Moments of $k$-hop counts in the random-connection model

TL;DR

This paper derives moment identities for multiparameter stochastic integrals in a random-connection model, simplifying the calculation of moments for k-hop counts and advancing theoretical understanding of such models.

Contribution

It introduces new moment identities for multiparameter processes in a random-connection model, reducing complexity in deriving moments of k-hop counts.

Findings

Derived general moment identities for multiparameter processes.

Simplified the derivation of moments for k-hop counts.

Provided a unified framework for moment calculations in the model.

Abstract

We derive moment identities for the stochastic integrals of multiparameter processes in a random-connection model based on a point process admitting a Papangelou intensity. Those identities are written using sums over partitions, and they reduce to sums over non-flat partition diagrams in case the multiparameter processes vanish on diagonals. As an application, we obtain general identities for the moments of $k$-hop counts in the random-connection model, which simplify the derivations available in the literature.

Equations

$$\Omega^{X}=\big{\{}\omega=\{x_{i}\}_{i\in I}\subset X\ :\ \#(A\cap\omega)<\infty\mbox{ for all compact }A\in{\cal B}(X)\big{\}}$$

$$C(A\times B):=\mathbb{E}\left[{\int }_A\text{mathds}{1}_B(\omega \setminus \{x\})\omega (dx)\right],A\in \mathcal{B}(X),B\in \mathcal{F},$$

$$\mathbb{E}\left[{\int }_{X}u(x;\omega )\omega (dx)\right]={\int }_{{\Omega }^X}{\int }_{X}u(x;\omega \cup x)C(dx,d\omega ),$$

$$C(dx,d\omega )=c(x;\omega )\lambda (dx)P(d\omega ),$$

$${\hat{\lambda }}^n(d\text{eufrak}{x}_n)=\hat{c}(\text{eufrak}{x}_n;\omega )\lambda (d{x}_1)\cdots \lambda (d{x}_n),$$

$$\hat{c}(\{x_1,\dots ,x_n,y\};\omega ):=c(y;\omega )\hat{c}(\{x_1,\dots ,x_n\};\omega \cup \{y\}),n\ge 0,$$

$$\mathrm{IE}[Z_{\alpha }^n]=\sum _{k=0}^n{\alpha }^{k}S(n,k),n\in \mathrm{I}\mathrm{N},$$

$$\mathrm{IE}\left[{\left({\int }_{X}f(x)\omega (dx)\right)}^n\right]=n!\sum _{\genfrac{}{}{0ex}{}{r_1+2r_2+\cdots +n{r}_n=n}{r_1,\dots ,r_n\ge 0}}\prod _{k=1}^n\left(\frac{1}{(k!{)}^{r_k}{r}_k!}{\left({\int }_X{f}^k(x)\lambda (dx)\right)}^{r_k}\right)$$

$$({\epsilon }_{\text{eufrak}{z}_n}^{+}F)(\omega )=F(\omega \cup \{z_1,\dots ,z_n\}),\omega \in \Omega ,$$

$$\mathbb{E}\left[{\left({\int }_{X}u(x;\omega )\omega (dx)\right)}^n\right]=\sum _{\rho \in \Pi [n]}\mathbb{E}\left[{\int }_{X^{|\rho |}}{ϵ}_{\text{eufrak}{z}_{|\rho |}}^{+}\prod _{l=1}^{|\rho |}{u}^{|{\rho }_l|}(z_l){\hat{\lambda }}^{|\rho |}(d\text{eufrak}{z}_{|\rho |})\right],$$

$$\Delta_{n\times r}:=\{1,\ldots,n\}\times\{1,\ldots,r\}=\big{\{}(k,l)\ :\ k=1,\ldots,n,\ l=1,\ldots,r\big{\}},$$

$${\zeta }^{\rho }(k,l)=p\text{mbox}ifandonlyif(k,l)\in {\rho }_p,$$

$$\epsilon^{+}_{\eufrak{z}_{k}}u(z_{1},\ldots,z_{k};\omega):=u\big{(}z_{1},\ldots,z_{k};\omega\cup\{z_{1},\ldots,z_{k}\}\big{)},\quad\eufrak{z}_{n}=(z_{1},\ldots,z_{n})\in X^{n},$$

$$\mathbb{E}\left[\left(\int_{X^{r}}u(z_{1},\ldots,z_{r};\omega)\omega(dz_{1})\cdots\omega(dz_{r})\right)^{n}\right]=\sum_{\rho\in\Pi[n\times r]}\mathbb{E}\left[\int_{X^{|\rho|}}\varepsilon^{+}_{\eufrak{z}_{|\rho|}}\prod_{k=1}^{n}u\big{(}z^{\rho}_{\pi_{k}}\big{)}\hat{\lambda}^{|\rho|}(d\eufrak{z}_{|\rho|})\right]$$

$$\mathbb{E}\left[{\int }_{X^r}u(z_1,\dots ,z_r;\omega )\omega (d{z}_1)\cdots \omega (d{z}_r)\right]=\sum _{\rho \in \Pi [1\times r]}\mathbb{E}\left[{\int }_{X^r}{\epsilon }_{\text{eufrak}{z}_{|\rho |}}^{+}u(z_{{\zeta }^{\rho }(1,1)},\dots ,z_{{\zeta }^{\rho }(1,r)};\omega ){\hat{\lambda }}^{|\rho |}(d\text{eufrak}{z}_{|\rho |})\right].$$

$$△=\{(1,2),(2,1),(2,2),(3,3),(4,2)\},$$

$$\text{ocircle}=\{(1,1),(3,1),(4,4),(5,3)\},$$

$$\square =\{(1,3),(2,4),(3,3),(4,1),(5,4)\},$$

$$\text{pentagon}=\{(1,4),(2,2)\},$$

$$\times =\{(2,3),(3,4),(4,2),(5,1)\}$$

$${\pi }_k=\{(k,1),(k,2),(k,3),(k,4),(k,5)\},k=1,2,3,4,5.$$

$$\mathbb{E}\left[\left(\int_{X^{r}}u(z_{1},\ldots,z_{r};\omega)\omega(dz_{1})\cdots\omega(dz_{r})\right)^{n}\right]=\sum_{\begin{subarray}{c}\rho\in\Pi[n\times r]\\ \rho\wedge\pi=\hat{0}\end{subarray}}\mathbb{E}\left[\int_{X^{|\rho|}}\epsilon^{+}_{\eufrak{z}_{|\rho|}}\prod_{k=1}^{n}u\big{(}z^{\rho}_{\pi_{k}}\big{)}\hat{\lambda}^{|\rho|}(d\eufrak{z}_{|\rho|})\right].$$

$$\mathbb{E}\left[{\int }_{X^r}u(z_1,\dots ,z_r;\omega )\omega (d{z}_1)\cdots \omega (d{z}_r)\right]$$

$$\mathbb{E}\left[\left(\int_{X^{r}}u(z_{1},\ldots,z_{r};\omega)\omega(dz_{1})\cdots\omega(dz_{r})\right)^{2}\right]=\sum_{\begin{subarray}{c}\rho\in\Pi[2\times r]\\ \rho\wedge\pi=\hat{0}\end{subarray}}\mathbb{E}\left[\int_{X^{|\rho|}}\epsilon^{+}_{\eufrak{z}_{|\rho|}}u\big{(}z^{\rho}_{\pi_{1}}\big{)}u\big{(}z^{\rho}_{\pi_{2}}\big{)}\hat{\lambda}^{|\rho|}(d\eufrak{z}_{|\rho|})\right],$$

$$\mathbb{E}\left[{\left({\int }_{X^r}u(z_1,\dots ,z_r;\omega )\omega (d{z}_1)\cdots \omega (d{z}_r)\right)}^2\right]$$

$$\displaystyle=\sum_{A\subset\pi_{1}}\frac{1}{(r-|A|)!}\sum_{\gamma:\pi_{2}\to A\cup\{r+|A|+1,\ldots,2r\}}\mathbb{E}\left[\int_{X^{2r-|A|}}\epsilon^{+}_{\eufrak{z}_{2r-|A|}}u\big{(}z_{\pi_{1}}\big{)}u\big{(}z_{\gamma(r+1)},\ldots,z_{\gamma(2r)}\big{)}\hat{\lambda}^{2r-|A|}(d\eufrak{z}_{2r-|A|})\right],$$

$$\rho=\big{\{}\{i,\gamma(i)\}\}_{i\in A}\cup\{\{i\}\}_{i\notin A}\cup\{\{i\}\}_{i\notin\gamma(A)}\big{\}},$$

$$\displaystyle\mathbb{E}\left[\left(\int_{X^{2}}u(z_{1},z_{2};\omega)\omega(dz_{1})\omega(dz_{2})\right)^{2}\right]=\sum_{\begin{subarray}{c}\rho\in\Pi[n\times 2]\\ \rho\wedge\pi=\hat{0}\end{subarray}}\mathbb{E}\left[\int_{X^{|\rho|}}\epsilon^{+}_{\eufrak{z}_{|\rho|}}\prod_{k=1}^{n}u\big{(}z_{\zeta^{\rho}(k,1)},z_{\zeta^{\rho}(k,2)}\big{)}\hat{\lambda}^{|\rho|}(d\eufrak{z}_{|\rho|})\right]$$

$$\mathbb{E}\left[{\left({\int }_{X^3}u(z_1,z_1,z_3;\omega )\omega (d{z}_1)\omega (d{z}_2)\omega (d{z}_3)\right)}^2\right]$$

$$\displaystyle=\sum_{\begin{subarray}{c}A\subset\{1,2,3\}\\ \gamma:\{4,5,6\}\to A\cup\{4+|A|,\ldots,6\}\end{subarray}}\frac{1}{(3-|A|)!}\mathbb{E}\left[\int_{X^{5}}\epsilon^{+}_{\epsilon^{+}_{\eufrak{z}_{5}}}u(z_{1},z_{2},z_{3})u\big{(}z_{\gamma(4)},z_{\gamma(5)},z_{\gamma(6)}\big{)}\hat{\lambda}^{5}(d\eufrak{z}_{5})\right]$$

$$=\mathbb{E}\left[{\int }_{X^6}{ϵ}_{\text{eufrak}{z}_6}^{+}u(z_1,z_2,z_3)u(z_4,z_5,z_6){\hat{\lambda }}^6(d\text{eufrak}{z}_6)\right]$$

$$\displaystyle\quad+\frac{1}{2}\sum_{\gamma:\{4,5,6\}\to\{1,5,6\}}\mathbb{E}\left[\int_{X^{5}}\epsilon^{+}_{\eufrak{z}_{5}}u(z_{1},z_{2},z_{3})u\big{(}z_{\gamma(4)},z_{\gamma(5)},z_{\gamma(6)}\big{)}\hat{\lambda}^{5}(d\eufrak{z}_{5})\right]$$

$$\displaystyle\quad+\frac{1}{2}\sum_{\gamma:\{4,5,6\}\to\{2,5,6\}}\mathbb{E}\left[\int_{X^{5}}\epsilon^{+}_{\eufrak{z}_{5}}u(z_{1},z_{2},z_{3})u\big{(}z_{\gamma(4)},z_{\gamma(5)},z_{\gamma(6)}\big{)}\hat{\lambda}^{5}(d\eufrak{z}_{5})\right]$$

$$\displaystyle\quad+\frac{1}{2}\sum_{\gamma:\{4,5,6\}\to\{3,5,6\}}\mathbb{E}\left[\int_{X^{5}}\epsilon^{+}_{\eufrak{z}_{5}}u(z_{1},z_{2},z_{3})u\big{(}z_{\gamma(4)},z_{\gamma(5)},z_{\gamma(6)}\big{)}\hat{\lambda}^{5}(d\eufrak{z}_{5})\right]$$

$$\displaystyle\quad+\sum_{\gamma:\{4,5,6\}\to\{1,2,6\}}\mathbb{E}\left[\int_{X^{4}}\epsilon^{+}_{\eufrak{z}_{4}}u(z_{1},z_{2},z_{3})u\big{(}z_{\gamma(4)},z_{\gamma(5)},z_{\gamma(6)}\big{)}\hat{\lambda}^{4}(d\eufrak{z}_{4})\right]$$

$$\displaystyle\quad+\sum_{\gamma:\{4,5,6\}\to\{1,3,6\}}\mathbb{E}\left[\int_{X^{4}}\epsilon^{+}_{\eufrak{z}_{4}}u(z_{1},z_{2},z_{3})u\big{(}z_{\gamma(4)},z_{\gamma(5)},z_{\gamma(6)}\big{)}\hat{\lambda}^{4}(d\eufrak{z}_{4})\right]$$

$$\displaystyle\quad+\sum_{\gamma:\{4,5,6\}\to\{2,3,6\}}\mathbb{E}\left[\int_{X^{4}}\epsilon^{+}_{\eufrak{z}_{4}}u(z_{1},z_{2},z_{3})u\big{(}z_{\gamma(4)},z_{\gamma(5)},z_{\gamma(6)}\big{)}\hat{\lambda}^{4}(d\eufrak{z}_{4})\right]$$

Full text

Key words: Point processes, moments, random-connection model, random graph, $k$-hops.

Mathematics Subject Classification (2010): 60G57; 60G55.

1 Introduction

The random-connection model, see e.g. Chapter 6 of [12], is a classical model in continuum percolation. It consists in a random graph built on the vertices of a point process on ${\mathord{{\rm I\kern-3.0ptR}}}^{d}$, by adding edges between two distinct vertices $x$ and $y$ with probability $H(\|x-y\|)$. In the case of the Rayleigh fading $H_{\beta}(\|x-y\|)=\mathrm{e}^{-\beta\|x-y\|}$ with $x,y\in{\mathord{{\rm I\kern-3.0ptR}}}^{2}$, the mean value of the number $N^{x,y}_{k}$ of $k$-hop paths connecting $x\in{\mathord{{\rm I\kern-3.0ptR}}}^{d}$ to $y\in{\mathord{{\rm I\kern-3.0ptR}}}^{d}$ has been computed in [9], together with the variance of $3$-hop counts. However, this argument does not extend to $k\geq 3$ as the proof of the variance identity for $3$-hop counts in [9] relies on the known Poisson distribution of the $2$-hop count. As shown by [9], the knowledge of moments can provide accurate numerical estimates of the probability $P(N^{x,y}_{k}>0)$ of at least one $k$-hop path, by expressing it as a series of factorial moments, and the need for a general theory of such moments has been pointed out therein.

On the other hand, moment identities for Poisson stochastic integrals with random integrands have been obtained in [18] based on moment identities for the Skorohod integral on the Poisson space, see [16, 17], and also [19] for a review. These moment identities have been extended to point processes with Papangelou intensities by [5], and to multiparameter processes by [2]. Factorial moments have also been computed by [4] for point processes with Papangelou intensities.

In this paper we derive closed-form expressions for the moments of the number of $k$-hop paths in the random-connection model. In Proposition 3.1 the moment of order $n$ of the $k$-hop count is given as a sum over non-flat partitions of $\{1,\ldots,nk\}$ in a general random-connection model based on a point process admitting a Papangelou intensity. Those results are then specialized to the case of Poisson point processes, with an expression for the variance of the $k$-hop count given in Corollary 3.2 using a sum over integer sequences. Finally, in the case of Rayleigh fadings we show that some results of [9], such as the computation of variance for $3$-hop counts, can be recovered via a shorter argument, see Corollary 5.3.

We proceed as follows. After presenting some background notation on point processes and Campbell measures, see [8], in Section 2 we review the derivation of moment identities for stochastic integrals using sums over partitions. In the multiparameter case we rewrite those identities for processes vanishing on diagonals, based on non-flat partition diagrams. In Section 3 we apply those results to the computation of the moments of $k$-hop counts in the random-connection model, and we specialize such computations to the case of Poisson point processes in Section 4. Section 5 is devoted to explicit computations in the case of Rayleigh fadings, which result into simpler derivations in comparison with the current literature on moments in the random-connection model.

Notation on point processes

Let $X$ be a Polish space with Borel $\sigma$-algebra ${\cal B}(X)$, equipped with a $\sigma$-finite non-atomic measure $\lambda(dx)$. We let

[TABLE]

denote the space of locally finite configurations on $X$, whose elements $\omega\in\Omega^{X}$ are identified with the Radon point measures $\displaystyle\omega=\sum_{x\in\omega}\epsilon_{x}$, where $\epsilon_{x}$ denotes the Dirac measure at $x\in X$. A point process is a probability measure $P$ on $\Omega^{X}$ equipped with the $\sigma$-algebra ${\cal F}$ generated by the topology of vague convergence.

Point processes can be characterized by their Campbell measure $C$ defined on ${\cal B}(X)\otimes{\cal F}$ by

[TABLE]

which satisfies the Georgii-Nguyen-Zessin [14] identity

[TABLE]

for all measurable processes $u:X\times\Omega^{X}\to{\mathord{{\rm I\kern-3.0ptR}}}$ such that both sides of (1.1) make sense.

In the sequel we deal with point processes whose Campbell measure $C(dx,d\omega)$ is absolutely continuous with respect to $\lambda\otimes P$, i.e.

[TABLE]

where the density $c(x;\omega)$ is called the Papangelou density. We will also use the random measure $\hat{\lambda}^{n}(d\eufrak{x}_{n})$ defined on $X^{n}$ by

[TABLE]

where $\hat{c}(\eufrak{x}_{n};\omega)$ is the compound Campbell density $\hat{c}:\Omega_{0}^{X}\times\Omega^{X}\longrightarrow{\mathord{{\rm I\kern-3.0ptR}}}_{+}$ defined inductively on the set $\Omega_{0}^{X}$ of finite configurations in $\Omega^{X}$ by

[TABLE]

see Relation (1) in [5]. In particular, the Poisson point process with intensity $\lambda(dx)$ is a point process with Campbell measure $C=\lambda\otimes P$ and $c(x;\omega)=1$, and in this case the identity (1.1) becomes the Slivnyak-Mecke formula [20], [11]. Determinantal point processes are examples of point processes with Papangelou intensities, see e.g. Theorem 2.6 in [6], and they can be used for the modeling of wireless networks with repulsion, see e.g. [7], [13], [10].

2 Moment identities

The moment of order $n\geq 1$ of a Poisson random variable $Z_{\alpha}$ with parameter $\alpha>0$ is given by

[TABLE]

where $S(n,k)$ is the number of ways to partition a set of $n$ objects into $k$ non-empty subsets, see e.g. Proposition 3.1 of [3]. Regarding Poisson stochastic integrals of deterministic integrands, in [1] the moment formula

[TABLE]

has been proved for deterministic functions $f\in\bigcap_{p\geq 1}L^{p}(X,\lambda)$.

The identity (2.2) has been rewritten in the langage of sums over partitions, and extended to Poisson stochastic integrals of random integrands in Proposition 3.1 of [18], and further extended to point processes admitting a Panpangelou intensity in Theorem 3.1 of [5], see also [4]. In the sequel, given $\eufrak{z}_{n}=(z_{1},\ldots,z_{n})\in X^{n}$, we will use the shorthand notation $\varepsilon^{+}_{\eufrak{z}_{n}}$ for the operator

[TABLE]

where $F$ is any random variable on $\Omega^{X}$. Given $\rho=\{\rho_{1},\ldots,\rho_{k}\}\in\Pi[n]$ a partition of $\{1,\ldots,n\}$ of size $|\rho|=k$, we let $|\rho_{i}|$ denote the cardinality of each block $\rho_{i}$, $i=1,\ldots,k$.

Proposition 2.1

Let $u:X\times\Omega^{X}\longrightarrow{\mathord{{\rm I\kern-3.0ptR}}}$ be a (measurable) process. For all $n\geq 1$ we have

[TABLE]

where the sum runs over all partitions $\rho$ of $\{1,\ldots,n\}$ with cardinality $|\rho|$.

Proposition 2.1 has also been extended, together with joint moment identities, to multiparameter processes $(u_{z_{1},\ldots z_{r}})_{(z_{1},\ldots z_{r})\in X^{r}}$, see Theorem 3.1 of [2]. For this, let $\Pi[n\times r]$ denote the set of all partitions of the set

[TABLE]

identified to $\{1,\ldots,nr\}$, and let $\pi:=(\pi_{1},\ldots,\pi_{n})\in\Pi[n\times r]$ denote the partition made of the $n$ blocks $\pi_{k}:=\{(k,1),\ldots,(k,r)\}$ of size $r$, for $k=1,\ldots,n$. Given $\rho=\{\rho_{1},\ldots,\rho_{m}\}$ a partition of $\Delta_{n\times r}$, we let $\zeta^{\rho}:\Delta_{n\times r}\longrightarrow\{1,\ldots,m\}$ denote the mapping defined as

[TABLE]

$k=1,\ldots,n$, $l=1,\ldots,r$, $p=1,\ldots,m$. In other words, $\zeta^{\rho}(k,l)$ denotes the index $p$ of the block $\rho_{p}\subset\Delta_{n\times r}$ to which $(k,l)$ belongs.

Next, we restate Theorem 3.1 of [2] by noting that, in the same way as in Proposition 2.1, it extends to point processes admitting a Papangelou intensity using the arguments of [5], [4]. When $(u(z_{1},\ldots,z_{k};\omega))_{z_{1},\ldots,z_{k}\in X}$ is a multiparameter process, we will write

[TABLE]

and in this case we may drop the variable $\omega\in\Omega^{X}$ by writing $\epsilon^{+}_{\eufrak{z}_{k}}u(z_{1},\ldots,z_{k};\omega)$ instead of $\epsilon^{+}_{\eufrak{z}_{k}}u(z_{1},\ldots,z_{k};\omega)$.

Proposition 2.2

Let $u:X^{r}\times\Omega^{X}\longrightarrow{\mathord{{\rm I\kern-3.0ptR}}}$ be a (measurable) $r$-process. We have

[TABLE]

with $z^{\rho}_{\pi_{k}}:=(z_{\zeta^{\rho}(k,1)},\ldots,z_{\zeta^{\rho}(k,r)})$, $k=1,\ldots,n$.

Proof. The main change in the proof argument of [2] is to rewrite the proof of Lemma 2.1 therein by applying (1.2) recursively as in the proof of Theorem 3.1 of [5], while the main combinatorial argument remains identical. $\square$

When $n=1$, Proposition 2.2 yields a multivariate version of the Georgii-Nguyen-Zessin identity (1.1), i.e.

[TABLE]

Non-flat partitions

In the sequel we write $\nu\preceq\sigma$ when a partition $\nu\in\Pi[n\times r]$ is finer than another partition $\sigma\in\Pi[n\times r]$, i.e. when every block of $\nu$ is contained in a block of $\sigma$. We also let $\hat{0}:=\{\{1,1\},\ldots,\{n,r\}\}$ denote the partition of $\Delta_{n\times r}$ made of singletons, and we write $\rho\wedge\nu=\hat{0}$ when $\mu=\hat{0}$ is the only partition $\mu\in\Pi[n\times r]$ such that $\mu\preceq\nu$ and $\mu\preceq\rho$, i.e. $|\nu_{k}\cap\rho_{l}|\leq 1$, $k=1,\ldots,n$, $l=1,\ldots,|\rho|$. In this case the partition diagram of $\Gamma(\nu,\rho)$ of $\nu$ and $\rho$ is said to be non-flat, see Chapter 4 of [15].

A partition $\rho$ is non-flat if the partition diagram of $\Gamma(\pi,\rho)$ of $\rho$ and the partition $\pi:=(\pi_{1},\ldots,\pi_{n})\in\Pi[n\times r]$ with $\pi_{k}:=\{(k,1),\ldots,(k,r)\}$, $k=1,\ldots,n$, is non-flat. The following figure shows an example of a non-flat partition

\pi_{1}$$\pi_{2}$$\pi_{3}$$\pi_{4}$$\pi_{5}

with $n=5$, $r=4$, and

[TABLE]

Processes vanishing on diagonals

The next consequence of Proposition 2.2 shows that when $u(z_{1},\ldots,z_{r};\omega)$ vanishes on the diagonals in $X^{r}$, the moments of $\int_{X^{r}}u(z_{1},\ldots,z_{r};\omega)\omega(dz_{1})\cdots\omega(dz_{r})$ reduce to sums over non-flat partition diagrams.

Proposition 2.3

Assume that $u(z_{1},\ldots,z_{r};\omega)=0$ whenever $z_{i}=z_{j}$, $1\leq i\not=j\leq r$, $\omega\in\Omega^{X}$. Then we have

[TABLE]

When $n=1$, the first moment in Proposition 2.3 yields the Georgii-Nguyen-Zessin identity

[TABLE]

see Lemma IV.1 in [9] and Lemma 2.1 in [2] for different versions based on the Poisson point process. In the case of second moments, we find

[TABLE]

and since the non-flat partitions in $\Pi[2\times r]$ are made of pairs and singletons, this identity can be rewritten as the following consequence of Proposition 2.3, in which for simplicity of notation we write $\pi_{1}=\{1,\ldots,r\}$ and $\pi_{2}=\{r+1,\ldots,2r\}$.

Corollary 2.4

Assume that $u(z_{1},\ldots,z_{r};\omega)=0$ whenever $z_{i}=z_{j}$, $1\leq i\not=j\leq r$, $\omega\in\Omega^{X}$. Then the second moment of the integral of $k$-processes is given by

[TABLE]

where the above sum is over all bijections $\gamma:\pi_{2}\to A\cup\{r+|A|+1,\ldots,2r\}$.

Proof. We express the partitions $\rho\in\Pi[n\times r]$ with non-flat diagrams $\Gamma(\pi,\rho)$ in Proposition 3.1 as the collections of pairs and singletons

[TABLE]

for all subsets $A\subset\pi_{1}=\{1,\ldots,r\}$ and bijections $\gamma:\{1,\ldots,r\}\to A\cup\{r+|A|+1,\ldots,2r\}$. $\square$

In the case of $2$-processes, Corollary 2.4 shows that

[TABLE]

Similarly, in the case of $3$-processes we find

[TABLE]

3 Random-connection model

Two point process vertices $x\not=y$ are connected in the random-connection graph with the probability $H(x,y)$ independently of $\omega(dx)$, where $H:X\times X\longrightarrow[0,1]$. In particular, the $1$-hop count $\mathds{1}_{\{x\leftrightarrow y\}}$ is a Bernoulli random variable with parameter $H(x,y)$, and we have the relation

[TABLE]

for distinct $x_{1},\ldots,x_{r},y_{1},\ldots,y_{r-1}\in X$, where $x\leftrightarrow y$ means that $x\in X$ is connected to $y\in X$.

Given $x,y\in X$, the number of $(r+1)$-hop sequences $z_{1},\ldots,z_{r}\in\omega$ of vertices connecting $x$ to $y$ in the random graph is given by the multiparameter stochastic integral

[TABLE]

of the $r$-process

[TABLE]

which vanishes on the diagonals in $X^{r}$, with $z_{0}:=x$ and $z_{r+1}:=y$. In addition, for any distinct $z_{1},\ldots,z_{r}\in X$ and $u(z_{1},\ldots,z_{r};\omega)$ given by (3.1) we have

[TABLE]

therefore the first order moment of the $(r+1)$-hop count between $x\in X$ and $y\in X$ is given as

[TABLE]

as a consequence of the Georgii-Nguyen-Zessin identity (2.3).

In the next proposition we compute the moments of all orders of $r$-hop counts as sums over non-flat partition diagrams. The role of the powers $1/n^{\rho}_{l,i}$ in (3.4) is to ensure that all powers of $H(x,y)$ in (3.4) are equal to one, since all powers of $\mathds{1}_{\{z\leftrightarrow z^{\prime}\}}$ in (3) below are equal to $\mathds{1}_{\{z\leftrightarrow z^{\prime}\}}$.

Proposition 3.1

The moment of order $n$ of the $(r+1)$-hop count between $x\in X$ and $y\in X$ is given by

[TABLE]

where $z_{0}=x$, $z_{r+1}=y$, $\zeta^{\rho}(l,0)=0$, $\zeta^{\rho}(l,r+1)=r+1$, and

[TABLE]

$1\leq l\leq n$, $0\leq i\leq r$.

Proof. Since $u(z_{1},\ldots,z_{r};\omega)$ vanishes whenever $z_{i}=z_{j}$ for some $1\leq i<j\leq r$, by Proposition 2.3 we have

[TABLE]

where we applied (3.2). $\square$

As in Corollary 2.4 we have the following consequence of Proposition 3.1, which is obtained by expressing the partitions $\rho\in\Pi[n\times r]$ with non-flat diagrams $\Gamma(\pi,\sigma)$ as a collection of pairs and singletons.

Corollary 3.2

The second moment of the $(r+1)$-hop count between $x\in X$ and $y\in X$ is given by

[TABLE]

where the above sum is over all bijections $\gamma:\{1,\ldots,r\}\to A\cup\{r+|A|+1,\ldots,2r\}$ with $\gamma(0)=0$, $\gamma(r+1)=r+1$, $z_{0}=x$, $z_{r+1}=y$, and

[TABLE]

[TABLE]

$0\leq i\leq r$.

Variance of $3$-hop counts

When $n=2$ and $r=2$, Corollary 3.2 allows us to express the variance of the $3$-hop count between $x\in X$ and $y\in X$ as follows:

[TABLE]

Variance of $4$-hop counts

When $r=3$ and $n=2$, Corollary 3.2 yields

[TABLE]

4 Poisson case

In this section and the next one, we work in the Poisson random-connection model, using a Poisson point process on $X={\mathord{{\rm I\kern-3.0ptR}}}^{d}$ with intensity $\lambda(dx)$ on ${\mathord{{\rm I\kern-3.0ptR}}}^{d}$. We let

[TABLE]

The $2$-hop count between $x\in{\mathord{{\rm I\kern-3.0ptR}}}^{d}$ and $y\in{\mathord{{\rm I\kern-3.0ptR}}}^{d}$ is given by the first order stochastic integral

[TABLE]

and its moment of order $n$ is

[TABLE]

therefore, from (2.1), the $2$-hop count between $x\in{\mathord{{\rm I\kern-3.0ptR}}}^{d}$ and $y\in{\mathord{{\rm I\kern-3.0ptR}}}^{d}$ is a Poisson random variable with mean

[TABLE]

By (3.3), the first order moment of the $r$-hop count is given by

[TABLE]

Corollary 4.1

The variance of the $r$-hop count between $x\in{\mathord{{\rm I\kern-3.0ptR}}}^{d}$ and $y\in{\mathord{{\rm I\kern-3.0ptR}}}^{d}$ is given by

[TABLE]

with $k_{0}=l_{0}=0$, $k_{p+1}=l_{p+1}=r$, $\sigma(0)=0$, and $\sigma(r)=r$, where the above sum if over all permutations $\sigma\in\Sigma[p]$ of $\{1,\ldots,p\}$.

Proof. We rewrite the result of Corollary 3.2 by denoting the set $A\subset\pi_{1}$ as $A=\{k_{1},\ldots,k_{p}\}$, for $1\leq k_{1}<\cdots<k_{p}\leq r-1$, and we identify $\gamma(A)\subset A\cup\{r+|A|+1,\ldots,2r\}$ to $\{l_{1},\ldots,l_{p}\}$, which requires a sum over the permutations of $\{1,\ldots,p\}$ since $1\leq l_{1}<\cdots<l_{p}\leq r-1$, where $1\leq p\leq r-1$. In addition, the multiple integrals over contiguous index sets in $A^{c}$ are evaluated using (4.1). $\square$

Variance of $3$-hop counts

When $n=2$ and $r=3$ Corollary 4.1 allows us to compute the variance of the $3$-hop count between $x\in{\mathord{{\rm I\kern-3.0ptR}}}^{d}$ and $y\in{\mathord{{\rm I\kern-3.0ptR}}}^{d}$, as follows:

[TABLE]

Variance of $4$-hop counts

By Corollary 4.1 we have

[TABLE]

5 Rayleigh fading

In this section we consider a Poisson point process on $X={\mathord{{\rm I\kern-3.0ptR}}}^{d}$ with flat intensity $\lambda(dx)=\lambda dx$ on ${\mathord{{\rm I\kern-3.0ptR}}}^{d}$, $\lambda>0$, and a Rayleigh fading function of the form

[TABLE]

Lemmas 5.1 and 5.2 can be used to evaluate the integrals appearing in Corollary 4.1 and in the variance (4.3) of $4$-hop counts.

Lemma 5.1

For all $n\geq 1$, $y_{1},\ldots,y_{n}\in{\mathord{{\rm I\kern-3.0ptR}}}^{d}$ and $\beta_{1},\ldots,\beta_{n}>0$ we have

[TABLE]

Proof. We start by showing that for all $n\geq 1$ we have

[TABLE]

Clearly, this relation holds for $n=1$. In addition, at the rank $n=2$ we have

[TABLE]

Next, assuming that (5) holds at the rank $n\geq 1$, we have

[TABLE]

As a consequence, we find

[TABLE]

$\square$

In particular, applying Lemma 5.1 for $n=2$ yields

[TABLE]

and the $2$-hop count between $x\in{\mathord{{\rm I\kern-3.0ptR}}}^{d}$ and $y\in{\mathord{{\rm I\kern-3.0ptR}}}^{d}$ is a Poisson random variable with mean

[TABLE]

By an induction argument similar to that of Lemma 5.1, we obtain the following lemma.

Lemma 5.2

For all $n\geq 1$, $x_{0},\ldots,x_{n}\in{\mathord{{\rm I\kern-3.0ptR}}}^{d}$ and $\beta_{1},\ldots,\beta_{n}>0$ we have

[TABLE]

Proof. Clearly, the relation holds at the rank $n=1$. Assuming that it holds at the rank $n\geq 1$ and using (5), we have

[TABLE]

$\square$

In particular, the first order moment of the $r$-hop count between $x_{0}\in{\mathord{{\rm I\kern-3.0ptR}}}^{d}$ and $x_{r}\in{\mathord{{\rm I\kern-3.0ptR}}}^{d}$ is given by

[TABLE]

Variance of $3$-hop counts

Corollary 4.1 and Lemma 5.2 allow us to recover Theorem II.3 of [9], for the variance of $3$-hop counts by a shorter argument, while extending it from the plane $X={\mathord{{\rm I\kern-3.0ptR}}}^{2}$ to $X={\mathord{{\rm I\kern-3.0ptR}}}^{d}$.

Corollary 5.3

The variance of the $3$-hop count between $x\in{\mathord{{\rm I\kern-3.0ptR}}}^{d}$ and $y\in{\mathord{{\rm I\kern-3.0ptR}}}^{d}$ is given by

[TABLE]

Proof. By (5.3) and Lemma 5.2 we have

[TABLE]

and we conclude by (4.2). $\square$