{n}})\rangle)\right]\\\
\qquad\qquad=\mathbb{E}\left[f_{1}(\langle(\widetilde{G}^{(1)},X^{\widetilde{G}^{(1)},\widetilde{\xi}^{(1)}})\rangle)f_{2}(\langle(\widetilde{G}^{(2)},X^{\widetilde{G}^{(2)},\widetilde{\xi}^{(2)}})\rangle)\right]\\\
\qquad\qquad=\mathbb{E}\left[f_{1}(\langle(\widetilde{G}^{(1)},X^{\widetilde{G}^{(1)},\widetilde{\xi}^{(1)}})\rangle)\right]\mathbb{E}\left[f_{2}(\langle(\widetilde{G}^{(2)},X^{\widetilde{G}^{(2)},\widetilde{\xi}^{(2)}})\rangle)\right]\\\
\qquad\qquad=\mathbb{E}\left[f_{1}(\langle(G,X^{G,\xi})\rangle)\right]\mathbb{E}\left[f_{2}(\langle(G,X^{G,\xi})\rangle)\right].\end{array}$

This allows us to conclude the desired result:

$\lim_{n\to\infty}\text{Cov}\left(f_{1}(\langle\mathscr{C}_{o_{n}^{1}}(G_{n},X^{G_{n},\xi^{n}})\rangle),f_{2}(\langle\mathscr{C}_{o_{n}^{2}}(G_{n},X^{G_{n},\xi^{n}})\rangle)\right)=0.$

∎

######  Remark 7.4.

From the proof of Theorem 4.5, it is not hard to see that in fact the
conclusion of Theorem 4.5 holds under Assumptions 1′ and 2′ and the spatial
localization property in place of Assumptions 1 and 2 and the finite
dissociability property respectively.

##  Appendix A Counterexample: When the SDE is not Well-Posed

There are two ways the SDE (3.3) can fail to be well-posed. Either there exist
no solutions, or there exist multiple solutions. To construct an IPS with no
solutions, it is a simple matter of constructing unbounded jump rate functions
that increase fast enough that all stochastic processes $X$ for which (3.3)
holds a.s. with $X^{G,\xi}:=X$ explode (have infinitely many discontinuities
on some compact interval in ${\mathbb{R}}_{+}$) with positive probability. In
this section, we show that well-posedness is non-trivial even if we restrict
ourselves to Markov processes with bounded jump rate functions by constructing
a graph and a collection of jump rate functions with respect to which the SDE
(3.3) has multiple strong solutions with different laws.

######  Proposition A.1.

Let ${\mathcal{T}}$ be the tree such that for every $k\in{\mathbb{N}}$, all
vertices $v$ in the $k$th generation of ${\mathcal{T}}$ have $4^{k}$ children.
Using the notation of Example 4.8, suppose ${\mathcal{X}}=\\{0,1\\}$,
${\mathcal{J}}=\\{1\\}$ and for $z\in{\mathcal{X}}^{V}$ and $v\in V$, let
$\widetilde{r}_{1}^{{\mathcal{T}}[\text{cl}_{v}],v}(z)=1$ if both $z_{v}=0$
and $\sum_{u\in{\mathcal{N}}_{v}}z_{u}>0$, and let
$\widetilde{r}_{1}^{{\mathcal{T}}[\text{cl}_{v}],v}(z)=0$ otherwise. Then the
corresponding SDE (3.3) has multiple strong solutions for the initial state
$\xi_{v}=X_{v}(0)=0,v\in V$ and these solutions also have different laws.

The proof of this proposition requires us to introduce the notion of infinite
causal chains.

######  Definition A.2 (Infinite causal chains).

For any locally finite graph $G$, $T\in(0,\infty)$ and associated Poisson
processes $\mathbf{N}^{G,T}$ defined as in (5.9), an infinite
$(G,\mathbf{N}^{G,T})$-causal chain ending at a vertex $u$ is a path
$\Gamma=(u=u_{0},u_{1},u_{2},\dots)$ such that for every $n\in{\mathbb{N}}$,
$(u_{n},u_{n-1},\dots,u_{1},u_{0})$ is a $(G,\mathbf{N}^{G,T})$-causal chain
(see Definition 5.18).

To prove Proposition A.1, we first show that the tree ${\mathcal{T}}$ has an
infinite causal chain almost surely.

######  Lemma A.3.

Let ${\mathcal{T}}=(V,E,\o)$ be the tree defined in Proposition A.1 and let
$\mathbf{N}^{{\mathcal{T}}}$ be the corresponding driving noise. Fix any
$T\in(0,\infty)$ and a strictly positive, non-decreasing family of constants
$\\{C_{k,T}\\}_{k\in{\mathbb{N}}}$, and let $\mathbf{N}^{{\mathcal{T}},T}$ be
the associated Poisson processes defined in (5.9). Then there exists an
infinite $({\mathcal{T}},\mathbf{N}^{{\mathcal{T}},T})$-causal chain with
probability 1.

###### Proof.

Define the path $\Gamma:=\\{\o:=v_{0},v_{1},\dots\\}$ using the following
recursive construction. For $n\geq 1$, first check if there exists $u\in
c_{v_{n}}({\mathcal{T}})$ such that
$\mathbf{N}^{{\mathcal{T}},T}_{u}(2^{-n-2},2^{-n-1})>0$. If this condition is
satisfied, then set $v_{n+1}=u$ (if multiple children satisfy this then we
choose one arbitrarily from amongst them). If not, then we choose any child of
$v_{n}$ arbitrarily and set it to be $v_{n+1}$.

Define $U:=\\{v_{i},i\in\mathcal{I}\\}\subseteq\Gamma$, where
$\mathcal{I}:=\\{n\in\mathbb{N}_{0}:\mathbf{N}_{v_{n}}^{{\mathcal{T}},T}(2^{-n-1},2^{-n})>0\\}$.
We now claim that $U$ contains an infinite path a.s.. To see why, we prove the
equivalent claim that $U$ a.s. contains all but finitely many vertices in
$\Gamma$. Let $\alpha_{n}:={\mathbb{P}}(v_{n+1}\notin U)$. Note that
$\overline{C}_{k,T}\geq C:=\overline{C}_{1,T}>0$ for all $k$. Thus, for each
$u\in U$,

${\mathbb{P}}\left(\mathbf{N}^{{\mathcal{T}},T}_{u}(2^{-n-1},2^{-n})=0\right)=e^{-2^{-n-1}\overline{C}_{|\text{cl}_{u}|,T}}\leq
e^{-2^{-n-1}C}.$

The independence of the Poisson processes
$(\mathbf{N}_{u}^{{\mathcal{T}},T})_{u\in V}$ then implies that

$\displaystyle\alpha_{n}$ $\displaystyle=\prod_{u\in
c_{v_{n}}({\mathcal{T}})}{\mathbb{P}}\left(\mathbf{N}^{{\mathcal{T}},T}_{u}(2^{-n-1},2^{-n})=0\right)\leq\left(e^{-2^{-n-1}C}\right)^{4^{n}}=e^{-2^{n-1}C}.$

Because $\alpha_{n}$ decreases super-exponentially fast,
$\sum_{n=1}^{\infty}\alpha_{n}<\infty$. By the Borel-Cantelli lemma, it
follows that with probability 1, all but finitely vertices in
$\\{v_{n}\\}_{n\in{\mathbb{N}}_{0}}$ are contained in $U$, thus proving the
claim. Note that for any path
$\Gamma^{\prime}=(u_{n},u_{n-1},\dots,u_{0})\subseteq U$ where $u_{0}$ is
closest to the root, there exists a decreasing sequence of times
$\\{t_{k}\in(2^{-(d_{\mathcal{T}}(u_{0},\o)+k)},2^{-(d_{\mathcal{T}}(u_{0},\o)+k)+1})\\}$
such that $\mathbf{N}^{{\mathcal{T}},T}_{u_{k}}(\\{t_{k}\\})=1$. Thus,
$\Gamma^{\prime}$ is a $({\mathcal{T}},\mathbf{N}^{{\mathcal{T}},T})$-causal
chain, and we have shown that $U$ contains an infinite
$({\mathcal{T}},\mathbf{N}^{{\mathcal{T}},T})$-causal chain a.s.. ∎

We can now prove Proposition A.1.

###### Proof of Proposition A.1:.

Although there are infinitely many strong solutions for the initial state
$\xi\equiv 0$, we explicitly construct two of them. First, the process
$X^{{\mathcal{T}},\xi}_{v}(t)=0$ for all $v\in V$ and $t\in{\mathbb{R}}_{+}$
is clearly a strong solution to the SDE (3.3) for the initial data
$({\mathcal{T}},\xi)$ with
$r^{{\mathcal{T}},v}_{1}(t,x):=\tilde{r}_{1}^{{\mathcal{T}}[cl_{v}]}(x(t-))$,
with $\tilde{r}$ as given in the proposition.

Fix $T<\infty$. Note that for each $v\in V$, by (5.9),
$\mathbf{N}^{{\mathcal{T}},T}_{v}$ is a unit rate Poisson process. Then for
$(v,t)\in V\times(0,T)$, we say $(v,t)$ has an infinite causal chain if there
exists an infinite $({\mathcal{T}},\mathbf{N}^{{\mathcal{T}},T})$-causal chain
ending at $v$ on the interval $[0,t]$. Then for $v\in V$ and $t\in[0,T]$,
define

$\widetilde{X}^{{\mathcal{T}},\xi}_{v}(t):=\begin{cases}1&\text{ if
}(v,t)\text{ has an infinite causal chain,}\\\ 0&\text{
otherwise.}\end{cases}$ (A.1)

We claim this is also a solution to the same SDE. Fix any $v\in V$ and define

$\widehat{X}_{v}(t):=\xi_{v}+\int_{(0,t]\times{\mathbb{R}}_{+}\times{\mathcal{J}}}\mathbb{I}_{\left\\{r\leq
r_{1}^{{\mathcal{T}},v}(s,\widehat{X}_{v},\widetilde{X}^{{\mathcal{T}},\xi}_{{\mathcal{N}}_{v}({\mathcal{T}})})\right\\}}\,\mathbf{N}^{{\mathcal{T}}}_{v}(ds,dr,dj),\quad
t\in{\mathbb{R}}_{+}.$ (A.2)

If $(v,t)\in V\times{\mathbb{R}}_{+}$ has no infinite causal chain, then by
Definition A.2, for any $s\leq t$, either
$\mathbf{N}^{{\mathcal{T}},T}_{v}([s,t])=\mathbf{N}^{{\mathcal{T}}}_{v}([s,t]\times(0,1]\times\\{1\\})=0$,
or there does not exist a $u\in{\mathcal{N}}_{v}({\mathcal{T}})$ for which
$(u,s)$ has an infinite causal chain. It immediately follows that for every
event $\tau$ in $\mathbf{N}^{{\mathcal{T}},T}_{v}$ in the interval $(0,t]$,
$r^{{\mathcal{T}},v}_{1}(\tau,\widehat{X}_{v},\widetilde{X}^{{\mathcal{T}},\xi}_{{\mathcal{N}}_{v}({\mathcal{T}})})=0$
so that $\widehat{X}_{v}(t)=\widetilde{X}^{{\mathcal{T}},\xi}_{v}(t)=0$. If
$(v,t)$ has an infinite causal chain, then there exists a minimal event
$\tau\leq t$ in $\mathbf{N}^{{\mathcal{T}},T}_{v}$ such that there exists a
neighbor $u$ of $v$ for which $(u,\tau)$ has an infinite causal chain. By
minimality of $\tau$, $(v,s)$ does not have an infinite causal chain on the
event $\\{s\leq\tau\\}$, and hence, $\widehat{X}_{v}(\tau-)=0$ by the previous
case. Thus,
$r^{{\mathcal{T}},v}_{1}(\tau,\widehat{X}_{v},\widetilde{X}^{{\mathcal{T}},\xi}_{{\mathcal{N}}_{v}({\mathcal{T}})})=1$,
so $\widehat{X}_{v}(\tau)=1$. It follows that
$\widehat{X}_{v}(t)=\widetilde{X}_{v}(t)$.

Thus, $\widehat{X}_{v}=\widetilde{X}^{{\mathcal{T}},\xi}_{v}$. Because $v$ and
$t$ were chosen arbitrarily, it follows that
$\widetilde{X}^{{\mathcal{T}},\xi}$ satisfies (3.3). Finally, because
$({\mathcal{T}},\mathbf{N}^{{\mathcal{T}},T})$-causal chains are
${\mathbb{F}}^{{\mathcal{T}},\xi,\mathbf{N}^{\mathcal{T}}}$-adapted,
$\widetilde{X}^{{\mathcal{T}},\xi}$ must also be
${\mathbb{F}}^{{\mathcal{T}},\xi,\mathbf{N}^{\mathcal{T}}}$-adapted by (A.1).
Thus, $\widetilde{X}^{{\mathcal{T}},\xi}$ is also a strong solution to (3.3),
and by Lemma A.3, $X^{{\mathcal{T}},\xi}\neq\widetilde{X}^{{\mathcal{T}},\xi}$
a.s.. ∎

##  Appendix B Measurable Representatives of Graph Isomorphism Classes

In this section, we establish measurability of marked graph representatives of
random isomorphism classes, culminating in the proofs of Lemmas 6.2 and 6.6.
Along the way, we introduce a Polish space of canonical representative graphs
that may be of independent interest. We start with preliminaries in Section
B.1 and then establish the main measurable selection results in Section B.2.

Throughout the section, we let $\overline{{\mathcal{Z}}}$ and ${\mathcal{Z}}$
be Polish spaces denoting the respective spaces in which the edge and vertex
marks lie, and let $d_{\bar{{\mathcal{Z}}}}$ and $d_{{\mathcal{Z}}}$ be
associated metrics that induce the respective topologies. Let $\odot$ be an
arbitrary point not lying in $\overline{{\mathcal{Z}}}\cup{\mathcal{Z}}$,
define the spaces
$\overline{{\mathcal{Z}}}_{\odot}:=\overline{{\mathcal{Z}}}\cup\\{\odot\\}$
and ${\mathcal{Z}}_{\odot}:={\mathcal{Z}}\cup\\{\odot\\}$, and endow them with
the corresponding Polish topologies from $\overline{{\mathcal{Z}}}$ and
${\mathcal{Z}}$, respectively, with $\odot$ being an isolated point.
Throughout the section, we often implicitly denote the components of (possibly
random) $[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$-graphs by
$G:=(V,E,\o,\overline{\vartheta},\vartheta)$ and
$G_{n}:=(V_{n},E_{n},\o_{n},\overline{\vartheta}^{n},\vartheta^{n})$.

######  Definition B.1.

Given a Polish space ${\mathcal{Z}}^{\prime}$ and a measurable space
$(\Omega,{\mathcal{F}})$, a mapping
$F:\Omega\mapsto\mathrm{clo}({\mathcal{Z}}^{\prime})$, where
$\mathrm{clo}({\mathcal{Z}}^{\prime})$ denotes the set of closed subsets of
${\mathcal{Z}}^{\prime}$, is said to be an ${\mathcal{F}}$-random closed
subset of ${\mathcal{Z}}^{\prime}$ if it is weakly measurable in the following
sense: for every open set $U\subseteq{\mathcal{Z}}^{\prime}$, the set
$\\{\omega\in\Omega:U\cap F(\omega)\neq\emptyset\\}$ lies in ${\mathcal{F}}$.
Moreover, $F$ is said to be non-empty if $F(\omega)\neq\emptyset$ for every
$\omega\in\Omega$.

Several results in this appendix make use of the following measurable
selection theorem.

######  Theorem B.2 (Kuratowski & Ryll-Nardzewski Measurable Selection
Theorem).

Suppose ${\mathcal{Z}}^{\prime}$ is a Polish space, $(\Omega,{\mathcal{F}})$
is a measurable space and
$F:\Omega\mapsto\mathrm{clo}({\mathcal{Z}}^{\prime})$ is a non-empty
${\mathcal{F}}$-random closed subset of ${\mathcal{Z}}^{\prime}$. Then there
exists an ${\mathcal{F}}$-measurable function
$Z:\Omega\to{\mathcal{Z}}^{\prime}$ such that $Z(\omega)\in F(\omega)$ for all
$\omega\in\Omega$.

###### Proof.

This is simply a restatement of [Bog07, Theorem 6.9.3] in our notation, in
particular with $X,{\mathcal{B}}$ and $\Psi$ in [Bog07] replaced by
${\mathcal{Z}}^{\prime}$, ${\mathcal{F}}$ and $F$, respectively. ∎

###  B.1 A Canonical Subspace of Rooted Graphs and its Properties

In this section, we construct a canonical subspace of the space of
$[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$-graphs introduced in Section 2.4
and equip it with a topology that is compatible with the topology of
${\mathscr{G}}_{*}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$. In the ensuing
definition, we use the following standard notion of convergence of subsets of
${\mathbb{N}}$. Given $S_{n}\subseteq{\mathbb{N}},n\in{\mathbb{N}},$ and
$S\subseteq{\mathbb{N}}$, we write $S_{n}\to S$ if and only if

$S=\bigcup_{n\in{\mathbb{N}}}\bigcap_{n^{\prime}>n}S_{n^{\prime}}=\bigcap_{n\in{\mathbb{N}}}\bigcup_{n\in{\mathbb{N}}}S_{n^{\prime}}.$

Equivalently, $S_{n}\to S$ if and only if for every $k\in S$, there exist only
finitely many $n\in{\mathbb{N}}$ such that $k\notin S_{n}$ and for every
$k^{\prime}\notin S$ there exist only finitely $n^{\prime}\in{\mathbb{N}}$
such that $k^{\prime}\in S_{n^{\prime}}$. In addition, we equip ${\mathbb{N}}$
with the discrete topology. Lastly, recall the definition of
$\text{cl}_{v}(G)$ from Section 2.1.

######  Definition B.3.

We equip the canonical space
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ of
rooted $[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$-graphs defined in (6.1) by

$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]:=\\{[\overline{{\mathcal{Z}}},{\mathcal{Z}}]\text{-graphs
}G=(V,E,\o,\overline{\vartheta},\vartheta)\text{ s.t.
}V\subseteq{\mathbb{N}}\\},$ (B.1)

with the following notion of convergence: $G_{n}\to G$ in
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ if and
only if

  1. 1.

$\lim_{n\to\infty}V_{n}=V$;

  2. 2.

$\lim_{n\to\infty}E_{n}=E$;

  3. 3.

$\lim_{n\to\infty}\o_{n}=\o$;

  4. 4.

$\lim_{\begin{subarray}{c}n\to\infty\\\ e\in
E_{n}\end{subarray}}\overline{\vartheta}^{n}_{e}=\overline{\vartheta}_{e}$ for
all $e\in E$;

  5. 5.

$\lim_{\begin{subarray}{c}n\to\infty\\\ v\in
V_{n}\end{subarray}}\vartheta^{n}_{v}=\vartheta_{v}$ for all $v\in V$.

  6. 6.

for each $v\in V$, $\lim_{\begin{subarray}{c}n\to\infty\\\ v\in
V_{n}\end{subarray}}\max\\{u\in\text{cl}_{v}(G_{n})\\}=\max\\{u\in\text{cl}_{v}(G)\\}$.

For
$G,G^{\prime}\in\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$,
notions such as graph distance $d_{G}(\cdot,\cdot)$, truncations $B_{m}(G)$,
sets of isomorphisms $I(G,G^{\prime})$ and isomorphism classes $\langle
G\rangle$ are all defined as in Sections 2.1 and 2.4.

######  Remark B.4.

The least intuitive condition is perhaps Condition 6 of Definition B.3, but it
is necessary for the topology on
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ to be
compatible with the topology of
${\mathscr{G}}_{*}[\overline{{\mathcal{Z}},{\mathcal{Z}}}]$ (in the sense made
precise in Lemma B.6). This is best illustrated via an example of how
compatibility could fail without Condition 6. Let
$\overline{{\mathcal{Z}}}={\mathcal{Z}}=\\{1\\}$ be trivial and suppose that
$V\subseteq\\{2k:k\in{\mathbb{N}}\\}$, $\o:=2$ and $E$ is any set of distinct
pairs of $V$ such that $G=(V,E,\o)$ is locally finite. If for each
$n\in{\mathbb{N}}$, $G_{n}$ is defined by $\o_{n}:=2$,
$V_{n}:=V\cup\\{2n+1\\}$ and $E_{n}:=E\cup\\{\\{2,2n+1\\}\\}$, then Conditions
1-5 of Definition B.3 are all satisfied, and only Condition 6 fails. However,
note that $G_{n}\cong G_{n^{\prime}}\ncong G$ for all
$n,n^{\prime}\in{\mathbb{N}}$, so $\langle G_{n}\rangle\to\langle
G_{1}\rangle\neq\langle G\rangle$ in
${\mathscr{G}}_{*}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$.

In order to apply Theorem B.2 to find measurable representatives of
isomorphism classes, it is necessary to prove that the space
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ is
Polish. This can be done by direct verification using Definition B.3. We fill
in the details for completeness.

######  Lemma B.5.

The space
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ is
Polish.

###### Proof.

Define the Polish space
${\mathcal{R}}:=\left({\mathbb{N}}\times{\mathcal{Z}}_{\odot}\times(\overline{{\mathcal{Z}}}_{\odot})^{\mathbb{N}}\right)^{{\mathbb{N}}}\times{\mathbb{N}}$,
equipped with the product topology, and consider the map
$\psi:\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]\to{\mathcal{R}}$
defined by $\psi(G):=((\psi_{k}(G))_{k\in{\mathbb{N}}},\o)$, where for each
$k\in{\mathbb{N}}$,
$\psi_{k}(G)=(c^{\prime}_{k},\vartheta^{\prime}_{k},(\overline{\vartheta}^{\prime}_{\\{k,k^{\prime}\\}})_{k^{\prime}\in{\mathbb{N}}}),$
with

$c^{\prime}_{k}:=\begin{cases}\max\\{\text{cl}_{k}(G)\\}&\text{ if }k\in V,\\\
k&\text{otherwise,}\end{cases}\;\quad\vartheta^{\prime}_{k}:=\begin{cases}\vartheta_{k}&\text{
if }k\in V,\\\ \odot&\text{
otherwise,}\end{cases}\;\quad\overline{\vartheta}^{\prime}_{\\{k,k^{\prime}\\}}:=\begin{cases}\overline{\vartheta}_{\\{k,k^{\prime}\\}}&\text{
if }\\{k,k^{\prime}\\}\in E,\\\ \odot&\text{ otherwise.}\end{cases}$

We then have the following observations:

  * (i)

The map $\psi$ is a bijection from
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ to
$\widehat{\mathcal{R}}$, where $\widehat{\mathcal{R}}$ is the subset of
elements
$\zeta=((c^{\prime}_{k},\vartheta^{\prime}_{k},\\{\overline{\vartheta}^{\prime}_{\\{k,k^{\prime}\\}}\\}_{k^{\prime}\in{\mathbb{N}}})_{k\in{\mathbb{N}}},\o)$
in ${\mathcal{R}}$ that satisfy the following constraints:

    1. (a)

$\o\in V_{\zeta}:=\\{k\in{\mathbb{N}}:\vartheta^{\prime}_{k}\neq\odot\\}$;

    2. (b)

for every $k\notin V_{\zeta}$, $c^{\prime}_{k}=k$;

    3. (c)

One has $\\{k,k^{\prime}\\}\subseteq V_{\zeta}$ for every
$\\{k,k^{\prime}\\}\in E_{\zeta}:=\cup_{k\in{\mathbb{N}}}E_{\zeta}(k)$, where
$E_{\zeta}(k):=\\{\\{k,k^{\prime}\\}\subset{\mathbb{N}}:k\neq
k^{\prime},\overline{\vartheta}^{\prime}_{\\{k,k^{\prime}\\}}\neq\odot\\}$;

    4. (d)

for every $k\in{\mathbb{N}}$,
$c^{\prime}_{k}=\max[\\{k\\}\cup\\{k^{\prime}\in{\mathbb{N}}:\\{k,k^{\prime}\\}\in
E_{\zeta}(k)\\}]$.

It is trivial to check that for every
$G\in\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$,
$\psi(G)$ must satisfy conditions (a)–(d) above. Thus, the image of
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ under
$\psi$ is contained in $\widehat{\mathcal{R}}$. On the other hand, note that
any
$\zeta=((c^{\prime}_{k},\vartheta^{\prime}_{k},\\{\overline{\vartheta}^{\prime}_{\\{k,k^{\prime}\\}}\\}_{k^{\prime}\in{\mathbb{N}}})_{k\in{\mathbb{N}}},\o)\in\widehat{\mathcal{R}}$
has a unique inverse under $\psi$, described by
$\psi^{-1}(\zeta)=G_{\zeta}:=(V_{\zeta},E_{\zeta},\o_{\zeta},\overline{\vartheta}_{k}(\zeta),\overline{\vartheta}^{\prime}_{\\{k,k^{\prime}\\}}(\zeta))$,
where $V_{\zeta}$ and $E_{\zeta}$ are respectively defined as in (a) and (c)
above, $\o_{\zeta}=\o$,
$\vartheta_{k}(\zeta)=\vartheta^{\prime}_{k}\in{\mathcal{Z}}$ for $k\in
V_{\zeta}$, and
$\overline{\vartheta}_{\\{k,k^{\prime}\\}}(\zeta)=\overline{\vartheta}^{\prime}_{\\{k,k^{\prime}\\}}\in\bar{{\mathcal{Z}}}$
for $\\{k,k^{\prime}\\}\in E_{\zeta}$. Condition (c) also ensures that
$\zeta\in\widehat{\mathcal{R}}$ implies $|E_{\zeta}(k)|<\infty$ for every
$k\in V_{\zeta}$, and so the resulting graph $G_{\zeta}$ is locally finite.
Moreover, conditions (a) and (c) together ensure that the edge and vertex
marks lie in $\overline{{\mathcal{Z}}}$ and ${\mathcal{Z}}$, respectively,
thus showing that $G_{\zeta}$ is a
$[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$-graph. Lastly, it is easy to see
that for any
$G\in\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ and
$\zeta\in\widehat{\mathcal{R}}$, $\psi\circ\psi^{-1}(\zeta)=\zeta$ and
$\psi^{-1}\circ\psi(G)=G$, thus proving $\psi$ is a bijection between
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ and
$\widehat{\mathcal{R}}$.

  * (ii)

$\widehat{\mathcal{R}}$ is a closed subset of ${\mathcal{R}}$ under
componentwise convergence: suppose the sequence
$\zeta^{n}=((c^{\prime,n}_{k},\vartheta^{\prime,n}_{k},\\{\overline{\vartheta}^{\prime,n}_{\\{k,k^{\prime}\\}}\\}_{k^{\prime}\in{\mathbb{N}}})_{k\in{\mathbb{N}}},\o^{n})\in\widehat{\mathcal{R}},n\in{\mathbb{N}},$
converges to
$\zeta=((c^{\prime}_{k},\vartheta^{\prime}_{k},\\{\overline{\vartheta}^{\prime}_{\\{k,k^{\prime}\\}}\\}_{k^{\prime}\in{\mathbb{N}}})_{k\in{\mathbb{N}}},\o)$
pointwise. Then $\zeta\in{\mathcal{R}}$ because ${\mathcal{R}}$ is Polish. To
show $\zeta\in\widehat{\mathcal{R}}$, it suffices to show that $\zeta$
satisfies the constraints (a)–(d) in (i) above. To prove condition (a) we
argue by contradiction. Suppose $\vartheta^{\prime}_{\o}=\odot$. Then since
$\o^{n}\rightarrow\o$ it follows that there exists $N<\infty$ such that
$\o^{n}=\o$ for all $n\geq N$. Since
$\vartheta^{\prime,n}_{\o}\rightarrow\vartheta^{\prime}_{\o}$ and $\odot$ is
isolated, this implies that for all sufficiently large $n$,
$\vartheta^{\prime,n}_{\o_{n}}=\vartheta^{\prime,n}_{\o}=\odot$, which implies
$\o_{n}\not\in V_{\zeta^{n}}$ and thus contradicts the assumption that
$\zeta^{n}\in\widehat{\mathcal{R}}$. Thus, this proves that $\zeta$ satisfies
condition (a). Condition (c) can be established in an exactly analogous
fashion. Next, suppose $k\notin V_{\zeta}$. Then since $\zeta$ satisfies (a)
as shown above, $\vartheta_{k}=\odot$ and since $\odot$ is isolated and
$\vartheta^{\prime,n}_{k}\rightarrow\vartheta_{k}$, it follows that
$\vartheta^{\prime,n}_{k}=\odot$ for all sufficiently large $n$. In turn,
since $\zeta^{n}\in\widehat{\mathcal{R}}$, this implies $k\not\in
V_{\zeta^{n}}$ and hence that $c_{k}^{\prime,n}=k$. Since
$c_{k}^{\prime,n}\rightarrow c_{k}^{\prime}$, this implies $c_{k}^{\prime}=k$
and condition (b) follows for $\zeta$. Finally, condition (d) for $\zeta$,
which implies each $E_{\zeta}(k)$, $k\in{\mathbb{N}},$ is a finite set, can be
deduced by similarly observing that $c_{k}^{\prime,n}=c_{k}$ and therefore
$E_{\zeta^{n}}(k)=E_{\zeta}(k)$ for all $k\in{\mathbb{N}}$ and all
sufficiently large $n$, and the fact that each $\zeta^{n}$ satisfies condition
(d).

  * (iii)

The map $\psi$ is a homeomorphism from
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ to
$\widehat{\mathcal{R}}$: For each $n\in{\mathbb{N}}$, let
$G_{n}=(V_{n},E_{n},\o_{n},\overline{\vartheta}_{n},\vartheta_{n})\in\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$
and
$\psi(G_{n})=((c^{\prime,n}_{k},\vartheta^{\prime,n}_{k},\\{\overline{\vartheta}^{\prime,n}_{\\{k,k^{\prime}\\}}\\}_{k^{\prime}\in{\mathbb{N}}})_{k\in{\mathbb{N}}},\o^{n})\in\widehat{\mathcal{R}}$.
Then we wish to show that $G_{n}\to G=(V,E,\o,\vartheta,\overline{\vartheta})$
in $\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ if
and only if
$\psi(G_{n})\to\psi(G):=((c^{\prime}_{k},\vartheta^{\prime}_{k},\\{\overline{\vartheta}^{\prime}_{\\{k,k^{\prime}\\}}\\}_{k^{\prime}\in{\mathbb{N}}})_{k\in{\mathbb{N}}},\o)$
in $\widehat{\mathcal{R}}$.

(a)

If $G_{n}\to G$, then $\lim_{n\to\infty}\psi(G_{n})=\psi(G)$: for any
$k,k^{\prime}\in{\mathbb{N}}$, Conditions 1 and 2 of Definition B.3 imply that
if $k\in V$ and $\\{k,k^{\prime}\\}\in E$, then $k\in V_{n}$ and
$\\{k,k^{\prime}\\}\in E_{n}$ for $n$ sufficiently large. Then for any $k\in
V$, Condition 6 implies that
$\lim_{n\to\infty}c^{\prime,n}_{k}\to\lim_{\begin{subarray}{c}n\to\infty\\\
k\in
V_{n}\end{subarray}}\max\\{\text{cl}_{k}(G_{n})\\}=\max\\{\text{cl}_{k}(G)\\}=c^{\prime}_{k}$
and for $k\notin V$, $k\notin V_{n}$ for $n$ sufficiently large so
$\lim_{n\to\infty}c^{\prime,n}_{k}=k=c^{\prime}_{k}$. Conditions 4 and 5 imply
that for $k\in V$ and $\\{k,k^{\prime}\\}\in E$,
$\lim_{n\to\infty}(\vartheta^{\prime,n}_{k},\overline{\vartheta}^{\prime,n}_{\\{k,k^{\prime}\\}})=\lim_{\begin{subarray}{c}n\to\infty,(\vartheta^{\prime,n}_{k},\overline{\vartheta}^{\prime,n}_{\\{k,k^{\prime}\\}})\in
V_{n}\times
E_{n}\end{subarray}}(\vartheta^{n}_{k},\overline{\vartheta}^{n}_{\\{k,k^{\prime}\\}})=(\vartheta_{k},\overline{\vartheta}_{\\{k,k^{\prime}\\}})=(\vartheta^{\prime}_{k},\overline{\vartheta}^{\prime}_{\\{k,k^{\prime}\\}})$.
If $k\notin V$, then $k\notin V_{n}$ for $n$ sufficiently large and because
$\odot$ is an isolated point, this implies that
$\lim_{n\to\infty}\vartheta^{\prime,n}_{k}=\odot=\vartheta^{\prime}_{k}$. By
the same argument, if $\\{k,k^{\prime}\\}\notin E$, then
$\lim_{n\to\infty}\overline{\vartheta}^{\prime,n}_{\\{k,k^{\prime}\\}}=\odot=\overline{\vartheta}^{{}^{\prime}}_{\\{k,k^{\prime}\\}}$.
Lastly, $\o^{n}\to\o$ by Condition 3 of Definition B.3. Thus,
$\psi(G_{n})\to\psi(G)$, which establishes the continuity of $\psi$.

(b)

If $\psi(G_{n})\to\psi(G)$, then $G_{n}\to G$: Conditions 1 and 2 of
Definition B.3 follow from the convergence of $\vartheta^{\prime,n}_{k}$ to
$\vartheta^{\prime}_{k}$ and the convergence of
$\overline{\vartheta}^{\prime,n}_{\\{k,k^{\prime}\\}}$ to
$\overline{\vartheta}^{{}^{\prime}}_{\\{k,k^{\prime}\\}}$ and the fact that
$\odot$ is isolated in ${\mathcal{Z}}_{\odot}$ and
$\overline{{\mathcal{Z}}}_{\odot}$. This directly implies that every vertex
$k\in V$ and edge $\\{k,k^{\prime}\\}\in E$ is in $V_{n}$ and $E_{n}$
respectively for all sufficiently large $n$, and likewise every non-vertex
$k\in{\mathbb{N}}\setminus V$ and non-edge
$\\{k,k^{\prime}\\}\in{\mathbb{N}}\setminus E$ is not in $V_{n}$ or $E_{n}$
respectively for all sufficiently large $n$. Condition 3 follows from the
convergence of $\o^{n}$ to $\o$. To prove Condition 4, recall that we have
already shown that any $k\in V$ is in $V_{n}$ for $n$ sufficiently large.
Thus, $\lim_{\begin{subarray}{c}n\to\infty\\\ k\in
V_{n}\end{subarray}}\vartheta^{n}_{k}=\lim_{\begin{subarray}{c}n\to\infty\\\
k\in
V_{n}\end{subarray}}\vartheta^{\prime,n}_{k}=\lim_{n\to\infty}\vartheta^{\prime,n}_{k}=\vartheta^{\prime}_{k}=\vartheta_{k}$.
The proof of Condition 5 is exactly analogous except we apply the fact that
$\\{k,k^{\prime}\\}\in E$ implies $\\{k,k^{\prime}\\}\in E_{n}$ for $n$
sufficiently large and then apply the convergence of
$\overline{\vartheta}^{\prime,n}_{\\{k,k^{\prime}\\}}$ to
$\overline{\vartheta}^{\prime}_{\\{k,k^{\prime}\\}}$. Lastly, for each $k\in
V$, because $c^{\prime,n}_{k}=c^{\prime}_{k}$ for sufficiently large $n$, this
implies that $E_{\zeta}(k)\cap\\{c^{\prime}_{k}+1,\dots\\}=\emptyset$ for $n$
sufficiently large so $c^{\prime}_{k}\geq\max\text{cl}_{k}(G)$. However,
$\\{k,c^{\prime}_{k}\\}\in E$ if and only if $\\{k,c^{\prime}_{k}\\}\in E_{n}$
for all $n$ sufficiently large, and because $\\{k,c^{\prime,n}_{k}\\}\in
E_{n}$ and $c^{\prime,n}_{k}=c^{\prime}_{k}$ for $n$ sufficiently large, this
implies that $\\{k,c^{\prime}_{k}\\}\in E$ so
$c^{\prime}_{k}=\max\text{cl}_{k}(G)$. Thus, Condition 6 holds.

Because $\psi$ is a one-to-one map onto $\widehat{\mathcal{R}}$, this proves
the claim.

Since $\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ is
homeomorphic to the closed subset $\widehat{\mathcal{R}}$ of the Polish space
${\mathcal{R}}$, it is also Polish. ∎

Next, in Lemma B.6 below, we show that the map $G\mapsto\langle G\rangle$ is
continuous and its set-valued inverse map $\langle
G\rangle\mapsto\\{G\in\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]:G\in\langle
G\rangle\\}$ is lower semicontinuous in the sense of [AubFra09, Definition
1.4.2].

######  Lemma B.6.

If $G_{n}\to G$ in
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$, then
the isomorphism classes also converge locally, that is, $\langle
G_{n}\rangle\to\langle G\rangle$ in
${\mathscr{G}}_{*}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$. Moreover, given
the limit $\langle G_{n}\rangle\to\langle G\rangle$ in
${\mathscr{G}}_{*}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ and any
representative $G\in\langle G\rangle$ such that
$G\in\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$,
there exists a sequence of representatives $G_{n}\in\langle G_{n}\rangle$,
$n\in{\mathbb{N}},$ such that $G_{n}\to G$ in
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$. In
other words, the correspondence
${\mathscr{G}}_{*}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]\ni\langle
G\rangle\mapsto\\{G\in\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]:G\in\langle
G\rangle\\}$ is lower semicontinuous.

###### Proof.

For convenience of notation, let $[n]:=\\{1,\dots,n\\}$. We start by proving
the first statement. Fix any $m\in{\mathbb{N}}$, and let $M_{m}:=\max\\{v\in
B_{m}(G)\\}$. Then Conditions 1 and 2 of Definition B.3 imply that for
sufficiently large $n$, $V\cap[M_{m}]=V_{n}\cap[M_{m}]$ and
$E\cap[M_{m}]^{2}=E_{n}\cap[M_{m}]^{2}$. Condition 3 implies that $\o_{n}=\o$
for $n$ sufficiently large. Thus, the subgraphs of $B_{m}([G_{*}])$ and
$B_{m}([G_{n,*}])$ induced by the set $[M_{m}]$ exactly match for sufficiently
large $n$. Then by Condition 6,
$\max\\{\text{cl}_{v}(G_{n})\\}=\max\\{\text{cl}_{v}(G_{n})\\}$ for all $v\in
V\cap[M_{m}]$ and $n$ sufficiently large. Because $B_{m}(G)\subseteq[M_{m}]$,
these statements imply that all of the vertices in $B_{m}(G_{n})$ also fall
inside $[M_{m}]$, and hence, $B_{m}([G_{*}])=B_{m}([G_{n,*}])$ for all
sufficiently large $n$, that is
$N_{m}:=\min\\{n\in{\mathbb{N}}:B_{m}([G_{n^{\prime},*}])=B_{m}([G_{*}])\text{
for all }n^{\prime}\geq n\\}$ is finite (where the minimum of an empty set is
taken to be $\infty$). For $n>N_{m}$, let $\varphi_{n,m}:B_{m}(G)\to
B_{m}(G_{n})$ be the identity isomorphism. Then for each $v\in B_{m}(G)$ and
$e\in E_{B_{m}(G)}$, Conditions 4 and 5 of Definition B.3 imply
$\lim_{\begin{subarray}{c}n\to\infty\\\
n>N_{m}\end{subarray}}\vartheta^{n}_{\varphi_{n,m}(v)}=\lim_{\begin{subarray}{c}n\to\infty\\\
v\in V_{n}\end{subarray}}\vartheta^{n}_{v}=\vartheta_{v},$ and
$\lim_{\begin{subarray}{c}n\to\infty\\\
n>N_{m}\end{subarray}}\overline{\vartheta}^{n}_{\varphi_{n,m}(e)}=\lim_{\begin{subarray}{c}n\to\infty\\\
e\in
E_{n}\end{subarray}}\overline{\vartheta}^{n}_{e}=\overline{\vartheta}_{e}$. By
Definition 2.1, this proves that $\langle G_{n}\rangle\to\langle G\rangle$.

To prove the second statement, first consider the case when the representative
$G\in\langle G\rangle$ has a vertex set that is canonical in the sense that
$V=[|V|]=\\{1,\dots,|V|\\}$. It is easy to see that one can always choose
representatives
$G^{\prime}_{n}=(V^{\prime}_{n},E^{\prime}_{n},\o^{\prime}_{n},\overline{\vartheta}^{\prime,n},\vartheta^{\prime,n})$
of $\langle G_{n}\rangle,$ $n\in{\mathbb{N}},$ such that

$\displaystyle V^{\prime}_{n}$
$\displaystyle=[|V^{\prime}_{n}|]=\\{1,\ldots,|V^{\prime}_{n}|\\}\text{ for
}n\in{\mathbb{N}},$ (B.2)

where we interpret $\\{1,\dots,\infty\\}$ as ${\mathbb{N}}$. By construction,
each $V_{n}$ is also in canonical form. Then by Definition 2.1, for each
$m\in{\mathbb{N}}_{0}$, there exist $n_{m}<\infty$ and a collection of
isomorphisms $\varphi_{n,m}\in I(B_{m}([G_{*}]),B_{m}([G^{\prime}_{n,*}]))$,
$m\in{\mathbb{N}}_{0},n>n_{m},$ such that for each $m\in{\mathbb{N}},$ the
inclusions $v\in B_{m}(G)$ and $e\in E_{B_{m}(G)}$ imply

$\lim_{\begin{subarray}{c}n\to\infty\\\
n>n_{m}\end{subarray}}d_{\overline{{\mathcal{Z}}}}(\overline{\vartheta}^{\prime,n}_{\varphi_{n,m}(e)},\overline{\vartheta}_{e})=0\quad\text{
and }\quad\lim_{\begin{subarray}{c}n\to\infty\\\
n>n_{m}\end{subarray}}d_{{\mathcal{Z}}}(\vartheta^{\prime,n}_{\varphi_{n,m}(v)},\vartheta_{v})=0.$
(B.3)

Hence, there exists a sequence of non-decreasing integers
$M_{n},n\in{\mathbb{N}},$ converging to infinity such that for each
$n\in{\mathbb{N}}$, the inclusions $v\in B_{M_{n}}(G)$ and $e\in
E_{B_{M_{n}}(G)}$ imply
$d_{{\mathcal{Z}}}(\vartheta^{\prime,n}_{\varphi_{n,M_{n}}(v)},\vartheta_{v})<2^{-M_{n}}$
and
$d_{\overline{{\mathcal{Z}}}}(\overline{\vartheta}^{\prime,n}_{\varphi_{n,M_{n}}(e)},\overline{\vartheta}_{e})<2^{-M_{n}}$
when $M_{n}>0$. Set $\varphi_{n}:=\varphi_{n,M_{n}}$, and for each
$n\in{\mathbb{N}},$ define
$\overline{\varphi}_{n}:{\mathbb{N}}\to{\mathbb{N}}$ by

$\overline{\varphi}_{n}(v)=\begin{cases}\varphi_{n}(v)&\text{ if }v\in
B_{M_{n}}(G),\\\ w^{n}_{v}&\text{ otherwise, }\end{cases}$

where if $v$ is the $k$th smallest element of ${\mathbb{N}}\setminus
B_{M_{n}}(G)$, then $w^{n}_{v}$ is the $k$th smallest element of
${\mathbb{N}}\setminus B_{M_{n}}(G_{n}^{\prime})$. Now for each
$n\in{\mathbb{N}}$, define

$G_{n}:=(V_{n},E_{n},\o_{n},\overline{\vartheta}^{n},\vartheta^{n}):=\left(\overline{\varphi}^{-1}_{n}(V^{\prime}_{n}),\overline{\varphi}^{-1}_{n}(E^{\prime}_{n}),\overline{\varphi}_{n}^{-1}(\o^{\prime}_{n}),(\overline{\vartheta}^{\prime,n}_{\overline{\varphi}_{n}(e)})_{e\in
E_{n}},(\vartheta^{\prime,n}_{\overline{\varphi}_{n}(v)})_{v\in
V_{n}}\right).$

By construction, for each $n\in{\mathbb{N}}$,
$V_{n}=V^{\prime}_{n}=\\{1,\dots,|V_{n}|\\}$, just as in (B.2). Then by
definition, for each $m\in{\mathbb{N}}$ and noting that (i) when $m\leq
M_{n}$, $\overline{\varphi}_{n}|_{B_{m}(G)}=\varphi_{n,M_{n}}|_{B_{m}(G)}$
(which implies
$\overline{\varphi}^{-1}_{n}|_{B_{m}(G_{n})}=\varphi_{n,M_{n}}^{-1}|_{B_{m}(G_{n})}$)
and (ii) $M_{n}$ increases to infinity, one has

$B_{m}([G_{n,*}])=\varphi^{-1}_{n,M_{n}}(B_{m}([G^{\prime}_{n,*}]))=B_{m}([G_{*}])\text{
for }n\text{ sufficiently large.}$ (B.4)

For every $v\in V$ and $e\in E$, there exists $m\in{\mathbb{N}}$ such that
$v\in B_{m}(G)$, $e\in E_{B_{m}(G)}$. Then by (B.4), $v\in V_{n}$ and $e\in
E_{n}$ for $n$ sufficiently large. On the other hand, suppose $v\notin V$.
Then because $V$ is in canonical form, $|V|<v$ is finite. Thus, $G$ has radius
$M<\infty$. Setting $m=M+1$ in (B.4),
$B_{M+1}([G_{n,*}])=B_{M+1}([G_{*}])=[G_{*}]$ for $n$ sufficiently large.
Furthermore, because $B_{M+1}([G_{n,*}])$ has radius $M$, it follows that
$[G_{n,*}]=B_{M+1}([G_{n,*}])=[G_{*}]$. Thus, $v\notin V_{n}$ for $n$
sufficiently large which implies that $V_{n}\to V$. Now, suppose
$e=\\{u,v\\}\notin E$. Then either (without loss of generality) $u\notin V$,
in which case $u\notin V_{n}$ for sufficiently large $n$ and $\\{u,v\\}\notin
E_{n}$ for such $n$, or $u,v\in V$. In the latter case, we can fix $m$ so that
$u,v\in B_{m}(G)$. Then for $n$ sufficiently large,
$B_{m}([G_{n,*}])=B_{m}([G_{*}])$ by (B.4), so $\\{u,v\\}\notin E_{n}$. Thus,
$E_{n}\to E$. Setting $m=0$,
$\o_{n}=\overline{\varphi}_{0}^{-1}(\o^{\prime}_{n})=\o$ for all
$n\in{\mathbb{N}}$. Lastly, for each $v\in V$, fix $m$ such that $v\in
B_{m-1}(G)$. Then by (B.4), $\text{cl}_{v}(G_{n})=\text{cl}_{v}(G)$ for $n$
sufficiently large. Thus, Conditions 1-3 and 6 of Definition B.3 hold. Now for
each $v\in V$ and $e\in E$, there must exist $m\in{\mathbb{N}}$ such that
$v\in B_{m}(G)$ and $e\in E_{B_{m}(G)}$. Then by the definition of $M_{n}$, it
follows that

$\displaystyle\lim_{n\to\infty}d_{{\mathcal{Z}}}(\vartheta^{n}_{v},\vartheta_{v})$
$\displaystyle=\lim_{n\to\infty}d_{{\mathcal{Z}}}(\vartheta^{{}^{\prime}n}_{\varphi_{n,M_{n}}(v)},\vartheta_{v})=\lim_{n\to\infty}2^{-M_{n}}=0$
$\displaystyle\lim_{n\to\infty}d_{{\mathcal{Z}}}(\overline{\vartheta}^{n}_{e},\overline{\vartheta}_{e})$
$\displaystyle=\lim_{n\to\infty}d_{{\mathcal{Z}}}(\overline{\vartheta}^{{}^{\prime}n}_{\varphi_{n,M_{n}}(e)},\overline{\vartheta}_{e})=\lim_{n\to\infty}2^{-M_{n}}=0.$

Thus, Conditions 4 and 5 of Definition B.3 also hold, proving that $G_{n}\to
G$ in $\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$.

We finish the proof of the second assertion by considering the general case in
which $G$ is an arbitrary representative of $\langle G\rangle$ with no
restriction on the vertex set. In this case, there exists
$G^{\prime}:=(V^{\prime},E^{\prime},\o^{\prime},\overline{\vartheta}^{\prime},\vartheta^{\prime})\cong
G$ whose vertex set $V^{\prime}:=V_{G^{\prime}}$ is in canonical form. Using
the argument above, construct the sequence $\\{G^{\prime}_{n}\\}$ whose vertex
sets $V^{\prime}_{n}:=V_{G^{\prime}_{n}}$ are in canonical form and such that
$G^{\prime}_{n}\to G^{\prime}$ in
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$. Given
any isomorphism $\varphi\in I(G^{\prime},G)$, define

$\overline{\varphi}(v):=\begin{cases}\varphi(v)&\text{ if }v\in
V_{G^{\prime}},\\\ w_{v}&\text{ otherwise,}\end{cases}$

where if $v$ is the $k$th smallest element of ${\mathbb{N}}\setminus
V^{\prime}$, then $w_{v}$ is the $k$th smallest element of
${\mathbb{N}}\setminus V$. Then
$\overline{\varphi}:{\mathbb{N}}\to{\mathbb{N}}$ is a bijection, so for each
$n\in{\mathbb{N}}$,

$G_{n}:=\overline{\varphi}(G_{n}):=(\overline{\varphi}(V^{\prime}_{n}),\overline{\varphi}(E^{\prime}_{n}),\overline{\varphi}(\o^{\prime}_{n}),(\overline{\vartheta}^{{}^{\prime}n}_{\overline{\varphi}^{-1}(e)})_{e\in\overline{\varphi}(E^{\prime}_{n})},(\vartheta^{{}^{\prime}n}_{\overline{\varphi}^{-1}(v)})_{v\in\overline{\varphi}(V^{\prime}_{n})}),$

is isomorphic to $G^{\prime}_{n}$ and $G_{n}\to G$. Thus, the correspondence
$\langle
G\rangle\mapsto\\{G^{\prime}\in\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]:G^{\prime}\in\langle
G\rangle\\}$ is lower semicontinuous by [AubFra09, Definition 1.4.2]. ∎

###  B.2 Existence of Measurable Selections

The goal of this section is to establish Lemmas 6.2 and 6.6. We begin this
section by showing that every random isomorphism class of rooted graphs has a
measurable random representative.

######  Lemma B.7.

Given any ${\mathscr{G}}_{*}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$-random
element $\langle G\rangle$, there exists a $\sigma(\langle
G\rangle)$-measurable representative
$[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$-random graph $G$ in
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$.

###### Proof.

For every $\langle
H\rangle\in{\mathscr{G}}_{*}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$, define

$\Psi_{*}(\langle
H\rangle):=\\{H^{\prime}\in\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]:H^{\prime}\in\langle
H\rangle\\}.$ (B.5)

Then the set $\Psi_{*}(\langle H\rangle)$ is non-empty since it contains $H$,
and is also closed due to the continuity of the map $H\mapsto\langle H\rangle$
established in Lemma B.6. Thus $\Psi_{*}$ maps any isomorphism class in
${\mathscr{G}}_{*}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ to the closed set
in
$\mathrm{clo}(\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}])$
that contains all graphs in
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ that lie
in that isomorphism class. Given the
${\mathscr{G}}_{*}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$-random element
$\langle G\rangle$, we first argue that to prove the lemma it suffices to
prove the claim that for every open set
$U\subseteq\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$,
the following set is open in
${\mathscr{G}}_{*}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$:

$\Lambda_{*}^{U}:=\\{\langle
H\rangle\in{\mathscr{G}}_{*}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]:\Psi_{*}(\langle
H\rangle)\cap U\neq\emptyset\\}=\bigcup_{H\in U}\\{\langle H\rangle\\}.$

Indeed, since $\langle G\rangle$ is a
${\mathscr{G}}_{*}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$-random element,
the claim implies that $\psi_{*}(\langle G\rangle)$ is a non-empty
$\sigma(\langle G\rangle)$-random closed subset of
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ in the
sense of Definition B.1, and the lemma follows on applying Theorem B.2 with
${\mathcal{Z}}^{\prime}=\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$,
$F=\psi_{*}(\langle G\rangle)$ and ${\mathcal{F}}=\sigma(\langle G\rangle)$.

We now turn to the proof of the claim. If $U$ is empty then so is
$\Lambda_{*}^{U}$. Now, fix
$U\subseteq\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$
non-empty. Then $\Lambda_{*}^{U}$ is trivially non-empty as well. For $\langle
H\rangle\in\Lambda_{*}^{U}$, suppose there exists a sequence $\\{\langle
H_{n}\rangle\\}_{n\in{\mathbb{N}}}$ of (deterministic) elements of
${\mathscr{G}}_{*}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ converging to
$\langle H\rangle$. Select $H^{\prime}\in\Psi_{*}(\langle H\rangle)\cap U$
(which is non-empty since $\Lambda_{*}^{U}\neq\emptyset$). By Lemma B.6, there
exists a sequence of representative graphs $H^{\prime}_{n}\in\langle
H_{n}\rangle$, $n\in{\mathbb{N}},$ that converges to $H^{\prime}$ in
${\mathcal{Z}}^{\prime}$. Because $H^{\prime}\in U$ and $U$ is open,
$H^{\prime}_{n}\in U$ for all but finitely many $n$. By the definition of
$\Lambda_{*}^{U}$, this implies $\langle H_{n}\rangle\in\Lambda_{*}^{U}$ for
all but finitely many $n$. Because this is true for any sequence converging to
an element in $\Lambda_{*}^{U}$, and because
${\mathscr{G}}_{*}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$ is a Polish space
(in which convergence is equivalent to sequential convergence)
$\Lambda_{*}^{U}$ is open. This concludes the proof. ∎

######  Remark B.8.

For $W_{1},W_{2}\subseteq{\mathbb{N}}$, let ${\mathcal{M}}(W_{1},W_{2})$ be
the space of mappings from $W_{1}$ to $W_{2}$, and define
${\mathcal{M}}:=\bigcup_{W_{1},W_{2}\subseteq{\mathbb{N}}}{\mathcal{M}}(W_{1},W_{2})$.
Then any map $f\in{\mathcal{M}}(W_{1},W_{2})\subset{\mathcal{M}}$ can be
embedded in ${\mathbb{N}}_{0}^{{\mathbb{N}}}$ (equipped with the discrete
product topology) via the bijective map

$\beta(f):=(\beta_{n}(f))_{n\in{\mathbb{N}}},\qquad\mbox{ where
}\quad\beta_{n}(f)=\left\\{\begin{array}[]{ll}f(n)&\mbox{ if }n\in W_{1},\\\
0&\mbox{ otherwise. }\end{array}\right.\quad\mbox{ for
}f\in{\mathcal{M}}(W_{1},W_{2}).$ (B.6)

We equip ${\mathcal{M}}$ with the topology induced by $\beta$, that is, we
define a subset $U\subseteq{\mathcal{M}}$ to be open if and only if
$\beta(U)\subseteq{\mathbb{N}}_{0}^{{\mathbb{N}}}$ is open. With this
definition, $\beta$ is automatically a homeomorphism and ${\mathcal{M}}$ is a
Polish space. We now apply Theorem B.2 to select the isomorphisms in Lemma 6.2
in a measurable manner.

######  Lemma B.9.

Fix a probability space $(\Omega,{\mathcal{F}},{\mathbb{P}})$ that supports
finite, ${\mathcal{F}}$-measurable
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$-random
elements $G_{i}$, $i=1,2,$ that satisfy ${\mathbb{P}}(G_{1}\cong G_{2})>0$.
Then given any ${\mathcal{F}}$-random closed subset $A$ of ${\mathcal{M}}$
such that $A$ is a non-empty subset of $I(G_{1},G_{2})$ on the event
$\\{G_{1}\cong G_{2}\\}$, there exists an ${\mathcal{F}}$-measurable map
$\varphi\in{\mathcal{M}}(V_{1},V_{2})$ such that $\varphi\in A$ on the event
$\\{G_{1}\cong G_{2}\\}$.

###### Proof.

Given any ${\mathcal{F}}$-random closed subset $A$ of ${\mathcal{M}}$ such
that $A\subseteq I(G_{1},G_{2})$, consider the set-valued mapping
$B(\omega):={\mathcal{M}}(V_{1},V_{2})$ if $G_{1}\ncong G_{2}$ and
$B(\omega):=A$ otherwise. Note that by finiteness of $G_{1}$ and $G_{2}$, $B$
is also finite because $B\subseteq{\mathcal{M}}(V_{1},V_{2})$, and thus
$B(\omega)$ is closed for all $\omega\in\Omega$. Let
$\overline{B}=\\{\beta(f):f\in B\\}$, with $\beta$ as in (B.6), and note that
$\overline{B}(\omega)$ is similarly finite and therefore closed and non-empty
for all $\omega\in\Omega$. We claim that $\overline{B}(\omega)$ is weakly
${\mathcal{F}}$-measurable. That is, for every open
$U\subseteq{\mathbb{N}}_{0}^{{\mathbb{N}}}$, the set $\\{\overline{B}\cap
U\neq\emptyset\\}$ is ${\mathcal{F}}$-measurable. If the claim holds, then
$\overline{B}$ is a non-empty ${\mathcal{F}}$-random closed subset, so by
Theorem B.2 with ${\mathcal{Z}}^{\prime}={\mathbb{N}}_{0}^{{\mathbb{N}}}$ and
$F=\overline{B}$, there exists an ${\mathcal{F}}$-measurable random variable
$b$ such that $b(\omega)\in\overline{B}(\omega)$ for every $\omega\in\Omega$.
The lemma follows on setting $\varphi=\beta^{-1}(b)$.

To prove the claim, first note that the set $\\{\omega:G_{1}\cong G_{2}\\}$ is
${\mathcal{F}}$-measurable by assumption. Moreover, fixing any open set
$U\subseteq{\mathbb{N}}_{0}^{{\mathbb{N}}}$, note that $\\{G_{1}\cong
G_{2}\\}\cap\\{\overline{B}\cap U\neq\emptyset\\}=\\{G_{1}\cong
G_{2}\\}\cap\\{A\cap\beta^{-1}(U)\neq\emptyset\\}$, and also that
$U^{\prime}:=\beta^{-1}(U)\subset{\mathcal{M}}$ is open since $\beta$ is
continuous. Because $A$ is an ${\mathcal{F}}$-random closed subset, this
implies $\\{G_{1}\cong G_{2}\\}\cap\\{\overline{B}\cap U\neq\emptyset\\}$ is
${\mathcal{F}}$-measurable. Next, observe that $\\{G_{1}\ncong
G_{2}\\}\cap\\{\overline{B}\cap U\neq\emptyset\\}=\\{G_{1}\ncong
G_{2}\\}\cap\\{{\mathcal{M}}(V_{1},V_{2})\cap\beta^{-1}(U)\neq\emptyset\\}=\\{G_{1}\ncong
G_{2}\\}\cap\\{{\mathcal{M}}(V_{1},V_{2})\cap U^{\prime}\neq\emptyset\\}$.
However, note that

$\\{{\mathcal{M}}(V_{1},V_{2})\cap
U^{\prime}\neq\emptyset\\}=\bigcup_{\text{finite
}W_{1},W_{2}\subset{\mathbb{N}}}\\{V_{i}(\omega)=W_{i},i=1,2\\}\cap\\{U^{\prime}\cap{\mathcal{M}}(W_{1},W_{2})\neq\emptyset\\},$

which is a countable union of ${\mathcal{F}}$-measurable sets since
$U^{\prime}$ is open. Thus, $\\{\overline{B}\cap U\neq\emptyset\\}$ is
${\mathcal{F}}$-measurable for every open $U$, and the claim follows. ∎

The following lemma ensures driving maps on different graphs can be
constructed so as to be consistent with isomorphisms between those graphs,
which is required in Sections 6 and 7.

######  Lemma B.10.

Fix a probability space $(\Omega,{\mathcal{F}},{\mathbb{P}})$ that supports
the ${\mathcal{F}}$-measurable
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{Z}}},{\mathcal{Z}}]$-random
elements $G_{i}$, $i=1,2$. Let $M$ be an ${\mathcal{F}}$-measurable random
variable such that
$M(\omega)\in\\{m\in{\mathbb{N}}_{0}:B_{m}(G_{1}(\omega))\cong
B_{m}(G_{2}(\omega))\\}$ for every $\omega\in\Omega$. Then for any
${\mathcal{F}}$-measurable isomorphism $\varphi\in
I(B_{M}(G_{1}),B_{M}(G_{2}))$ and ${\mathcal{F}}$-measurable driving map
$\psi_{1}:V_{1}\to{\mathbb{N}}$ such that
${\mathbb{N}}\setminus\psi_{1}(V_{1})$ is infinite, there exists an
${\mathcal{F}}$-measurable driving map $\psi_{2}:V_{2}\to{\mathbb{N}}$ such
that $\psi_{1}(v)=\psi_{2}(\varphi(v))$ for every $v\in B_{M}(G_{1})$ and
$\psi_{1}(G_{1})\cap\psi_{2}(G_{2}\setminus B_{M}(G_{2}))=\emptyset$.

###### Proof.

Fix an ${\mathcal{F}}$-measurable random variable $M$ and
${\mathcal{F}}$-measurable isomorphism $\varphi\in
I(B_{M}(G_{1}),B_{M}(G_{2}))$ as in the statement of the lemma. Let
$\widetilde{{\mathcal{Z}}}:=({\mathbb{N}}_{0}^{{\mathbb{N}}})^{2}$, equipped
with the product topology, and consider the $\widetilde{{\mathcal{Z}}}$-random
element $b=(b_{1},b_{2})$ with $b_{i}=(b_{i,k})_{k\in{\mathbb{N}}}$ given, for
$i=1,2$ and $k\in{\mathbb{N}}$, by

$b_{i,k}=\begin{cases}0&\text{ if }k\notin G_{i},\\\ \psi_{1}(k)&\text{ if
}k\in V_{G_{1}},\\\ \psi_{1}(\varphi^{-1}(k))&\text{ if }k\in
B_{M}(G_{2})\text{ and }i=2,\\\ \alpha(k)&\text{ if }k\in V_{2}\setminus
B_{M}(G_{2})\text{ and }i=2,\end{cases}$

where $\alpha:{\mathbb{N}}\to{\mathbb{N}}$ is an injection mapping each $k$ to
the $k$th smallest element of ${\mathbb{N}}\setminus\psi_{1}(V_{1})$. Observe
that $\alpha$ is well defined because ${\mathbb{N}}\setminus\psi_{1}(V_{1})$
is infinite and is also an ${\mathcal{F}}$-measurable element of
${\mathcal{M}}$ because $\psi_{1}$ is ${\mathcal{F}}$-measurable. Thus, $b$ is
clearly also ${\mathcal{F}}$-measurable. Define $\psi_{i}:=\beta^{-1}(b_{i})$,
$i=1,2$, with $\beta$ as in (B.6), and note that since $\beta^{-1}$ is Borel
measurable, each $\psi_{i}$ is also ${\mathcal{F}}$-measurable. Because
$\psi_{1}(V_{1})$ and $\alpha({\mathbb{N}})$ are disjoint, $\psi_{2}$ is also
injective and therefore a driving map. Furthermore, for each $k\in
B_{M}(G_{1})$,
$\psi_{1}(k)=\psi_{1}(\varphi^{-1}(\varphi(k)))=\psi_{2}(\varphi(k))$, and
$\psi_{1}(G_{1})\subseteq\psi_{1}(V_{1})$ and $\psi_{2}(G_{2}\setminus
B_{M}(G_{2}))\subseteq\alpha({\mathbb{N}})$, so $\psi_{1}(G_{1})$ and
$\psi_{2}(G_{2}\setminus B_{M}(G_{2}))$ are disjoint. This completes the
proof. ∎

We finish the appendix with proofs of Lemmas 6.2 and 6.6.

###### Proof of Lemma 6.2.

The first assertion follows from Lemma B.7 and the fact that
$\sigma(\langle(\overline{G}_{n},\overline{\xi}^{n})\rangle)\subseteq\widehat{{\mathcal{F}}}$.
For the second assertion, let
$\\{(G_{n},\xi^{n})\\}_{n\in{\mathbb{N}}},(G,\xi)$ be any sequence of
measurable representatives satisfying Properties 1 and 2 of Definition 6.1. By
Definition 2.1, the convergence of $\langle(G_{n},\xi^{n})\rangle$ to
$\langle(G,\xi)\rangle$ in
${\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$
implies that for every $m\in{\mathbb{N}}$, $B_{m}([G_{*}])\cong
B_{m}([G_{n,*}])$ for sufficiently large $n$. Since ${\mathcal{X}}$ is
discrete, this further implies that there exists an a.s. finite random
variable $N_{m}$ such that $B_{m}([G_{*}],\xi)\cong B_{m}([G_{n,*}],\xi^{n})$
for all $n\geq N_{m}$ and $N_{m}$ is $\widehat{{\mathcal{F}}}$-measurable.
Then for any $\varphi\in{\mathcal{M}}$, define the following random variable:

$\Psi_{n,m}(\varphi):=\begin{cases}\sum_{v\in
B_{m}(G)}d_{{\mathcal{K}}}(\mathbf{\kappa}^{n}_{\varphi(v)},\mathbf{\kappa}_{v})+\sum_{e\in
E_{B_{m}(G)}}d_{\overline{{\mathcal{K}}}}(\overline{\mathbf{\kappa}}^{n}_{\varphi(e)},\overline{\mathbf{\kappa}}_{e})&\text{
if }\varphi\in I_{n,m},\\\ \infty&\text{ otherwise,}\end{cases}$

where $I_{n,m}:=I(B_{m}([G_{*}],\xi),B_{m}([G_{n,*}],\xi^{n}))$. Let
${\mathcal{M}}^{\Psi}_{n,m}=\text{argmin}_{\varphi\in{\mathcal{M}}}\Psi_{n,m}(\varphi)$,
where we define ${\mathcal{M}}^{\Psi}_{n,m}$ to be empty if
$\Psi_{n,m}(\varphi)=\infty$ for all $\varphi\in{\mathcal{M}}$. On the
$\widehat{{\mathcal{F}}}$-measurable set $\\{n\geq N_{m}\\}$, $I_{n,m}$ is
non-empty and finite, and so ${\mathcal{M}}^{\Psi}_{n,m}$ is also non-empty
and finite. We first show that to prove the following claim:

Key claim:  ${\mathcal{M}}^{\Psi}_{n,m}$ is a $\widehat{{\mathcal{F}}}$-random
closed subset of ${\mathcal{M}}$ that is non-empty on the event
$\\{I_{n,m}\neq\emptyset\\}$.

Deferring the proof of the claim, first note that given the claim, applying
Lemma B.9, with $G_{1}=B_{m}([G_{*}],\xi)$, $G_{2}=B_{m}([G_{n,*}],\xi^{n})$
and $A={\mathcal{M}}^{\Psi}_{n,m}$, there exists a sequence of
$\widehat{{\mathcal{F}}}$-measurable maps
$\\{\overline{\varphi}_{n,m}\\}_{n,m\in{\mathbb{N}}}$ such that
$\overline{\varphi}_{n,m}\in{\mathcal{M}}^{\Psi}_{n,m}$ for every
$n,m\in{\mathbb{N}}$ on the event $\\{n\geq N_{m}\\}$. Hence, by the
definition of ${\mathcal{M}}^{\Psi}_{n,m}$, the fact that
$\overline{\varphi}_{n,m}\in I_{n.m}$ and by Definition 2.1, for any
$m\in{\mathbb{N}}$,

$\lim_{n\to\infty,n>\overline{N}_{m}}\min_{\varphi\in
I_{n,m}}\Psi_{n,m}(\varphi)=\lim_{n\to\infty,n>N_{m}}\Psi_{n,m}(\overline{\varphi}_{n,m})=0.$

This implies that for every $m\in{\mathbb{N}}$ there exists a
$\widehat{{\mathcal{F}}}$-measurable integer $\widehat{N}_{m}$ such that
$\Psi_{n,m}(\overline{\varphi}_{n,m})<2^{-m}$ for all $n\geq\widehat{N}_{m}$.
Moreover, $\widehat{N}_{m}$ is non-decreasing and a.s. finite and so
$M_{n}:=\max\\{m\in{\mathbb{N}}:n\geq\widehat{N}_{m}\\}$ is
$\widehat{{\mathcal{F}}}$-measurable and increases to infinity. Therefore,
Property 3 of Definition 6.1 is satisfied. For each $n\in{\mathbb{N}}$ and
$m\leq M_{n}$, define
$\varphi_{n,m}:=\overline{\varphi}_{n,M_{n}}|_{B_{m}(G)}$. It follows that
$\\{\varphi_{n,m}\\}_{n\in{\mathbb{N}},m\leq M_{n}}$ is also an
$\widehat{{\mathcal{F}}}$-measurable sequence and satisfies Property 4 of
Definition 6.1. Furthermore, Properties 5 and 6 follow directly from the fact
that for $v\in B_{m}(G)$ and $e\in E_{B_{m}(G)}$,

$\max\\{d_{{\mathcal{K}}}(\mathbf{\kappa}^{n}_{\varphi_{n,m}(v)},\mathbf{\kappa}_{v}),d_{\overline{{\mathcal{K}}}}(\overline{\mathbf{\kappa}}^{n}_{\varphi_{n,m}(e)},\overline{\mathbf{\kappa}}_{e}))\leq\Psi_{n,M_{n}}(\overline{\varphi}_{n,M_{n}})<2^{-M_{n}}\to
0\quad\text{as}\quad n\to\infty.$

We now turn to the proof of the key claim. Fix $n,m\in{\mathbb{N}}$. We first
prove the following:

Sub-Claim 1: For each $\varphi\in{\mathcal{M}}$, $\Psi_{n,m}(\varphi)$ is
$\widehat{{\mathcal{F}}}$-measurable.

_Proof of Sub-Claim 1:_ Fix $\varphi\in{\mathcal{M}}$. Define
${\mathcal{A}}^{\varphi}_{m}:={\mathcal{A}}_{m,1}\cap{\mathcal{A}}^{\varphi}_{m,2}$,
where

$\displaystyle{\mathcal{A}}_{m,1}$ $\displaystyle:=$
$\displaystyle\left\\{(H_{1},H_{2})\in\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]^{2}:\max_{i=1,2,v\in
V_{H_{i}}}d_{H_{i}}(\o,v)\leq m\right\\},$
$\displaystyle{\mathcal{A}}^{\varphi}_{m,2}$ $\displaystyle:=$
$\displaystyle\left\\{(H_{1},H_{2})\in\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]^{2}:\varphi\in
I(\theta(H_{1}),\theta(H_{2}))\right\\},$

with
$\theta:\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]\to\widehat{{\mathscr{G}}_{*}}[1,{\mathcal{X}}]$
being the mapping that takes
$H=(V_{H},E_{H},\o_{H},\overline{\mathbf{\kappa}}^{H},\mathbf{\kappa}^{H},\xi^{H})\in\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$
to
$(V_{H},E_{H},\o_{H},\xi^{H})\in\widehat{{\mathscr{G}}_{*}}[1,{\mathcal{X}}]$,
which is the rooted representative graph $H$ with only the
${\mathcal{X}}$-valued vertex marks retained. Then consider the mapping
$\Theta_{m}^{\varphi}:(\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}])^{2}\to{\mathbb{R}}_{+}\cup\\{\infty\\}$
given by

$\displaystyle\Theta_{m}^{\varphi}(H_{1},H_{2}):=\begin{cases}\sum_{v\in
V_{H_{1}}}d_{{\mathcal{K}}}(\mathbf{\kappa}^{H_{2}}_{\varphi(v)},\mathbf{\kappa}^{H_{1}}_{v})+\sum_{e\in
E_{H_{1}}}d_{\overline{{\mathcal{K}}}}(\overline{\mathbf{\kappa}}^{H_{2}}_{\varphi(e)},\overline{\mathbf{\kappa}}^{H_{1}}_{e})&\text{
if }(H_{1},H_{2})\in{\mathcal{A}}^{\varphi}_{m},\\\ \infty&\text{ otherwise.
}\end{cases}$

Since
$\Psi_{n,m}(\varphi)=\Theta_{m}^{\varphi}(B_{m}(G,\xi),B_{m}(G_{n},\xi^{n}))$,
to prove Sub-Claim 1, it suffices to show the following:

Sub-Claim 2: The map $\Theta_{m}^{\varphi}$ is continuous.

_Proof of Sub-Claim 2:_ Fix $\varphi\in{\mathcal{M}}(W_{1},W_{2})$ and
$m\in{\mathbb{N}}$. If $W_{1}$ is infinite,
$\Theta^{\varphi}_{m}\equiv\infty$, and so is trivially continuous. Next,
suppose $|W_{1}|<\infty$. Then, since $\theta$ and $H\mapsto\max_{v\in
V_{H}}d_{H}(\o,v)$ are continuous and
$\\{(H_{1},H_{2})\in(\widehat{{\mathscr{G}}_{*}}[1,{\mathcal{X}}])^{2}:\varphi\in
I(H_{1},H_{2})\\}$ is closed, it follows that ${\mathcal{A}}_{m,1}$ and
${\mathcal{A}}^{\varphi}_{m,2}$ are closed, so ${\mathcal{A}}_{m}^{\varphi}$
is likewise closed. Let
$\\{(H^{n}_{1},H^{n}_{2})\\}_{n\in{\mathbb{N}}}\subseteq\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}]\subset{\mathcal{A}}^{\varphi}_{m}$
be any sequence that is convergent in
$\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}]$, and let
$(H^{\infty}_{1},H^{\infty}_{2})$ denote its limit. Then
$(H^{\infty}_{1},H^{\infty}_{2})\in{\mathcal{A}}^{\varphi}_{m}$ since
${\mathcal{A}}^{\varphi}_{m}$ is closed. Moreover, by Conditions 1 and 2 of
Definition B.3, $V_{H^{n}_{1}}=V_{H^{\infty}_{1}}=W_{1}$ and
$E_{H^{n}_{1}}=E_{H^{\infty}_{1}}$ for all sufficiently large $n$. Then the
convergence of $\Theta^{\varphi}_{m}(H^{n}_{1},H^{n}_{2})$ to
$\Theta^{\varphi}_{m}(H^{\infty}_{1},H^{\infty}_{2})$ is an immediate
consequence of the definition of $\Theta^{\varphi}_{m}$ and Conditions 4 and 5
of Definition B.3. Lastly note that the map
$\theta_{2}:(\widehat{{\mathscr{G}}_{*}}[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}])^{2}\to(\widehat{{\mathscr{G}}_{*}}[1,{\mathcal{X}}])^{2}$
defined by $\theta_{2}(H_{1},H_{2}):=(\theta(H_{1}),\theta(H_{1}))$ is
continuous and that
$A^{\varphi}_{m}=\theta_{2}^{-1}(\overline{A}^{\varphi}_{m})$ where
$\overline{A}^{\varphi}\subseteq(\widehat{{\mathscr{G}}_{*}}[1,{\mathcal{X}}])^{2}$
consists entirely of pairs of graphs with radius at most $m$. However, it is
easily verified by Definition B.3 that (since ${\mathcal{X}}$ is a discrete
space) any finite graph is an isolated point in
$\widehat{{\mathscr{G}}_{*}}[1,{\mathcal{X}}]$, which implies that
$\overline{A}^{\varphi}_{m}$, and therefore $A^{\varphi}_{m}$, must also be
open. Sub-Claim 2 then follows on noting that $\Theta^{\varphi}_{m}$ is
identically equal to infinity and thus trivially continuous on the closed set
$({\mathcal{A}}^{\varphi}_{m})^{c}$.

Next, define
$\Psi^{\text{min}}_{n,m}:=\min_{\varphi\in{\mathcal{M}}}\Psi_{n,m}(\varphi)$
where $\Psi^{\text{min}}_{n,m}=\infty$ when $I_{n,m}=\emptyset$. Then
$\Psi^{\text{min}}_{n,m}$ always exists because $I_{n,m}$ is finite. Note that
since $|B_{m}(G)|+|B_{m}(G_{n})|<\infty$, it follows that

$\Psi^{\text{min}}_{n,m}=\min_{\begin{subarray}{c}\varphi\in{\mathcal{M}}(W_{1},W_{2}):\\\
|W_{1}|+|W_{2}|<\infty\end{subarray}}\Psi_{n,m}(\varphi),$

is a minimum over a countable collection of
$\widehat{{\mathcal{F}}}$-measurable random variables. Sub-Claim 1 then shows
that $\Psi^{\text{min}}_{n,m}$ is also $\widehat{{\mathcal{F}}}$-measurable.
For any open $U\subseteq{\mathcal{M}}$,

$\\{{\mathcal{M}}^{\Psi}_{n,m}\cap
U\neq\emptyset\\}=\bigcup_{\begin{subarray}{c}\varphi\in{\mathcal{M}}(W_{1},W_{2})\cap
U:\\\
|W_{1}|+|W_{2}|<\infty\end{subarray}}\\{\Psi_{n,m}(\varphi)=\Psi^{\text{min}}_{n,m}\\}\cap\\{\Psi^{\text{min}}_{n,m}<\infty\\}.$

Since this is a countable union of $\widehat{{\mathcal{F}}}$-measurable sets,
the key claim follows from Definition B.1. ∎

###### Proof of Lemma 6.6:.

Let $\\{\langle(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}}$ be a random
sequence on $(\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}})$
that converges a.s. to $\langle(G,\xi)\rangle$ in
${\mathscr{G}}_{*}[\overline{{\mathcal{K}}},{\mathcal{K}}]$. By Lemma 6.2,
there exists a rep-con sequence
$(\\{(G_{n},\xi^{n}),M_{n}\\}_{n\in{\mathbb{N}}}$,
$(G,\xi),\\{\varphi_{n,m}\\}_{n\in{\mathbb{N}},m\leq M_{n}})$ of
$(\\{\langle(G_{n},\xi^{n})\rangle\\}_{n\in{\mathbb{N}}},\langle(G,\xi)\rangle)$
defined on the same probability space
$(\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}})$. Let $\psi$
be a $\widehat{{\mathcal{F}}}$-measurable driving map
$\psi:V_{G}\to{\mathbb{N}}$ such that ${\mathbb{N}}\setminus\psi(G)$ is
infinite (e.g., consider the map $\psi(k)=2k$). Then, invoking Properties 3
and 4 of Definition 6.1 and repeatedly applying Lemma B.10, with $M=M_{n}$,
$G_{1}=([G_{*}],\xi)$, $G_{2}=([G_{n,*}],\xi^{n})$,
${\mathcal{F}}=\widehat{{\mathcal{F}}}$, $\varphi=\varphi_{n,M_{n}}$ and
$\psi_{1}=\psi$, for each $n\in{\mathbb{N}}$ we can construct a
$\widehat{{\mathcal{F}}}$-measurable driving map
$\psi_{n}:G_{n}\to{\mathbb{N}}$ such that for every $m\leq M_{n}$ and $v\in
B_{m}(G)$, $\psi_{n}(\varphi_{n,m}(v))=\psi(v)$. Then extending the space
$(\widehat{\Omega},\widehat{\mathcal{F}},\widehat{{\mathbb{P}}})$ to add a
countable sequence of i.i.d. Poisson processes
$\\{\mathbf{N}_{k}\\}_{k\in{\mathbb{N}}}$ and using the driving maps
$\psi_{n}$, $n\in{\mathbb{N}}$, and $\psi$ to generate the respective
${\mathbb{F}}$-driving noise $(G_{n},\mathbf{N}^{G_{n}})$, $n\in{\mathbb{N}},$
and $(G,\mathbf{N}^{G})$ as in Definition 6.4, we obtain a consistent rep-con
extension. ∎

##  Appendix C Well-Posedness for Finite Initial Data

Under Assumption 1, well-posedness of (3.3) is common knowledge when the
initial data is finite, but we establish it here for completeness. In this
case, we also show that the trajectories also satisfy the following additional
regularity property. Recall the definition of the discontinuity set ${\rm
Disc}_{t}\left(x\right)$ given in (2.2).

######  Definition C.1 (Proper trajectories).

Given a countable set $W$ and $t\in[0,\infty)$, we say
$x\in{\mathcal{D}}_{t}^{W}$ is _proper_ if ${\rm
Disc}_{t}\left(x_{w}\right)\cap{\rm Disc}_{t}\left(x_{v}\right)=\emptyset$ for
all distinct $v,w\in W$. Moreover, we say a trajectory $x\in{\mathcal{D}}^{W}$
is proper if its restriction $x[t]$ to $[0,t]$ is proper for all
$t\in{\mathbb{R}}_{+}$.

###### Proof of Proposition 4.1 and trajectories being a.s. proper:.

Let $(G,\xi)=(V,E,\o,\overline{\mathbf{\kappa}},\mathbf{\kappa},\xi)$ any a.s.
finite $[\overline{{\mathcal{K}}},{\mathcal{K}}\times{\mathcal{X}}]$-random
graph. By Lemma 3.8, it suffices to prove that (3.3) is strongly well-posed
for $(G,\xi)$ with a.s. proper solutions under the additional assumption that
$(G,\xi)$ is deterministic.

Let $({\mathbb{F}},\mathbf{N}^{G})$ be a filtration-Poisson process pair in
the sense of Remark 3.5. First note that for any
$({\mathbb{F}},\mathbf{N}^{G})$-weak solution $(G,X)$ to (3.3), any distinct
vertices $u,v\in V$ and any $T>0$,

${\rm Disc}_{T}\left(X_{v}\right)\cap{\rm
Disc}_{T}\left(X_{u}\right)\subseteq\\{s\in[0,T]:\mathbf{N}^{G}_{u}(\\{s\\}\times(0,C_{k,T}]\times{\mathcal{J}})\mathbf{N}^{G}_{v}(\\{s\\}\times(0,C_{k,T}]\times{\mathcal{J}})=1\\}=\emptyset\text{
a.s.,}$

by (3.3) and Assumption 1 where $\\{C_{k,T}\\}$ is the family of constants
from Assumption 1 and $k:=\max\\{|\text{cl}_{u}|,|\text{cl}_{v}|\\}$. Since
this holds for all $T$, $(G,X)$ is a.s. proper.

Next, fix a filtration-Poisson process pair $({\mathbb{F}},\mathbf{N}^{G})$.
Define the finite collection,

${\mathcal{E}}:=\\{(\tau_{n},r_{n},j_{n},v_{n})\in[0,T]\times[0,C_{K,T}]\times{\mathcal{J}}\times
V:\mathbf{N}^{G}_{v_{n}}(\\{(\tau_{n},r_{n},j_{n})\\})=1\\},$

$K=\max_{v\in V}|\text{cl}_{v}|$ and $\\{\tau_{n}\\}_{n=1}^{|{\mathcal{E}}|}$
is increasing. Note that $\\{\tau_{n}\\}$ is, in fact, strictly increasing.
define the ${\mathcal{D}}^{V}_{T}$-random element $X[T]$ by,

$X_{v}(t)=\begin{cases}\xi_{v}&\text{ if }0\leq t<\tau_{1},\\\
X_{v}(\tau_{n})&\text{ if }\tau_{n}\leq t<\tau_{n+1},n<|{\mathcal{E}}|,\\\
X_{v}(\tau_{n})&\text{ if }\tau_{n}\leq t\leq T,n=|{\mathcal{E}}|,\\\
X_{v}(\tau_{n})&\text{ if
}t=\tau_{n+1},r_{n+1}>r^{G,v}_{j_{n+1}}(t_{n+1},X),\\\
X_{v}(\tau_{n})+j_{n+1}&\text{ if
}t=\tau_{n+1},r_{n+1}>r^{G,v}_{j_{n+1}}(t_{n+1},X).\end{cases}$

Then note that for $t\in[0,T]$, $X(t)$ is clearly
${\mathcal{F}}^{G,\xi,\mathbf{N}^{G}}_{t}$-measurable so $X$ is a
$\mathbf{N}^{G}$-strong solution to (3.3) for $(G,\xi)$ on the interval
$[0,T]$. Furthermore, it is clear that any
$({\mathbb{F}},\mathbf{N}^{G})$-weak solution to (3.3) must satisfy the above
display, so all $({\mathbb{F}},\mathbf{N}^{G})$-weak solutions equal $X$ on
the interval $[0,T]$. Because $T$ is arbitrary, and for any $T^{\prime}>T$ the
corresponding solution $X^{\prime}$ satisfies $X^{\prime}[T]=X[T]$ a.s., it
follows that there exists a $\mathbf{N}^{G}$-strong solution to (3.3) and that
solution is pathwise unique. Therefore (3.3) is strongly well-posed for
$(G,\xi)$. ∎

##  Appendix D Characterization of Strong Well-Posedness on Random Graphs

###### Proof of Lemma 3.8.

The key issue here is to show that conditioning on the initial data does not
change the driving noise structure. Note that this is slightly non-standard as
the driving noise is indexed by the vertices of the graph and thus is not
completely independent of the initial data. Let
$(\Omega,{\mathcal{F}},{\mathbb{F}},{\mathbb{P}})$ be a complete, filtered
probability space that supports $(G,\xi)$ and a filtration-Poisson process
pair $({\mathbb{F}},\mathbf{N}^{G})$ such that $(G,X)$ and $(G,Y)$ are two
$({\mathbb{F}},\mathbf{N}^{G})$-weak solutions of (3.3) for $(G,\xi)$. To
prove the lemma, it suffices to prove the following claim: for
${\mathbb{P}}$-a.s. $\omega\in\Omega$, setting
$(H,\xi^{H}):=(G(\omega),\xi(\omega))$, there exists a filtration-Poisson
process pair $({\mathcal{F}}^{H},\mathbf{N}^{H})$ on some probability space
$(\Omega^{H},{\mathcal{F}}^{H},{\mathbb{F}}^{H},{\mathbb{P}}^{H})$ and two
$({\mathcal{F}}^{H},\mathbf{N}^{H})$-weak solutions $(H,X^{H})$ and
$(H,Y^{H})$ to (3.3) for $(H,\xi^{H})$ such that
$\mathcal{L}((G,X,Y,\mathbf{N}^{G})|{\mathcal{F}}_{0})(\omega)=\mathcal{L}((H,X^{H},Y^{H},\mathbf{N}^{H}))$.
Indeed, if the claim holds, then a.s. strong well-posedness of (3.3) for every
realization of the random graph $(G,\xi)$ implies that
${\mathbb{P}}(X=Y|{\mathcal{F}}_{0})(\omega)={\mathbb{P}}^{H}(X^{H}=Y^{H})=1$
for ${\mathbb{P}}$-a.s. $\omega\in\Omega$. Hence ${\mathbb{P}}(X=Y)=1$, which
proves strong well-posedness. This claim can be proved via direct verification
of Definitions 3.3, 3.4. We include the details for completion.

To prove the claim, let $(H,X^{H},Y^{H},\mathbf{N}^{H})$ be a random element
with law $\mathcal{L}((G,X,Y,\mathbf{N}^{G})|{\mathcal{F}}_{0})(\omega)$ and
fix a complete, filtered probability space
$(\Omega^{H},{\mathcal{F}}^{H},{\mathbb{F}}^{H},{\mathbb{P}}^{H})$ that
supports $(H,X^{H},Y^{H},\mathbf{N}^{H})$ where ${\mathbb{F}}^{H}$ is the
minimal filtration satisfying the usual conditions such that $X^{H},Y^{H}$ and
$\mathbf{N}^{H}$ are all ${\mathbb{F}}^{H}$-adapted in the sense that for any
$v\in V_{H}$, $X^{H}_{v},Y^{H}_{v}$ and $\mathbf{N}^{H}_{v}$ are all