# Scaling of the clustering function in
spatial inhomogeneous random graphs

Remco van der Hofstad  Department of Mathematics and Computer
Science,Eindhoven University of Technology  Pim van der Hoorn  Department of
Mathematics and Computer Science,Eindhoven University of Technology  Neeladri
Maitra  Department of Mathematics and Computer Science,Eindhoven University of
Technology

###### Abstract

We consider an infinite spatial inhomogeneous random graph model with an
integrable connection kernel that interpolates nicely between existing spatial
random graph models. Key examples are versions of the weight-dependent random
connection model, the infinite geometric inhomogeneous random graph, and the
age-based random connection model. These infinite models arise as the local
limit of the corresponding finite models. For these models we identify the
asymptotics of the _local clustering_ as a function of the degree of the root
in different regimes in a unified way. We show that the scaling exhibits
_phase transitions_ as the interpolation parameter moves across different
regimes. This allows us to draw conclusions on the geometry of a _typical_
triangle contributing to the clustering in the different regimes.

##  1 Introduction

The last two decades has seen the rise of models for complex networks that use
geometry [6, 9, 11, 14, 22]. The main idea is that geometry can encode
similarity of nodes and thus serves as a component for deciding whether an
edge will be present. Random graph models with underlying geometry have proven
to be very successful in reproducing several key features of real-world
networks. As such, geometry of complex networks has become an important
research domain [1, 2, 27, 29, 32]. One key feature properly captured by
geometric models is the presence of triangles, triples of connected nodes,
measured in terms of their _clustering_. In the literature, three different
measures for clustering are often considered. The first is the so-called
global clustering coefficient, defined as three times the ratio of the number
of triangles to the number of paths of length two in the graph. The second is
the local clustering coefficient, which for each node counts the fraction of
triangles it participates in and then averages this over all nodes. Finally,
the _clustering function_ , sometimes called the clustering spectrum, is the
local clustering function restricted to nodes of a given degree $k$. That is,
it is a function that takes as input $k\in\mathbb{N}$, and outputs the average
number of triangles among all possible triangles that are present among degree
$k$ vertices in the graph, see (1.5) below for a definition.

This paper studies the behaviour of the _clustering function_ of a _typical
vertex_ , in a model of infinite spatial inhomogeneous random graphs. Spatial
inhomogeneous random graphs can be seen as _spatial_ versions of inhomogeneous
random graphs, as studied in [3], where nodes have a position in a geometric
space, see also [31, Remark 1.5] for a discussion. In a typical construction
of these graphs, one considers vertices as atoms of some marked Poisson point
process, where the edges are included conditionally independently given the
vertices and their marks, according to a connection function $\kappa$. These
graphs also generalize infinite geometric inhomogeneous random graphs [4, 5,
6] and hyperbolic random graphs [21]. Graphs of similar flavour have been
considered in [11, 12].

For an infinite graph, a functional similar to the clustering function can be
considered under some conditions. In particular, if a sequence of finite
random graphs converges locally to an infinite graph, then the associated
sequence of clustering functions also converge to a limiting functional of the
infinite graph, see [15, Exercise 2.32]. This limit is expressed as the
clustering function of a _typical vertex_ of this graph, see (1.6) below. We
study how the clustering function behaves as the degree of the typical vertex
diverges. Problems of similar flavor have been studied in [8, 30] for the
hyperbolic random graph. In particular, a phase transition in the clustering
function is established in [8, Proposition 1.4], which disproved the
conjectured inverse linear scaling of the clustering function of these graphs
[21]. For spatial versions of preferential attachment networks, convergence of
the average and global clustering coefficients are shown in [17], while
scalings of the clustering function for a different spatial preferential
attachment model have been studied in [16]. Typically, the clustering function
vanishes polynomially, as the degree of a typical vertex tends to infinity,
for sparse network models from the literature. For example, for the models
considered in [23, 28], the authors see an inverse linear decay, while a
polynomial decay is observed, for example, in [7, 25, 29, 33]. The main
intuition behind the prediction of an inverse linear decay is that most
triangles are edge-disjoint, and there can be at most $k/2$ such edge-disjoint
triangles incident to a vertex if it has degree $k$.

In this paper, we see that under suitable choices of the model parameters,
even for sparse random graphs, the clustering function can decay
logarithmically, or not decay at all, see for example Theorem 1.1 below. This
last behavior is to be understood as that _there are rare vertices of
arbitrarily large degrees, and for these vertices, a constant proportion of
their neighbor pairs are also neighbors_ , i.e., the neighborhood of such rare
high-degree vertices are _almost_ cliques - a phenomenon that has not yet been
observed in the literature for sparse random graphs.

The rest of this section is organized as follows: In Sections 1.1 and 1.2 we
define our random graph models. In Section 1.3 we discuss the clustering
function. We state our main result in Section 1.4, followed by a discussion on
different examples covered under our setup in Section 1.5. Finally in Section
1.6 we discuss the proof strategy of our main result.

###  1.1 The random graph model

The random graphs we study are spatial inhomogeneous random graphs, or SIRGs
[31]. The vertex set is either $[n]=\\{1,\dots,n\\}$ (finite model) or
$\mathbb{N}\cup\\{0\\}$ (infinite model). Each vertex $i$ has a location
$X_{i}$ in $\mathbb{R}^{d}$, where for the finite model the locations are
i.i.d. uniform in the box

$I_{n}:=\left[-\frac{n^{1/d}}{2},\frac{n^{1/d}}{2}\right]^{d}.$

For the infinite model, all the locations $(X_{i})_{i\geq 1}$ except that of
$0$, form the collection of atoms of a rate $1$ homogeneous Poisson point
process $\Gamma$ on $\mathbb{R}^{d}$, and we let the vertex $0$ have the
location $X_{0}:=\mathbf{0}\in\mathbb{R}^{d}$. In particular, the vertex
locations of $\mathbb{G}^{\infty}$ form a palm version
$\Gamma\cup\\{\mathbf{0}\\}$ of the Poisson process $\Gamma$. Moreover, each
vertex $i$ has a random positive real weight $W_{i}$ associated to it, where
$(W_{i})_{i\geq 1}$ is a sequence of i.i.d. copies of a random variable $W$.
Edges between pairs of vertices $i$ and $j$, conditionally on $(X_{i},W_{i})$
and $(X_{j},W_{j})$, are placed independently with probability

$\kappa(\|X_{i}-X_{j}\|,W_{i},W_{j}),$

where $\kappa:\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}\to[0,1]$ is
assumed to be symmetric in its last two arguments. We denote the finite model
as $\mathbb{G}^{(n)}(\kappa,W)$, and the infinite model as
$\mathbb{G}^{\infty}(\kappa,W)$. The infinite rooted graph
$(\mathbb{G}^{\infty},0)$ arises as the local limit in probability of the
finite model, see [31, Theorem 1.2].

For the rest of the paper, we fix the sequence of weights
$\mathbb{W}=(W_{i})_{i\in S}$, where $S$ is either $[n]$ or
$\mathbb{N}\cup\\{0\\}$, corresponding respectively to the finite and infinite
models, to be a sequence of i.i.d. copies from a $\text{Pareto}(\beta-1,1)$
distribution. That is, each $W$ has density function

$f_{W}(t)=\frac{\beta-1}{t^{\beta}}\mathbbm{1}_{\left\\{t>1\right\\}},$ (1.1)

with $\beta>2$, i.e., the weights have finite expectation. For such choices of
power-law weights, with appropriate choice of the connection function
$\kappa$, the degree distribution of a vertex also obeys a power law, so that
they naturally give rise to scale-free spatial random graphs.

###  1.2 The interpolation kernel

We now discuss the connection function $\kappa$ we are concerned with. For a
_long-range_ parameter $\alpha\in(0,\infty]$, we consider

$\displaystyle\kappa(r,s,t)=\kappa_{\alpha}(r,s,t)=\begin{cases}\mathbbm{1}_{\left\\{r<g(s,t)^{1/d}\right\\}}+\left(\frac{g(s,t)}{r^{d}}\right)^{\alpha}\mathbbm{1}_{\left\\{r\geq
g(s,t)^{1/d}\right\\}}&\mbox{if}\;\;\alpha\in(0,\infty),\\\
\mathbbm{1}_{\left\\{r<g(s,t)^{1/d}\right\\}}&\mbox{if}\;\;\alpha=\infty,\end{cases}$
(1.2)

for some symmetric function $g:\mathbb{R}\times\mathbb{R}\to\mathbb{R}_{+}$.
We refer to $g$ as the _weight function_ of $\kappa$. For vertices $i$ and
$j$, this is to be interpreted as follows: when $\alpha<\infty$, the edge
$\\{i,j\\}$ is present with probability $1$ if
$\|X_{i}-X_{j}\|<g(W_{i},W_{j})^{1/d}$, otherwise it is present with
probability
$\left(\frac{g(W_{i},W_{j})}{\|X_{i}-X_{j}\|^{d}}\right)^{\alpha}$; while when
$\alpha=\infty$, the edge $\\{i,j\\}$ is present precisely when
$\|X_{i}-X_{j}\|<g(W_{i},W_{j})^{1/d}$.

Let us introduce the following notation which we use throughout the paper: for
any two reals $r_{1}$ and $r_{2}$,

$r_{1}\vee r_{2}:=\max\\{r_{1},r_{2}\\},\;\;\text{and}\;\;r_{1}\wedge
r_{2}=\min\\{r_{1},r_{2}\\}.$

In this paper, we work with the weight function

$g(s,t)=g_{a}(s,t)=(s\vee t)(s\wedge t)^{a},$ (1.3)

for a parameter $a\in[0,\infty)$. Different values of the interpolation
parameter $a$ in (1.3) capture different spatial random graphs from the
literature, see Section 1.5 for details.

###  1.3 The clustering function

Let $G=(V(G),E(G))$ be a locally finite graph. For $v\in V(G)$, we denote its
degree in $G$ by $d_{v}(G)$ and further define

$\Delta_{v}(G):=\sum_{v_{1},v_{2}\in
V(G)}\mathbbm{1}_{\left\\{\\{v,v_{1}\\}\in
E(G)\right\\}}\mathbbm{1}_{\left\\{\\{v,v_{2}\\}\in
E(G)\right\\}}\mathbbm{1}_{\left\\{\\{v_{1},v_{2}\\}\in E(G)\right\\}},$ (1.4)

so that $\Delta_{v}(G)$ is the number of triangles in $G$ containing $v$ as a
vertex. In addition, we let $N_{k}(G)$ be the number of vertices in $G$ with
$d_{v}(G)=k$.

######  Definition 1.1 (Clustering function).

The clustering function $\mathrm{CC}_{G}:\mathbb{N}\to\mathbb{R}_{+}$ of $G$
is defined as

$\mathrm{CC}_{G}(k):=\begin{cases}\frac{1}{N_{k}(G)}\sum_{v\in
V(G):d_{v}(G)=k}{\frac{\Delta_{v}(G)}{k(k-1)}}&\mbox{if}\;\;0<N_{k}(G)<\infty,\\\
0&\mbox{otherwise.}\end{cases}$ (1.5)

Thus, the quantity $\mathrm{CC}_{G}(k)$ measures the proportion of two-paths
in $G$ that are triangles, where the middle vertex of the two-path has degree
$k$.

It is not immediately clear how to define such a notion for an infinite graph
$G$, since $N_{k}(G)$ can be infinite. However, if $G$ arises as the local
limit of a growing sequence $(G_{n})_{n\geq 1}$ of finite graphs, then the
corresponding sequence of clustering functions has a limit, which can then be
defined as the clustering function of $G$. Indeed, it was shown in [31,
Corollary 2.10] that, for any $k\in\mathbb{N}$,

$\displaystyle\mathrm{CC}_{\mathbb{G}^{(n)}}(k)\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}\gamma(k),$

where

$\gamma(k)=\frac{1}{\binom{k}{2}}\mathbb{E}\left[\left.\Delta_{0}(\mathbb{G}^{\infty})\right|d_{0}(\mathbb{G}^{\infty})=k\right].$
(1.6)

We call $\gamma(k)$ the _clustering function_ of $\mathbb{G}^{\infty}$. In
this paper we analyse the asymptotics of $\gamma(k)$ for large $k$.

###  1.4 Main result

Let us denote the Lebesgue measure of a ball of unit radius in
$\mathbb{R}^{d}$ by $\omega_{d}$. For any $\alpha\in(1,\infty]$ we then define

$\xi_{\alpha}=\xi_{\alpha,\beta,d}:=\frac{\alpha\omega_{d}(\beta-1)}{\alpha-1},$
(1.7)

where $\xi_{\infty}:=\lim_{\alpha\to\infty}\xi_{\alpha}=\omega_{d}(\beta-1)$.
For any $w_{1},w_{2}\in(1,\infty)$, we define the functions

$\displaystyle I(w_{1},w_{2})$
$\displaystyle:=\iint_{(\mathbb{R}^{d})^{2}}\kappa(\|\mathbf{x}\|,1,w_{1})\kappa(\|\mathbf{x}\|,1,w_{2})\kappa(\|\mathbf{y}\|,w_{1},w_{2})d\mathbf{x}d\mathbf{y},\text{
and}$ $\displaystyle\mathcal{S}_{1}(w_{1},w_{2})$
$\displaystyle:=\iint_{(\mathbb{R}^{d})^{2}}\kappa(\|\mathbf{x}\|,1,w_{1})\kappa(\|\mathbf{y}\|,1,w_{2})\kappa(\|\mathbf{x}-\mathbf{y}\|,w_{1},w_{2})d\mathbf{x}d\mathbf{y}.$

Further, let $\mathrm{S}_{k}(a,\beta)$ be defined as

$\mathrm{S}_{k}(a,\beta)=\begin{cases}k^{-1}&\mbox{if}\;\;\beta>(a+\frac{3}{2})\vee
2,a\in(0,\infty),\\\
k^{-1}\log{k}&\mbox{if}\;\;\beta=a+\frac{3}{2},a\in(\frac{1}{2},\infty),\\\
k^{2a+2-2\beta}&\mbox{if}\;\;\beta\in(a+1,a+\frac{3}{2}),a\in(\frac{1}{2},\infty),\\\
(\log{k})^{-2}&\mbox{if}\;\;\beta=a+1,a\in(1,\infty),\\\
1&\mbox{if}\;\;\beta\in(2,a+1),a\in(1,\infty).\end{cases}$ (1.8)

Next for $\mathbf{w}=(w_{1},w_{2})\in\mathbb{R}^{2}$, using the notation

$f_{W}(\mathbf{w})d\mathbf{w}:=f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}$ (1.9)

for the product measure on $\mathbb{R}^{2}$ with density
$f_{W}(w_{1})f_{W}(w_{2})$, let $\Gamma(a,\alpha,\beta,d)$ be defined as

$\Gamma(a,\alpha,\beta,d)=\begin{cases}\frac{\beta-a-1}{\xi_{\alpha}}\hskip
46.0pt\iint_{1}^{\infty}I(w_{1},w_{2})f_{W}(\mathbf{w})d\mathbf{w}&\mbox{if}\;\;\beta>\left(a+\frac{3}{2}\right)\vee
2,a>0,\\\ \frac{2a+1}{4\xi_{\alpha}}\hskip
54.0pt{(\beta-1)^{2}\int_{0}^{\infty}I(1,r)r^{-\beta}dr}&\mbox{if}\;\;\beta=a+\frac{3}{2},a>\frac{1}{2},\\\
\left(\frac{\beta-a-1}{\xi_{\alpha}}\right)^{4+2a-2\beta}\iint_{0}^{\infty}\mathcal{S}_{1}(w_{1},w_{2})f_{W}(\mathbf{w})d\mathbf{w}&\mbox{if}\;\;\beta\in(a+1,a+\frac{3}{2}),a>\frac{1}{2},\\\
\xi_{\alpha}^{-1},\hskip
58.0pt\iint_{0}^{\infty}\mathcal{S}_{1}(w_{1},w_{2})f_{W}(\mathbf{w})d\mathbf{w}&\mbox{if}\;\;\beta=a+1,a>1,\\\
\left(\frac{(1+a-\beta)(\beta-2)}{(a-1)\xi_{\alpha}}\right)^{2}\iint_{0}^{\infty}\mathcal{S}_{1}(w_{1},w_{2})f_{W}(\mathbf{w})d\mathbf{w}&\mbox{if}\;\;\beta\in(2,a+1),a>1.\end{cases}$
(1.10)

With these definitions we are ready to state our main result about the scaling
of the clustering function:

######  Theorem 1.1.

Let $\alpha>1$, $\beta>2$ and let $W$ be a random variable with density (1.1).
Furthermore, let $\kappa$ be defined as in (1.2) with $g$ as in (1.3).
Consider the infinite SIRG $\mathbb{G}^{\infty}(\kappa,W)$ and associated
clustering function $\gamma(k)$ as defined in (1.6). Then, as $k\to\infty$,

$\displaystyle\frac{\gamma(k)}{\mathrm{S}_{k}(a,\beta)}\to\Gamma(a,\alpha,\beta,d)\in(0,\infty),$

where $\mathrm{S}_{k}(a,\beta)$ is as in (1.8), with
$\Gamma(a,\alpha,\beta,d)$ is as in (1.10).

The main message of Theorem 1.1 is that the clustering function of the
infinite SIRG $\mathbb{G}^{\infty}$ has _three_ different scaling regimes,
depending on the parameters $a$ and $\beta$, characterized by the function
$\mathrm{S}_{k}(a,\beta)$.

In Figure 1, we provide the phase diagram of the scaling function
$\mathrm{S}_{k}(a,\beta)$ in terms of $a$ and $\beta$. The lines $a=0$, $a=1$
and $\beta=a+2$ all correspond to known random graph models, see Section 1.5
for a discussion on the different examples of random graph models covered
under our set up.

$a$$a=0$$\beta$$(\frac{1}{2},0)$$(1,0)$$a=1$$(0,2)$$\beta=a+2$$\beta=a+\frac{3}{2}$$\beta=a+1$$\beta=\frac{a+3}{2}$$\mathrm{S}_{k}(a,\beta)=k^{-1}$$\mathrm{S}_{k}(a,\beta)=k^{-1}\log{k}$$\mathrm{S}_{k}(a,\beta)=k^{2a+2-2\beta}$$\mathrm{S}_{k}(a,\beta)=\frac{1}{\log{k}^{2}}$$\mathrm{S}_{k}(a,\beta)=\text{constant}$

Figure 1: Phase diagram of the scaling function $\mathrm{S}_{k}(a,\beta)$.

######  Remark 1.1 (Range of parameters).

Let us discuss our choice of parameters $\beta>2$ and $\alpha>1$. When
$\alpha\leq 1$, the conditional degree of $0$ given $\\{W_{0}=w\\}$, is almost
surely infinite for any $w>0$ by Lemma 2.1, which implies that
$d_{0}(\mathbb{G}^{\infty})$ is almost surely infinite. Similarly, when
$\alpha>1$ and $\beta\leq 2$, $d_{0}(\mathbb{G}^{\infty})$ is almost surely
infinite, again by Lemma 2.1. In particular our choice of the range of
parameters $\beta>2$, $\alpha>1$ are so that the random variable
$d_{0}(\mathbb{G}^{\infty})$ is finite almost surely, and the quantity of
interest $\gamma(k)$ makes sense.

######  Remark 1.2 (Phase transitions).

We note that there are three different phases for the scaling of the
clustering function $\gamma(k)$: $\beta>(a+\frac{3}{2})\vee 2$,
$\beta\in(a+1,a+\frac{3}{2})$ and $\beta\in(2,a+1)$, with two phase transition
boundaries at $\beta=a+\frac{3}{2}$ and $\beta=a+1$. The scalings of
$\gamma(k)$ at $\beta=a+1$ and $\beta\in(2,a+1)$ are new, especially the
constant scaling of $\gamma(k)$ in the regime $\beta\in(2,a+1)$ is surprising.
The phase transition at $\beta=a+\frac{3}{2}$ has already been observed for a
particular case: namely $a=1$ and $d=1$, with $\alpha=\infty$. In this case
the corresponding infinite SIRG $\mathbb{G}^{\infty}$ is a good approximation
to the local limit of the threshold hyperbolic random graph model with
appropriate parameters (see e.g. [31, Section 2.1.3]), using arguments as
appearing in [20, Section 9]. For these random graphs, the phase transition at
the point $(1,5/2)$ of Figure 1 was observed in [8, Proposition 1.4].

######  Remark 1.3 (Choice of Pareto weights).

Our choice of the weight distribution (1.1) is purely for technical reasons.
One can also consider regularly-varying distributions with tail parameter
$\beta-1$, but Pareto weights make the arguments and constants cleaner. We
expect Theorem 1.1 to go through when one has regularly varying weights with
similar proofs, albeit more technical, except in the critical regimes
$\beta=a+\frac{3}{2}$ and $\beta=a+1$. Here we expect the particular choice of
the regularly varying distribution to introduce additional slowly-varying
corrections, which change the scaling. The choice of Pareto weights also helps
us draw a cleaner link between the models of SIRG we consider and existing
random graph models from the literature, as discussed in Section 1.5. All
these random graph models have a mixed-Poisson degree distribution, and hence
have the same power-law tail as the mixing random variable (e.g. see [31,
Section 2.2])

###  1.5 Examples

We now discuss different spatial random graph models from the literature that
different values of $a$ in (1.3) give rise to.

##### Product kernels.

When one takes $a=1$ in (1.3), $g(s,t)$ becomes the product $st$. This
corresponds to the models having a product structure in their weights, most
notably Geometric Inhomogeneous Random Graphs (GIRGs) [4, 5, 6], Weight
Dependent Random Connection Models (WDRCMs) [12], and Hyperbolic Random Graphs
(HRGs) [21]. Let us discuss these examples briefly.

The infinite GIRG (see [31, Section 2.1.2]) has as its vertex set a
homogeneous Poisson process on $\mathbb{R}^{d}$. Each vertex has a power-law
weight associated to it, where the collection of weights form an i.i.d.
collection. Conditionally on everything, the edges are placed independently
with a connection function of the form (1.2) with $g(s,t)=st$. When restricted
to $d=1$, these graphs give a version of HRGs [21] \- for a discussion of
this, see [20, Section 9]. Here we comment that typically in GIRGs or HRGs,
the connection function does not exactly have the form (1.2), but can easily
be seen to be bounded from above and below up to constant factors by (1.2). In
particular, assuming the form (1.2) for the connection function does not
affect the conclusions of Theorem 1.1, but comes with less unnecessary
technicalities, which is why we prefer (1.2).

The $a=1$ case can also be seen as a special case of the _weight-dependent
random connection model_ [12]. Here the vertices are again a homogeneous
Poisson point process on $\mathbb{R}^{d}$, but each vertex now has a uniformly
distributed $[0,1]$ mark associated to it, as opposed to power-law weights in
the case of GIRGs, and each edge between two vertices $(\mathbf{x},s)$ and
$(\mathbf{y},t)$ is included conditionally independently with probability
$\rho(g(s,t)\|\mathbf{x}-\mathbf{y}\|^{d})$, where the function $\rho(\cdot)$
is non-increasing and integrable. In particular, let the _profile function_
$\rho(g(s,t)\|\mathbf{x}-\mathbf{y}\|^{d})$ in [12, (1.2)] be equal to
$\kappa_{\alpha}(\|\mathbf{x}-\mathbf{y}\|,s,t)$, with weight function $g$ as
in (1.3) with $a=1$. This corresponds to taking $\beta=1$ in the product
kernel [12, (1.6)] of these models, and taking $1/\gamma=\beta-1$, where now
$\beta$ is the tail parameter of the Pareto distribution coming from (1.1).
This can be seen using the standard fact that if $U$ is uniform on $(0,1)$,
then $U^{-\gamma}$ has the density (1.1) with $1/\gamma=\beta-1$.

##### The max kernel.

When one takes $a=0$ in (1.3), the function $g(s,t)$ becomes $s\vee t$. This
also corresponds to a special case of the _weight-dependent random connection
model_ [12]. To realize this case, we take $\beta=1$ for the _min kernel_
$g^{\mathrm{min}}(s,t)=(s\wedge t)/\beta$ as in [12, p3], and let
$1/\gamma=\beta-1$ as before. This transforms the connection function
$\rho(g(s,t)\|\mathbf{x}-\mathbf{y}\|)$ into the connection function
$\kappa_{\alpha}$ as in (1.2), with $a=0$ and $1/\gamma=\beta-1$, with weight
function $g$ as in (1.3) with $a=0$.

##### Age-dependent random connection models.

When one takes $a=\beta-2$, the corresponding infinite model
$\mathbb{G}^{\infty}$ gives rise to the _age-dependent random connection
model_ , which comes out as the local limit of the _age-based spatial
preferential attachment model_ , as studied in [11]. This model also has a
homogeneous Poisson point process as its vertex set, with i.i.d. marks
associated to them that are uniformly distributed on $[0,1]$. The connection
function (see [11, Section 3]) for this model has the form
$\varphi(\beta^{-1}(s\vee u)^{1-\gamma}(s\wedge
u)^{\gamma}\|\mathbf{x}-\mathbf{y}\|^{d})$ (see e.g. the bottom of [11,
p315]), for parameters $\beta\in(0,\infty)$, $\gamma\in(0,1)$, $s,t$ being
i.i.d. uniforms on $(0,1)$, and a non-increasing, integrable profile function
$\varphi$. Let $\beta=1$, and consider the profile function
$\varphi(r)=\mathbbm{1}_{\left\\{r<1\right\\}}+\left(\frac{1}{r}\right)^{\alpha}\mathbbm{1}_{\left\\{r\geq
1\right\\}}$. For these choices, we can write the above connection function as
$\kappa_{\alpha}(r,s^{\prime},t^{\prime})$, as appearing in (1.2), with the
weight function $g_{a}(s^{\prime},t^{\prime})$ being $(s^{\prime}\vee
t^{\prime})(s^{\prime}\wedge t^{\prime})^{a}$, where $s^{\prime}=s^{-\gamma}$,
$t^{\prime}=t^{-\gamma}$, and $a=1/\gamma-1$. As before, using the standard
fact that if $U$ is uniform on $(0,1)$, then $W=U^{-\gamma}$ is a Pareto
random variable with scale parameter $1$ and shape parameter $1/\gamma$, if we
let $1/\gamma=\beta-1$, then using $\gamma\in(0,1)$, $\beta>2$, both
$s^{\prime}$ and $t^{\prime}$ have the density (1.1), and the weight function
$g_{a}$ is (1.3) with $a=\beta-2$. This completes the link between the
infinite SIRGs we consider, and the age-based spatial preferential attachment
networks.

A similar kernel was also considered in [13], where the authors look at
percolation questions for one dimensional models, and in the recent work [19],
the authors look at a similar interpolating kernel, and study cluster-size
decay of these models.

###  1.6 Proof strategy

In this section we discuss the main ideas to prove Theorem 1.1. First, let us
recall some standard notations. Throughout the paper, for real sequences
$(f_{k})_{k\geq 1}$ and $(g_{k})_{k\geq 1}$, we use the standard Landau
notations $f_{k}=O\left(g_{k}\right)$ and $f_{k}=\Omega\left(g_{k}\right)$, as
$k\to\infty$, to mean respectively that there exist real constants $C,c$ such
that

$\displaystyle\limsup_{k\to\infty}\left|\frac{f_{k}}{g_{k}}\right|\leq
C,\;\;\text{and}\;\;\liminf_{k\to\infty}\left|\frac{f_{k}}{g_{k}}\right|\geq
c.$

We write $f_{k}=\Theta(g_{k})$ when both $f_{k}=O\left(g_{k}\right)$ and
$f_{k}=\Omega\left(g_{k}\right)$. Further, we write
$f_{k}=o\left(g_{k}\right)$ and $f_{k}=\omega(g_{k})$ to mean respectively
that

$\displaystyle\limsup_{k\to\infty}\frac{f_{k}}{g_{k}}=0\;\;\text{and}\;\;\liminf_{k\to\infty}\frac{f_{k}}{g_{k}}=\infty.$

The first step is to reduce the expression of $\gamma(k)$ using properties of
the locations and weights of the vertices of the graph $\mathbb{G}^{\infty}$.
Let us for now denote the conditional density of the weight $W_{0}$ of the
root $0$ given that its degree $d_{0}$ equals $k$ by $w\mapsto f_{k}(w)$.
Using this conditional density, we obtain the following representation of
$\gamma(k)$,

$\gamma(k)=\int_{\mathbb{R}_{+}}\frac{T(w)}{\mathcal{M}(w)^{2}}f_{k}(w)dw,$

where the quantity ${T(w)}/{\mathcal{M}(w)^{2}}$ is a bounded function of $w$.
Here $T(w)$ can be understood as the expected number of triangles containing
$0$ given $W_{0}=w$, and $\mathcal{M}(w)$ is the expected degree of $0$ given
$W_{0}=w$. This representation is obtained in Proposition 2.2. We then use
concentration properties of Poisson random variables to show that
asymptotically, as $k\to\infty$, the density $f_{k}(w)$ behaves like a Dirac
density, with a point mass at $\mathcal{M}^{-1}(k)$. This is the content of
Proposition 2.3. As a consequence,

$\gamma(k)=(1+o(1))\frac{T(\mathcal{M}^{-1}(k))}{k^{2}},$ (1.11)

as $k\to\infty$, and thus it suffices to show that

$\frac{T(\mathcal{M}^{-1}(k))}{k^{2}\mathrm{S}_{k}(a,\beta)}\to\Gamma(a,\alpha,\beta,d).$

We then proceed in two phases. First in Section 3 we establish the exact
scaling of $\mathcal{M}^{-1}(k)$. More precisely, if we define

$\sigma(w)=\begin{cases}w&\mbox{if }\beta>\left(a+\frac{3}{2}\right)\vee
2,a>0,\\\ w\log{\left(w\right)}&\mbox{if
}\beta=a+\frac{3}{2},a>\frac{1}{2},\\\ w^{4+2a-2\beta}&\mbox{if
}\beta\in(2,a+\frac{3}{2}),a>\frac{1}{2}.\end{cases}$

then our result implies that
$\sigma(\mathcal{M}^{-1}(k))/(k^{2}\mathrm{S}_{k}(a,\beta))$ converges to a
constant, see Corollary 3.4.

For the final part we write

$\frac{T(\mathcal{M}^{-1}(k))}{k^{2}\mathrm{S}_{k}(a,\beta)}=\frac{\sigma(\mathcal{M}^{-1}(k))}{k^{2}\mathrm{S}_{k}(a,\beta)}\frac{T(\mathcal{M}^{-1}(k))}{\sigma(\mathcal{M}^{-1}(k))}.$

The first term converges by Corollary 3.4, so that all that is left is to
study the limit $T(w)/\sigma(w)$ as $w\to\infty$ in more detail. To prepare
for this, we establish general upper and lower bounds on $T(w)$ and study
their asymptotic behavior in Section 4. Finally, in Section 5, we establish
the limits $T(w)/\sigma(w)$ for the three different cases and wrap up the
proof of Theorem 1.1.

###  1.7 More on the phase transitions

As Theorem 1.1 shows, the clustering function undergoes two phase transitions.
While the exact scaling of the clustering function in the different regimes,
as well as the exact location of the transition, depends on the choice of the
function $g$ in $\kappa$, there is something general we can say about the
driving force behind both phase transitions.

We start with the first transition. To discuss this, for $w,{w}_{1},{w}_{2}>0$
we introduce the notation

$h(w,{w}_{1},{w}_{2}):=g(w_{1},{w}_{2})[g(w,{w}_{1})\wedge g(w,{w}_{2})].$
(1.12)

Recall from (1.11) that the scaling of $\gamma(k)$ is determined by that of
$T(\mathcal{M}^{-1}(k))$. In Section 1.6 we mentioned that we will establish
general upper and lower bounds on $T(w)$.

In Lemma 4.1, we show that

$\displaystyle T(w)$
$\displaystyle\leq\xi_{\alpha}^{2}\int_{1}^{\infty}\int_{1}^{\infty}h(w,{w}_{1},{w}_{2})\mathbbm{1}_{\left\\{{w}_{1}\wedge{w}_{2}\leq
w\right\\}}({w}_{1}{w}_{2})^{-\beta}d{w}_{1}d{w_{2}}$ $\displaystyle\hskip
10.0pt+\xi_{\alpha}^{2}\int_{1}^{\infty}\int_{1}^{\infty}g(w,{w}_{1})g(w,{w}_{2})\mathbbm{1}_{\left\\{{w}_{1}\wedge{w}_{2}>w\right\\}}({w}_{1}{w}_{2})^{-\beta}d{w}_{1}d{w}_{2},$

while in Lemma 4.3, we obtain

$\displaystyle
T(w)\geq\omega_{d}^{2}\int_{1}^{\infty}\int_{1}^{\infty}\mathbbm{1}_{\left\\{g({w}_{1},{w}_{2})<g(w,{w}_{2})\right\\}}h(w,{w}_{1},{w}_{2})$
$\displaystyle\hskip
100.0pt\times\left[1-\left(\frac{g({w}_{1},{w}_{2})}{(g(w,{w}_{1})\wedge
g(w,{w}_{2})}\right)^{1/d}\right]^{d}({w}_{1}{w}_{2})^{-\beta}d{w}_{1}d{w}_{2}.$

We remark that when $w$ is sufficiently large, up to constant factors, the
first term of the upper bound, as well as the lower bound, are both truncated
integrals of the function $h(w,{w}_{1},{w}_{2})$. For the first term of the
upper bound, this is immediate, while for the lower bound we use that, for $w$
sufficiently large, the term

$\left[1-\left(\frac{g({w}_{1},{w}_{2})}{(g(w,{w}_{1})\wedge
g(w,{w}_{2})}\right)^{1/d}\right]^{d}$

is of constant order. In particular, as we see in the proof of Theorem 1.1,
the upper and lower bounds are of the same order for large $w$, which means
that the scaling of $T(w)$, for large $w$, is guided by truncated integrals of
the function $h(w,{w}_{1},{w}_{2})$. As we see in the proof of Theorem 1.1, it
is the phase transition in the integrability of the function
$h(w,{w}_{1},{w}_{2})$ that causes the first phase transition in the scaling
of $\gamma(k)$ at $\beta=a+\frac{3}{2}$, see Theorem 1.1. In particular, when
$\beta>a+\frac{3}{2}$, the function $h(w,{w}_{1},{w}_{2})$, when multiplied by
the densities of ${w}_{1}$ and ${w}_{2}$ having the form (1.1), is integrable
in ${w}_{1}$ and ${w}_{2}$, and the obtained function of $w$ decays linearly
in $w$, which is the main cause for the inverse linear scaling of $\gamma(k)$
in this regime. Instead when $\beta\leq a+\frac{3}{2}$, the function
$h(w,{w}_{1},{w}_{2})$ is not integrable anymore, and its truncated integral
diverges logarithmically when $\beta=a+\frac{3}{2}$, and polynomially when
$\beta<a+\frac{3}{2}$.

Finally, we discuss the occurence of the second phase transition at
$\beta=a+1$. Here we understand the phase transition in terms of how the
scaling of $\mathcal{M}(w)$ changes. In particular, as $w\to\infty$, one
observes the following transition in behaviour of the scaling of the mean
degree $\mathcal{M}(w)$ of $0$ given $W_{0}=w$, about $\beta=a+1$:

$\displaystyle\mathcal{M}(w)=\begin{cases}\Theta(w)&\mbox{if}\;\;\beta>(a+1)\vee
2,a\in[0,\infty),\\\
\Theta(w\log{w})&\mbox{if}\;\;\beta=a+1,a\in(1,\infty),\\\
\Theta(w^{2+a-\beta})&\mbox{if}\;\;\beta\in(2,a+1),a\in(1,\infty),\end{cases}$

see Figure 2 and Section 3 for more details on the behaviour of
$\mathcal{M}(w)$.

In particular, whenever $\beta<a+\frac{3}{2}$,
${T(\mathcal{M}^{-1}(k))}/{k^{2}}$ always scales like
${\mathcal{M}^{-1}(k)^{4+2a-2\beta}}/{k^{2}},$ as we will prove in Section 3.
Thus, the transition in behaviour of the scaling of $\mathcal{M}^{-1}(k)$
guides the transition in scaling of ${T(\mathcal{M}^{-1}(k))}/{k^{2}}$ around
$\beta=a+1$.

##### Organization of the paper.

Section 2 gives preliminary simplifications to analyse $\gamma(k)$. In Section
3, we analyse the scaling of $\mathcal{M}^{-1}(k)$. In Section 4, we develop
some bounding techniques which aid us in analysing upper and lower asymptotics
of $\gamma(k)$. Using results from Sections 3 and 4, we prove Theorem 1.1 in
Section 5. Section 6 is devoted to discussion. Appendix A contains some
standard results which are used to derive a result of Section 2. Appendix B
contains the proof of a technical result used in Section 3.

##  2 Preliminaries

This section is devoted to two simplifications of the clustering function
$\gamma(k)$ from (1.6). First, in Proposition 2.2 we reduce its expression to
an integral form as given in (2.5). Next, in Proposition 2.3 we prove a
concentration result which enables us to argue that asymptotically, this
integral is really an integral with respect to a Dirac measure. This
simplifies the analysis for its asymptotics.

###  2.1 Simplifying $\gamma(k)$

Let us denote

$\displaystyle\mathbb{U}:=\mathbb{R}^{d}\times\mathbb{R}_{+}$

for the parameter space of the vertices: the first coordinate denotes the
location and the second the weight of a vertex. We first prove that
conditionally on the weight of a typical vertex, the neighbors of the typical
vertex in any infinite SIRG form an inhomogeneous Poisson point process on
$\mathbb{U}$:

######  Lemma 2.1 (Neighbors of $0$).

Consider an infinite SIRG $\mathbb{G}^{\infty}$, where the vertex locations
are given by a Palm version $\Gamma\cup\\{\mathbf{0}\\}$ of a homogeneous
Poisson process $\Gamma$ on $\mathbb{R}^{d}$, the weights are i.i.d. with
density $f_{W}$, and the connection function
$\kappa:\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}\to[0,1]$ is symmetric
in its last two arguments. Let us denote the weight of $0$ by $W_{0}$. Then,
conditionally on $\\{W_{0}=w\\}$, the locations and weights of the neighbors
of $0$ in the graph $\mathbb{G}^{\infty}$ are distributed as the atoms of an
inhomogeneous Poisson point process $\mathcal{N}_{0}^{(w)}$ on $\mathbb{U}$,
with intensity function, for any $\mathbf{p}=(\mathbf{x},x)\in\mathbb{U}$,
given by

$\rho_{w}(\mathbf{p})=\rho_{w}(\mathbf{x},x):=\kappa(\|\mathbf{x}\|,w,x)f_{W}(x).$
(2.1)

The proof of Lemma 2.1 is a consequence of standard theory of markings and
thinnings of Poisson point processes. The required necessary concepts and the
proof is postponed to Appendix A.

Recall $\gamma(k)$ from (1.6). Let us use Lemma 2.1 to simplify the expression
of $\gamma(k)$, when $\mathbb{G}^{\infty}$ has independent weights distributed
according to the density (1.1), and with connection function (1.2), for some
symmetric weight function $g$. Denoting in this case the integral of the above
intensity function by $\mathcal{M}(w)$, we have,

$\mathcal{M}(w)=\int_{\mathbb{U}}\rho_{w}(\mathbf{p})d\mathbf{p}=\int_{1}^{\infty}\int_{\mathbb{R}^{d}}\kappa_{\alpha}(\|\mathbf{x}\|,w,x)d\mathbf{x}f_{W}(x)dx.$
(2.2)

Moreover, Lemma 2.1 implies that the number of neighbors of $0$ conditionally
on the event $\\{W_{0}=w\\}$ has distribution

$d_{0}(\mathbb{G}^{\infty})\big{|}_{W_{0}=w}\stackrel{{\scriptstyle
d}}{{=}}\text{Poi}\left(\mathcal{M}(w)\right).$ (2.3)

Using (2.3), a simple application of the law of total probability shows that
the conditional density function $f_{k}(w)$ of the weight $W_{0}$ of $0$,
conditionally on the event $\\{d_{0}(\mathbb{G}^{\infty})=k\\}$ has the form

$f_{k}(w):=\frac{\mathbb{P}\left(\text{Poi}(\mathcal{M}(w))=k\right)f_{W}(w)}{\int_{1}^{\infty}\mathbb{P}\left(\text{Poi}(\mathcal{M}(r))=k\right)f_{W}(r)dr}=\frac{e^{-\mathcal{M}(w)}\mathcal{M}(w)^{k}f_{W}(w)}{\int_{1}^{\infty}e^{-\mathcal{M}(r)}\mathcal{M}(r)^{k}f_{W}(r)dr}.$
(2.4)

From now on, to simplify notation, we write
$d\underline{\mathbf{x}}=d\mathbf{x}_{1}d\mathbf{x}_{2}$ for the Lebesgue
measure on $(\mathbb{R}^{d})^{2}$. Also recall the notation
$f_{W}(\underline{\mathbf{w}})d\underline{\mathbf{w}}$ from (1.9).

######  Proposition 2.2 (Simplifying $\gamma(k)$).

Consider an infinite SIRG $\mathbb{G}^{\infty}$ with vertex locations
distributed as $\Gamma\cup\\{\mathbf{0}\\}$ where $\Gamma$ is a homogeneous
Poisson point process on $\mathbb{R}^{d}$, with independent vertex weights
distributed according to the density (1.1), and with connection function (1.2)
for some symmetric weight function $g$. Assume that the integral on the RHS of
(2.2) is finite. Then the clustering function $\gamma(k)$ of
$\mathbb{G}^{\infty}$ has the form

$\gamma(k)=\int_{1}^{\infty}\frac{T(w)}{\mathcal{M}(w)^{2}}f_{k}(w)dw,$ (2.5)

where $f_{k}(w)$ is as in (2.4), and

$T(w):=\iint_{0}^{\infty}\iint_{(\mathbb{R}^{d})^{2}}\kappa_{\alpha}(\|\mathbf{x}_{1}\|,{w}_{1},w)\kappa_{\alpha}(\|\mathbf{x}_{2}\|,{w}_{2},w)\kappa_{\alpha}(\|\mathbf{x}_{1}-\mathbf{x}_{2}\|,{w}_{1},{w}_{2})d\underline{\mathbf{x}}\,f_{W}(\underline{\mathbf{w}})d\underline{\mathbf{w}}.$
(2.6)

###### Proof.

Recall the Poisson process $\mathcal{N}^{(w)}_{0}$ from Lemma 2.1. Let
$\mathcal{N}_{0}^{(w,k)}$ be the conditional point process on $\mathbb{U}$
defined as

$\displaystyle\mathcal{N}_{0}^{(w,k)}:=\mathcal{N}^{(w)}_{0}\big{|}_{\mathcal{N}^{(w)}_{0}(\mathbb{U})=k}.$

Let $(U_{\\{i,j\\}})_{i,j=1}^{k}$ be a symmetric array of uniformly
distributed random variables on $[0,1]$, independent of everything else.

Denoting the atoms of $\mathcal{N}_{0}^{(w,k)}$ by
$\\{\mathbf{p}_{i}=(\mathbf{x}_{i},{x}_{i})\in\mathbb{U}:1\leq i\leq k\\}$, we
note that

$\displaystyle\mathbb{E}\left[\left.\Delta_{0}(\mathbb{G}^{\infty})\right|d_{0}(\mathbb{G}^{\infty})=k,W_{0}=w\right]=\mathbb{E}\left[\sum_{\mathbf{p}_{i},\mathbf{p}_{j}\in\mathcal{N}_{0}^{(w,k)}}^{\neq}\mathbbm{1}_{\left\\{U_{\\{i,j\\}}<\kappa_{\alpha}(\|\mathbf{x}_{i}-\mathbf{x}_{j}\|,{x}_{i},{x}_{j})\right\\}}\right],$

where $\sum_{\mathbf{p}_{i},\mathbf{p}_{j}\in\mathcal{N}_{0}^{(w,k)}}^{\neq}$
denotes the sum over distinct pairs
$\mathbf{p}_{i},\mathbf{p}_{j}\in\mathcal{N}_{0}^{(w,k)}$ and the expectation
is taken both with respect to the point process and the uniform random
variables. This implies that

$\mathbb{E}\left[\left.\Delta_{0}(\mathbb{G}^{\infty})\right|d_{0}(\mathbb{G}^{\infty})=k\right]=\int_{1}^{\infty}\mathbb{E}\left[\sum_{\mathbf{p}_{i},\mathbf{p}_{j}\in\mathcal{N}_{0}^{(w,k)}}^{\neq}\mathbbm{1}_{\left\\{U_{\\{i,j\\}}<\kappa_{\alpha}(\|\mathbf{x}_{i}-\mathbf{x}_{j}\|,{x}_{i},{x}_{j})\right\\}}\right]f_{k}(w)dw.$
(2.7)

Using the notation $\mathbf{p}_{i}=(\mathbf{x}_{i},x_{i})$, we note that the
collection of $\mathbb{U}$-valued random variables
$\\{\mathbf{p}_{i}\\}_{\mathbf{p}_{i}\in\mathcal{N}_{0}^{(w,k)}}$ form an
i.i.d. collection of random variables with density given by

$\mathbf{p}\mapsto
h_{w}(\mathbf{p})=h_{w}(\mathbf{x},x)=\frac{\kappa(\|\mathbf{x}\|,w,x)f_{W}(x)}{\int_{1}^{\infty}\int_{\mathbb{R}^{d}}\kappa(\|\mathbf{x}\|,w,x)f_{W}(x)d\mathbf{x}\,dx}=\frac{\kappa(\|\mathbf{x}\|,w,x)f_{W}(x)}{\mathcal{M}(w)}.$
(2.8)

In particular, the expectation of the sum in the integrand in (2.7) evaluates
to

$\displaystyle\binom{k}{2}\int_{\mathbb{U}}\int_{\mathbb{U}}\kappa_{\alpha}(\|\mathbf{x}_{1}-\mathbf{x}_{2}\|,x_{1},x_{2})h_{w}(\mathbf{p}_{1})h_{w}(\mathbf{p}_{2})d(\mathbf{p}_{1})d(\mathbf{p}_{2}),$
$\displaystyle=\frac{\binom{k}{2}}{\mathcal{M}(w)^{2}}\int_{1}^{\infty}\int_{1}^{\infty}\iint_{(\mathbb{R}^{d})^{2}}\kappa_{\alpha}(\|\mathbf{x}_{1}\|,w_{1},w)\kappa_{\alpha}(\|\mathbf{x}_{2}\|,w_{2},w)\kappa_{\alpha}(\|\mathbf{x}_{1}-\mathbf{x}_{2}\|,w_{1},w_{2})d\underline{\mathbf{x}}\,f_{W}(\underline{\mathbf{w}})d\underline{\mathbf{w}}$
$\displaystyle=\frac{\binom{k}{2}T(w)}{\mathcal{M}(w)^{2}},$

where
$\mathbf{p}_{1}=(\mathbf{x}_{1},x_{1}),\mathbf{p}_{2}=(\mathbf{x}_{2},x_{2})$,
and to obtain the first equality we have simply changed variables
$(x_{1},x_{2})\mapsto(w_{1},w_{2})$. Since
$\gamma(k)=\binom{k}{2}^{-1}\mathbb{E}\left[\left.\Delta_{0}(\mathbb{G}^{\infty})\right|d_{0}(\mathbb{G}^{\infty})=k\right]$
from (1.6), the result follows. ∎

###  2.2 Concentration of the weight of $0$ when $d_{0}(\mathbb{G}^{\infty})$
is large

The main goal of this section is to argue that the conditional density
$w\mapsto f_{k}(w)$ of (2.4) is asymptotically like a Dirac mass at
$\mathcal{M}^{-1}(k)$. Informally, the argument goes as follows: Conditionally
on the Poisson variable (2.3) being large, standard Poisson concentration
arguments imply that its mean $\mathcal{M}(w)$ also has to be large. In
particular, if $\mathcal{M}(w)$ is a nice monotone function of $w$, this means
that $w$ also has to be large. That is, the degree of $0$ in
$\mathbb{G}^{\infty}$ cannot be large unless its weight is also large. In this
section, we formalise this by arguing that when the degree of $0$ is $k$, and
$k\to\infty$, the weight of $0$ is highly concentrated, and diverges as
$\mathcal{M}^{-1}(k)$. As a consequence, the conditional density function
$f_{k}(w)$ appearing in the integral (2.5) behaves like a Dirac mass with an
atom at $\mathcal{M}^{-1}(k)$. Consequently, using (2.5), the $k\to\infty$
asymptotic behaviour of $\gamma(k)$ is the same as that of
${T(\mathcal{M}^{-1}(k))}/{k^{2}}$.

Recall the notations $T(w)$ and $\mathcal{M}(w)$ from (2.2) and (2.6). First,
we need a regularity condition for diverging sequences, that ensures that they
are robust under certain lower order perturbations:

######  Definition 2.1 (Fluctuation-robust sequences).

A sequence $(\psi(k))_{k\geq 1}$ diverging to infinity is called _fluctuation-
robust_ , if for any $C>0$, as $k\to\infty$,

$\displaystyle\frac{\left(\psi(k)+C\sqrt{\psi(k)\log{\psi(k)}}\right)\vee\psi\left(k+C\sqrt{k\log{k}}\right)}{\left(\psi(k)-C\sqrt{\psi(k)\log{\psi(k)}}\right)\wedge\psi\left(k-C\sqrt{k\log{k}}\right)}\to
1.$

Examples of fluctuation-robust sequences are positive powers of $k$, or some
positive power of $k$ times some logarithmic factor: $k\log{k}$,
$k^{\epsilon}(\log{k})^{\delta}$ for any $\epsilon>0$ and
$\delta\in\mathbb{R}$, etc. It is easy to verify that for any constant $C>0$
and $L\in\mathbb{N}$, all of $C\psi(k)$, $\psi(Ck)$, $\psi^{(L)}(k)$ are
fluctuation-robust, if $\psi(k)$ is so, where $\psi^{(L)}(\cdot)$ denotes the
$L-$fold composition of $\psi(\cdot)$. Note that the sequence $\psi(k)=e^{k}$
is not fluctuation-robust. Observe that if $\psi(k)$ is fluctuation-robust,
then any sequence $\phi(k)$ with ${\psi(k)}/{\phi(k)}\to 1$ is also
fluctuation-robust.

Let us now state the main result of this section. Recall the connection
function (1.2). For the following proposition, we do not need the form (1.3)
of $g$, and it is true as soon as the weight function $g$ satisfies certain
assumptions:

######  Proposition 2.3.

Let $\mathbb{G}^{\infty}$ be an infinite SIRG with vertex locations
distributed as $\Gamma\cup\\{\mathbf{0}\\}$ for a homogeneous Poisson point
process $\Gamma$ on $\mathbb{R}^{d}$, independent vertex weights having
density (1.1), and with the connection function (1.2) for some symmetric
function $g$. Recall also $\mathcal{M}(w)$ from (2.2). Assume the following:

  * 1.

For fixed $s\in\mathbb{R}$, the function $w\mapsto g(s,w)$ is monotone
increasing in $w$.

  * 2.

$\mathcal{M}(w)<\infty$ for all $w>0$, with $\mathcal{M}(w)\to\infty$ as
$w\to\infty$. Further, there exists $K_{0}>0$ and $\zeta\in\mathbb{R}$ such
that $\mathcal{M}(w)$ is strictly increasing and differentiable in
$(K_{0},\infty)$, with $\mathcal{M}^{\prime}(w)\leq w^{\zeta}$ for all
$w>K_{0}$.

  * 3.

There exists $\xi\in\mathbb{R}$ such that $\mathcal{M}^{-1}(k)\leq k^{\xi}$,
as $k\to\infty$.

  * 4.

$T(\mathcal{M}^{-1}(k))=\Omega(k^{-\eta})$ for some $\eta>0$, as $k\to\infty$.

Under the above mentioned assumptions,

  * (a.)

if further $T(\mathcal{M}^{-1}(k))=\Theta(\psi(k))$ for some fluctuation-
robust sequence $(\psi(k))_{k\in\mathbb{N}}$, then as $k\to\infty$,

$\displaystyle\gamma(k)=\Theta(T(\mathcal{M}^{-1}(k))k^{-2}).$

  * (b.)

if further the sequence $(T(\mathcal{M}^{-1}(k)))_{k\in\mathbb{N}}$ is itself
fluctuation-robust, then, as $k\to\infty$,

$\displaystyle\frac{\gamma(k)}{T(\mathcal{M}^{-1}(k))k^{-2}}\to 1.$

Thus, under the conditions of Proposition 2.3, asymptotically the density
$w\mapsto f_{k}(w)$ appearing in the integral (2.5) behaves like a Dirac mass
at $\mathcal{M}^{-1}(k)$, as $k\to\infty$, provided the sequence
$(T(\mathcal{M}^{-1}(k)))_{k\in\mathbb{N}}$ scales like a fluctuation-robust
sequence.

Proposition 2.3 is used in the proof of Theorem 1.1 in the following way: Note
that the sequence $\mathrm{S}_{k}(a,\beta)$ in the statement of Theorem 1.1 is
fluctuation robust, for given $a$ and $\beta$. Consequently, so is the
sequence $k^{2}\mathrm{S}_{k}(a,\beta)$. Hence it is enough to show that
$\frac{T(\mathcal{M}^{-1}(k))}{k^{2}\mathrm{S}_{k}(a,\beta)}\to\Gamma(a,\alpha,\beta,d)$,
since this automatically implies that $T(\mathcal{M}^{-1}(k))$ is fluctuation-
robust, and hence the appropriately rescaled limit of both
$T(\mathcal{M}^{-1}(k))/k^{2}$ and $\gamma(k)$ is the same, due to Proposition
2.3 (b).

Before going into the proof, we first state some useful results. We begin with
the standard Chernoff concentration bound for Poisson random variables, e.g.
see [8, (2.12)], as

$\mathbb{P}\left(|\text{Poi}(\lambda)-\lambda|\geq x\right)\leq
2\exp{\left(-\frac{x^{2}}{2(\lambda+x)}\right)},$ (2.9)

for any $x>0$. In particular, for any $C>0$, and
$\lambda=\lambda_{n}\to\infty$,

$\mathbb{P}\left(|\text{Poi}(\lambda_{n})-\lambda_{n}|\geq
C\sqrt{\lambda_{n}\log{\lambda_{n}}}\right)=O\left(\lambda_{n}^{-\frac{C^{2}}{2}}\right).$
(2.10)

We also recall the standard stochastic domination property of Poisson random
variables: for $\lambda_{1}>\lambda_{2}$ and any $x>0$,

$\mathbb{P}\left(\text{Poi}(\lambda_{2})>x\right)\leq\mathbb{P}\left(\text{Poi}(\lambda_{1})>x\right).$
(2.11)

Next, we state a lemma which will be useful in the proof of Proposition 2.3:

######  Lemma 2.4.

Under the assumptions of Proposition 2.3, there exists $K_{1}>0$ such that for
all $k>K_{1}$,

$\int_{1}^{\infty}\mathbb{P}\left(\text{Poi}(\mathcal{M}(w))=k\right)w^{-\beta}dw\geq
k^{-\xi\beta-\zeta\xi-2}.$

We are now ready to prove Proposition 2.3.

###### Proof of Proposition 2.3.

Let $I_{k}^{-}:=k-C\sqrt{k\log{k}}$, and define $I_{k}^{+}$ analogously, where
$C$ is a sufficiently large constant to be chosen later.

Note that $J_{k}^{-}:=\mathcal{M}^{-1}(I_{k}^{-})$ and
$J_{k}^{+}:=\mathcal{M}^{-1}(I_{k}^{+})$ are well defined for all large $k$,
since $\mathcal{M}(w)$ is continuous and strictly increasing on
$(K_{0},\infty)$.

Further, note that if $w<J_{k}^{-}$, i.e., $\mathcal{M}(w)<I_{k}^{-}$, then

$\displaystyle\mathbb{P}\left(\mathrm{Poi}(\mathcal{M}(w))=k\right)$
$\displaystyle\leq\mathbb{P}\left(\mathrm{Poi}(\mathcal{M}(w))>I_{k}^{-}+C\sqrt{I_{k}^{-}\log{I_{k}^{-}}}\right)$
$\displaystyle\leq\mathbb{P}\left(\mathrm{Poi}(I_{k}^{-})>I_{k}^{-}+C\sqrt{I_{k}^{-}\log{I_{k}^{-}}}\right)=O\left(k^{-C^{2}/2}\right),$

as $k\to\infty$, where to obtain the first inequality, we have used that
$I_{k}^{-}+C\sqrt{I_{k}^{-}\log{I_{k}^{-}}}<k$ for all large $k$, and to
obtain the last inequality, we have used $\mathcal{M}(w)<I_{k}^{-}$ along with
(2.10). A similar argument shows that, for $w>J_{k}^{+}$,

$\displaystyle\mathbb{P}\left(\mathrm{Poi}(\mathcal{M}(w))=k\right)=O\left(k^{-C^{2}/2}\right).$

We conclude that if $w\notin[J_{k}^{-},J_{k}^{+}]$,

$\mathbb{P}\left(\mathrm{Poi}(\mathcal{M}(w))=k\right)=O\left(k^{-C^{2}/2}\right)$
(2.12)

Hence, we can write

$\frac{\gamma(k)}{T(\mathcal{M}^{-1}(k))k^{-2}}=\frac{\overline{\gamma}(k)}{T(\mathcal{M}^{-1}(k))k^{-2}}+\varepsilon_{k},$
(2.13)

where

$\overline{\gamma}(k):=\int_{J_{k}^{-}}^{J_{k}^{+}}\frac{T(w)}{\mathcal{M}(w)^{2}}f_{k}(w)dw,$
(2.14)

and

$\displaystyle\varepsilon_{k}$
$\displaystyle:=\frac{1}{T(\mathcal{M}^{-1}(k))k^{-2}}\int_{(1,\infty)\setminus[J_{k}^{-},J_{k}^{+}]}\frac{T(w)}{\mathcal{M}(w)^{2}}f_{k}(w)dw$
$\displaystyle\leq\frac{1}{T(\mathcal{M}^{-1}(k))k^{-2}}\frac{1}{\int_{1}^{\infty}\mathbb{P}\left(\text{Poi}(\mathcal{M}(r))=k\right)r^{-\beta}dr}\int_{(1,\infty)\setminus[J_{k}^{-},J_{k}^{+}]}\mathbb{P}\left(\text{Poi}(\mathcal{M}(w))=k\right)w^{-\beta}dw.$

Here to obtain the first inequality, we have used the easily checked upper
bound of $1$ on $T(w)/{\mathcal{M}(w)^{2}}$. Observe that using Assumption
(4), Lemma 2.4, and (2.12), for all large $k$,

$\displaystyle\varepsilon_{k}=O\left(k^{2-\eta+\xi\beta+\zeta\xi+2-C^{2}/2}\right)=o\left(1\right),$
(2.15)

by choosing $C>0$ sufficiently large.

Next we aim to bound $\overline{\gamma}(k)$ from above and below. Recall that
we assume that $w\mapsto g_{s}(w)$ is monotone increasing in $w$. As a
consequence, for any $x\in\mathbb{R}^{d}$ and $s\in\mathbb{R}_{+}$,
$w\mapsto\kappa_{\alpha}(\|\mathbf{x}\|,s,w)$ is also monotone increasing in
$w$. In particular, this implies that $w\mapsto T(w)$ is monotone increasing
in $w$. Also, $w\mapsto\mathcal{M}(w)$ is strictly increasing in $w$ whenever
$w$ is sufficiently large by assumption. Using the monotonicity of $T(w)$ and
$\mathcal{M}(w)$, and recalling $\overline{\gamma}(k)$ from (2.14), we get can
an upper bound on it by simply bounding by the supremum value of the
integrand,

$\overline{\gamma}(k)\leq\frac{T(J_{k}^{+})}{\mathcal{M}(J_{k}^{-})^{2}}\int_{J_{k}^{-}}^{J_{k}^{+}}f_{k}(w)dw=\frac{T(\mathcal{M}^{-1}(I_{k}^{+}))}{(I_{k}^{-})^{2}}(1-\delta_{k}),$

where

$\displaystyle 1-\delta_{k}:=\int_{J_{k}^{-}}^{J_{k}^{+}}f_{k}(w)dw$
$\displaystyle=\frac{\int_{J_{k}^{-}}^{J_{k}^{+}}\text{Poi}(\mathcal{M}(w)=k)w^{-\beta}dw}{\int_{1}^{\infty}\text{Poi}(\mathcal{M}(r)=k)r^{-\beta}dr}$
$\displaystyle=1-\frac{\int_{(1,\infty)\setminus[J_{k}^{-},J_{k}^{+}]}\text{Poi}(\mathcal{M}(r)=k)r^{-\beta}dr}{\int_{1}^{\infty}\text{Poi}(\mathcal{M}(w)=k)w^{-\beta}dw}$
$\displaystyle=1+O\left(k^{(-C^{2}/2)+{\xi\beta+\zeta\xi+2}}\right)\to 1,$
(2.16)

as $k\to\infty$, by choosing $C>0$ large, where we have used (2.12) and Lemma
2.4 to obtain (2.16).

Similarly, we can also obtain a lower bound on $\overline{\gamma}(k)$, by
bounding it from below by the infimum value of the integrand. In particular,
we obtain the following chain of inequalities,

$\frac{k^{2}}{T(\mathcal{M}^{-1}(k))}\frac{T(\mathcal{M}^{-1}(I_{k}^{-}))}{(I_{k}^{+})^{2}}(1-\delta_{k})\leq\frac{\overline{\gamma}(k)}{T(\mathcal{M}^{-1}(k))k^{-2}}\leq\frac{k^{2}}{T(\mathcal{M}^{-1}(k))}\frac{T(\mathcal{M}^{-1}(I_{k}^{+}))}{(I_{k}^{-})^{2}}(1-\delta_{k}).$
(2.17)

Now let us prove (a.). Since $T(\mathcal{M}^{-1}(k))=\Theta(\psi(k))$, there
are constants $C_{1},c_{1}>0$ such that whenever $k$ is sufficiently large,

$\displaystyle c_{1}\leq\frac{T(\mathcal{M}^{-1}(k))}{\psi(k)}\leq C_{1}.$

Hence,

$\displaystyle\frac{T(\mathcal{M}^{-1}(I_{k}^{-}))}{T(\mathcal{M}^{-1}(k))}\leq\frac{C_{1}\psi(I_{k}^{-})}{c_{1}\psi(k)},$

which implies, as $\psi(k)$ is fluctuation-robust and consequently
${\psi(I_{k}^{-})}/{\psi(k)}\to 1$, that
${T(\mathcal{M}^{-1}(I_{k}^{-}))}/{T(\mathcal{M}^{-1}(k))}=O\left(1\right)$,
as $k\to\infty$. A similar argument shows that
${T(\mathcal{M}^{-1}(I_{k}^{-}))}/{T(\mathcal{M}^{-1}(k))}=\Omega(1)$, as
$k\to\infty$. Hence, as $k\to\infty$,
${T(\mathcal{M}^{-1}(I_{k}^{-}))}/{T(\mathcal{M}^{-1}(k))}=\Theta(1)$, and
similarly,
${T(\mathcal{M}^{-1}(I_{k}^{+}))}/{T(\mathcal{M}^{-1}(k))}=\Theta(1)$.
Further, note from the definitions of $I_{k}^{+}$ and $I_{k}^{-}$ that both
${k}/{I_{k}^{+}},{k}/{I_{k}^{-}}\to 1$ as $k\to\infty$. Also, (2.16) shows
that $\delta_{k}\to 0$. Hence, both the left-most and the right-most
expressions of (2.17) are $\Theta(1)$, as $k\to\infty$. Hence, by (2.17),

$\frac{\overline{\gamma}(k)}{T(\mathcal{M}^{-1}(k))k^{-2}}=\Theta(1).$ (2.18)

Finally, recall (2.13). Since (2.15) shows that $\varepsilon_{k}\to 0$, we
have both that ${\overline{\gamma}(k)}/{(T(\mathcal{M}^{-1}(k))k^{-2})}$ and
${\gamma(k)}/{(T(\mathcal{M}^{-1}(k))k^{-2})}$ are of the same order, which
concludes the proof of (a), using (2.18).

Now, we move on to proving (b). The philosophy is quite similar to that of the
proof of (a). In this case, recall that $T(\mathcal{M}^{-1}(k))$ is itself
log-robust. This implies that
${T(\mathcal{M}^{-1}(I_{k}^{-}))}/{T(\mathcal{M}^{-1}(k))}\to 1$ and
${T(\mathcal{M}^{-1}(I_{k}^{+}))}/{T(\mathcal{M}^{-1}(k))}\to 1$. Thus arguing
similarly as before, both the left-most and the right-most expressions of
(2.17) converge to $1$, as $k\to\infty$. So that also
${\overline{\gamma}(k)}/{(T(\mathcal{M}^{-1}(k))k^{-2})}\to 1$ as
$k\to\infty$. Recalling (2.13) and (2.15), we obtain that
${\gamma(k)}/{(T(\mathcal{M}^{-1}(k))k^{-2})}\to 1$, as $k\to\infty$, which
concludes the proof of (b). ∎

Finally, let us give the proof of Lemma 2.4:

###### Proof of Lemma 2.4.

Let $K_{1}>0$ be sufficiently large so that $\mathcal{M}(w)$ is strictly
increasing and differentiable on $(K_{1},\infty)$, and further,
$\mathcal{M}^{\prime}(w)\leq w^{\zeta}$ and $\mathcal{M}^{-1}(k)\leq 2k^{\xi}$
for all $w\in(K_{1},\infty)$. Note that the existence of such a $K_{1}$ is
guaranteed by Assumptions (2.) and (3.) of Proposition 2.3. Restricting the
domain of integration to $(K_{1},\infty)$ and then changing variables as
$t\mapsto\mathcal{M}(w)$, we get a lower bound on
$\int_{1}^{\infty}\mathbb{P}\left(\text{Poi}(\mathcal{M}(w)=k)\right)w^{-\beta}dw$
as,

$\displaystyle\int_{\mathcal{M}(K_{1})}^{\infty}\frac{t^{k}}{k!}e^{-t}(\mathcal{M}^{-1}(t))^{-\beta}\frac{dt}{\mathcal{M}^{\prime}(\mathcal{M}^{-1}(t))}$
$\displaystyle\geq\frac{2^{-\beta-\zeta}}{k!}\int_{\mathcal{M}(K_{1})}^{\infty}e^{-t}t^{(k-\xi\beta-\xi\zeta-1)+1}dt$
$\displaystyle=\Omega\left(\frac{\Gamma(k-\xi\beta-\xi\zeta-1,\mathcal{M}(K_{1}))}{\Gamma(k+1)}\right),$

where $\Gamma(t,x)=\int_{x}^{\infty}e^{-r}r^{t-1}dr$ is the upper incomplete
gamma function, and $\Gamma(t)=\Gamma(t,0)$ is the gamma function. Next we
recall standard scaling relations between gamma functions: for any fixed $x>0$
and $a\in\mathbb{R}$, as $t\to\infty$,

$\displaystyle\frac{\Gamma(t+a,x)}{\Gamma(t)}=\frac{\Gamma(t+a)}{\Gamma(t)}-\frac{\int_{0}^{x}e^{-r}r^{t+a-1}dr}{\Gamma(t)}=\frac{\Gamma(t+a)}{\Gamma(t)}+o\left(1\right)=t^{a}+o\left(1\right).$

Taking $t=k+1$ and $a=-\xi\beta-\xi\zeta-2$ finishes the proof of the lemma. ∎

##  3 Scaling of $\mathcal{M}^{-1}(k)$

In the previous section, we have reduced the analysis of the clustering
function $\gamma(k)$ to analyzing $T(\mathcal{M}^{-1}(k))/k^{2}$. The first
step is to study $\mathcal{M}^{-1}(k)$, which is the content of this section.
Recall the definition from (2.2). Then by (1.1),

$\displaystyle\mathcal{M}(w)$
$\displaystyle=\int_{1}^{\infty}\int_{\mathbb{R}^{d}}\kappa_{\alpha}(\|\mathbf{x}\|,w,x)d\mathbf{x}f_{W}(x)dx$
$\displaystyle=\begin{cases}\int_{1}^{\infty}\int_{\mathbb{R}^{d}}\left[\mathbbm{1}_{\left\\{\|\mathbf{x}\|<g(x,w)^{1/d}\right\\}}+\left(\frac{g(x,w)}{\|\mathbf{x}\|^{d}}\right)^{\alpha}\mathbbm{1}_{\left\\{\|\mathbf{x}\|\geq
g(x,w)^{1/d}\right\\}}\right]d\mathbf{x}\,x^{-\beta}dx,&\text{if
}\alpha\in(1,\infty),\\\
\int_{1}^{\infty}\int_{\mathbb{R}^{d}}\mathbbm{1}_{\left\\{\|\mathbf{x}\|<g(x,w)^{1/d}\right\\}}d\mathbf{x}\,x^{-\beta}dx,&\text{if
}\alpha=\infty.\end{cases}$
$\displaystyle=\begin{cases}\frac{\alpha\omega_{d}(\beta-1)}{\alpha-1}\int_{1}^{\infty}g(x,w)x^{-\beta}dx,&\text{if
}\alpha\in(1,\infty),\\\
\omega_{d}(\beta-1)\int_{1}^{\infty}g(x,w)x^{-\beta}dx,&\text{if
}\alpha=\infty.\end{cases}$

Next, recalling
$\xi_{\alpha}=\xi_{\alpha,\beta,d}:=\frac{\alpha\omega_{d}(\beta-1)}{\alpha-1}$
from (1.7), with $\xi_{\infty}=\lim_{\alpha\to\infty}\xi_{\alpha}$, and from
(1.3) that $g(w_{1},w_{2})=(w_{1}\vee w_{2})(w_{1}\wedge w_{1})^{a}$, we
conclude that

$\mathcal{M}(w)=\xi_{\alpha}\left(\int_{1}^{w}wx^{a-\beta}dx+\int_{w}^{\infty}x^{1-\beta}w^{a}dx\right).$
(3.1)

Direct computations of this integral yield the following result about the
asymptotic behavior of $\mathcal{M}(w)$:

######  Lemma 3.1.

Let $\mathcal{M}(w)$ be defined as in (2.2) and define

$m(w)=\begin{cases}w&\text{if }\beta>(a+1)\vee 2,a\geq 0,\\\ w\log w&\text{if
}\beta=a+1,a>1,\\\ w^{2+a-\beta}&\text{if }\beta\in(2,a+1),a>1.\end{cases}$

Then,

$\lim_{w\to\infty}\frac{\mathcal{M}(w)}{m(w)}=\begin{cases}\frac{\xi_{\alpha}}{\beta-a-1}&\text{if
}\beta>(a+1)\vee 2,a\geq 0,\\\ \xi_{\alpha}&\text{if }\beta=a+1,a>1,\\\
\frac{(a-1)\xi_{\alpha}}{(\beta-2)(a+1-\beta)}&\text{if
}\beta\in(2,a+1),a>1.\end{cases}$

######  Remark 3.1 (Infinite mean degree).

Observe that if $\beta\in(2,\frac{a+3}{2})$ then $2+a-2\beta<-1$. Hence, since
in this regime $\mathcal{M}(w)=\Theta\left(w^{2+a-\beta}\right)$, it follows
that $\mathcal{M}(w)$ is no longer integrable with respect to the density
$f_{W}(w)=\Theta\left(w^{-\beta}\right)$. This implies that
$\mathbb{E}\left[d_{0}(\mathbb{G}^{\infty})\right]=\infty$ in this regime.

The above result shows that, asymptotically, $\mathcal{M}(w)$ is some
polynomial of combinations of $w$ and $\log(w)$. The next step is then to use
the asymptotic behavior of $\mathcal{M}(w)$ to derive the behavior of
$\mathcal{M}^{-1}(k)$. Let us define certain functions which have the form of
$\mathcal{M}(w)$. To begin with, let us call a real function _eventually
strictly increasing_ if it is strictly increasing on an interval of the form
$(K_{0},\infty)$ for some $K_{0}\in\mathbb{R}$. Then we define:

######  Definition 3.1 (Almost power functions).

A function $f:\mathbb{R}_{+}\to\mathbb{R}$ is called an _almost power
function_ , if it is an eventually strictly increasing continuous function,
such that

$\lim_{x\to\infty}\frac{f(x)}{x^{a}\log(x)^{b}}=c\in(0,\infty),$ (3.2)

for some $a>0$ and $b\geq 0$.

We use a lemma that relates the scaling of almost power functions to the
scaling of their inverses:

######  Lemma 3.2.

Let $f:\mathbb{R}_{+}\to\mathbb{R}$ be an almost power function of the form
(3.2). Then, for all large $y$, $f^{-1}(y)$ is well defined, and as
$y\to\infty$,

$f^{-1}(y)=\left(\frac{a^{b}y}{c(\log{y})^{b}}\right)^{1/a}(1+o\left(1\right)).$

Lemma 3.2 is proved in Appendix B. For now we observe that, by (3.1) and Lemma
3.1, the function $w\mapsto\mathcal{M}(w)$ satisfies the conditions of the
lemma. Hence, we obtain the scaling behavior of the inverse
$\mathcal{M}^{-1}(k)$. Figure 2 shows the two scaling regimes of
$\mathcal{M}^{-1}(k)$ in the $(\beta,a)$ space.

######  Proposition 3.3.

Let $\mathcal{M}(w)$ be defined as in (2.2) and define

$m^{\prime}(k)=\begin{cases}k&\text{if }\beta>(a+1)\vee 2,a\geq 0,\\\ k/\log
k&\text{if }\beta=a+1,a>1,\\\ k^{1/(2+a-\beta)}&\text{if
}\beta\in(2,a+1),a>1.\end{cases}$

Then,

$\lim_{k\to\infty}\frac{\mathcal{M}^{-1}(k)}{m^{\prime}(k)}=\begin{cases}\frac{\beta-a-1}{\xi_{\alpha}}&\text{if
}\beta>(a+1)\vee 2,a\geq 0,\\\ \xi_{\alpha}^{-1}&\text{if }\beta=a+1,a>1,\\\
\left(\frac{(\beta-2)(a+1-\beta)}{(a-1)\xi_{\alpha}}\right)^{1/(2+a-\beta)}&\text{if
}\beta\in(2,a+1),a>1.\end{cases}$

$a$$\beta$$(1,0)$$(0,2)$$\beta=a+1$$\beta=\frac{a+3}{2}$$\mathcal{M}^{-1}(k)=\Theta(k)$$\mathcal{M}^{-1}(k)=\Theta(k/\log{k})$$\mathcal{M}^{-1}(k)=\Theta(w^{1/(2+a-\beta)})$

Figure 2: Phase diagram of the scaling of $\mathcal{M}^{-1}(k)$ as
$k\to\infty$.

Next define

$\sigma(w)=\begin{cases}w&\mbox{if}\;\;\beta>(a+\frac{3}{2})\vee
2,a\in(0,\infty),\\\
w\log{\left(w\right)}&\mbox{if}\;\;\beta=a+\frac{3}{2},a\in(\frac{1}{2},\infty),\\\
w^{4+2a-2\beta}&\mbox{if}\;\;\beta\in(2,a+\frac{3}{2}),a\in(\frac{1}{2},\infty).\end{cases}$
(3.3)

Then, combining Proposition 3.3 with the definition of
$\mathrm{S}_{k}(a,\beta)$ yields the following corollary, which is the first
step to proving Theorem 1.1:

######  Corollary 3.4.

As $k\to\infty$,

$\frac{\sigma(\mathcal{M}^{-1}(k))}{k^{2}\mathrm{S}_{k}(a,\beta)}\to\begin{cases}\frac{\beta-a-1}{\xi_{\alpha}}&\text{if
}\beta>\left(a+\frac{3}{2}\right)\vee 2,a>0,\\\
\frac{1}{2\xi_{\alpha}}&\text{if }\beta=a+\frac{3}{2},a>\frac{1}{2},\\\
\left(\frac{(\beta-a-1)}{\xi_{\alpha}}\right)^{4+2a-2\beta}&\mbox{if
}\beta\in(a+1,a+\frac{3}{2}),a>\frac{1}{2},\\\ \xi_{\alpha}^{-2}&\mbox{if
}\beta=a+1,a>1,\\\
\left(\frac{(1+a-\beta)(\beta-2)}{(a-1)\xi_{\alpha}}\right)^{2}&\mbox{if
}{\beta\in(2,a+1),a>1.}\end{cases}$

##  4 Scaling of $T(w)$

Having established the asymptotic behavior of $\mathcal{M}^{-1}(k)$, in this
section we study the scaling of the integral $T(w)$ as $w\to\infty$. In
particular, we will show that $T(w)=\Theta\left(\sigma(w)\right)$, where
$\sigma(w)$ is defined in (3.3). For this, we first obtain upper and lower
bounds on $T(w)$ that are sharp, up to constants, for sufficiently large $w$.
It is important to note that the bounds we obtain in Lemmas 4.1 and 4.3 are
valid for any symmetric function $g$ and can thus be used, together with
Proposition 2.3, to study the scaling of the clustering function for different
types of $g$’s.

To study the asymptotics of these bounds we will use the specific form of $g$
as in (1.3) and show that for this function $T(w)=O\left(\sigma(w)\right)$ and
$T(w)=\Omega(\sigma(w))$. We start with the upper bound, as the lower bound
requires a bit more computations.

To aid in our analysis we define the spatial integral

$\mathcal{S}_{w}({w}_{1},{w}_{2}):=\int_{(\mathbb{R}^{d})^{2}}\kappa_{\alpha}(\|\mathbf{x}\|,w,{w}_{1})\kappa_{\alpha}(\|\mathbf{y}\|,w,{w}_{2})\kappa_{\alpha}(\|\mathbf{x}-\mathbf{y}\|,{w}_{1},{w}_{2})d\mathbf{x}d\mathbf{y},$
(4.1)

so that

$T(w)=\int_{1}^{\infty}\int_{1}^{\infty}\mathcal{S}_{w}({w}_{1},{w}_{2})f_{W}({w}_{1})d{w}_{1}f_{W}({w}_{2})d{w}_{2}.$

###  4.1 The upper bound

We obtain an upper bound on $T(w)$ by splitting the integral depending on
whether $w_{1}\wedge w_{2}$ is smaller or larger than $w$. For each regime we
then use a different upper bound for the integrand
$\mathcal{S}_{w}(w_{1},w_{2})$. Then we study each term separately:

######  Lemma 4.1.

Let $T(w)$ be defined as in (2.6) with $\kappa_{\alpha}$ from (1.2) for some
symmetric weight function $g$. Then,

$\displaystyle T(w)$
$\displaystyle\leq\xi_{\alpha}^{2}\iint_{1}^{\infty}h(w,w_{1},w_{2})\mathbbm{1}_{\left\\{w_{1}\wedge
w_{2}\leq w\right\\}}(w_{1}w_{2})^{-\beta}dw_{1}d{w_{2}}$ (4.2)
$\displaystyle\hskip
10.0pt+\xi_{\alpha}^{2}\iint_{1}^{\infty}g(w,w_{1})g(w,w_{2})\mathbbm{1}_{\left\\{w_{1}\wedge
w_{2}>w\right\\}}(w_{2}w_{2})^{-\beta}dw_{1}dw_{2},$ (4.3)

where $h(w,w_{1},w_{2})$ is defined in (1.12).

###### Proof.

Note that for an upper bound on $T(w)$, it is sufficient to focus on the case
$\alpha\in(1,\infty)$, since for any $\alpha\in(1,\infty)$, from (1.2) one has
$\kappa_{\alpha}(\cdot)\geq\kappa_{\infty}(\cdot)$ point-wise. Hence for the
rest of the proof we assume $\alpha\in(1,\infty)$.

Next we begin by providing two upper bounds on $\mathcal{S}_{w}(w_{1},w_{2})$.
Bounding $\kappa_{\alpha}(\|\mathbf{x}\|,w,w_{1})$ from above by $1$, we get
that

$\displaystyle\mathcal{S}_{w}(w_{1},w_{2})\leq\left(\frac{\alpha\omega_{d}}{\alpha-1}\right)^{2}g(w,w_{2})g(w_{1},w_{2}).$

Similarly,

$\mathcal{S}_{w}(w_{1},w_{2})\leq\left(\frac{\alpha\omega_{d}}{\alpha-1}\right)^{2}g(w,w_{1})g(w_{1},w_{2}).$
(4.4)

Thus using the non-negativity of $g$,

$\displaystyle\mathcal{S}_{w}(w_{1},w_{2})$
$\displaystyle\leq\left(\frac{\alpha\omega_{d}}{\alpha-1}\right)^{2}\left[g(w,w_{1})\wedge
g(w,w_{2})\right]g(w_{1},w_{2})=\left(\frac{\alpha\omega_{d}}{\alpha-1}\right)^{2}h(w,w_{1},w_{2}).$
(4.5)

Also note that by bounding
$\kappa_{\alpha}(\|\mathbf{x}-\mathbf{y}\|,w_{1},w_{2})\leq 1$,

$\mathcal{S}_{w}(w_{1},w_{2})\leq\left(\frac{\alpha\omega_{d}}{\alpha-1}\right)^{2}g(w,w_{2})g(w,w_{1}).$
(4.6)

Finally, we split the integral

$\displaystyle T(w)$
$\displaystyle=(\beta-1)^{2}\int_{1}^{\infty}\int_{1}^{\infty}\mathcal{S}_{w}(w_{1},w_{2})\mathbbm{1}_{\left\\{w_{1}\wedge
w_{2}\leq w\right\\}}(w_{1}w_{2})^{-\beta}dw_{1}dw_{2}$ $\displaystyle\hskip
10.0pt+(\beta-1)^{2}\int_{1}^{\infty}\int_{1}^{\infty}\mathcal{S}_{w}(w_{1},w_{2})\mathbbm{1}_{\left\\{w_{1}\wedge
w_{2}>w\right\\}}(w_{1}w_{2})^{-\beta}dw_{1}dw_{2}.$ (4.7)

Using the bounds (4.5) and (4.6), respectively, for
$\mathcal{S}_{w}(w_{1},w_{2})$ in the first and second terms of the right hand
side (RHS) of (4.7) gives the result. ∎

######  Remark 4.1 (Bounding $\mathcal{S}_{w}(w_{1},w_{2})$ differently).

Note that depending on whether $w_{1}\wedge w_{2}$ is larger or smaller than
$w$, we bound $\mathcal{S}_{w}(w_{1},w_{2})$ differently. As already discussed
in Section 1.6, when $w_{1}$ and $w_{2}$ have density (1.1), the function
$h(w,w_{1},w_{2})$ is not integrable in $(w_{1},w_{2})$ if $\beta\leq
a+\frac{3}{2}$. Consequently, using the bound (4.5) for $w_{1}\wedge w_{2}>w$,
for $\beta\leq a+\frac{3}{2}$, would only give us a trivial upper bound of
$\infty$ on $T(w)$, which is not useful. Keeping this in mind, we were
required to bound $\mathcal{S}_{w}(w_{1},w_{2})$ differently. When
$\beta>a+\frac{3}{2}$, $h(w,w_{1},w_{2})$ is integrable when $w_{1}$ and
$w_{2}$ have density (1.1), and indeed for this case this distinction does not
make a difference: as we will see, when $\beta>a+\frac{3}{2}$, the term (4.3)
is of negligible order compared to the term (4.2).

With this bound in hands we can now use the definition of our interpolation
function $g$ to study the scaling of $T(w)$.

######  Lemma 4.2.

Let $T(w)$ be defined as in (2.6) with $\kappa$ and $g$ as in (1.2) and (1.3),
respectively. Then, as $w\to\infty$,

$T(w)=\begin{cases}O\left(w\right)&\mbox{if}\;\;\beta>\left(a+\frac{3}{2}\right)\vee
2,a>0,\\\
O\left(w\log{\left(w\right)}\right)&\mbox{if}\;\;\beta=a+\frac{3}{2},a>\frac{1}{2},\\\
O\left((w)^{4+2a-2\beta}\right)&\mbox{if}\;\;\beta\in(2,a+\frac{3}{2}),a>\frac{1}{2}.\end{cases}$
(4.8)

###### Proof.

We show that the first term (4.2) from Lemma 4.1 has the exact same scaling as
in the statement of the lemma. For the second term (4.3) we then use the
definition of $g$ from (1.3) to get, as $w\to\infty$,

$\int_{w}^{\infty}\int_{w}^{\infty}g(w,w_{1})g(w,w_{2})f_{W}(\underline{\mathbf{w}})d\mathbf{w}=\left(w\right)^{2a}\left((\beta-1)\int_{w}^{\infty}w_{1}^{1-\beta}dw_{1}\right)^{2}=O\left(\left(w\right)^{4+2a-2\beta}\right).$

Noting that $w^{4+2a-2\beta}$ is bounded by each term on the RHS of (4.8), in
its respective case, then finishes the proof.

For (4.2), we again use the definition of $g$ from (1.3) to get

$\displaystyle\int_{1}^{\infty}\int_{1}^{\infty}h(w,w_{1},w_{2})\mathbbm{1}_{\left\\{w_{1}\wedge
w_{2}\leq w\right\\}}(w_{1}w_{2})^{-\beta}dw_{1}dw_{2}$
$\displaystyle=\int_{1}^{\infty}\int_{1}^{\infty}\left[g(w,w_{1})\wedge
g(w,w_{2})\right]g(w_{1},w_{2})\mathbbm{1}_{\left\\{w_{1}\wedge w_{2}\leq
w\right\\}}(w_{1}w_{2})^{-\beta}dw_{1}dw_{2}$
$\displaystyle=\int_{1}^{\infty}\int_{1}^{\infty}w(w_{1}\wedge
w_{2})^{a}(w_{1}\vee w_{2})(w_{1}\wedge
w_{2})^{a}\mathbbm{1}_{\left\\{w_{1}\wedge w_{2}\leq
w\right\\}}(w_{1}w_{2})^{-\beta}dw_{1}dw_{2}$
$\displaystyle=2w\int_{1}^{w}\int_{w_{1}}^{\infty}{w_{1}}^{2a-\beta}{w_{2}}^{1-\beta}dw_{2}dw_{1}$
$\displaystyle=\frac{2}{\beta-2}w\int_{1}^{w}w_{1}^{2+2a-2\beta}dw_{1},$

where to obtain the second last inequality we have used the symmetry in
$w_{1}$ and $w_{2}$. Now, the intended result follows by noting that as
$w\to\infty$,

$\displaystyle\int_{1}^{w}w_{1}^{2+2a-2\beta}dw_{1}=\begin{cases}O\left(1\right)&\mbox{if}\;\;\beta>\left(a+\frac{3}{2}\right)\vee
2,a>0,\\\
O\left(\log{\left(w\right)}\right)&\mbox{if}\;\;\beta=a+\frac{3}{2},a>\frac{1}{2},\\\
O\left((w)^{3+2a-2\beta}\right)&\mbox{if}\;\;\beta\in(2,a+\frac{3}{2}),a>\frac{1}{2}.\end{cases}$

∎

###  4.2 The lower bound

Next, we prove a lower bound on $T(w)$. For this we rely on geometric
techniques. Recall that $\omega_{d}$ denotes the Lebesgue measure of a ball of
unit radius in $\mathbb{R}^{d}$.

######  Lemma 4.3.

Consider the connection function (1.2), with some symmetric weight function
$g$. Recall $h(w,w_{1},w_{2})$ from (1.12). Then,

$\displaystyle
T(w)\geq\omega_{d}^{2}\int_{1}^{\infty}\int_{1}^{\infty}\mathbbm{1}_{\left\\{g(w_{1},w_{2})<g(w,w_{2})\right\\}}h(w,w_{1},w_{2})$
$\displaystyle\hskip
100.0pt\times\left[1-\left(\frac{g(w_{1},w_{2})}{(g(w,w_{1})\wedge
g(w,w_{2})}\right)^{1/d}\right]^{d}f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}.$
(4.9)

###### Proof.

Note that for a lower bound on $T(w)$, we need only consider the case
$\alpha=\infty$ since for any $\alpha\in(0,\infty)$,
$\kappa_{\alpha}>\kappa_{\infty}$ point-wise. Thus, we assume $\alpha=\infty$
for the rest of the proof. For $\mathbf{x}\in\mathbb{R}^{d}$ and $r>0$, we
also use the notation

$B(\mathbf{x},r):=\\{\mathbf{y}\in\mathbb{R}^{d}:\|\mathbf{x}-\mathbf{y}\|<r\\}$

to denote the open ball of radius $r$ around $\mathbf{x}$ in $\mathbb{R}^{d}$.

For $\alpha=\infty$,

$\displaystyle
T(w)=\int_{1}^{\infty}\int_{1}^{\infty}\int_{(\mathbb{R}^{d})^{2}}\mathbbm{1}_{\left\\{\|\mathbf{x}\|<g(w,w_{1})^{1/d}\right\\}}\mathbbm{1}_{\left\\{\|\mathbf{y}\|<g(w,w_{2})^{1/d}\right\\}}$
$\displaystyle\hskip
118.0pt\times\mathbbm{1}_{\left\\{\|\mathbf{x}-\mathbf{y}\|<g(w_{1},w_{2})^{1/d}\right\\}}d\mathbf{x}d\mathbf{y}\;f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2},$

where $g$ is the weight function (1.3). A simple change of variable
$\mathbf{z}=\mathbf{x}-\mathbf{y}$ yields

$\displaystyle
T(w)=\int_{1}^{\infty}\int_{1}^{\infty}\int_{B(\mathbf{0},g(w,w_{1})^{1/d})}\int_{B(\mathbf{x},g(w,w_{2})^{1/d})}\mathbbm{1}_{\left\\{\|\mathbf{z}\|<g(w_{1},w_{2})^{1/d}\right\\}}d\mathbf{z}d\mathbf{x}\;f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}.$

Note that if $\|\mathbf{x}\|>g(w,w_{1})^{1/d}+g(w_{1},w_{2})^{1/d}$, then the
set $B(\mathbf{x},g(w,w_{2})^{1/d})$ is disjoint from the open ball of radius
$g(w_{1},w_{2})^{1/d}$ around the origin $\mathbf{0}\in\mathbb{R}^{d}$, see
Figure 3(a).

This means in particular that if $\|\mathbf{z}\|<g(w_{1},w_{2})^{1/d}$, then
$\mathbf{z}\notin B(\mathbf{x},g(w,w_{2})^{1/d})$. Consequently in this case,

$\int_{B(\mathbf{x},g(w,w_{2})^{1/d})}\mathbbm{1}_{\left\\{\|\mathbf{z}\|<g(w_{1},w_{2})^{1/d}\right\\}}d\mathbf{z}=0.$
(4.10)

By this observation,

$\displaystyle
T(w)=\int_{1}^{\infty}\int_{1}^{\infty}\int_{B\left(\mathbf{0},(g(w,w_{1})^{1/d}\wedge[g(w,w_{2})^{1/d}+g(w_{1},w_{2})^{1/d}])\right)}$
$\displaystyle\hskip
110.0pt\times\int_{B(\mathbf{x},g(w,w_{2})^{1/d})}\mathbbm{1}_{\left\\{\|\mathbf{z}\|<g(w_{1},w_{2})^{1/d}\right\\}}d\mathbf{z}d\mathbf{x}\;f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}.$

Next, we obtain a generic lower bound on the left hand side (LHS) of (4.10),
in the complementary range, that is, for
$\|\mathbf{x}\|<g(w,w_{2})^{1/d}+g(w_{1},w_{2})^{1/d}$.

First observe that the integral (4.10) is the volume of intersection of the
two balls $B(\mathbf{x},g(w,w_{2})^{1/d})$ and
$B(\mathbf{0},g(w_{1},w_{2})^{1/d})$. A lower bound on this volume is the
volume of any ball that is completely contained in this intersection. It is
not too hard to figure out what the maximum diameter $\ell$ of such a ball,
completely contained in this intersection, can be (see Figure 3(b)):

$\mathbf{0}$$\mathbf{x}$$g(w,{w}_{2})^{1/d}$$g({w}_{1},{w}_{2})^{1/d}$

(a) When $\|\mathbf{x}\|>g(w,w_{1})^{1/d}+g(w_{1},w_{2})^{1/d}$,
$B(\mathbf{0},g(w,w_{2})^{1/d})$ is disjoint from
$B(\mathbf{x},g(w,w_{2})^{1/d})$, so that the integral in the LHS of (4.10)
equals $0$.

$\mathbf{0}$$\mathbf{x}$$g(w,{w}_{2})^{1/d}$$\ell$$g({w}_{1},{w}_{2})^{1/d}$

(b) When $\|\mathbf{x}\|<g(w,w_{1})^{1/d}+g(w_{1},w_{2})^{1/d}$,
$B(\mathbf{0},g(w,w_{2})^{1/d})\cap
B(\mathbf{x},g(w,w_{2})^{1/d})\neq\emptyset$. The maximum possible diameter
$\ell$ of a ball completely contained in $B(\mathbf{x},g(w,w_{2})^{1/d})\cap
B(\mathbf{0},g(w_{1},w_{2})^{1/d})$ is shown in red.

Figure 3:

It is

$\displaystyle\ell=\begin{cases}0&\text{if
}B(\mathbf{0},g(w_{1},w_{2})^{1/d})\cap
B(\mathbf{x},g(w,w_{2})^{1/d})=\emptyset,\\\ 2g(w_{1},w_{2})^{1/d}&\text{if
}B(\mathbf{0},g(w_{1},w_{2})^{1/d})\subset B(\mathbf{x},g(w,w_{2})^{1/d}),\\\
2g(w,w_{2})^{1/d}&\text{if }B(\mathbf{0},g(w_{1},w_{2})^{1/d})\supset
B(\mathbf{x},g(w,w_{2})^{1/d}),\\\
g(w,w_{2})^{1/d}+g(w_{1},w_{2})^{1/d}-\|\mathbf{x}\|&\text{otherwise.}\end{cases}$

In particular, when $g(w,w_{2})>g(w_{1},w_{2})$, since it can never hold that
$B(\mathbf{0},g(w_{1},w_{2})^{1/d})\supset B(\mathbf{x},g(w,w_{2})^{1/d})$,

$\displaystyle\int_{B(\mathbf{x},g(w,w_{2})^{1/d})}\mathbbm{1}_{\left\\{\|\mathbf{z}\|<g(w_{1},w_{2})^{1/d}\right\\}}d\mathbf{z}$
$\displaystyle\geq\omega_{d}\left[\frac{g(w,w_{2})^{1/d}+g(w_{1},w_{2})^{1/d}-\|\mathbf{x}\|}{2}\wedge
g(w_{1},w_{2})^{1/d}\right]^{d}\mathbbm{1}_{\left\\{g(w,w_{2})>g(w_{1},w_{2})\right\\}}.$

Further bounding from below by multiplying by the indicator
$\mathbbm{1}_{\left\\{\|\mathbf{x}\|<g(w,w_{2})^{1/d}-g(w_{1},w_{2})^{1/d}\right\\}}$,

$\displaystyle\int_{B(\mathbf{x},g(w,w_{2})^{1/d})}\mathbbm{1}_{\left\\{\|\mathbf{z}\|<g(w_{1},w_{2})^{1/d}\right\\}}d\mathbf{z}$
$\displaystyle\geq\omega_{d}\mathbbm{1}_{\left\\{g(w_{1},w_{2})<g(w,w_{2})\right\\}}\mathbbm{1}_{\left\\{\|\mathbf{x}\|<g(w,w_{2})^{1/d}-g(w_{1},w_{2})^{1/d}\right\\}}g(w_{1},w_{2}).$
(4.11)

At this point we introduce some short-hand notations to keep things neat. Let
us denote

$\mathcal{R}_{\pm}=g(w,w_{1})^{1/d}\wedge[g(w,w_{2})^{1/d}\pm
g(w_{1},w_{2})^{1/d}].$ (4.12)

Using the lower bound (4.11) and using the notation (4.12) (recall the
notation $f_{W}(\mathbf{w})d\mathbf{w}$ from (1.9)), we have

$\displaystyle T(w)$
$\displaystyle\geq\int\int\int_{B(\mathbf{0},\mathcal{R}_{+})}\left(\omega_{d}\mathbbm{1}_{\left\\{g(w_{1},w_{2})<g(w,w_{2})\right\\}}\mathbbm{1}_{\left\\{\|\mathbf{x}\|<g(w,w_{2})^{1/d}-g(w_{1},w_{2})^{1/d}\right\\}}g(w_{1},w_{2})\right)d\mathbf{x}f_{W}(\mathbf{w})d\mathbf{w}$
$\displaystyle=\omega_{d}\int\int
g(w_{1},w_{2})\mathbbm{1}_{\left\\{g(w_{1},w_{2})<g(w,w_{2})\right\\}}\left(\int_{B(\mathbf{0},\mathcal{R}_{-})}d\mathbf{x}\right)f_{W}(\mathbf{w})d\mathbf{w}$
$\displaystyle=\omega_{d}^{2}\int\int
g(w_{1},w_{2})\mathbbm{1}_{\left\\{g(w_{1},w_{2})<g(w,w_{2})\right\\}}\left[g(w,w_{1})^{1/d}\wedge[g(w,w_{2})^{1/d}-g(w_{1},w_{2})^{1/d}]\right]^{d}f_{W}(\mathbf{w})d\mathbf{w}.$
(4.13)

Finally, noting that $g(w,w_{1})^{1/d}>g(w,w_{1})^{1/d}-g(w_{1},w_{2})^{1/d}$,
we have from (4.13),

$\displaystyle T(w)$ $\displaystyle\geq\omega_{d}^{2}\int\int
g(w_{1},w_{2})\mathbbm{1}_{\left\\{g(w_{1},w_{2})<g(w,w_{2})\right\\}}\left[g(w,w_{1})^{1/d}\wedge[g(w,w_{2})^{1/d}-g(w_{1},w_{2})^{1/d}]\right]^{d}f_{W}(\mathbf{w})d\mathbf{w}$
$\displaystyle\geq\omega_{d}^{2}\int\int
g(w_{1},w_{2})\mathbbm{1}_{\left\\{g(w_{1},w_{2})<g(w,w_{2})\right\\}}$
$\displaystyle\hskip
40.0pt\times\left[[g(w,w_{1})^{1/d}-g(w_{1},w_{2})^{1/d}]\wedge[g(w,w_{2})^{1/d}-g(w_{1},w_{2})^{1/d}]\right]^{d}f_{W}(\mathbf{w})d\mathbf{w}$
$\displaystyle=\omega_{d}^{2}\int\int
g(w_{1},w_{2})\mathbbm{1}_{\left\\{g(w_{1},w_{2})<g(w,w_{2})\right\\}}\left[[g(w,w_{1})^{1/d}\wedge
g(w,w_{2})^{1/d}]-g(w_{1},w_{2})^{1/d}\right]^{d}f_{W}(\mathbf{w})d\mathbf{w}$
$\displaystyle=\omega_{d}^{2}\int\int
g(w_{1},w_{2})\mathbbm{1}_{\left\\{g(w_{1},w_{2})<g(w,w_{2})\right\\}}$
$\displaystyle\hskip 40.0pt\times\left(g(w,w_{1})\wedge
g(w,w_{2})\right)\left[1-\left(\frac{g(w_{1},w_{2})}{(g(w,w_{1})\wedge
g(w,w_{2})}\right)^{1/d}\right]^{d}f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}$
$\displaystyle=\omega_{d}^{2}\int\int\mathbbm{1}_{\left\\{g(w_{1},w_{2})<g(w,w_{2})\right\\}}h(w,w_{1},w_{2})\left[1-\left(\frac{g(w_{1},w_{2})}{(g(w,w_{1})\wedge
g(w,w_{2})}\right)^{1/d}\right]^{d}f_{W}(\mathbf{w})d\mathbf{w},$

which is (4.9) recalling (1.1). ∎

######  Lemma 4.4.

Let $T(w)$ be defined as in (2.6) with $\kappa$ and $g$ as in (1.2) and (1.3),
respectively. Then, as $w\to\infty$,

$T(w)=\begin{cases}\Omega\left(w\right)&\mbox{if}\;\;\beta>\left(a+\frac{3}{2}\right)\vee
2,a>0,\\\
\Omega\left(w\log{\left(w\right)}\right)&\mbox{if}\;\;\beta=a+\frac{3}{2},a>\frac{1}{2},\\\
\Omega\left(w^{4+2a-2\beta}\right)&\mbox{if}\;\;\beta\in(2,a+\frac{3}{2}),a>\frac{1}{2}.\end{cases}$

###### Proof.

Using the lower bound from Lemma 4.3, it suffices to show that as
$w\to\infty$,

$\displaystyle\omega_{d}^{2}\int_{1}^{\infty}\int_{1}^{\infty}\mathbbm{1}_{\left\\{g(w_{1},w_{2})<g(w,w_{2})\right\\}}h(w,w_{1},w_{2})\left[1-\left(\frac{g(w_{1},w_{2})}{(g(w,w_{1})\wedge
g(w,w_{2})}\right)^{1/d}\right]^{d}f_{W}(\underline{\mathbf{w}})d\underline{\mathbf{w}}$
$\displaystyle=\begin{cases}\Omega\left(w\right)&\mbox{if}\;\;\beta>(a+\frac{3}{2})\vee
2,a\in(0,\infty),\\\
\Omega\left(w\log{\left(w\right)}\right)&\mbox{if}\;\;\beta=a+\frac{3}{2},a\in(1/2,\infty),\\\
\Omega\left(w^{4+2a-2\beta}\right)&\mbox{if}\;\;\beta\in(2,a+\frac{3}{2}),a\in(1/2,\infty).\end{cases}$
(4.14)

Observe that if both $w_{1},w_{2}<{\mathcal{M}^{-1}(k)}/{2}$, then using the
definition of $g$ from (1.3), one has
$\mathbbm{1}_{\left\\{g(w_{1},w_{2})<g(\mathcal{M}^{-1}(k),w_{2})\right\\}}=1$.
Furthermore,

$\displaystyle\left[1-\left(\frac{g(w_{1},w_{2})}{(g(\mathcal{M}^{-1}(k),w_{1})\wedge
g(\mathcal{M}^{-1}(k),w_{2})}\right)^{1/d}\right]^{d}$
$\displaystyle=\left[1-\left(\frac{(w_{1}\vee w_{2})(w_{1}\wedge
w_{2})^{a}}{\mathcal{M}^{-1}(k)(w_{1}\wedge{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}w_{2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}})^{a}}\right)^{1/d}\right]^{d}$
$\displaystyle\geq\left[1-\left(\frac{1}{2}\right)^{1/d}\right]^{d}=:\underline{K}.$

Consequently, we can bound the LHS of (4.14) from below by restricting the
integration domain from $(1,\infty)\times(1,\infty)$ to
$\left(1,{\mathcal{M}^{-1}(k)}/{2}\right)\times\left(1,{\mathcal{M}^{-1}(k)}/{2}\right)$,
to get the following lower bound on it:

$\underline{K}\omega_{d}^{2}(\beta-1)^{2}\int_{1}^{\frac{\mathcal{M}^{-1}(k)}{2}}\int_{1}^{\frac{\mathcal{M}^{-1}(k)}{2}}h(\mathcal{M}^{-1}(k),w_{1},w_{2})(w_{1}w_{2})^{-\beta}dw_{1}dw_{2}.$
(4.15)

Hence to conclude Lemma 4.4, it is enough to check that the double integral in
(4.15) is at least the RHS of (4.14), which we proceed to do. Note that

$\displaystyle\int_{1}^{\frac{\mathcal{M}^{-1}(k)}{2}}\int_{1}^{\frac{\mathcal{M}^{-1}(k)}{2}}h(\mathcal{M}^{-1}(k),w_{1},w_{2})(w_{1}w_{2})^{-\beta}dw_{1}dw_{2}$
$\displaystyle=\int_{1}^{\frac{\mathcal{M}^{-1}(k)}{2}}\int_{1}^{\frac{\mathcal{M}^{-1}(k)}{2}}\left[g(\mathcal{M}^{-1}(k),w_{1})\wedge
g(\mathcal{M}^{-1}(k),w_{2})\right]g(w_{1},w_{2})(w_{1}w_{2})^{-\beta}dw_{1}dw_{2}$
$\displaystyle=\mathcal{M}^{-1}(k)\int_{1}^{\frac{\mathcal{M}^{-1}(k)}{2}}\int_{1}^{\frac{\mathcal{M}^{-1}(k)}{2}}(w_{1}\wedge
w_{2})^{a}(w_{1}\vee w_{2})(w_{1}\wedge
w_{2})^{a}(w_{1}w_{2})^{-\beta}dw_{1}dw_{2}$
$\displaystyle=2\mathcal{M}^{-1}(k)\int_{1}^{\frac{\mathcal{M}^{-1}(k)}{2}}\int_{w_{1}}^{\frac{\mathcal{M}^{-1}(k)}{2}}w_{1}^{2a-\beta}w_{2}^{1-\beta}{dw_{2}dw_{1}}$
$\displaystyle=2{K}\underline{K}\mathcal{M}^{-1}(k)\int_{1}^{\frac{\mathcal{M}^{-1}(k)}{2}}w_{1}^{2a+2-2\beta}dw_{1}$
$\displaystyle\hskip
10.0pt-2{K}\underline{K}\mathcal{M}^{-1}(k)\int_{1}^{\frac{\mathcal{M}^{-1}(k)}{2}}w_{1}^{2a-\beta}(\mathcal{M}^{-1}(k)/2)^{2-\beta}dw_{1},$
(4.16)

where $K={1}/{(\beta-2)}>0$, and in the second last step we have used the
symmetry in $w_{1}$ and $w_{2}$.

To conclude Lemma 4.4, it is enough to check that (4.16) is at least the RHS
of (4.14). To keep notation clear, let us write $R$ for the constant
$K\underline{K}$. We do a case-by-case analysis:

##### Case 1: $\beta\geq(a+\frac{3}{2})\vee 2$.

In this case, observe that the first term of (4.16) equals

$\displaystyle
2{R}\mathcal{M}^{-1}(k)\times\begin{cases}\Omega\left(1\right)&\mbox{if}\;\;\beta>(a+3/2)\vee
2,a\in(0,\infty),\\\
\Omega\left(\log{\left(\mathcal{M}^{-1}(k)\right)}\right)&\mbox{if}\;\;\beta=a+3/2,a\in(1/2,\infty),\end{cases}$

while the subtracted second term of (4.16) equals

$\displaystyle
2{R}\mathcal{M}^{-1}(k)\int_{1}^{\frac{\mathcal{M}^{-1}(k)}{2}}w_{1}^{2a-\beta}(\mathcal{M}^{-1}(k)/2)^{2-\beta}dw_{1}=\Theta((\mathcal{M}^{-1}(k))^{4+2a-2\beta}).$

In particular, since $\beta>a+\frac{3}{2}$, the first term of (4.16) is the
dominating term, and is indeed at least the RHS of (4.14).

##### Case 2: $2<\beta<a+\frac{3}{2}$.

In this case, we note that the first term of (4.16) equals

$\displaystyle\frac{2{R}}{3+2a-2\beta}\mathcal{M}^{-1}(k)\left[\left(\frac{\mathcal{M}^{-1}(k)}{2}\right)^{3+2a-2\beta}-1\right].$
(4.17)

Simplifying the subtracted second term in (4.16) gives

$\displaystyle
2^{\beta-1}R\mathcal{M}^{-1}(k)^{3-\beta}\int_{1}^{\frac{\mathcal{M}^{-1}(k)}{2}}w_{1}^{2a-\beta}dw_{1}\leq
2^{\beta-1}R\mathcal{M}^{-1}(k)^{3-\beta}\frac{1}{1+2a-\beta}\left(\frac{\mathcal{M}^{-1}(k)}{2}\right)^{1+2a-\beta},$
(4.18)

since $\beta<a+3/2<1+2a$ i.e. $1+2a-\beta>0$, as $a>1/2$ (note that we need
$a>1/2$ so that $a+3/2>2$). Using (4.17) and (4.18), we note that the term
(4.16) is at least

$\displaystyle\frac{2{R}}{3+2a-2\beta}\mathcal{M}^{-1}(k)\left[\left(\frac{\mathcal{M}^{-1}(k)}{2}\right)^{3+2a-2\beta}-1\right]-\frac{2^{\beta-1}{R}}{1+2a-\beta}\mathcal{M}^{-1}(k)^{3-\beta}\left(\frac{\mathcal{M}^{-1}(k)}{2}\right)^{1+2a-\beta}$
$\displaystyle=\frac{2^{2\beta-2a-2}{R}}{3+2a-2\beta}\mathcal{M}^{-1}(k)\left[\mathcal{M}^{-1}(k)^{3+2a-2\beta}-2^{3+2a-2\beta}\right]-\frac{2^{2\beta-2a-2}{R}}{1+2a-\beta}\mathcal{M}^{-1}(k)^{4+2a-2\beta}$
$\displaystyle=2^{2\beta-2a-2}{R}\mathcal{M}^{-1}(k)^{4+2a-2\beta}\left[\frac{1}{3+2a-2\beta}-\frac{1}{1+2a-\beta}\right]+\Omega(\mathcal{M}^{-1}(k)).$

Observing that $\left[{(3+2a-2\beta)^{-1}}-{(1+2a-\beta)^{-1}}\right]>0$ since
$\beta>2$, and $4+2a-2\beta>1$ since $\beta<a+\frac{3}{2}$, the last
expression is indeed $\Omega(\mathcal{M}^{-1}(k)^{4+2a-2\beta})$, which
completes the proof for the case $1+2a-\beta>0$.

∎

Note that combining Lemmas 4.2 and 4.4 we get the scaling of $T(w)$, as
$w\to\infty$:

$T(w)=\begin{cases}\Theta\left(w\right)&\mbox{if}\;\;\beta>(a+3/2)\vee
2,a\in(0,\infty)\\\
\Theta\left(w\log{w}\right)&\mbox{if}\;\;\beta=a+3/2,a\in(1/2,\infty)\\\
\Theta\left(w^{4+2a-2\beta}\right)&\mbox{if}\;\;\beta\in(2,a+3/2),a\in(1/2,\infty).\end{cases}$
(4.19)

At this point we make the observation that the assumptions of Proposition 2.3
are indeed satisfied. Indeed, assumption (1.) is immediate from the choice of
$g$ in (1.3). It is clear from the expression (3.1) that assumption (2.) is
satisfied. Assumption (3.) follows from Proposition 3.3. Finally, assumption
(4.) follows from (4.19). We make no more mention of this, and we apply
Proposition 2.3 in the next section where we prove Theorem 1.1, wherever
necessary.

##  5 Proof of Theorem 1.1

In this section we prove Theorem 1.1. Recall the definition of $T(w)$ from
(2.6), and note that using the notation $\mathcal{S}_{w}(w_{1},w_{2})$ from
(4.1), we have

$T(\mathcal{M}^{-1}(k))=\int_{1}^{\infty}\int_{1}^{\infty}\mathcal{S}_{\mathcal{M}^{-1}(k)}(w_{1},w_{2})f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}$
(5.1)

Observe that the sequence $(\mathrm{S}_{k}(a,\beta))_{k\geq 1}$ is
fluctuation-robust, and thus so is $(k^{2}\mathrm{S}_{k}(a,\beta))_{k\geq 1}$.
Hence by Proposition 2.3 it suffices to show that

$\lim_{k\to\infty}\frac{T(\mathcal{M}^{-1}(k))}{k^{2}\mathrm{S}_{k}(a,\beta)}=\Gamma(a,\alpha,\beta,d).$

Recall the definition of $\sigma$ from (3.3). Following the strategy we laid
out, we write

$\frac{T(\mathcal{M}^{-1}(k))}{k^{2}\mathrm{S}_{k}(a,\beta)}=\frac{\sigma(\mathcal{M}^{-1}(k))}{k^{2}\mathrm{S}_{k}(a,\beta)}\frac{T(\mathcal{M}^{-1}(k))}{\sigma(\mathcal{M}^{-1}(k))}.$

The first factor converges by Corollary 3.4. Comparing the constants in
Corollary 3.4 to $\Gamma(a,\alpha,\beta,d)$ in Theorem 1.1, it suffices to
show that

$\lim_{w\to\infty}\frac{T(w)}{\sigma(w)}=\begin{cases}\int_{1}^{\infty}\int_{1}^{\infty}I(w_{1},w_{2})f_{W}(w_{1})dw_{1}f_{W}(w_{2})w_{1}&\text{if
}\beta>\left(a+\frac{3}{2}\right)\vee 2,a>0,\\\
\left(a+\frac{1}{2}\right)\int_{1}^{\infty}I(1,r)f_{W}(r)dr&\text{if
}\beta=a+\frac{3}{2},a>\frac{1}{2},\\\
\int_{0}^{\infty}\int_{0}^{\infty}\mathcal{S}_{1}(w_{1},w_{2})f_{W}(w_{1})w_{1}f_{W}(w_{2})w_{2}&\text{if
}\beta\in(2,a+\frac{3}{2}),a>\frac{1}{2}.\end{cases}$ (5.2)

This is achieved in Lemmas 5.2, 5.3 and 5.4.

The general strategy is to employ a specific change of variables
$(x,y)=(\phi_{1}(x^{\prime}),\phi_{2}(y^{\prime}))$ and
$(w_{1},w_{2})=(\psi_{1}(w_{1}^{\prime}),\psi_{2}(w_{2}^{\prime}))$ to the
integral expression of $T(w)$ so that

$T(w)=k^{2}\mathrm{S}_{k}(a,\beta)\int_{1}^{\infty}\int_{1}^{\infty}\hat{\mathcal{S}}_{w}(\psi_{1}(w_{1}^{\prime}),\psi_{2}(w_{2}^{\prime}))d\psi_{1}(w_{1}^{\prime})d\psi_{2}(w_{2}^{\prime}),$

where
$\hat{\mathcal{S}}_{w}(\psi_{1}(w_{1}^{\prime}),\psi_{2}(w_{2}^{\prime}))$ is
the integral $\mathcal{S}_{w}(w_{1},w_{2})$ after the change of variables and
without the leading order scaling. We then analyze the new integral and use
dominated convergence to prove that it converges to a constant as
$k\to\infty$. This is also where we make use of the upper bounds established
for $T(w)$ in Section 4.

The right choice of the change of variables depends on the regime of $a$ and
$\beta$. In addition, for the third regime: $2<\beta<(a+\frac{3}{2})$, the
entire integration domain $w_{1},w_{2}\in(1,\infty)$ in $T(w)$ contributes to
the constant. Instead in the other two regimes the main contribution comes
from the integration over the domain $w_{1},w_{2}\leq w$, which is summarized
in the following result:

######  Lemma 5.1.

Let $a>0$ and $\beta>(a+\frac{3}{2})\vee 2$ or $\beta=a+\frac{3}{2}>2$. Then,
as $w\to\infty$,

$T(w)=(1+o(1)\int_{1}^{w}\int_{1}^{w}\mathcal{S}_{w}(w_{1},w_{2})f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{1}.$

###### Proof.

Lemmas 4.2 and 4.4 together imply that in these regimes, $T(w)$ scales either
as $w$ or $w\log(w)$. Hence, it suffices to show that

$\left|T(w)-\int_{1}^{w}\int_{1}^{w}\mathcal{S}_{w}(w_{1},w_{2})f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}\right|=o\left(w\log(w)\right).$

Note that the LHS is nothing more than the integral over the domain $w_{1}\vee
w_{2}>w$. We must therefore show that

$\int_{1}^{\infty}\int_{1}^{\infty}\mathcal{S}_{w}(w_{1},w_{2})\mathbbm{1}_{\left\\{w_{1}\vee
w_{2}>w\right\\}}f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}=o\left(w\log(w)\right).$

We first note that by symmetry the full integral is equal to twice the
integral where $w_{1}\leq w_{2}$. We then split the inner integral over the
two cases $1<w_{1}<w$ and $w<w_{1}<w_{2}$ to obtain:

$\displaystyle\int_{1}^{\infty}\int_{1}^{\infty}\mathcal{S}_{w}(w_{1},w_{2})\mathbbm{1}_{\left\\{w_{1}\vee
w_{2}>w\right\\}}f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}$ $\displaystyle\leq
2\int_{w}^{\infty}\int_{1}^{w}\mathcal{S}_{w}(w_{1},w_{2})f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}$
(5.3) $\displaystyle\hskip
10.0pt+2\int_{w}^{\infty}\int_{w}^{w_{2}}\mathcal{S}_{w}(w_{1},w_{2})f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}.$
(5.4)

We first show that the second integral in (5.4) is
$O\left(w^{4+2a-2\beta}\right)$ which is $o(w\log(w))$ for the two regimes of
$a$ and $\beta$. Using the bound (4.6) we get

$\displaystyle
2\int_{w}^{\infty}\int_{w}^{w_{2}}\mathcal{S}_{w}(w_{1},w_{2})f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}$
$\displaystyle\leq
2\left(\frac{\alpha\omega_{d}}{\alpha-1}\right)^{2}\int_{w}^{\infty}\int_{w}^{w_{2}}g(w,w_{1})g(w,w_{2})f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}$
$\displaystyle=O\left(1\right)\left(\int_{w}^{\infty}g(w,w_{1})w_{1}^{-\beta}dw_{1}\right)^{2}=O\left(w^{4+2a-2\beta}\right).$

We proceed with the first integral in (5.3). Note that $w_{1}\leq w_{2}$ which
implies that $g(w,w_{1})\leq g(w,w_{2})$. In addition, recall the function
$h(w,w_{1},w_{2})$ from (1.12) and observe that because $w_{1}\leq w\leq
w_{2}$,

$h(w,w_{1},w_{2})=[g(w,w_{1})\wedge
g(w,w_{2})]g(w_{1},w_{2})=g(w,w_{1})g(w_{1},w_{2})=ww_{1}^{2a}w_{2}.$

Next, by (4.5),

$\displaystyle
2\int_{w}^{\infty}\int_{1}^{w}\mathcal{S}_{w}(w_{1},w_{2})f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}$
$\displaystyle\leq
2\left(\frac{\alpha\omega_{d}}{\alpha-1}\right)^{2}w\int_{w}^{\infty}\int_{1}^{w}w_{1}^{2a}w_{2}f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}$
$\displaystyle=O\left(1\right)w\int_{w}^{\infty}w_{2}^{1-\beta}dw_{2}\int_{1}^{w}w_{1}^{2a-\beta}dw_{1}$
$\displaystyle=\begin{cases}O\left(w^{3-\beta}\right)&\text{if }\beta>2a+1,\\\
O\left(w^{2-2a}\log w\right)&\text{if }\beta=2a+1,\\\
O\left(w^{4+2a-2\beta}\right)&\text{if }\beta<2a+1.\end{cases}$

Since each term on the RHS is $o(w\log(w))$ when $\beta>(a+\frac{3}{2})\vee 2$
and $a>0$ or $\beta=a+\frac{3}{2}>2$ with $a>\frac{1}{2}$ the result follows.
∎

###  5.1 The case $\beta>(a+\frac{3}{2})\vee 2$ and $a\in[0,\infty)$

By Lemma 5.1 it suffices to show that

$\lim_{w\to\infty}\frac{\overline{T}(w)}{w}=\int_{1}^{\infty}\int_{1}^{\infty}I(w_{1},w_{2})f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2},$
(5.5)

where

$\displaystyle\overline{T}(w)$
$\displaystyle:=\int_{1}^{w}\int_{1}^{w}\mathcal{S}_{w}(w_{1},w_{2})f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}=2\int_{1}^{w}\int_{1}^{w_{2}}\mathcal{S}_{w}(w_{1},w_{2})f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}$

using the symmetry in $w_{1}$ and $w_{2}$.

The following observation will prove useful throughout the whole section: for
any $t,s>0$ it holds that

$\kappa(\|t^{1/d}s^{a/d}\mathbf{x}\|,tw_{1},sw_{2})=\kappa(\|\mathbf{x}\|,w_{1},w_{2}).$
(5.6)

######  Lemma 5.2.

Let $\beta>(a+\frac{3}{2})\vee 2$, with $a\geq 0$. Then, as $w\to\infty$,

$\frac{\overline{T}(w)}{w}\to\int_{1}^{\infty}\int_{1}^{\infty}I(w_{1},w_{2})f_{W}({w}_{1})d{w}_{1}f_{W}({w}_{2})d{w}_{2}.$

###### Proof.

Recall that

$\mathcal{S}_{w}(w_{1},w_{2})=\iint_{{(\mathbb{R}^{d})^{2}}}\kappa(\|\mathbf{x}\|,w,w_{1})\kappa(\|\mathbf{y}\|,w,w_{2})\kappa(\|\mathbf{x}-\mathbf{y}\|,w_{1},w_{2})d\mathbf{x}d\mathbf{y}.$

We now apply the following change of variables:

$\mathbf{x}=w^{1/d}\mathbf{x}^{\prime}\quad\text{and}\quad\mathbf{y}^{\prime}=\mathbf{y}-\mathbf{x},$
(5.7)

to obtain

$\displaystyle\mathcal{S}_{w}(w_{1},w_{2})$
$\displaystyle=w\iint_{{(\mathbb{R}^{d})^{2}}}\kappa(\|w^{1/d}\mathbf{x}^{\prime}\|,w,w_{1})\kappa(\|\mathbf{y}^{\prime}+w^{1/d}\mathbf{x}^{\prime}\|,w,w_{2})\kappa(\|\mathbf{y}^{\prime}\|,w_{1},w_{2}){d\mathbf{x}^{\prime}d\mathbf{y}^{\prime}}$
$\displaystyle=w\iint_{{(\mathbb{R}^{d})^{2}}}\kappa(\|\mathbf{x}^{\prime}\|,1,w_{1})\kappa(\|\mathbf{x}^{\prime}+w^{-1/d}\mathbf{y}^{\prime}\|,1,w_{2})\kappa(\|\mathbf{y}^{\prime}\|,w_{1},w_{2})d\mathbf{x}^{\prime}d\mathbf{y}^{\prime},$

where we also use the scaling relation (5.6) for the first two $\kappa$ terms
in the integrand.

We now apply dominated convergence twice. For this we first note that the
integrand above is dominated by
$\kappa(\|\mathbf{x}^{\prime}\|,1,w_{1})\kappa(\|\mathbf{y}^{\prime}\|,w_{1},w_{2})$.
Hence, using straightforward calculations based on the definition of $\kappa$,
we get

$\displaystyle\frac{\mathcal{S}_{w}(w_{1},w_{2})}{w}$
$\displaystyle\leq\iint_{{(\mathbb{R}^{d})^{2}}}\kappa(\|\mathbf{x}\|,1,w_{1})\kappa(\|\mathbf{y}\|,w_{1},w_{2})d\mathbf{x}d\mathbf{y}=\xi_{\alpha}^{2}{w_{1}}(w_{1}\vee
w_{2})(w_{1}\wedge w_{2})^{a},$

which is finite for any finite $w_{1},w_{2}$. Therefore since
$\kappa(\|\mathbf{x}+w^{-1/d}\mathbf{y}\|,1,w_{2})\to\kappa(\|\mathbf{x}\|,1,w_{2})$
pointwise, it follows that

$\frac{\mathcal{S}_{w}(w_{1},w_{2})}{w}\to\iint_{{(\mathbb{R}^{d})^{2}}}\kappa(\|\mathbf{x}\|,1,w_{1})\kappa(\|\mathbf{x}\|,1,w_{2})\kappa(\|\mathbf{y}\|,w_{1},w_{2})d\mathbf{x}d\mathbf{y}=I(w_{1},w_{2}),$

by dominated convergence.

For the last step we first observe that

$\displaystyle\frac{\overline{T}(w)}{w}$
$\displaystyle=\int_{1}^{\infty}\int_{1}^{\infty}\frac{\mathcal{S}_{w}(w_{1},w_{2})}{w}\mathbbm{1}_{\left\\{w_{1},w_{2}\leq
w\right\\}}f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}$
$\displaystyle\leq\xi_{\alpha}^{2}\int_{1}^{\infty}\int_{1}^{\infty}w_{1}^{a}(w_{1}\vee
w_{2})(w_{1}\wedge w_{2})^{a}f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}$
$\displaystyle=2\xi_{\alpha}^{2}(\beta-1)^{2}\int_{1}^{\infty}\int_{w_{1}}^{\infty}w_{1}^{2a}w_{2}(w_{1}w_{2})^{-\beta}{dw_{2}dw_{1}}.$

It is straightforward to see that the above integral is finite. Hence, since
$\frac{\mathcal{S}_{w}(w_{1},w_{2})}{w}\mathbbm{1}_{\left\\{w_{1},w_{2}\leq
w\right\\}}$ converges pointwise to $I(w_{1},w_{2})$, which is integrable, we
apply dominated convergence to conclude

$\frac{\overline{T}(w)}{w}\to\int_{1}^{\infty}\int_{1}^{\infty}I(w_{1},w_{2})f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2},$

as required. This proves Theorem 1.1 for $\beta>(a+\frac{3}{2})\vee 2$ and
$a\in(0,\infty)$. ∎

###  5.2 The case $\beta=a+\frac{3}{2}$ and $a\in(\frac{1}{2},\infty)$

The proof for this case follows a similar approach to that for
$\beta>(a+\frac{3}{2})\vee 2$. Again, by Lemma 5.1 we only need to consider
$\overline{T}(w)$ instead of $T(w)$:

######  Lemma 5.3.

Let $\beta=a+\frac{3}{2}$, with $a>\frac{1}{2}$. Then, as $w\to\infty$,

$\frac{\overline{T}(w)}{w\log(w)}\to{(\beta-1)^{2}\int_{0}^{\infty}I(1,r)r^{-\beta}dr.}$

###### Proof.

Note that a similar change of variables as applied in Lemma 5.2 extracts a
scaling factor $w$ outside the integral. However, for this case we need to
extract an additional factor $\log(w)$. This is achieved by also scaling the
variables $w_{1}$ and $w_{2}$. The proof thus proceeds as follows. We first
apply a change of variable on $w_{1}$ and $w_{2}$ and then on $\mathbf{x}$ and
$\mathbf{y}$. After this we get that $\overline{T}(w)$ is equal to $w\log(w)$
times some integral. We then use dominated convergence, in a similar way as in
the proof of Lemma 5.2, to show that this integral converges.

Apply the change of variables $(w_{1},w_{2})\mapsto(u,r)$ given by

$w_{1}=w^{u}\quad\text{and}\quad w_{2}=w^{u}r.$

Then $dw_{1}=\log(w)w^{u}du$ and $dw_{2}=w^{u}dr$ so that

$\displaystyle\int_{1}^{w}\int_{1}^{w}\mathcal{S}_{w}(w_{1},w_{2})f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}$
$\displaystyle=\int_{0}^{1}\int_{w^{-u}}^{w^{1-u}}\mathcal{S}_{w}(w^{u},w^{u}r)w^{2u}\log(w)f_{W}(w^{u})f_{W}(w^{u}r)drdu$
$\displaystyle=\int_{0}^{1}\int_{1}^{w^{1-u}}\mathcal{S}_{w}(w^{u},w^{u}r)w^{2u-2u\beta}\log(w)f_{W}(1)f_{W}(r)drdu$
$\displaystyle\hskip
10.0pt+(\beta-1)\int_{0}^{1}\int_{w^{-u}}^{1}\mathcal{S}_{w}(w^{u},w^{u}r)w^{2u-2u\beta}\log(w)f_{W}(1)r^{-\beta}drdu.$
(5.8)

Next consider

$\mathcal{S}_{w}(w^{u},w^{u}r)=\iint_{{(\mathbb{R}^{d})^{2}}}\kappa(\|\mathbf{x}\|,w,w^{u})\kappa(\|\mathbf{y}\|,w,w^{u}r)\kappa(\|\mathbf{x}-\mathbf{y}\|,w^{u},w^{u}r)d\mathbf{x}d\mathbf{y}.$

Here, we apply the change of variables

$\mathbf{x}=w^{\frac{au+1}{d}}\mathbf{x}^{\prime}\quad\text{and}\quad\mathbf{y}=w^{\frac{(a+1)u}{d}}\mathbf{y}^{\prime}+w^{\frac{au+1}{d}}\mathbf{x}^{\prime},$

to get

$\displaystyle\mathcal{S}_{w}(w^{u},w^{u}r)$
$\displaystyle=w^{2au+u+1}\iint_{{(\mathbb{R}^{d})^{2}}}\kappa\left(\|w^{\frac{au+1}{d}}\mathbf{x}^{\prime}\|,w,w^{u}\right)\kappa\left(\|w^{\frac{au+1}{d}}(\mathbf{x}^{\prime}+w^{\frac{1-u}{d}}\mathbf{y}^{\prime})\|,w^{u},w^{u}r\right)$
$\displaystyle\hskip
200.0pt\times\kappa\left(\|w^{\frac{au+u}{d}}\mathbf{y}^{\prime}\|,w,w^{u}r\right)d\mathbf{x}^{\prime}d\mathbf{y}^{\prime}$
$\displaystyle=w^{2au+u+1}\iint_{{(\mathbb{R}^{d})^{2}}}\kappa\left(\|\mathbf{x}^{\prime}\|,1,1\right)\kappa\left(\|\mathbf{x}^{\prime}+w^{-\frac{1-u}{d}}\mathbf{y}^{\prime}\|,1,r\right)\kappa\left(\|\mathbf{y}^{\prime}\|,1,r\right)d\mathbf{x}^{\prime}d\mathbf{y}^{\prime},$

where we have used (5.6) twice to go from the second to the third line.
Observe that the integrand is bounded by
$\kappa\left(\|\mathbf{x}^{\prime}\|,1,1\right)\kappa\left(\|\mathbf{y}^{\prime}\|,1,r\right)$,
which is integrable. Hence, by dominated convergence,

$\frac{\mathcal{S}_{w}(w^{u},w^{u}r)}{w^{2au+u+1}}\to\iint_{\mathbb{R}^{2}}\kappa\left(\|\mathbf{x}^{\prime}\|,1,1\right)\kappa\left(\|\mathbf{x}^{\prime}\|,1,r\right)\kappa\left(\|\mathbf{y}^{\prime}\|,1,r\right)d\mathbf{x}^{\prime}d\mathbf{y}^{\prime}=I(1,r).$

Recall the the split of $\overline{T}(w)$ in (5.8). Consider the first term on
the RHS of (5.8). We have

$\displaystyle\int_{0}^{1}\int_{1}^{w^{1-u}}\mathcal{S}_{w}(w^{u},w^{u}r)w^{2u-2u\beta}\log(w)f_{W}(1)f_{W}(r)drdu$
$\displaystyle=w\log(w)f_{W}(1)\int_{0}^{1}\int_{1}^{w^{1-u}}\frac{\mathcal{S}_{w}(w^{u},w^{u}r)}{w^{2au+u+1}}w^{2au+3u-2u\beta}f_{W}(r)drdu$
$\displaystyle=w\log(w)f_{W}(1)\int_{0}^{1}\int_{1}^{w^{1-u}}\frac{\mathcal{S}_{w}(w^{u},w^{u}r)}{w^{2au+u+1}}f_{W}(r)drdu,$

where we used that $2a+3-2\beta=0$, since $\beta=a+\frac{3}{2}$.

We now apply dominated convergence. For this we recall that
$\frac{\mathcal{S}_{w}(w^{u},w^{u}r)}{w^{2au+u+1}}\to I(1,r)$ and that

$I(1,r)\leq\left(\int_{\mathbb{R}^{d}}\kappa(\|\mathbf{x}\|,1,r)d\mathbf{x}\right)^{2}$

where the RHS is integrable over $(1,\infty)$ with respect to $f_{W}(r)$.
Therefore, by dominated convergence, the first term on the RHS of (5.8) when
divided by $w\log{(w)}$, converges as $w\to\infty$ to

$f_{W}(1)\int_{0}^{1}\int_{1}^{\infty}I(1,r)f_{W}(r)drdu=(\beta-1)\int_{1}^{\infty}I(1,r)f_{W}(r)dr.$

Now consider the second term on the RHS of (5.8). Arguing similarly, we note
that when divided by $w\log{(w)}$, this term becomes

$\displaystyle
f_{W}(1)(\beta-1)\int_{0}^{1}\int_{w^{-u}}^{1}\frac{\mathcal{S}_{w}(w^{u},w^{u}r)}{w^{2au+u+1}}r^{-\beta}drdu.$

We want to apply dominated convergence again. Here we use a different
representation of the integral $\mathcal{S}_{w}(w^{u},w^{u}r)$. We apply the
change of variables

$\mathbf{y}=w^{\frac{au+1}{d}}\mathbf{y}^{\prime}\quad\text{and}\quad\mathbf{x}=w^{\frac{(a+1)u}{d}}\mathbf{x}^{\prime}+w^{\frac{au+1}{d}}\mathbf{y}^{\prime},$

to obtain in a similar fashion as before,

$\displaystyle\frac{\mathcal{S}_{w}(w^{u},w^{u}r)}{w^{2au+u+1}}$
$\displaystyle=\iint_{(\mathbb{R}^{d})^{2}}\kappa(\|w^{\frac{(a+1)u}{d}}\mathbf{x}^{\prime}+w^{\frac{au+1}{d}}\mathbf{y}^{\prime}\|,w,w^{u})\kappa(\|\mathbf{y}^{\prime}\|,1,r)\kappa(\|\mathbf{x}^{\prime}\|,1,r)d\mathbf{x}^{\prime}d\mathbf{y}^{\prime}.$
(5.9)

In particular, we note that
$\frac{\mathcal{S}_{w}(w^{u},w^{u}r)}{w^{2au+u+1}}\mathbbm{1}_{\left\\{w^{-u}<r<1\right\\}}\to
I(1,r)\mathbbm{1}_{\left\\{0<r<1\right\\}}$, as $w\to\infty$. Further using
the representation (5.9), we note that
$\frac{\mathcal{S}_{w}(w^{u},w^{u}r)}{w^{2au+u+1}}\mathbbm{1}_{\left\\{w^{-u}<r<1\right\\}}$
is bounded from above by
$\left(\int_{\mathbb{R}^{d}}\kappa(\|\mathbf{x}\|,1,r)d\mathbf{x}\right)^{2}\mathbbm{1}_{\left\\{0<r<1\right\\}}=\overline{C}r^{2a}\mathbbm{1}_{\left\\{0<r<1\right\\}}$,
when $0<r<1$, using the definition of $\kappa$, for some constant
$\overline{C}>0$. Using that $\beta=a+3/2$ and $a>1/2$, it is easily checked
that the integral $(\beta-1)\int_{0}^{1}\int_{0}^{1}r^{2a}r^{-\beta}drdu$ is
finite. Hence, by dominated convergence, the second term on the RHS of (5.8)
when divided by $w\log{(w)}$, converges to

$\displaystyle
f_{W}(1)(\beta-1)\int_{0}^{1}\int_{0}^{1}I(1,r)r^{-\beta}drdu=(\beta-1)^{2}\int_{0}^{1}I(1,r)r^{-\beta}dr.$

Overall, in this case, the quantity $\frac{\overline{T}(w)}{w\log{(w)}}$
converges to $(\beta-1)^{2}\int_{0}^{\infty}I(1,r)r^{-\beta}dr$, as
$w\to\infty$. This completes the proof of Theorem 1.1 for
$\beta=a+\frac{3}{2}$ and $a\in(\frac{1}{2},\infty)$. ∎

###  5.3 The case $\beta\in(2,a+\frac{3}{2})$ and $a\in(\frac{1}{2},\infty)$

######  Lemma 5.4.

Let $\beta\in(2,a+\frac{3}{2})$, with $a\in(\frac{1}{2},\infty)$. Then as
$w\to\infty$,

$\displaystyle\frac{T(w)}{w^{4+2a-2\beta}}\to\int_{0}^{\infty}\int_{0}^{\infty}\mathcal{S}_{1}(w_{1},w_{2})f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}.$

###### Proof.

Recall the expression for $T(w)$ from (2.6). Furthermore, note that for any
$t>0$, $\mathbf{x}\in\mathbb{R}^{d}$ and $w_{1},w_{2}>0$,

$\kappa(\|t^{(1+a)/d}\mathbf{x}\|,tw_{1},tw_{2})=\kappa(\|\mathbf{x}\|,w_{1},w_{2}).$

We now apply the change of variables

$\mathbf{x}^{\prime}=\frac{\mathbf{x}}{w^{(1+a)/d}},\quad\mathbf{y}^{\prime}=\frac{\mathbf{y}}{w^{(1+a)/d}},\quad
w_{1}^{\prime}=\frac{w_{1}}{w},\quad\text{and}\quad
w_{2}^{\prime}=\frac{w_{2}}{w}.$ (5.10)

Straightforward calculations imply that

$\displaystyle
T(w)=w^{4+2a-2\beta}\int_{1/w}^{\infty}\int_{1/w}^{\infty}\mathcal{S}_{1}(w_{1}^{\prime},w_{2}^{\prime})f_{W}(w_{1}^{\prime})w_{1}^{\prime}f_{W}(w_{2}^{\prime})dw_{2}^{\prime}.$

To prove that the integral converges we first note that, by (4.19),

$\int_{1/w}^{\infty}\int_{1/w}^{\infty}\mathcal{S}_{1}(w_{1}^{\prime},w_{2}^{\prime})f_{W}(w_{1}^{\prime})dw_{1}^{\prime}f_{W}(w_{2}^{\prime})w_{2}^{\prime}=\frac{T(w)}{w^{4+2a-2\beta}}=\Theta\left(1\right).$

Hence, by monotone convergence,

$\int_{1/w}^{\infty}\int_{1/w}^{\infty}\mathcal{S}_{1}(w_{1}^{\prime},w_{2}^{\prime})f_{W}(w_{1}^{\prime})dw_{1}^{\prime}f_{W}(w_{2}^{\prime})dw_{2}^{\prime}\to\int_{0}^{\infty}\int_{0}^{\infty}\mathcal{S}_{1}(w_{1}^{\prime},w_{2}^{\prime})f_{W}(w_{1}^{\prime})dw_{1}^{\prime}f_{W}(w_{2}^{\prime})dw_{2}^{\prime}<\infty,$

and thus

$\lim_{w\to\infty}\frac{T(w)}{w^{4+2a-2\beta}}=\int_{0}^{\infty}\int_{0}^{\infty}\mathcal{S}_{1}(w_{1},w_{2})f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}$
(5.11)

This completes the proof of Theorem 1.1 for $\beta\in(2,a+\frac{3}{2})$ and
$a\in(\frac{1}{2},\infty)$. ∎

##  6 Discussion

This section consists of discussions of our proof, possible extensions, and
simulations.

###  6.1 Geometry of typical triangles

Note that because of the Poisson point process representation of the neighbors
of $0$ in $\mathbb{G}^{\infty}$ as obtained in Lemma 2.1, the fraction
$T(w)/\mathcal{M}(w)^{2}$ can be seen as the probability that two randomly
sampled neighbors of $0$ have an edge between them, given that $W_{0}=w$.
Consequently, since the density $f_{k}(w)$ behaves as a Dirac mass at
$\mathcal{M}^{-1}(k)$ thanks to Proposition 2.3, the fraction
$T(\mathcal{M}^{-1}(k))/k^{2}$ can be interpreted as the probability that two
randomly sampled neighbors of $0$ are also neighbors, given
$d_{0}(\mathbb{G}^{\infty})=k$. The change of variables that we encounter in
Section 5 gives us interesting insights about the locations and weights
$(\mathbf{x},w_{1}),(\mathbf{y},w_{2})\in\mathbb{R}^{d}\times\mathbb{R}_{+}$
of two such randomly sampled neighbors. This further gives us insight into,
what a randomly sampled triangle between $0$ and two such neighbors look like.
Figure 4 shows an abstract depiction of such a typical triangle. The quantity
$\|\mathbf{x}-\mathbf{y}\|$ is the spatial distance from $\mathbf{x}$ to
$\mathbf{y}$ in $\mathbb{R}^{d}$. We will next explain how this distance and
the weights of each node in a typical triangle behaves in the three different
regimes of $\beta$ and $a$.

To begin, recall Corollary 3.4 and (5.2). Our main idea is to analyse the
change of variables encountered in Lemmas 5.2, 5.3 and 5.4, to get the
convergence of $T(w)/\sigma(w)$, for $w=\mathcal{M}^{-1}(k)$.

$\|\mathbf{x}-\mathbf{y}\|$$(\|\mathbf{y}\|,{w}_{2})$$(X_{0}=\mathbf{0},W_{0})$$(\|\mathbf{x}\|,{w}_{1})$

Figure 4: The triangle between $0$ and its two randomly sampled neighbors
$(\mathbf{x},w_{1})$ and $(\mathbf{y},w_{2})$.

####  6.1.1 Case $\beta>(a+\frac{3}{2})\vee 2$ and $a\in[0,\infty)$

Let us carefully dissect the variable change in (5.7). Clearly, when
$w=\mathcal{M}^{-1}(k)$, by (5.7), $\|\mathbf{x}\|$ scales like
$\mathcal{M}^{-1}(k)^{1/d}$, which scales like $k^{1/d}$ (recall Proposition
3.3). On the other hand, we did not need to scale
$\mathbf{y}-\mathbf{x}=\mathbf{y}^{\prime}$, i.e. $\|\mathbf{x}-\mathbf{y}\|$
remains bounded, which implies that $\mathbf{y}$ stays in a ball of finite
radius about $\mathbf{x}$. Thus, $\|\mathbf{y}\|$ is also of the order
$k^{1/d}$, and is of constant order spatial distance away from $\mathbf{x}$.
Finally, note that we did not have to rescale the respective weights $w_{1}$
and $w_{2}$ corresponding to $x$ and $y$, which means that they are of
constant order.

####  6.1.2 Case $\beta=a+\frac{3}{2}$ and $a\in(\frac{1}{2},\infty)$.

Arguing as in the previous case, and observing that $\mathcal{M}^{-1}(k)$
scales like $k$ using Proposition 3.3, we note from the proof of Lemma 5.3
that when $w=\mathcal{M}^{-1}(k)$, the relevant change of variables yield
$\mathbf{x}=C_{1}\mathbf{x}^{\prime}k^{1/d}=C_{2}\overline{\mathbf{x}}k^{(1+au^{\prime})/d}$,
for some constants $C_{1},C_{2}>0$ and $u^{\prime}\in(0,1)$. Consequently,
$\|\mathbf{x}\|=\Theta(k^{(1+au^{\prime})/d})$. Next note that
$\mathbf{y}-\mathbf{x}=\mathbf{y}^{\prime}=C_{3}\overline{\mathbf{y}}k^{(au^{\prime}+u^{\prime})/d}$,
where $C_{3}>0$ is some constant. This implies that
$\|\mathbf{x}-\mathbf{y}\|=\Theta(k^{(au^{\prime}+u^{\prime})/d})$. Note that
the distance $\|\mathbf{x}-\mathbf{y}\|$ between $\mathbf{x}$ and $\mathbf{y}$
is of smaller order than $\|\mathbf{x}\|$, since $u^{\prime}\in(0,1)$. This
means that $\|\mathbf{y}\|$ is also of the order $\|\mathbf{x}\|$. Finally,
note that for this case we had to rescale the weights as well. In particular,
we changed variables as $w_{1}=e^{u}=C_{4}k^{u^{\prime}}$, for some constant
$C_{4}>0$, which means that the weight $w_{1}$ corresponding to $\mathbf{x}$
is $\Theta(k^{u^{\prime}})$, with $u^{\prime}\in(0,1)$. We also had to do a
variable change as $\mathbf{w}={w_{2}}/{C_{4}k^{u^{\prime}}}={w_{2}}/{w_{1}}$,
i.e. the weight $w_{2}$ corresponding to $\mathbf{y}$ is of the same order as
$w_{1}$, the weight of $\mathbf{x}$.

####  6.1.3 Case $\beta\in(2,a+\frac{3}{2})$ and $a\in(\frac{1}{2},\infty)$.

Note that for this case, for $w=\mathcal{M}^{-1}(k)$, the variable change in
(5.10) suggests that both $\mathbf{x}$ and $\mathbf{y}$ are
$\Theta((\mathcal{M}^{-1}(k))^{(1+a)/d})$, while the respective weights
$w_{1}$ and $w_{2}$ are both $\Theta(\mathcal{M}^{-1}(k))$. Finally, note that
for this case we did not have to rescale $\mathbf{x}-\mathbf{y}$, which means
that the distance $\|\mathbf{x}-\mathbf{y}\|$ between $\mathbf{x}$ and
$\mathbf{y}$ is also of the order $(\mathcal{M}^{-1}(k))^{(1+a)/d}$. Depending
on how $\mathcal{M}^{-1}(k)$ scales in $k$ (recall Proposition 3.3), the
orders of the different quantities
$\|\mathbf{x}\|,\|\mathbf{y}\|,w_{1},w_{2},\|\mathbf{x}-\mathbf{y}\|$ differ
in different regimes of $\beta$ and $a$. The relationship and the behaviour is
summarized in Table 1.

Relation between $\beta$ and $a$ | $W_{0}$ |  $(\|\mathbf{x}\|,w_{1})$ and $(\|\mathbf{y}\|,w_{2})$ | $\|\mathbf{x}-\mathbf{y}\|$
---|---|---|---
$\beta>(a+\frac{3}{2})\vee 2,a\in[0,\infty)$ | $\Theta(k)$ | $(\Theta(k^{1/d}),\Theta(1))$ | $\Theta(1)$
$\beta=a+\frac{3}{2},a\in(\frac{1}{2},\infty)$ | $\Theta(k)$ | $\left(\Theta\left(k^{\frac{1+au}{d}}\right),\Theta\left(k^{u}\right)\right)$ | $\Theta\left(k^{\frac{au+u}{d}}\right)$
|  | for $u\in(0,1)$ | for $u\in(0,1)$
$\beta\in(a+1,a+\frac{3}{2}),a\in(\frac{1}{2},\infty)$ | $\Theta(k)$ | $\left(\Theta\left(k^{\frac{1+a}{d}}\right),\Theta\left(k\right)\right)$ | $\Theta\left(k^{\frac{1+a}{d}}\right)$
$\beta=a+1,a\in(1,\infty)$ | $\Theta\left(\frac{k}{\log{k}}\right)$ | $\left(\Theta\left(\left(\frac{k}{\log{k}}\right)^{\frac{1+a}{d}}\right),\Theta\left(\frac{k}{\log{k}}\right)\right)$ | $\Theta\left(\left(\frac{k}{\log{k}}\right)^{\frac{1+a}{d}}\right)$
$\beta\in(2,a+1),a\in(1,\infty)$ | $\Theta\left(k^{\frac{1}{2+a-\beta}}\right)$ | $\left(\Theta\left(k^{\frac{1+a}{d(2+a-\beta)}}\right),\Theta\left(k^{\frac{1}{2+a-\beta}}\right)\right)$ | $\Theta\left(k^{\frac{1+a}{d(2+a-\beta)}}\right)$
Table 1: Summary of the behavior of the typical triangle between $0$ and its
two randomly sampled neighbors $(\mathbf{x},w_{1})$, $(\mathbf{y},w_{2})$.

###  6.2 The Boolean model

In this section, we discuss a popular related model from the literature.
Consider a homogeneous Poisson point process $\Gamma$ on $\mathbb{R}^{d}$, and
a non-negative random variable $W$. Form the open ball
$B(\mathbf{x},W_{\mathbf{x}})$ around each point $\mathbf{x}\in\Gamma$, where
the collection $(W_{\mathbf{x}})_{\mathbf{x}\in\Gamma}$ is a collection of
i.i.d. copies of $W$. The random graph formed by placing an edge between
distinct pairs $\mathbf{x},\mathbf{y}\in\Gamma$ whenever
$B(\mathbf{x},W_{\mathbf{x}})\cap B(\mathbf{y},W_{\mathbf{y}})\neq\emptyset$
is called the _Boolean_ model [10, 26]. Let $W$ have distribution (1.1).
Clearly, this random graph model is a SIRG, with connection function (1.2)
with $\alpha=\infty$ and $g(s,t)=(s+t)^{d}$. One can also consider the
corresponding $\alpha\in(1,\infty)$ version. For this model,
$\mathbb{E}\left[\left.d_{0}(\mathbb{G}^{\infty})\right|W_{0}=w\right]=\mathcal{M}(w)$
is integrable with respect to the density (1.1) when $\beta>d$.

A related model was studied in [12], where the authors consider a weight
function of the form
$g^{\mathrm{sum}}(s,t)=\beta^{-1}(s^{-\gamma/d}+t^{-\gamma/d})^{-d}$, with $s$
and $t$ now uniformly distributed on $(0,1)$, and $\beta\in(0,\infty)$,
$\gamma\in[0,1)$ parameters, see [12, (1.5)]. A particular case of this model
can be considered, when one takes $\beta=1$, the profile function $\rho$ as
considered in [12, (1.2)] to be
$\kappa_{\alpha}(\|\mathbf{x}-\mathbf{y}\|,s^{\prime},t^{\prime})$ as in
(1.2), where $s^{\prime}=s^{-\gamma}$, and $t^{\prime}=t^{-\gamma}$, with the
weight function $g(s^{\prime},t^{\prime})=(s^{\prime}+t^{\prime})^{d}$. Note
that $s^{\prime}$ and $t^{\prime}$ now have density (1.1) with
$\beta-1=1/\gamma$. Again, as in the last paragraph,
$\mathbb{E}\left[\left.d_{0}(\mathbb{G}^{\infty})\right|W_{0}=w\right]=\mathcal{M}(w)$
is integrable with respect to the density (1.1) when $\beta>d$, i.e.
$\gamma<1/(d-1)$.

These models does not directly come as a particular case of the weight
function (1.3) that we consider. However, following the same proof strategy
and calculations as in the proof of Theorem 1.1, it can be shown that the
corresponding clustering function $\gamma(k)$ has an inverse linear scaling:
$\lim_{k\to\infty}k\gamma(k)=\Gamma\in(0,\infty)$. The calculations are very
similar to the $a=0$ and $\beta>d$ case for the weight function (1.3) that we
consider, and so we omit these calculations. In particular, we state the
following theorem without proof:

######  Theorem 6.1.

Consider the infinite SIRG $\mathbb{G}^{\infty}(\kappa,W)$, where $W$ has
distribution (1.1) and $\kappa=\kappa_{\alpha}$ is of the form (1.2), with
$g(s,t)=(s+t)^{d}$. Assume $\beta>d$. Then, with $\gamma(k)$ denoting the
clustering function of this graph,

$\lim_{k\to\infty}k\gamma(k)=\Gamma,$

where (recall (1.7) and the definitions below it)

$\Gamma=\frac{\beta-\alpha-1}{\xi_{\alpha}}\int_{1}^{\infty}\int_{1}^{\infty}I(w_{1},w_{2})f_{W}(w_{1})dw_{1}f_{W}(w_{2})dw_{2}.$

As remarked before, we need the condition $\beta>d$ to ensure that
$\mathbb{E}\left[d_{0}(\mathbb{G}^{\infty})\right]<\infty$.

###  6.3 On finite models

Note that Theorem 1.1 gives results on how $\gamma(k)$ scales. However
$\gamma(k)$ is the $n\to\infty$ limit of the sequence of clustering functions
$\left(\mathrm{CC}_{\mathbb{G}^{(n)}}(k)\right)_{n\geq 1}$ of the finite
models $\mathbb{G}^{(n)}$. What happens if we let $k=k_{n}$ depend on $n$,
with $k_{n}\to\infty$, for the graph $\mathbb{G}^{(n)}$? That is, let us
consider the sequence of random variables
$\mathrm{CC}_{\mathbb{G}^{(n)}}(k_{n})$, and ask the question: do results
similar as in the theorems above hold with high probability as $n\to\infty$?
We expect if $k_{n}\to\infty$ sufficiently slowly, then we have a positive
answer to the question. We formulate this as a conjecture in this section. We
focus on the finite SIRG $\mathbb{G}^{(n)}$, where the locations are i.i.d. on
$\left[-{n^{1/d}}/{2},{n^{1/d}}/{2}\right]^{d}$. As remarked earlier, the
finite model locally converges to the infinite model $\mathbb{G}^{\infty}$
(see [31, Theorem 1.2]). For the finite model $\mathbb{G}^{(n)}$, recall the
clustering function $\mathrm{CC}_{\mathbb{G}^{(n)}}(k)$ from (1.5). Recall for
fixed $k$,
$\mathrm{CC}_{\mathbb{G}^{(n)}}(k)\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}\gamma(k)$.
For $k_{n}\to\infty$, denote the sequence $\phi(k_{n})\to\infty$ by

$\displaystyle\phi(k_{n}):=\begin{cases}\frac{k_{n}}{\log{k_{n}}}&\mbox{if}\;\;\beta=a+1,a\in(1,\infty),\\\
k_{n}&\mbox{otherwise,}\end{cases}$

and denote the sequence $\psi(n)\to\infty$ as

$\displaystyle\psi(n)=\begin{cases}{n^{\frac{1}{\beta-1}}}&\mbox{if}\;\;\beta\geq(a+1)\vee
2,a\in[0,\infty),\\\
{n^{\frac{2+a-\beta}{\beta-1}}}&\mbox{otherwise.}\end{cases}$

We conjecture the following limit laws for finite models:

######  Conjecture 6.2 (Scaling for finite graphs).

Consider the finite SIRG $\mathbb{G}^{(n)}$.

  * a.

Let $k_{n}\to\infty$ satisfy $\phi(k_{n})=o\left(\psi(n)\right)$. Then, as
$n\to\infty$,
$\frac{\mathrm{CC}_{\mathbb{G}^{(n)}}(k_{n})}{\gamma(k_{n})}\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}1$.

  * b.

Let $k_{n}\to\infty$ satisfy $\phi(k_{n})=(1+o\left(1\right))\psi(n)$, as
$n\to\infty$. Then, as $n\to\infty$,
$\frac{\mathrm{CC}_{\mathbb{G}^{(n)}}(k_{n})}{\gamma(k_{n})}\stackrel{{\scriptstyle
d}}{{\rightarrow}}L_{0}\text{Poi}(\lambda)$, for some $L_{0},\lambda>0$.

  * c.

Let $k_{n}\to\infty$ satisfy $\phi(k_{n})=\omega\left({\psi(n)}\right)$. Then
as $n\to\infty$,
$\frac{\mathrm{CC}_{\mathbb{G}^{(n)}}(k_{n})}{\gamma(k_{n})}\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}0$.

##### Heuristic arguments for Conjecture 6.2.

Let $W$ be a random variable with density (1.1). Observe that by (2.3),
because $\mathbb{G}^{(n)}$ converges locally to $\mathbb{G}^{\infty}$, the
expected number of vertices with degree at least $k_{n}$ in $\mathbb{G}^{(n)}$
$\sim n\mathbb{P}\left(\mathrm{Poi}(\mathcal{M}(W))>k_{n}\right)\sim
n\mathbb{P}\left(\mathcal{M}(W)>k_{n}\right)=n\mathbb{P}\left(W>\mathcal{M}^{-1}(k_{n})\right)$
as $W\to\infty$. Then (1.1) and Proposition 3.3 ensure that when $k_{n}$
satisfies $\phi(k_{n})=o\left(\psi(n)\right)$, this expected value does not
decay to $0$. In other words, _as soon as it is possible to choose vertices of
degree $k_{n}$ in the graph $\mathbb{G}^{(n)}$, the ratio of
$\mathrm{CC}_{\mathbb{G}^{(n)}}(k_{n})$ and $\gamma(k_{n})$ should converge in
probability to $1$, i.e., they are asymptotically the same, as claimed by
Conjecture 6.2 (a)._ There are two main difficult steps in proving Conjecture
6.2 (a). Firstly, to rigorously prove that as soon as
$\phi(k_{n})=o\left(\psi(n)\right)$, the expected number of vertices in
$\mathbb{G}^{(n)}$ with degree $k_{n}$, does not decay to $0$. Secondly, to
prove $\mathrm{CC}_{\mathbb{G}^{(n)}}(k_{n})/\gamma(k_{n})$ converges to $1$,
as soon as $\phi(k_{n})=o\left(\psi(n)\right)$. Parts (b) and (c) of
Conjecture 6.2 are to be understood as quantifying the finite-size effects in
our models. That is, when $\phi(k_{n})=\omega(\psi(n))$ as in Conjecture 6.2
(c), there are not many vertices of degree $k_{n}$ in $\mathbb{G}^{(n)}$, and
even when there are, they simply do not have enough edges among their
neighbors to contribute to the clustering in comparison to their degrees.
Conjecture 6.2 (b) claims that at the boundary of the finite-size effects,
which we call the _finite-size threshold_ , the limit is random, and in fact
Poissonian. Parts (b) and (c) require a careful and tedious analysis of
boundary effects around the finite-size threshold, which constitutes its main
difficulty. A result of similar flavor for Hyperbolic Random Graphs, which
corresponds to taking $d=1$ and $\alpha=\infty$ in our model, was proved in
[8, Theorem 1.5]. This proof relied heavily upon the particular choice of the
model and its parameters, and does not appear to be easily generalizable to
our setting.

As we have seen in Theorem 1.1, there are two phase transitions in the scaling
of $\gamma(k)$, so three different phases. In Figure 6, we see simulations of
the clustering as a function of the degree in the finite graph
$\mathbb{G}^{(n)}$, in these three different phases of $\beta$ and $a$. Here
for $k\in\\{d_{v}(\mathbb{G}^{(n)}):v\in V(\mathbb{G}^{(n)})\\}$, the log-log
plot of $k\to\mathrm{CC}_{\mathbb{G}^{(n)}}(k)$ is shown. Since in all these
three phases, $\gamma(k)=\Theta(k^{-p})$ as $k\to\infty$, for some $p\geq 0$,
we see a straight line in the log-log plot, before the decay becomes much
faster due to finite-size effects. The (approximate) finite-size threshold is
indicated by the green dashed vertical line. In the plots of Figure 6, the
number of vertices $n$ of $\mathbb{G}^{(n)}$ is fixed to be $n=22000$, $a=2$,
$\alpha=1$ and $d=2$ are fixed. $\beta$ is varied to obtain random graphs in
the three different regimes: $\beta>a+\frac{3}{2}$,
$\beta\in(a+1,a+\frac{3}{2})$ and $\beta\in(\frac{a+3}{2},a+1)$. Here for the
third phase, we do not go all the way to $\beta=2$, since, in the regime
$\beta\in(2,\frac{a+3}{2})$,
$\mathbb{E}\left[d_{0}(\mathbb{G}^{\infty})\right]=\infty$, which we recall
from Section 3.

(a) Clustering function in the regime $\beta>a+3/2$. Here $n=22000$, $a=2$ and
$\beta=4$.

(b) Clustering function in the regime $\beta\in(a+1,a+\frac{3}{2})$. Here
$n=22000$, $a=2$ and $\beta=3.01$.

(a) Clustering function in the regime $\beta\in(2,a+1)$. Here $n=22000$, $a=2$
and $\beta=2.6$.  Figure 6: Plots of clustering against degree in the log-log
scale. The vertical green dashed line indicates the approximate finite-size
thresholds.

As can be seen in Figure 6, as the weight tail $\beta$ becomes heavier, the
decay of the clustering function, as the degree becomes large, becomes slower
and slower. The outliers in the simulations for extremely high-degree vertices
(extreme right of the plots) show low clustering because of finite-size
effects. Note that the clustering contribution from extremely low-degree
vertices (extreme left of the plots) are not of interest to us, since we are
interested in the clustering behaviour as the degree diverges.

In Figure 7, for visualization, we have included simulations of the graph
$\mathbb{G}^{(n)}$ in the three different regimes $\beta>a+\frac{3}{2}$,
$\beta\in(a+1,a+\frac{3}{2})$ and $\beta\in(2,a+1)$. In all these simulations,
$n=8000$ and $\alpha=1$ are fixed. Here larger sized vertices have more
degree. A uniform bond percolation parameter of $p=1/7000$ is first applied on
all the three graphs for better visibility of edges. Further, bond percolation
is applied to the graphs in the regimes $\beta\in(a+1,a+\frac{3}{2})$ and
$\beta\in(2,a+1)$ so that they have (approximately) the same average degree,
as the graph in the regime $\beta>a+\frac{3}{2}$. In Figure 7(c), we see the
graph $\mathbb{G}^{(n)}$ in the regime $\beta\in(2,a+1)$. Here $\beta$ is also
chosen such that $\beta>(a+3)/2$, so that
$\mathbb{E}\left[d_{0}(\mathbb{G}^{\infty})\right]<\infty$. However, the
clustering $\gamma(k)$ of the corresponding infinite model converges to a
constant, see Theorem 1.1. As a consequence, the graph admits vertices of very
high degrees, and these vertices have neighborhoods that are almost cliques.
Could these vertices be part of a large dense subgraph? A future research
direction for us is to study other properties of the graphs in this _exotic_
regime.

(a) The graph $\mathbb{G}^{(n)}$ in the regime $\beta>a+3/2$. Here $n=8000$,
$a=2$, $\beta=4$.

(b) The graph $\mathbb{G}^{(n)}$ in the regime $a+3/2>\beta>a+1$. Here
$n=8000$, $a=2$, $\beta=3.01$.

(c) The graph $\mathbb{G}^{(n)}$ in the regime $a+1>\beta>2$. Here $n=8000$,
$a=2$, $\beta=2.6$.

Figure 7: In Figures 7(b) and 7(c), bond percolation is applied so that the
average degree is approximately the same as the graph in Figure 7(a).

##  Appendix A Markings and thinnings: proof of Lemma 2.1

Before proving Lemma 2.1, we first recall the concepts of marking and thinning
of a point process. For more on what follows, we refer the reader to [24,
Chapter 5].

######  Definition A.1 (Probability kernel).

For measurable spaces $(\mathbb{X},\mathscr{X})$ and
$(\mathbb{Y},\mathscr{Y})$, a probability kernel $K$ from $\mathbb{X}$ to
$\mathbb{Y}$ is a map $K:\mathbb{X}\times\mathscr{Y}\to[0,1]$, such that for
every fixed $x\in\mathbb{X}$, $K(x,\cdot)$ is a probability measure on
$\mathbb{Y}$, and for every fixed $A\in\mathscr{Y}$, $K(\cdot,A)$ is an
$\mathscr{X}$-measurable function on $\mathbb{X}$.

######  Definition A.2 (Proper point process).

A point process $\eta$ on a measurable space $(\mathbb{X},\mathscr{X})$ is
called proper, if there exists $\mathbb{X}$-valued random variables
$X_{1},X_{2},\dots$ and an $\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$-valued
random variable $\mathscr{N}$, such that almost surely,

$\displaystyle\eta=\sum_{i=1}^{\mathscr{N}}\delta_{X_{i}}.$

It is a well-known fact that any Poisson point process is proper, see [24,
Corollaries 3.7 and 6.5].

######  Definition A.3 (Marking).

Let $\eta=\sum_{i=1}^{\mathscr{N}}\delta_{X_{i}}$ be a proper point process on
$\mathbb{X}$. Let $K$ be a probability kernel from $\mathbb{X}$ to
$\mathbb{Y}$ for some measurable space $(\mathbb{Y},\mathscr{Y})$. Let
$Y_{1},Y_{2},\dots$ be $\mathbb{Y}$-valued random variables and let
$(Y_{i})_{i\leq m}$, given $\mathscr{N}=m$ and $(X_{i})_{i\leq m}$, be
independent random variables with distribution $K(X_{i},\cdot)$, $i\leq m$.
Then the $\mathbb{X}\times\mathbb{Y}$-valued point process

$\displaystyle\zeta=\sum_{i=1}^{\mathscr{N}}\delta_{X_{i},Y_{i}}$

is called a _$K$ -marking_ of $\eta$.

If there is a probability measure $\mathbb{Q}$ on $\mathbb{Y}$ such that
$K(x,\cdot)=\mathbb{Q}(\cdot)$ for all $x\in\mathbb{X}$, then $\zeta$ is
called an _independent $\mathbb{Q}$-marking_ of $\eta$. The following is the
main theorem on markings of Poisson point processes (see [24, Theorem 5.6])
which we state without proof. Recall an $s$-finite measure is a countable sum
of finite measures:

######  Theorem A.1 (Marking theorem).

Let $\zeta$ be a $K$-marking of a proper Poisson point process $\eta$ with
$s$-finite intensity measure $\lambda$. Then $\zeta$ is a Poisson point
process on $\mathbb{X}\times\mathbb{Y}$ with intensity measure $\lambda\otimes
K$, where, for any $C\in\mathscr{X}\times\mathscr{Y}$,

$\displaystyle\lambda\otimes
K(C):=\int_{\mathbb{X}\times\mathbb{Y}}\mathbbm{1}_{\left\\{(x,y)\in
C\right\\}}K(x,dy)\lambda(dx).$

######  Definition A.4 (Thinning).

Let $p:\mathbb{X}\to[0,1]$ be measurable, and consider the probability kernel
$K$ from $\mathbb{X}$ to $[0,1]$ defined by

$\displaystyle K_{p}(x,\cdot)=(1-p(x))\delta_{0}+p(x)\delta_{1}.$ (A.1)

Then if $\zeta$ is a $K_{p}$-marking of a proper point process $\eta$, the
point process $\zeta(\cdot\times\\{1\\})$ on $\mathbb{X}$ is called a
$p$-thinning of $\eta$.

The following is the main theorem on thinnings of Poisson point process (see
[24, Theorem 5.8]), which we state without proof:

######  Theorem A.2 (Thinning theorem).

Let $\zeta$ be a $K_{p}$-marking of a proper Poisson process $\eta$, where
$K_{p}$ is as in (A.1). Then $\zeta_{i}:=\zeta(\cdot\times\\{i\\})$ for
$i=1,0$ are independent Poisson point processes on $\mathbb{X}$.

Note that since the intensity measure of $\zeta$ is $\lambda\otimes K_{p}$,
the intensity measures $\zeta_{0}$ and $\zeta_{1}$ of respectively
$\zeta(\cdot\times\\{0\\})$ and $\zeta(\cdot\times\\{1\\})$ are

$\displaystyle\zeta_{0}(dx)=(1-p(x))\lambda(dx),\;\;\;\zeta_{1}(dx):=p(x)\lambda(dx).$
(A.2)

We now have the necessary ingredients to give the proof of Lemma 2.1.

###### Proof of Lemma 2.1.

Let $\eta$ denote a Poisson point process on $\mathbb{U}$ with intensity
measure given by $\lambda_{d}(d\mathbf{x})\times f_{W}(x)dx$, where we recall
that $\lambda_{d}$ denotes the Lebesgue measure on $\mathbb{R}^{d}$. Let
$\eta_{0}$ be the point process $\eta+\delta_{(\mathbf{0},w)}$, i.e.
$\eta_{0}$ is $\eta$ with the point $(\mathbf{0},w)\in\mathbb{U}$ added - a
Palm version of $\eta$. Observe that the collection $\\{(X_{i},W_{i})\\}_{i\in
V(\mathbb{G}^{\infty})}$ of tuples of the locations and weights of the
vertices of $\mathbb{G}^{\infty}$, are distributed as the atoms of $\eta_{0}$.
Thus, we view the atoms of $\eta_{0}$ as the vertices of
$\mathbb{G}^{\infty}$.

For $w>0$ fixed, let $p_{w}:\mathbb{U}\to[0,1]$ be defined by

$\displaystyle
p_{w}(\mathbf{p})=p_{w}(\mathbf{x},x):=\kappa(\|\mathbf{x}\|,w,x).$

That is, given $\mathbf{p}=(\mathbf{x},x)\in\eta$, $p_{w}(\mathbf{p})$ is the
probability that there is an edge between the vertices $\mathbf{p}$ and
$(\mathbf{0},w)$ in $\mathbb{G}^{\infty}$.

Consider the probability kernel $K_{p_{w}}$ from $\mathbb{U}$ to $[0,1]$
defined by

$\displaystyle
K_{p_{w}}(\mathbf{p},\cdot):=(1-p(\mathbf{p}))\delta_{0}(\cdot)+p(\mathbf{p})\delta_{1}(\cdot).$

Let $\eta^{(p_{w})}$ be the $K_{p_{w}}$-marking of $\eta$, and consider
$\eta^{(p_{w})}_{1}:=\eta^{(p_{w})}(\cdot\times\\{1\\})$, the $p_{w}$-thinning
of $\eta$. Observe that the neighbors of $(\mathbf{0},w)$ in
$\mathbb{G}^{\infty}$ have the same distribution as the atoms of
$\eta^{(p_{w})}_{1}$. From (A.2), the intensity measure of the Poisson point
process $\eta^{(p_{w})}_{1}$ is

$\displaystyle\kappa(\|\mathbf{x}\|,w,x)\lambda_{d}(d\mathbf{x})\times
f_{W}(x)dx.$

In other words, the neighbors of $(\mathbf{0},w)$ are distributed as a Poisson
point process on $\mathbb{U}$, with intensity function $\rho_{w}(\mathbf{p})$
as in (2.1). This completes the proof.∎

######  Remark A.1.

Note that our proof shows that the point process of the locations and weights
of the non-neighbors of $0$, given it has weight $W_{0}=w$, also form a
Poisson point process, namely
$\eta^{p_{w}}_{0}:=\eta^{(p_{w})}(\cdot\times\\{0\\})$.

##  Appendix B Inverse scaling of almost power functions: proof of Lemma 3.2

We prove the scaling of $f^{-1}(k)$ by contradiction. To this end we assume
that there exists some $\delta>0$, and a sequence of reals $(k_{n})_{n\geq
1}$, with $k_{n}\to\infty$ as $n\to\infty$, such that, for all $n\geq 1$,

$\displaystyle\left|\frac{f^{-1}(k_{n})}{\left(\frac{a^{b}k}{c(\log{k})^{b}}\right)^{1/a}}-1\right|>\delta.$

The above inequality implies that along a subsequence

$\displaystyle
f^{-1}(k_{n})>(1+\delta)\left(\frac{a^{b}k}{c(\log{k})^{b}}\right)^{1/a}:=y_{n}^{+},$

or

$\displaystyle
f^{-1}(k_{n})<(1-\delta)\left(\frac{a^{b}k}{c(\log{k})^{b}}\right)^{1/a}:=y_{n}^{-}.$

Without loss of generality, let us assume the first case. The second case can
be dealt with in a similar fashion (reversing all inequalities).

Since $f$ is strictly increasing on $(t,\infty)$, and
$f^{-1}(k_{n})\to\infty$, for all large $n\geq 1$,

$k_{n}>f^{-1}(y_{n}^{+}).$ (B.1)

Since $f(k)=ck^{a}(\log{k})^{b}+o\left(ck^{a}(\log{k})^{b}\right)$ as
$k\to\infty$, and

$\displaystyle c(y_{n}^{+})^{a}(\log y_{n}^{+})^{b}$
$\displaystyle=c\left((1+\delta)\left(\frac{a^{b}k}{c(\log{k})^{b}}\right)^{1/a}\right)^{a}\left(\log{\left((1+\delta)\left(\frac{a^{b}k}{c(\log{k})^{b}}\right)^{1/a}\right)}\right)^{b}$
$\displaystyle=(1+\delta)(k_{n}+o\left(k_{n}\right)),$

as $k_{n}\to\infty$, from (B.1), we have

$\displaystyle k_{n}>k_{n}(1+\delta)(1+o(1)),$

which is a contradiction. This completes the proof.

##### Acknowledgements.

The work of RvdH is supported in part by the Netherlands Organisation for
Scientific Research (NWO) through Gravitation-grant NETWORKS-024.002.003. NM
thanks Martijn Gösgens for help with the pictures, and Joost Jorritsma for
help with the simulations, based on the code [18]. We thank an anonymous
reviewer for a meticulous reading of the submitted version, whose observations
greatly improved the paper.

##### Availability of data and materials.

The code for the simulations that appear in Section 6 have been built upon the
code [18]. The exact code for the simulations in Section 6 are available from
the corresponding author on reasonable request.

##### Conflict of interest.

The authors declare that they have no conflict of interest.

## References

  * [1] M. Bloznelis and L. Leskelä.  Clustering and percolation on superpositions of bernoulli random graphs.  _Random Structures & Algorithms_, n/a(n/a).  doi:https://doi.org/10.1002/rsa.21140.  URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.21140.
  * Boguñá et al. [2021] M. Boguñá, I. Bonamassa, M. De Domenico, S. Havlin, D. Krioukov, and M. Serrano.  Network geometry.  _Nature Reviews Physics_ , 3(2):114–135, 2021\.
  * Bollobás et al. [2007] B. Bollobás, S. Janson, and O. Riordan.  The phase transition in inhomogeneous random graphs.  _Random Structures Algorithms_ , 31(1):3–122, 2007.  ISSN 1042-9832.  doi:10.1002/rsa.20168.  URL https://doi.org/10.1002/rsa.20168.
  * Bringmann et al. [2016] K. Bringmann, R. Keusch, and J. Lengler.  Average distance in a general class of scale-free networks with underlying geometry.  2016\.  URL https://arxiv.org/abs/1602.05712.
  * Bringmann et al. [2017] K. Bringmann, R. Keusch, and J. Lengler.  Sampling geometric inhomogeneous random graphs in linear time.  In _25th European Symposium on Algorithms_ , volume 87 of _LIPIcs. Leibniz Int. Proc. Inform._ , pages Art. No. 20, 15. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2017.
  * Bringmann et al. [2019] K. Bringmann, R. Keusch, and J. Lengler.  Geometric inhomogeneous random graphs.  _Theoret. Comput. Sci._ , 760:35–54, 2019.  ISSN 0304-3975.  doi:10.1016/j.tcs.2018.08.014.  URL https://doi.org/10.1016/j.tcs.2018.08.014.
  * Csányi and Szendrői [2004] G. Csányi and B. Szendrői.  Structure of a large social network.  _Phys. Rev. E_ , 69:036131, Mar 2004.  doi:10.1103/PhysRevE.69.036131.  URL https://link.aps.org/doi/10.1103/PhysRevE.69.036131.