Abstract
In two recent papers by Veitch and Roy and by Borgs, Chayes, Cohn, and Holden, a new class of sparse random graph processes based on the concept of graphexes over \(\sigma \)-finite measure spaces has been introduced. In this paper, we introduce a metric for graphexes that generalizes the cut metric for the graphons of the dense theory of graph convergence. We show that a sequence of graphexes converges in this metric if and only if the sequence of graph processes generated by the graphexes converges in distribution. In the course of the proof, we establish a regularity lemma and determine which sets of graphexes are precompact under our metric. Finally, we establish an identifiability theorem, characterizing when two graphexes are equivalent in the sense that they lead to the same process of random graphs.
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Notes
- 1.
To see that this setting is indeed more general than the assumption of bounded marginals for (unsigned) graphexes we recall that by Proposition 2.4, a graphex with bounded marginals is integrable. Using this, and the fact that by definition, the graphon part of a graphex is bounded, the claim is easy to verify.
References
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Acknowledgements
László Miklós Lovász thanks Microsoft Research New England for an internship in the summer of 2016, when most of the research part of this work was done. László Miklós Lovász was also supported by NSF Postdoctoral Fellowship Award DMS 1705204 for part of this work. All of us thank Svante Janson and Nina Holden for various discussions about the work presented here, and the anonymous referee for comments and suggestions.
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Appendices
Appendices
A.1: Local Finiteness
In this appendix, we prove Proposition 2.4.
Throughout this appendix, \({\varvec{\Omega }}=(\Omega ,\mathcal {F},\mu )\) will be a \(\sigma \)-finite measure space, \(S:\Omega \rightarrow \mathbb {R}_+\) will be measurable, \(W:\Omega \times \Omega \rightarrow [0,1]\) will be a symmetric, measurable function, \(\eta =\sum _{i}\delta _{x_i}\) will be a Poisson point process on \(\Omega \) with intensity \(\mu \), and
We start with the following lemma, which is the analogue of Lemma A.3.6 from [19] for general measure spaces. We use \(\mathbb {E}\) to denote expectations with respect to the Poisson point process and \(W\circ W\) to denote the function \((x,y)\mapsto \int W(x,z)W(z,y)d\mu (z)\).
Lemma A.1.1
Let \(\psi (x)=1-e^{-x}\). Then the following hold, with both sides of the various identities being possibly infinite:
-
(1)
\(\mathbb {E}[\eta (S)]=\Vert S\Vert _1\) and \(\mathbb {E}[\eta ^2(W)]=\Vert W\Vert _1\),
-
(2)
\(\mathbb {E}[\psi (\eta (S))]=\psi (\Vert \psi (S)\Vert _1)\), and
-
(3)
\(\mathbb {E}[(\eta ^2(W))^2]=\Vert W\Vert _1^2+4\Vert W\circ W\Vert _2+2\Vert W^2\Vert _1\).
Proof
We first assume that \(m=\mu (\Omega )\) is finite and S is bounded. Then \(\eta \) can be generated by first choosing N as a Poisson random variable with rate m and then choosing \(x_1,\dots ,x_N\) i.i.d. according to the distribution \(\frac{1}{m}\mu \). Conditioned on N, the expectations of \(\eta (S)\) and \(\eta ^2(W)\) are \(\frac{N}{m}\Vert S\Vert _1\) and \(\frac{N(N-1)}{m^2}\Vert W\Vert _1\), respectively, and the expectation of \(\psi (\eta (S))\) is
Therefore,
Also,
Finally,
To calculate the expectation of
we distinguish whether \(\{i,j\}\) and \(\{k,\ell \}\) intersect in 0, 1, or 2 elements, leading to the expression
Taking the expectation over N gives the expression in the lemma similarly. This completes the proof for spaces of finite measure and bounded functions S. The general case follows by the monotone convergence theorem. \(\square \)
Using Lemma A.1.1, we now prove the following proposition, which is the analogue of the relevant parts for us of Theorem A3.5 from [19] for general \(\sigma \)-finite measure spaces.
Proposition A.1.2
Let \(S:\Omega \rightarrow \mathbb {R}_+\) be measurable, and let \(W:\Omega \times \Omega \rightarrow [0,1]\) be symmetric and measurable. Then the following hold:
-
(1)
\(\eta (S) <\infty \) a.s. if and only if \(\Vert \min \{S,1\}\Vert _1<\infty \), and
-
(2)
\(\eta ^2(W)<\infty \) a.s. if and only if there exists a finite \(D>0\) such that the following three conditions hold:
-
(a)
\(D_W<\infty \) almost surely,
-
(b)
\(\mu (\{x\in \Omega : D_W(x)>D\})<\infty \), and
-
(c)
\(\Vert W|_{\{x\in \Omega : D_W(x)\le D\}}\Vert _1<\infty \).
-
(a)
Proof
Since \(\frac{1}{2}\min \{1,x\}\le \psi (x)\le \min \{1,x\}\), the condition \(\Vert \min \{S,1\}\Vert _1<\infty \) in (1) is equivalent to the statement that \(\Vert \psi (S)\Vert _1<\infty \), which is equivalent to the statement that \(\psi (\Vert \psi (S)\Vert _1)<1\). By Lemma A.1.1 (2), this is equivalent to saying that \(\mathbb {E}[\psi (\eta (S))]<1\), which holds if and only if \(\eta (S)<\infty \) with positive probability. By Kolmogorov’s zero-one law, we either have \(\eta (S)<\infty \) almost surely, or \(\eta (S)=\infty \) almost surely; therefore we have obtained that \(\Vert \min \{S,1\}\Vert _1<\infty \) if and only if \(\eta (S)<\infty \) almost surely.
To prove the second statement, assume first that the conditions (a)–(c) hold. Condition (a) then implies that a.s., no Poisson point falls into the set \(\{D_W=\infty \}\), which means we may replace \(\Omega \) by a space such that \(D_W(x)<\infty \) for all \(x\in \Omega \). Let \(\Omega _{>D}=\{x\in \Omega : D_W(x)>D\}\) and \(\Omega _{\le D}=\Omega \setminus \Omega _{>D}\). Since \(\Omega _{>D}\) has finite measure by assumption (b), we have that a.s., only finitely many Poisson points fall into this set, which in particular implies that the contribution of the points \(x_i,x_j\in \Omega _{>D}\) to \(\eta ^2(W)\) is a.s. finite. Next let us consider the contributions to \(\eta ^2(W)\) from pairs of points \(x_i,x_j\) such that one lies in \(\Omega _{>D}\) and the other one lies in \(\Omega _{\le D}\). Observing that the Poisson process in \(\Omega _{>D}\) and \(\Omega _{\le D}\) are independent, and that a.s., there are only finitely many points in \(\Omega _{>D}\), it will clearly be enough to show that for all \(x\in \Omega _{>D}\), a.s. with respect to the Poisson process in \(\Omega _{\le D}\),
But by Lemma A.1.1 (1) applied to the function \(S':\Omega _{\le D} \rightarrow \mathbb {R}_+\) defined by \(S'(y)=W(x,y)\), the expectation of this quantity is equal to
This is bounded by \(D_W(x)\) and hence finite, which proves that the sum is a.s. finite. We are thus left with estimating \(\eta ^2(W|_{\Omega _{\le D}})\). Again by Lemma A.1.1 (1), we have that \(\mathbb {E}[\eta ^2(W|_{\Omega _{\le D}})]=\Vert W|_{\Omega _{\le D}}\Vert _1\) which is finite by assumption (c), showing that \(\eta ^2(W|_{\Omega _{\le D}})\) is a.s. finite.
Conversely, let us assume that a.s., \(\eta ^2(W)<\infty \). First we will prove that this implies \(\mu (\{D_W=\infty \})=0\). Assume for a contradiction that this is not the case. Since \(\mu \) is \(\sigma \)-finite, we can find a measurable set \(N\subseteq \Omega \) such that \(D_W(x)=\infty \) for all \(x\in N\) and \(0<\mu (N)<\infty \). Consider the contribution to \(\eta ^2(W)\) by all Poisson points \((x_i,x_j)\) such that \(x_i\in N\) and \(x_j\in N^c=\Omega \setminus N\). Since the Poisson processes on N and \(N^c\) are independent, the finiteness of \(\eta ^2(W)\) implies that for almost all \(x\in N\), the sum \(\sum _{j:x_j\in N^c }W(x,x_j)\) is a.s. finite. Applying statement (1) of the current proposition to \(W(x,\cdot )\) (and recalling that W is bounded by 1), we conclude that for almost all \(x\in N\), \(\int _{N^c}W(x,y)\,d\mu (y)<\infty \), which implies that for almost all \(x\in N\), \(\int _NW(x,y)\,d\mu (y)=D_W(x)-\int _{N^c}W(x,y)\,d\mu (y)=\infty \). This is a contradiction since \(\mu (N)<\infty \) and \(W\le 1\).
We next prove (b) (for any value of D). Suppose for a contradiction that \(\mu (\{x \in \Omega :D_W(x)>D\})=\infty \). We then claim that almost surely, \(\eta ^2(W)=\infty \). After obtaining the Poisson process, color each point randomly red or blue, with equal probability, independently. We can then obtain the red and blue points equivalently by taking two independent Poisson processes, both with intensity \(\mu /2\). We claim that almost surely, the sum of W(x, y) just over red-blue pairs is already \(\infty \). We know that almost surely, there are an infinite number of red points \(x_i\) with \(D_W(x_i)>D\). Let \(x_n\) be such a sequence, and given \(y \in \Omega \), let \(S'(y)=\sum _{n=1}^\infty W(x_n,y)\). Then the sum of W over red-blue edges is equal to \(\eta (S')\) for the Poisson process with intensity \(\mu /2\). Therefore, it suffices to prove that \(\Vert \min \{S',1\}\Vert _{1,\mu /2}=\infty \). First, note that if either \(\mu (\{y \in \Omega :S'(y)=\infty \})>0\) or \(\mu (\{y \in \Omega :S'(y)>1\})=\infty \), then it clearly holds. Otherwise, we have that as \(D' \rightarrow \infty \), \(\mu (\{y \in \Omega :S'(y)>D'\}) \rightarrow 0\); therefore, there exists some \(D'\) (without loss of generality, we may assume \(D'\ge 1\)) such that \(\mu (\{y \in \Omega :S'(y)>D'\})<D/2\). Let \(\Omega '\) be the complement of \(\{y \in \Omega :S'(y)>D'\}\). We then have that for each \(x_n\),
We also have that
Therefore,
This contradiction completes the proof.
We are left with proving (c) (we will again prove it for any value of D). Assume the opposite, and let \(\Lambda _n\subseteq \Lambda \) be an increasing sequence such that \(\mu (\Lambda _n)<\infty \) and \(\bigcup _n\Lambda _n=\Omega _{\le D}\). Let \(U_n=W|_{\Lambda _n}\). Then \(\Vert U_n\Vert _1<\infty \), \(\Vert U_n\Vert _1\uparrow \Vert W|_{\Omega _{\le D}}\Vert _1=\infty \), and \(\Vert D_{U_n}\Vert _\infty \le D\), implying in particular that \(\Vert U_n\circ U_n\Vert _1=\Vert D_{U_n}\Vert _2^2\le D\Vert D_{U_n}\Vert _1=D\Vert U_n\Vert _1\). Given an arbitrary constant \(\lambda \), we claim that
provided n is large enough to ensure that \(\Vert U_n\Vert _1>\lambda \). Indeed, writing
we can bound the first term by \(\lambda \) and the second by \(\sqrt{\mathbb {E}[(\eta ^2(U_{n}))^2]\mathbb {P}[\eta ^2(U_n)>\lambda ]}\), using Cauchy’s inequality. We therefore obtain that
Rearranging, we obtain the bound
where we used Lemma A.1.1 (1) and (3) in the last step. Observing that
and bounding \(4\Vert U_n\circ U_n\Vert _1+\Vert U_n^2\Vert _1\) by \((4D+2)\Vert U_n\Vert _1\), we obtain (14). Since the right side of (14) goes to 1 as \(n\rightarrow \infty \), we get that with probability one, \(\eta ^2(W|_{\Omega _{\le D}})>\lambda \) for all \(\lambda \), which contradicts the assumption that \(\eta ^2(W|_{\Omega _{\le D}})<\infty \) a.s. \(\square \)
Proof
(Proposition 2.4) We first prove the equivalence of (A) – (E). Clearly \((B) \Rightarrow (C)\Rightarrow (A)\) and \((D) \Rightarrow (E)\). It is also not hard to see that \((E) \Rightarrow (A)\). Indeed, note first that for any D, the condition on S is equivalent to the condition that \(\min \{S,D\}\) is integrable (which implies that \(\mu (\{S>D\})<\infty \).) Set \(\Omega '=\{D_W\le D\}\cap \{S\le D\}\). Then (E) implies that
and \(\mu (\Omega \setminus \Omega ')\le \mu (\{D_W>D\})+\mu (\{S>D\})<\infty \), proving (A). So it will be enough to show \((A) \Rightarrow (B)\) and \((A) \Rightarrow (D)\).
Suppose that (A) holds, and let \(\Omega '\) be a set such that \(\mu (\Omega \setminus \Omega ')<\infty \), \(\mathbb {W}' = \mathbb {W}|_{\Omega '}\), and \(\Vert \mathbb {W}'\Vert _1 = C < \infty \). Let \(D>0\). First, assume that \(D>D_0=\mu (\Omega \setminus \Omega ')\). Then
Since \(\Vert D_{\mathbb {W}'}\Vert _1 \le \Vert \mathbb {W}'\Vert _1=C\), this set has measure at most
Now, let \(\mathbb {W}''=\mathbb {W}|_{\{x:D_\mathbb {W}(x) \le D\}}\). Then
We have thus proven that (B) holds for all D larger than some \(D_0\), and more generally for any D for which there exists an \(\Omega ' \subseteq \Omega \) with \(\mu (\Omega \setminus \Omega ')<D\) and \(\Vert \mathbb {W}|_{\Omega '}\Vert _1 <\infty \).
Note that if \(\mathbb {W}|_{\{x:D_\mathbb {W}(x) \le D\}}\) is integrable for \(D>D_0\), then it must remain integrable if we decrease D, since that is just a restriction to a subset. Therefore, this implies that \(\mathbb {W}|_{\{x:D_\mathbb {W}(x) \le D\}}\) is integrable for all D. Since \(D_\mathbb {W}<\infty \) almost everywhere, we further have that \(\mu (\{x \in \Omega : D_\mathbb {W}(x) \ge \lambda \})\) tends to 0 as \(\lambda \) tends to \(\infty \) (since we at least know that it is finite for large enough \(\lambda \)). Fixing \(D>0\), we can therefore take \(D'\) large enough so that \(\mu (\{x \in \Omega : D_\mathbb {W}(x) \ge D'\})<D\). Taking \(\Omega ':=\Omega \setminus \{x \in \Omega : D_\mathbb {W}(x) \ge D'\}\), we get a set \(\Omega '\) such that \(\mu (\Omega \setminus \Omega ')<D\) and \(\Vert \mathbb {W}|_{\Omega '}\Vert _1 <\infty \) proving that (B) holds for all \(D>0\).
On the other hand if (A) holds for some \(\Omega '\), then \(\Vert W|_{\Omega '}\Vert _1<\infty \) and \(\Vert S 1_{\Omega '}\Vert _1<\infty \). Proceeding exactly as above we conclude that for all D, \(\mu (\{D_W>D\})<\infty \) and \(\Vert W|_{\{D_W\le D\}}\Vert _1<\infty \), as well as \(\mu (\{S>D\})<\infty \) and \(\Vert S1_{\{S\le D\}}\Vert _1<\infty \). Since
the latter condition is equivalent to \(\Vert \min \{S,D\}\Vert _1<\infty \), as required.
We are left with proving that the local finiteness conditions in Definition 2.1 are necessary and sufficient for the almost sure finiteness of \(G_T(\mathbb {W})\) for all \(T<\infty \). It is easy to check that the local finiteness conditions are not affected if we multiply the underlying measure by T and S by T. We therefore assume that \(T=1\). Let \(\eta =\sum _{i}\delta _{x_i}\) be a Poisson process of intensity \(\mu \) on \(\Omega \), let \(Y_i\) be Poisson random variable with mean \(S(x_i)\), and let \(Y_{ij}\) be Bernoulli with mean \(W(x_i,x_j)\), all of them independent of each other. We will have to show that the local finiteness conditions on \(\mathbb {W}\) are equivalent to the a.s. finiteness of the sums
We next use the fact that a sum of independent, non-negative random variables \(\sum _k Z_k\) is a.s. finite if and only if \(\sum _i\mathbb {E}[\min \{Z_i,1\}]<\infty \). In the case of \(e_W\), \(Y_{i,j}\) is bounded, and therefore we immediately have that \(e_W\) is a.s. finite if and only if \(\eta ^2(W)\) is a.s. finite. Proposition A.1.2 (b) then proves this case. In the case of \(e_S\), setting \(S'=\min \{S,1\}\), applying Proposition A.1.2 to \(S'\), and noting that \(S'\) is bounded, we have that \(\sum _i S'(x_i)\) is almost surely finite if and only if \(\Vert S'\Vert _1<\infty \). This is exactly the condition on S. \(\square \)
A.2: Sampling with Loops
In this section, we discuss how to handle samples with loops. The sampling process is adjusted as follows. We follow the same process as for \(\mathcal {G}_T(\mathbb {W})\) and \(\mathcal {G}_\infty (\mathbb {W})\); however, for each vertex labeled as (t, x), with probability W(x, x), we add a loop to the vertex. Deleting isolated vertices as before, and then removing the feature labels from the vertices, we obtain a family \((\widetilde{\mathcal {G}}_T(\mathbb {W}))_{T\ge 0}\) of labelled graphs with loops, as well as the infinite graph \(\widetilde{\mathcal {G}}_\infty (\mathbb {W})=\bigcup _{T\ge 0}\widetilde{\mathcal {G}}_T(\mathbb {W})\).
Note that a vertex that was previously isolated may not be isolated anymore if it receives a loop, so a vertex may have been deleted from \(\mathcal {G}_T(\mathbb {W})\) but not from \(\widetilde{\mathcal {G}}_T(\mathbb {W})\). We add a further condition for local finiteness:
Note that if \(\mathbb {W}\) is atomless, then the values W(x, x) do not have an effect on \(\mathcal {G}_T\) and \(\mathcal {G}_\infty \), and the diagonal constitutes a zero measure set in \(\Omega \times \Omega \).
As stated, Theorem 2.5 is false for sampling with loops. Since the diagonal may be a zero measure set, almost everywhere equal pullbacks do not imply having the same looped samples. We could further add the condition that W(x, x) is equal to the pullback almost everywhere, but the theorem would still be false. This is demonstrated by the following example. Let \({\varvec{\Omega }}_1={\varvec{\Omega }}_2=[0,1]\). Take \(W_1\) to be constant 1/2 on \([0,1] \times [0,1]\), and let \(W_2\) be constant 1/2 off the diagonal, 0 if \(x<1/2\), 1 otherwise. Let \(\mathbb {W}_i=(W_i,0,0,{\varvec{\Omega }}_i)\). Then we claim that \(\widetilde{\mathcal {G}}_T(\mathbb {W}_1)\) and \(\widetilde{\mathcal {G}}_T(\mathbb {W}_2)\) have the same distribution. Indeed, both are equivalent to taking \({{\,\mathrm{Poisson}\,}}(T)\) vertices, adding a loop to each vertex with probability 1/2, independently, and also taking an edge between each pair of vertices with probability 1/2, independently over different pairs.
It turns out that in general, allowing diagonal values strictly between 0 and 1 is not necessary, because we could extend the feature space to determine whether each vertex has loops. For graphexes where the diagonal is 0 or 1, we can then conclude an analogous theorem from Theorem 2.5.
We first show the following:
Proposition A.2.1
For any graphex \(\mathbb {W}=(W,S,I,{\varvec{\Omega }})\), there exists a graphex \(\widetilde{\mathbb {W}}=(\widetilde{W},\widetilde{S},\widetilde{I},\widetilde{{\varvec{\Omega }}})\) on an atomless space \({\widetilde{{\varvec{\Omega }}}}\) such that on the diagonal, \({\widetilde{W}}\) is \(\{0,1\}\) valued and such that \(\widetilde{\mathcal {G}}_\infty (\widetilde{\mathbb {W}})\) and \(\widetilde{\mathcal {G}}_T(\widetilde{\mathbb {W}})\) are equivalent to \(\widetilde{\mathcal {G}}_\infty (\mathbb {W})\) and \(\widetilde{\mathcal {G}}_T(\mathbb {W})\), respectively.
Proof
Let \(\widetilde{{\varvec{\Omega }}}={\varvec{\Omega }}\times [0,1]\), and let \(\pi _1,\pi _2\) be the projection maps. Note that we can obtain a Poisson process on \(\widetilde{{\varvec{\Omega }}} \times \mathbb {R}_+\) by taking a Poisson process on \({\varvec{\Omega }}\times \mathbb {R}\), and independently labeling each point with a uniform random real number from [0, 1], which becomes the second coordinate. Clearly \(\widetilde{{\varvec{\Omega }}}\) is atomless, so the diagonal values only affect the generation of the loops. Define \(\widetilde{I}=I\), \(\widetilde{S}=S \circ \pi _1\), \(\widetilde{W}(x,y)=W(\pi _1(x),\pi _1(y))\) if \(x \ne y\), and
Then the sampling of edges between vertices is not affected by the second coordinate of a vertex. Note that the probability that there exist two vertices corresponding to the same point in \(\widetilde{{\varvec{\Omega }}}\) is zero, since \(\widetilde{{\varvec{\Omega }}}\) is atomless. For the loops, since we can obtain the vertices by first taking the Poisson process on \({\varvec{\Omega }}\times \mathbb {R}\) and then randomly labeling each vertex with a [0, 1] real number, we can see that for a point \(y \in \Omega \), if it ends up as a point, there is a W(y, y) probability that the point x corresponding to it has \(\widetilde{W}(x,x)=1\), and \(1-W(y,y)\) that \(\widetilde{W}(x,x)=0\), and this is independent over different points. Therefore, the distribution of loops is the same. \(\square \)
Using this proposition, sampling loops according to the diagonal is equivalent to the following theory. The objects are graphexes with special subsets \(\mathbb {W}=(W,S,I,{\varvec{\Omega }},A)\) where W, S, I, and \({\varvec{\Omega }}\) are as before, and the special set \(A \subseteq \Omega \) is a measurable subset with finite measure. We sample \(\widetilde{\mathcal {G}}_\infty (\mathbb {W})\) in the same way as \(\mathcal {G}_\infty (\mathbb {W})\), except that we add a loop to each vertex with a feature label in A. We then take the non-isolated vertices with time label at most T for \(\widetilde{\mathcal {G}}_T(\mathbb {W})\). We can extend the definition of measure-preserving map by requiring that points in the special set be mapped to points in the special set, and points not in the special set be mapped to points not in the special set. We also define \({{\,\mathrm{dsupp}\,}}\) as earlier, except it contain all points in A (even if otherwise they would not be included).
Theorem A.2.2
Let \(\mathbb {W}_1\) and \(\mathbb {W}_2\) be graphexes with special subsets as above. Then \(\widetilde{\mathcal {G}}_T(\mathbb {W}_1)\) and \(\widetilde{\mathcal {G}}_T(\mathbb {W}_2)\) have the same distribution for all \(T \in \mathbb {R}_+\) if and only if there exists a third graphex with special subset \(\mathbb {W}\) such that \(\mathbb {W}_1\) and \(\mathbb {W}_2\) are pullbacks of \(\mathbb {W}\).
Proof
It is clearly enough to prove the only if direction. Suppose therefore that \(\mathbb {W}_1\) and \(\mathbb {W}_2\) have the same distribution. Then for any \(0<c<1\), \(c\mathbb {W}_1\) and \(c \mathbb {W}_2\) have the same distributions (i.e., W, S, I are all multiplied by c, and the special set stays the same). Then let \(\widetilde{\mathbb {W}_i}\) be obtained by taking \( \mathbb {W}_i/2\), adding a set \(B_i\) of measure 1 to \(\Omega _i\), and extending \(W_i\) to be 1 on \(B_i \times B_i\), 1 between \(B_i\) and \(A_i\), and 0 between \(B_i\) and \(\Omega _i \setminus A_i\). Then we can obtain \(G_T(\widetilde{\mathbb {W}_i})\) from \(\widetilde{G}_T(\mathbb {W}_i)\) by the following process. We first keep each edge that is not a loop with probability 1/2, and delete it otherwise, independently. We keep all the loops. Then we take \({{\,\mathrm{Poisson}\,}}(T)\) new vertices, put an edge between every pair, and put an edge between each new vertex and each vertex that had a loop (and delete loops). It is clear that in this way, the distributions \(G_T(\widetilde{\mathbb {W}}_1)\) and \(G_T(\widetilde{\mathbb {W}}_2)\) are the same for every T. Therefore, there exists a graphex \(\widetilde{\mathbb {W}}=(\widetilde{W},\widetilde{S},\widetilde{I},\widetilde{{\varvec{\Omega }}})\) such that \(\widetilde{\mathbb {W}}_1\) and \(\widetilde{\mathbb {W}}_2\) are both pullbacks of \(\widetilde{\mathbb {W}}\). It is clear that \(\widetilde{\mathbb {W}}\) must have a set of measure 1, call it B, which has \(\widetilde{W}(x,y)=1\) if \(x,y \in B\), and \(\widetilde{W}(x,y)\) is either 0 or 1 if \(x \in B, y \notin B\), and only depends on y, and \(\widetilde{W}(x,y) \le 1/2\) if \(x,y \notin B\), and B must pullback to exactly \(B_1\) and \(B_2\). If we let A be the set of points x with \(\widetilde{W}(x,y)=1\) for any and all \(y \in B\), then A must pullback to \(A_1\) and \(A_2\). If we therefore let \(\mathbb {W}\) have underlying set \(\widetilde{\Omega } \setminus B\), and be equal to \(2\widetilde{\mathbb {W}}\) restricted to this set, and special set A, then \(\mathbb {W}\) pulls back to both \(\mathbb {W}_1\) and \(\mathbb {W}_2\). \(\square \)
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Borgs, C., Chayes, J.T., Cohn, H., Lovász, L.M. (2019). Identifiability for Graphexes and the Weak Kernel Metric. In: Bárány, I., Katona, G., Sali, A. (eds) Building Bridges II. Bolyai Society Mathematical Studies, vol 28. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-59204-5_3
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