# The Cut Method on Hypergraphs for the Wiener Index

Sandi Klavžar a,b,c Gašper Domen Romih a

###### Abstract

The cut method has been proved to be extremely useful in chemical graph
theory. In this paper the cut method is extended to hypergraphs. More
precisely, the method is developed for the Wiener index of $k$-uniform partial
cube-hypergraphs. The method is applied to cube-hypergraphs and hypertrees.
Extensions of the method to hypergraphs arising in chemistry which are not
necessary $k$-uniform and/or not necessary linear are also developed.

a Faculty of Mathematics and Physics, University of Ljubljana, Slovenia

<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

b Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia

c Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

Keywords: hypergraph; Wiener index; cut method; partial cube-hypergraph;
hypertree; phenylene; Clar structure

##  1 Introduction

The cut method, whose standard form was introduced in 1995 in [17], has had a
remarkable response in chemical graph theory. The method originally designed
for the Wiener index of partial cubes was later developed for many other
topological indices and has undergone many generalizations to more general
situations than partial cubes. This applies in itself to many applications in
mathematical chemistry where topological indices play important role. The
basic idea is to first find a partition of the edges of a (molecular) graph
and by removing parts of this partition construct smaller (weighted) graphs,
called quotient graphs. After that, we infer back to the original graph from
the quotient graphs. The state of research on the cut method up to 2015 is
summarized in the survey article [18]. The method is still the subject of
ongoing research, see [1, 3, 4, 6, 11, 31, 32] as well as references therein.

Hypergraphs form a structure that greatly generalizes the concept of a graph.
In chemical graph theory, the standard method of representing molecules is by
means of associated (chemical) graphs. However, some molecules are more
complicated than others and sometimes it is more convenient and more adequate
to represent them as hypergraphs, see [14, 20] for some chemical problems
dealing with hypergraph theory. As a result, various problems of importance in
mathematical chemistry have been investigated on hypergraphs, including
spectral aspects [2, 23, 29] and different topological indices [33, 34]. Very
recently, while investigating molecular representations in drug design, a
hypergraph-based topological framework was designed to characterize molecular
structures and interactions at atomic level [25]. Interestingly, in the very
same year when the cut method was introduced, Burosch and Ceccherini published
the paper [7] on isometric embeddings into hypergraphs, which is the second
main source for the present paper.

The Wiener index is one of the most researched topics in the whole field of
chemical graph theory. As already mentioned, the cut method was first designed
for the Wiener index of graphs. In the last few years, the Wiener index has
received a lot of attention also on hypergraphs. In [30] the authors
investigate, among others, $3$-uniform paths and lower bounds on the Wiener
index of $k$-uniform hypergraphs. In [12, 22] hypergraphs are constructed from
trees and their Wiener index investigated. The effect of some transformations
on the Wiener index of a hypergraph and extremal hypertrees with respect to
the Wiener index is studied in [24]. The $k$-uniform unicyclic hypergraphs
with maximum/minimum and second maximum/minimum Wiener index are determined in
[35], while the Wiener index of some composite hypergraphs and sunflower
hypergraphs is the topic of [5]. Finally, in [8] the concept of the $k$-Wiener
index is introduced and studied on the so called $k$-plex hypergraphs.

We proceed as follows. In the next section we introduce the mathematical
machinery on hypergraphs needed latter on. In particular, partial cube-
hypergraphs are defined and their characterizations recalled. In Section 3 we
develop the cut method for the Wiener index of a hypergraph. In the last
section we provide applications and extensions of the cut method including
cube-hypergraphs, hypertrees, and the so called linear phenylene hypergraphs.

##  2 Preliminaries

In this section, we set the scene for the hypergraph cut method. In the first
part, we introduce the necessary concepts about hypergraphs, focusing on
distance and their Cartesian products. We then introduce partial cube-
hypergraphs on which the cut method will operate and recall two of their
characterizations.

###  2.1 Hypergraphs

A hypergraph $H=(V(H),E(H))$ has the vertex set $V(H)$ and the edge set
$E(H)$, where each edge $e\in E(H)$ is a non-empty subset of $V(H)$. $H$ is
$k$-uniform if the size of every edge $e\in E(H)$ is $k$ and is linear if
$|e\cap e^{\prime}|\leq 1$ for every $e,e^{\prime}\in E(H)$, $e\neq
e^{\prime}$. Let $H$ and $H^{\prime}$ be hypergraphs. If
$V(H^{\prime})\subseteq V(H)$ and $E(H^{\prime})\subseteq E(H)$ we say that
$H^{\prime}\subseteq H$ is a subhypergraph of $H$. Clearly, if $H$ is
$k$-uniform, then $H^{\prime}$ is also $k$-uniform. If $F\subseteq E(H)$, then
$H-F$ denotes the subhypergraph of $H$ obtained from $H$ by removing all the
edges from $F$.

Let $u$ and $v$ be different vertices of $H$. A $u,v$-path of length $s\geq 1$
in $H$ is a sequence $u_{0}=u,e_{1},u_{1},\ldots,e_{s},u_{s}=v$, where $u_{i}$
are pairwise different vertices, $e_{i}$ are pairwise different edges, and
$\\{u_{i-1},u_{i}\\}\subseteq e_{i}$ for $i\in[s]=\\{1,\ldots,s\\}$. The
distance $d_{H}(u,v)$ between vertices $u$ and $v$ is the length of a shortest
$u,v$-path. We also set $d_{H}(u,u)=0$. A subhypergraph $H^{\prime}\subseteq
H$ is isometric if $d_{H^{\prime}}(u,v)=d_{H}(u,v)$ holds for all $u,v\in
V(H^{\prime})$. We further say that a set of vertices $X\subseteq V(H)$ is
convex in $H$ if for every $u,v\in X$ and every $z\in V(H)$, the equality
$d_{H}(x,z)+d_{H}(z,y)=d_{H}(x,y)$ implies $z\in X$. The Wiener index of a
hypergraph $H$ is defined as the sum of the distances between all unordered
pairs of vertices of $H$, that is,

$W(H)=\sum_{\\{u,v\\}\in\binom{V(H)}{2}}d_{H}(u,v).$

The Cartesian product $H\,\square\,H^{\prime}$ of hypergraphs $H$ and
$H^{\prime}$ is a hypergraph with the vertex set $V(H)\times V(H^{\prime})$
and the edge set

$\\{\\{u\\}\times e^{\prime}:\ u\in V(H),\ e^{\prime}\in
E(H^{\prime})\\}\cup\\{e\times\\{u^{\prime}\\}:\ e\in E(H),\ u^{\prime}\in
V(H^{\prime})\\}.$

Just as Cartesian products of graphs, Cartesian products of hypergraphs have
several nice properties, cf [15, 16]. In particular, if $H$ and $H^{\prime}$
are $k$-uniform hypergraphs, then $H\,\square\,H^{\prime}$ is also
$k$-uniform, and the Cartesian product operation is associative. For $k\geq
2$, let ${\cal Q}_{k}$ denote the hypergraph with $k$ vertices and a single
edge containing all the vertices. For $n\geq 1$, the $k$-uniform $n$-cube
${\cal Q}_{k}^{n}$ is the Cartesian product of $k$ copies of ${\cal Q}_{k}$.
See Fig. 1 where ${\cal Q}_{3}^{1}$, ${\cal Q}_{3}^{2}$, and ${\cal
Q}_{3}^{3}$ are presented.

${\cal Q}_{3}^{1}$

${\cal Q}_{3}^{2}$

${\cal Q}_{3}^{3}$

Figure 1: Cube-hypergraphs

The $k$-uniform $n$-cube ${\cal Q}_{k}^{n}$ can be equivalently described as
follows. Its vertex set is $\\{0,1,\ldots,k-1\\}^{n}$ and an edge consists of
all $n$-tuples which coincide on $n-1$ coordinates while the remaining
coordinate ranges over $\\{0,1,\ldots,k-1\\}$. It follows that $|V({\cal
Q}_{k}^{n})|=k^{n}$ and $|E({\cal Q}_{k}^{n})|=n\cdot k^{n-1}$. Note that
${\cal Q}_{2}^{n}$ is a $2$-uniform hypergraph which is as a graph known as
the $n$-cube.

###  2.2 Partial cube-hypergraphs

A $k$-uniform hypergraph $H$ is a partial cube-hypergraph if $H$ is an
isometric subhypergraph of some ${\cal Q}_{k}^{n}$.

A hypergraph $H$ is edge-gated if for any edge $e=\\{a_{1},\ldots,a_{k}\\}\in
E(H)$ and any vertex $x\in V(H)$ there exists $j\in[k]$ such that
$d_{H}(x,a_{i})=d_{H}(x,a_{j})+1$ for $i\in[k]$, $i\neq j$. We also say that
$a_{j}$ is the gate of $x$ in $e$. Note that if $x\in e$ then $x$ is its own
gate in $e$.

It is easy to see that in $2$-uniform hypergraphs (alias graphs) $H$ is edge-
gated if and only if $H$ is bipartite. From this reason, edge-gated
hypergraphs were named bipartite hypergraphs in [7], where this concept was
originally introduced. However, since there are numerous ways how bipartite
graphs can be extended to hypergraphs we decided to change the terminology.
The present terminology also mimics the established graph terminology, cf.
[10].

It is easy to see that if hypergraphs $H$ and $H^{\prime}$ are both edge-gated
then so is $H\,\square\,H^{\prime}$. Also, if $H^{\prime}$ is a connected
isometric subgraph of an edge-gated hypergraph $H$, then $H^{\prime}$ is edge-
gated as well. It follows that partial cube-hypergraphs and hence in
particular $k$-uniform $n$-cubes are edge-gated.

If $x$ and $y$ are two (adjacent) vertices of a hypergraph $H$, then let
$H(x,y)$ denote the set of vertices that are closer to $x$ than to $y$, that
is,

$H(x,y)=\\{z\in V(H):\ d_{H}(z,x)<d_{H}(z,y)\\}.$

Further, if $e=\\{a_{1},\ldots,a_{k}\\}\in E(H)$, then let

$H(a_{i},e)=\\{z\in V(H):\ d_{H}(z,a_{i})<d_{H}(z,a_{j}),\ j\neq i\\}.$

In addition, set

$H_{e}=\\{H(a_{1},e),\ldots,H(a_{k},e)\\}.$

Let now $H$ be an edge-gated hypergraph and $e=\\{a_{1},\ldots,a_{k}\\}\in
E(H)$. Since $a_{i}\in H(a_{i},e)$ we have the following important facts.

######  Lemma 2.1.

[7, Lemma 1(ii), Lemma 2] If $H$ is an edge-gated hypergraph and
$e=\\{a_{1},\ldots,a_{k}\\}\in E(H)$, then the following statements hold.

  1. (i)

$H_{e}$ is a partition of $V(H)$.

  2. (ii)

If $e^{\prime}\in E(H)$, then either $|e^{\prime}\cap H(a_{i},e)|=1$ for all
$i\in[k]$ or $e^{\prime}\subseteq H(a_{i},e)$ for some $i\in[k]$.

We next recall the following, key definition from [7]. If $H$ is a hypergraph,
then the binary relation $\Theta$ is defined on $E(H)$ as follows:

$e\Theta e^{\prime}\ \equiv\ \forall A\in H_{e}:\ e^{\prime}\cap
A\neq\emptyset.$

Note first that for any edge $e\in E(H)$ we have $e\Theta e$. If $H$ is edge-
gated, then $\Theta$ is also symmetric by Lemma 2.1(ii). Moreover, we recall
the following important fact.

######  Lemma 2.2.

[7, Lemma 3] If $H$ is an edge-gated hypergraph and for every $e\in E(H)$,
every $A\in H_{e}$ is convex, then $f\Theta f^{\prime}$ if and only if
$H_{f}=H_{f^{\prime}}$.

For hypergraphs which fulfil the conditions of Lemma 2.2, the relation
$\Theta$ is an equivalence relation where the transitivity is guaranteed by
Lemma 2.2. Partial cube-hypergraphs which are $k$-uniform can now be
characterized as follows.

######  Theorem 2.3.

[7, Theorem 1] A $k$-uniform hypergraph $H$ is a partial cube-hypergraph if
and only if $H$ is edge-gated and for every $e\in E(H)$, every $A\in H_{e}$ is
convex.

######  Theorem 2.4.

[7, Theorem 2] A $k$-uniform hypergraph $H$ is a partial cube-hypergraph if
and only if $H$ is edge-gated and $\Theta$ is transitive.

##  3 Cut method for hypergraphs

We now have all the tools needed for the main theorem of this article. But
before we can formulate it, we need two additional auxiliary results and the
following concepts.

If $H$ is a connected hypergraph, then $F\subseteq E(H)$ is a cut if the edges
from $F$ are pairwise disjoint and $H-F$ consists of at least two components.
We further say that the cut $F$ is a convex cut if the vertex set of each
component of $H-F$ is a convex set.

Let $H$ be a $k$-uniform partial cube-hypergraph. Theorems 2.3 and 2.4 imply
that the $\Theta$ relation is an equivalence relation on $E(H)$. We will
denote its equivalence classes by $F_{1},\ldots,F_{m}$. In addition, if $e\in
E(H)$, then the equivalence class with the representative $e$ will also be
denoted by $F_{e}$, that is, $F_{e}=\\{f\in E(H)\ :\ e\Theta f\\}$.

From Lemma 2.2 we infer that the hypergraph $H-F_{e}$ consists of components
whose vertex sets are precisely the sets from $H_{e}$. This yields the
following important fact.

######  Proposition 3.1.

Let $H$ be $k$-uniform partial cube-hypergraph and let $e\in E(H)$. Then
$H-F_{e}$ has exactly $k$ components.

We also need the following auxiliary result.

######  Proposition 3.2.

Let $H$ be $k$-uniform partial cube-hypergraph and let $e\in E(H)$. If $u$ and
$v$ are vertices from different components of $H-F_{e}$, then every shortest
$u,v$-path contains exactly one edge from $F_{e}$.

###### Proof.

By Proposition 3.1, $H-F_{e}$ contains $k$ components which we denote by
$H_{1},\ldots H_{k}$. We may without loss of generality assume that $u\in
H_{1}$ and $v\in H_{k}$. Furthermore, let $F_{e}=\\{e_{1},\ldots,e_{\ell}\\}$.
By Lemma 2.2(ii) the vertices $u_{i}$ and $v_{i}$ defined as

$\\{u_{i}\\}=V(H_{1})\cap e_{i}\quad\mbox{and}\quad\\{v_{i}\\}=V(H_{k})\cap
e_{i}$

are well-defined for every $i\in[\ell]$. From the edge-gated property of $H$
it follows that $d_{H}(u_{i},v)=d_{H}(v_{i},v)+1$. Since every $u,v$-path
contains at least one of the vertices $u_{i}$, every shortest $u,v$-path
contains exactly one of the edges $e_{i}$, $i\in[\ell]$. ∎

Let $H$ be a $k$-uniform partial cube-hypergraph and let $F_{1},\ldots,F_{m}$
be its $\Theta$-classes. By Proposition 3.1, $H-F_{i}$ has $k$ components, we
denote them in the sequel by $H_{1}(F_{i}),\ldots,H_{k}(F_{i})$. Set in
addition

$n_{j}(F_{i})=|V(H_{j}(F_{i}))|,\ j\in[k],\ i\in[m].$ (1)

The cut method for hypergraphs now reads as follows.

######  Theorem 3.3.

If $H$ is a $k$-uniform partial cube-hypergraph, $F_{1},\ldots,F_{m}$ are its
$\Theta$-classes, and integers $n_{j}(F_{i})$ are defined as in (1), then

$W(H)=\sum_{i=1}^{m}\sum_{\\{j,j^{\prime}\\}\in{[k]\choose
2}}n_{j}(F_{i})\cdot n_{j^{\prime}}(F_{i})\,.$

###### Proof.

Since $F_{1},\ldots,F_{m}$ form a partition of $E(H)$, the idea is to consider
the contribution of each edge to $W(H)$. Consider arbitrary vertices $u$ and
$v$ of $H$ and an arbitrary $u,v$-shortest path $P$. By Proposition 3.2, edges
from $P$ pairwise lie in different $\Theta$-classes of $E(H)$. If $e$ is an
edge of $P$, then the contribution of $F_{e}$ to the distance $d_{H}(u,v)$ is
exactly $1$. Consequently, the contribution of $F_{e}$ to $W(H)$ is exactly

$\sum_{\\{j,j^{\prime}\\}\in{[k]\choose 2}}n_{j}(F_{e})\cdot
n_{j^{\prime}}(F_{e}).$

Summing over all $\Theta$-classes the result follows. ∎

##  4 Some applications

In this section we give some examples and applications of Theorem 3.3.

###  4.1 Cube-hypergraphs

Cube-hypergraphs are partial cube-hypergraphs by definition. Hence Theorem 3.3
applies to them and leads to the following result.

######  Proposition 4.1.

If $n\geq 1$ and $k\geq 2$, then

$W({\cal Q}_{k}^{n})=n{k\choose 2}k^{2(n-1)}.$

###### Proof.

To apply Theorem 3.3, we first determine the $\Theta$-classes of ${\cal
Q}_{k}^{n}$. Let an edge $e\in E({\cal Q}_{k}^{n})$ be of the form
$\\{a_{i}=(i,0,\ldots,0)\ :\ i\in\\{0,1,\ldots,k-1\\}\\}$. Then $H(e,a_{i})$
contains the vertices $(i,v_{2},\ldots,v_{n})$, where
$(v_{2},\ldots,v_{n})\in\\{0,1,\ldots,k-1\\}^{n-1}$. By Theorem 2.3, sets
$H(e,a_{i})$ are convex and the subhypergraphs induced by them are isomorphic
to ${\cal Q}_{k}^{n-1}$. Then the $\Theta$-class $F_{e}=F_{1}$ contains all
the edges whose last $n-1$ coordinates are fixed and the first coordinate
ranges from $0$ to $k-1$. Using the same reasoning we get that every
$\Theta$-class is of the above form. Therefore ${\cal Q}_{k}^{n}$ has
$\Theta$-classes $F_{1},\ldots,F_{n}$ where ${\cal Q}_{k}^{n}-F_{i}$ has
components which are isomorphic to ${\cal Q}_{k}^{n-1}$ for $i\in[n]$. It then
follows that $n_{j}(F_{i})=k^{n-1}$ for every $j\in[k]$ and $i\in[n]$. From
Theorem 3.3 it follows that

$\displaystyle W({\cal Q}_{k}^{n})$
$\displaystyle=\sum_{i=1}^{n}\sum_{\\{j,j^{\prime}\\}\in{[k]\choose
2}}k^{n-1}\cdot k^{n-1}=n\binom{k}{2}k^{2(n-1)},$

which we wanted to show.

∎

Setting $k=2$, the hypergraph ${\cal Q}_{2}^{n}$ is the $n$-cube graph $Q_{n}$
and Proposition 4.1 implies a well-known result $W(Q_{n})=n4^{n-1}$, which can
in particular be deduced from the formula for the Wiener index of Cartesian
products [13].

###  4.2 Hypertrees

A hypergraph $T$ is a hypertree if it is connected, linear, and has no cycles.
Here a cycle in a hypergraph is defined just as we defined a path except that
the first and the last vertex from the corresponding sequence coincide. A
hypertree which is linear and $k$-uniform is a partial cube-hypergraph where
every edge $e$ is it own $\Theta$-class. Hence Theorem 3.3 as a special case
yields the following result.

######  Corollary 4.2.

If $T$ is a $k$-uniform hypertree, then

$W(T)=\sum_{e\in E(T)}\sum_{\\{j,j^{\prime}\\}\in\binom{[k]}{2}}n_{j}(e)\cdot
n_{j^{\prime}}(e),$

where $n_{j}(e)=n_{j}(F_{e})$.

Actually Corollary 4.2 holds also if we do not require that a hypertree is
uniform. For this sake one just needs to reformulate Proposition 3.1 such that
its conclusion asserts that for any edge $e\in E(T)$, the hypergraph $T-e$ has
exactly $|e|$ components. Moreover the second key auxiliary result,
Proposition 3.2, also holds by the fact that in a hypertree there is a unique
shortest path between two vertices. In this way Corollary 4.2 extends to

######  Theorem 4.3.

[28, Theorem 3] If $T$ is a hypertree, then

$W(T)=\sum_{e\in
E(T)}\sum_{\\{j,j^{\prime}\\}\in\binom{[|e|]}{2}}n_{j}(e)\cdot
n_{j^{\prime}}(e).$

For an example consider a hypertree $T_{1}$ from Figure 2. The hypertree
$T_{1}$ has seven vertices and four edges. We now apply Theorem 4.3. For
instance consider the edge $e=\\{a_{1},a_{2},a_{3}\\}$ as shown in the figure.
Then $n_{1}(e)=2$, $n_{2}(e)=1$ and $n_{3}(e)=4$. Therefore the contribution
of $e$ to the formula of Theorem 4.3 is $2\cdot 1+1\cdot 4+2\cdot 4$. Doing
similar computations for the other three edges (see the bottom line of Fig. 2)
we get

$W(T_{1})=1\cdot 6+(2\cdot 1+1\cdot 4+2\cdot 4)+(5\cdot 1+5\cdot 1+1\cdot
1)+6\cdot 1=37.$

$e$$a_{1}$$a_{2}$$a_{3}$ Figure 2: Hypertree $T_{1}$.

A limitation of Theorem 4.3 is that it only works for linear hypertrees. On
the other hand, there exist many different definitions of acyclicity in
hypergraphs, where some of them also allow for non-linear hypergraphs. See for
example [27]. We next show with an example that the main idea of Theorem 4.3
can sometimes be generalized to such cases as well.

Define the linear phenylene hypergraphs $LP_{n}$, $n\geq 2$, as follows. (For
some recent studies of phenylenes in mathematical chemistry see [9, 19, 21,
26].) $LP_{n}$ has vertex set $[6n]$. It has $2n-1$ hyperedges. The first $n$
of them are of the form $\\{6i+1,6i+2,\ldots,6i+6\\}$ where
$i\in\\{0,1,\ldots,n-1\\}$, and the remaining $n-1$ hyperedges edges are of
the form $\\{6i+5,6i+6,6i+7,6i+8\\}$, where $i\in\\{0,1,\ldots,n-2\\}$. In
Figure 3 the hypergraph $LP_{4}$ is drawn.

Figure 3: Hypergraph $LP_{4}$.

It is easy to see that every edge $e\in E(LP_{n})$ is a convex cut with the
following property. Taking any two vertices $u,v$ from different components of
$LP_{n}-e$, every shortest $u,v$-path contains $e$ (exactly once). Note,
however, that the two vertices which lie in the intersection of two hyperedges
are not separated by any of the cuts. But it is clear that the distance
between such two vertices is $1$. Together there are $2(n-1)$ such pairs and
therefore this number needs to be added to the Wiener index of $LP_{n}$. This
is enough to calculate the Wiener index of $LP_{n}$ using cut method as
follows.

Removing an edge of the form $\\{6i+1,6i+2,\ldots,6i+6\\}$, where $i\in[n-2]$,
produces four components where two of them contain a single vertex and the
remaining two have $6i+2$ and $6(n-i-1)+2$ vertices, respectively. The cases
when $i=0$ or $i=n-1$ give five components each, four of them contain a single
vertex, while the remaining one contains $6n-4$ vertices. On the other hand,
removing an edge of the form $\\{6i+5,6i+6,6i+7,6i+8\\}$ produces two
components with $6(i+1)$ and $6(n-i-1)$ vertices, respectively. Therefore, the
contribution of all these cuts to the Wiener index of $LP_{n}$ for $n>1$ is

$\displaystyle\sum_{i=1}^{n-2}\left[2(6i+2+6(n-i-1)+2)+(6i+2)(6n-6i-4)+1\right]$
$\displaystyle+$ $\displaystyle 2\left(\binom{4}{2}+4(6n-4)\right)$
$\displaystyle+$ $\displaystyle\sum_{i=0}^{n-2}\left[6(i+1)\cdot
6(n-i-1)\right]=12n^{3}+6n^{2}-5n+2,$

where the second line above comes from the contribution of the first hyperedge
and the last hyperedge containing six vertices. Adding to this expression the
contribution $2(n-1)$ from previous paragraph and performing a straightforward
computation we arrive to the following result.

######  Proposition 4.4.

If $n\geq 2$ then, $W(LP_{n})=12n^{3}+6n^{2}-3n$.

###  4.3 More elaborate example

The cut method as developed in Section 3 assumes that a hypergraph is a
$k$-uniform partial cube-hypergraph. In general this is a strong assumption.
We have just demonstrated in Section 4.2 that the method can be extended also
when the hypergraph is not $k$-uniform partial cube-hypergraph, provided that
Propositions 3.1 and 3.2 remain valid. In the subsequent example we further
elaborate this idea on a mulecular hypergraph $H$ of a Clar structure which is
shown in Figure 4(a) and in [14, Fig. 3].

(a) Hypergraph $H$

(b) Cut type I

(c) Cut type II

Figure 4: Hypergraph $H$ and its convex cuts.

There are two different types of cuts in $H$. The cut of type I consists of
the central $6$-edge and three $2$-edges that do not intersect it as can be
seen in Figure 4(b). A cut of type II consists of a non-central $6$-edge and
its opposite $2$-edge as can be seen in Figure 4(c). Both cuts are convex and
also the conclusion of Proposition 3.2 holds. This, together with the fact
that $E(H)$ partitions into one cut of type I and six cuts of type II, allows
us to use the cut method to calculate Wiener index of $H$ as

$\displaystyle W(H)=\binom{6}{2}7\cdot 7+6\left(\binom{4}{2}+4\cdot 7+4\cdot
31+7\cdot 31\right)=2985.$

## Acknowledgements

This work has been supported by the financial support from the Slovenian
Research Agency (research core funding P1-0297 and projects J1-2452 and
N1-0285).

## Declaration of interests

The authors declare that they have no conflict of interest.

## Data availability

Our manuscript has no associated data.

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# Free space optical link to a tethered balloon for frequency transfer and
chronometric geodesy

Nicolas Maron  1,2 Sébastien Fernandez  1,2 François-Xavier Esnault  2 Thomas
Lévèque  2 Tepuaonini Muzeau3 and Peter Wolf1  1LNE-SYRTE, Observatoire de
Paris, Université PSL, CNRS, Sorbonne Université, 61 avenue de l’Observatoire,
75014 Paris, France
2Centre National d’Etudes Spatiales, 18 avenue Edouard Belin, 31400 Toulouse,
France
3Institut National des Sciences Appliquées de Toulouse, 31400 Toulouse, France
<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

We present the results of an optical link to a corner cube on board a tethered
balloon at 300 m altitude including a Tip/Tilt compensation for the balloon
tracking. Our experiment measures the carrier phase of a 1542 nm laser, which
is the useful signal for frequency comparison of distant clocks. An active
phase noise compensation of the carrier is implemented, demonstrating a
fractional frequency stability of $8\times 10^{-19}$ after 16 s averaging,
which slightly (factor $\sim$ 3) improves on best previous links via an
airborne platform. This state-of-the-art result is obtained with a
transportable set-up that enables a fast field deployment.

##  1 Introduction

Optical clocks have demonstrated uncertainties in the low $10^{-18}$ region in
fractional frequency, after integration times of order $10^{3}$ seconds [1, 2,
3]. This makes them prime candidates for applications in navigation, geodesy,
and fundamental physics [4].

Currently, only optical methods using the carrier phase or femtosecond pulses,
allow comparing such clocks without degrading their performance. Such methods
have been implemented in optical fibre links [5, 6, 7, 8] and more recently
through free space [9, 10, 11, 12, 13, 14, 15, 16]. Many of the applications
mentioned above require intercontinental and/or satellite links, and efforts
to work towards space optical links for ultra-stable clock comparisons are on
the way [17, 18, 19].

In the meantime, ground links between clocks over shorter distances are
required, not only as demonstrators for space, but also for field measurement
of the local geopotential via the relativistic gravitational redshift of
clocks, known as chronometric geodesy [20, 21, 22]. This requires not only a
high performance clock (a fractional frequency change of $1\times 10^{-18}$
corresponds to a change in orthometric height of $\sim 1$ cm), but also a
practical method to compare the frequency of that clock to a reference clock
in a fixed location. The latter would be typically delivered at the nearest
access to a fibre network, in general at a distance of order $\sim 100$ km.

Thus, using clocks for centimetric mapping of the geopotential at multiple
points in regions with difficult access (mountains, coastline, …) [20], is
expected to be one of the first actual application of transportable optical
clocks [2, 23] and medium distance ($\sim 100$ km) free-space optical links.
Such links will rarely have direct point-to-point unobstructed lines of sight,
and work is on the way to establish links through airborne relays using drones
[12, 16]. Those experiments have demonstrated frequency stability of
$2.5\times 10^{-18}$ (in terms of modified Allan deviation, MDEV) for
integration times of a few seconds. The work presented in this paper is along
the same lines, with similar (marginally better) performance, but uses a
balloon, which brings advantages like slower motion and potentially much
higher altitude (up to stratospheric) allowing common views over much larger
baselines. Additionally balloons or stratospheric platforms persist over much
longer times (days or even weeks) allowing fine study of time varying effects
e.g. of tidal, hydrological or geological origin.

Free space optical-link technology for clock comparisons has developed rapidly
in the wake of optical clocks, from pioneering experiments demonstrating the
feasibility, potential, and main limitations [9], to near quantum limited
performance compatible with ground to geostationary orbit distances [19].
Three geometries have been used: (i) Genuine point to point links (with a
fibre link in parallel for performance evaluation) [13, 14]; (ii) Partially
folded links with separated emitter and receiver closely co-located [10, 11,
19]; (iii) Fully folded links with common emitter and receiver [15]. Strictly
speaking only links of type (i) allow a full, loop-hole free, demonstration of
the link performance. Indeed, the performance relies on reduction by several
orders of magnitude of the main phase noise sources (e.g. atmospheric
turbulence, motion of terminals, aperture related effects) in real time or
post-analysis. Such reduction requires reciprocity between the noise of the
outward and return paths [24], and folded links may introduce correlations
between the two paths that would not be there in a realistic scenario of point
to point clock comparisons.

Nonetheless, folded links still allow studying key aspects and technological
performance for optical clock comparisons, in particular when used in
conjunction with airborne relays. The corresponding technological
demonstration includes tracking of a moving target, effect of link outages,
phase noise compensation functionality and performance, transportability and
compactness.

All demonstrations to airborne platforms have so far used fully folded
architectures (type (iii) above) [12, 16], and our experiment is no exception.
Nevertheless, in contrast to previous work, our demonstration is based on a
transportable and quickly field deployable setup, which implements a link to a
tethered balloon rather than drones. Thus, our results are an essential
milestone for point to point ground to airborne links, and by extension for
ground to ground links via an airborne relay for intermediate distances.

Our article is organised as follows. In section 2 we describe the experimental
set-up with the results shown in section 3. We also show results from an
equivalent (same distance) folded link with a static target on the ground. In
section 4 we discuss our results, study the main noise sources, and put them
into context with respect to previous work. We conclude with an outlook to
future work in section 5.

##  2 Experimental scheme

Our experimental setup, depicted on Fig. 2 consists off a free-space phase-
compensated laser link between a transportable ground segment and an airborne
corner cube. The link stabilization is realized through a heterodyne Michelson
architecture and the system allows to perform a simultaneous out-of-loop phase
measurement. The ground segment of our system is split into a phase
compensation and measurement unit and an optical terminal. The airborne
segment consists of a corner cube reflector and four beacon laser diodes
attached to a tethered helium balloon. The complete setup was developed and
tested at CNES premises in Toulouse and then transported to the Aire sur
l’Adour balloon operation centre of CNES for field measurements. The ground
segment fits easily into a 6 m3 mini-van and can be deployed by two operators
within a short time (typ. $<$ 30 minutes between arrival and full lock).
Figure 2 shows images of the ground terminal, payload and balloon during
operation.

Figure 1: Schematic of the experiment

AOM: Acousto-Optic Modulator; Beac.: Beacon laser; BPF: 1310 nm Band Pass
Filter Dich.: Dichroic mirror; DPLL: Digital Phase Lock Loop; EDFA: Erbium
Doped fibre Amplifier (bidirectional); FM: Faraday Mirror; GBE: Galilean Beam
Expander; IR: Infrared camera; Meas.: out of loop phase Measurement; Opt.
Pow.: return Optical Power monitoring; Rx: Receiver; Tx: Transmitter; T/T:
Tip/Tilt mirror; Vis.: Visible camera; $\Delta\phi$: phase noise from
turbulences

###  2.1 Phase compensation and measurement unit

The phase compensation and measurement unit enables both to stabilize the
phase of the laser link and to perform an out-of-loop measurement of the link
stability. To do so, the optical signal is generated from a planar extended
cavity 1542 nm laser diode delivering 7 mW into a monomode fibre. The phase
noise compensation of the link is based of an unequal-arms Michelson
interferometer [25]. Its short fibre arm ($\sim 1$ m) spanning from the Tx
splitter to FM1 (see Fig. 2) serves as a local reference to which the long
folded arm (the link) spanning from the Tx splitter to FM2 is compared. The
comparison is done by a heterodyne detection on a photodiode at 160 MHz
achieved by two AOMs inserted into the long arm. The first AOM (Tx, controlled
by the DPLL) shifts the laser frequency by +38
MHz$-\frac{1}{2\pi}\frac{d\Delta\phi(t)}{dt}$ where $2\Delta\phi(t)$ is the
round trip phase noise in the long arm seen by the DPLL. The second AOM (Rx)
shifts the laser frequency by +42 MHz and enables to discriminate all possible
optical path in the folded architecture as well as intermediate parasitic
reflections. The 160 MHz beatnote on the Tx photodiode is isolated by a
bandpass filter and then divided down to 16 MHz in order to fit in the input
range of the DPLL. The DPLL is run on a RedPitaya FPGA board using the
software developed in [26]. The bandwidth of the full phase-noise compensation
loop is $\sim$15 kHz.

The laser source and the receiver being physically close allows for an out of
loop phase comparison between the two using small uncompensated fibres (yellow
dashed line in Fig. 2). The received laser frequency is shifted by 80 MHz
relative to the source. The two are optically beat on a photodiode and
generate a 80 MHz RF beatnote. In order for our phasemeter, a Microsemi
53100A, to measure the useful 80 MHz beatnote phase, we must isolate it from
neighbouring beatnotes at $\pm$4 MHz generated by alternative optical paths in
the folded link. This is achieved by downcoverting the 80 MHz signal (using a
mixer) down to 1 MHz and low pass filtering it. The phasemeter and the DPLL
are referenced onto a common local oscillator from a frequency generator.

###  2.2 Optical terminal

The optical terminal behaves as a transceiver in which the laser light is sent
and received through the same optical system. The science signal is received
from the phase compensation and measurement unit through a monomode fibre. It
is coupled to free space using a f=20 mm fibre collimator followed by a x5
magnification Galilean beam expander (GBE). This optical system generates a
Gaussian beam with a waist of $w=9.5$ mm with little clipping from the 35 mm
aperture of the GBE. The actuation of the optical terminal is based on a 2"
Piezo actuated Tip/Tilt mirror for fast target movements and first order
atmospheric turbulence compensation, and an astronomical equatorial mount for
high amplitude low frequency target motion tracking.

The target tracking relies on four 1310 nm (10° divergence) beacon laser
diodes attached to the airborne corner cube and aligned with its optical axis.
The beacon lasers are filtered by a dichroic mirror and detected on the
terminal by a 215 fps infrared (IR) monochromatic camera. The camera focal
length is 75 mm and it uses a 320x256 pixel sensor with 20 $\mu$m pixel size.
Taking into account the x5 GBE, this corresponds to a angular pixel size on
the sky of 53 $\mu$rad, and to a $\approx$ 5x4 m image at the balloon distance
of 300 m. The barycenter of each beacon image (spread over $<9$ pixels) is
individually detected in real time using a computer vision algorithm. Their
overall centroid position within the image stream is then stabilized by a
digital servo loop that controls the Tip/Tilt mirror. The stabilization
setpoint is chosen to maximize the science beam optical return into the fibre.
Despite the high frame rate, a delay of 16.5 ms between the camera and the
control computer was found to be the main limiting factor on our Tip/Tilt
correction bandwidth, sitting around 20 Hz. A second servo loop maintains the
Tip/Tilt mirror close to the middle of its actuation range (10 mrad) using the
astronomical mount. In typical operating conditions lock was never lost during
several hours of operation.

A wide field-of-view (6°), co-aligned, visible camera allows the user to
roughly aim the terminal at its target. Once the target enters the IR camera
field-of-view (1°), our control software automatically takes over and locks
onto the target. The optical terminal includes a monitoring photodiode that
receives 10 % of the return power injected in the fibre allowing fine tuning
and monitoring of the link transmission.

###  2.3 Balloon carrier and payload

The balloon is a 16 m3 Allsopp “Helikite” hybrid kite/Helium balloon that can
carry up to 8 kg payload including the weight of the tether. It is stabilized
by light wind, but more difficult to operate in strong wind conditions. For
our experiment, the tether length is set to 300 m.

Our payload is made of two parts mounted at both ends of a slim $\sim 1$ m
metal tube. The first is a plastic box containing the laser diode drivers
connected to a USB battery mounted on the balloon itself. The second is the
corner cube (2.5" diameter) and the four laser beacons at $\lambda$=1310 nm
assembled on a 3D printed bloc, mounted at a right angle onto the tube and
facing outward along its length. The metal tube is attached in-line with the
balloon and the tether, which passes inside the tube. This arrangement ensures
that the corner cube is facing the anchorage point on the ground, thereby
providing a passively maintained alignment with the closeby optical terminal
sufficient for the link operation, regardless of the balloon motion
(translation or rotation).

Figure 2: Picture of the experimental setup deployed at CNES in Aire sur
l’Adour before balloon take-off. The inset shows the corner-cube with the four
beacon diodes. The tether is anchored $\sim$ 2 m from the ground terminal.
Operators are located inside the van during flight.

##  3 Results

We report here the results of a measurement campaign at the CNES balloon
launch site in Aire sur l’Adour in the south-west of France in March 2023. The
measurements have been achieved over 2 days of favourable weather conditions
with light winds (typically $<5$ m/s) and no rain or fog that would hinder
visibility. The tether length was fixed to 300 m and the light winds resulted
in an almost vertical (elevation $\geq 70$ degrees) position of the balloon
above the ground terminal, i.e. close to 300 m in altitude. In addition to the
balloon link, a static link was established to an identical payload, installed
on a mount, 300 m away from the terminal and about 1.5 m above the ground.
This static link served as a reference for, and comparison to, the balloon
link.

The free space-link loss i.e. the fraction of emitted power injected back into
the fibre after a round-trip was about 0.025 (-16 dB) and was subject to
optimizing the set-point of the tip/tilt correction, and the adjustment of the
collimator and GBE focus. The latter was a compromise between maximum received
power (small divergence) and minimal power fluctuations (larger divergence).
For typical settings about 400 $\mu$W were launched into free space and the
power injected back into the fibre was about a factor 2 above the extinction
threshold of our measurement ($\sim 0.5\,\mu$W).

Figure 3: In-flight phase noise measurement PSD. Black curve, free-space link
uncompensated; red, free-space link compensated; green, AM/PM evaluation from
monitoring photodiode measurement; dark blue, noise floor due to the link
delay, evaluated from black curve; light blue, noise of a 300m compensated
fibre link. Figure 4: On ground phase noise measurement PSD. Black curve,
free-space link uncompensated; red, free-space link compensated; green, AM/PM
evaluation from monitoring photo-diode measurement; dark blue, noise floor due
to the link delay, evaluated from black curve; light blue, noise of a 300m
compensated fibre link.

Figures 3 and 4 show the results in terms of phase noise PSD in both
uncompensated and phase noise compensated configurations, for the balloon and
ground links. All reported phase measurements are taken with the out-of-loop
diode (“Meas.” on Fig. 2). Additionally we plot the “theoretical” compensated
performance calculated from the uncompensated data and the known link distance
(see Appendix A for details), which corresponds to our measurement floor as it
represents the expected performance with perfect phase noise compensation and
no additional noise sources. The total duration of the measurements were 80 s
(compensated in-flight), 120 s (uncompensated in-flight), 172 s (compensated
ground), 505 s (uncompensated ground) [27]. The compensated and uncompensated
acquisitions were carried out in quick succession to ensure similar
conditions, however the on-ground and in-flight measurements were taken on
different days. We also plot noise from our equivalent fibre link using the
same phase noise compensation system, and the expected amplitude to phase
noise (AM/PM) conversion in our out of loop measurement (see section 4).

Finally, we show the frequency stability (modified Allan deviation, MDEV) in
figure 5, demonstrating sub $10^{-18}$ performance after less than 20 s
averaging. Note, that when our set-up was optimised (focal settings, PID
gains, …) no signal outages were observed even during our longest runs ($>500$
s), thus all plotted MDEV are using continuous gap-free data.

Figure 5: Fractional frequency stability (modified Allan deviation). Same data
as on figures 3 and 4. Compensated (ON) and uncompensated (OFF) results for
balloon and ground links.

##  4 Discussion

We start by examining the uncompensated phase noise in flight. We note that
the peak in the PSD around 0.6 Hz and corresponding bump in the MDEV around
0.5 s [28] corresponds to oscillatory longitudinal motion of the payload,
observed during all flights, including some preliminary tests in 2022. From
the MDEV (or equivalently the PSD or directly from the raw phase data) we see
an oscillation of the distance along the line of sight of roughly 2 mm
amplitude. We also see prominent oscillations at 0.6 Hz of the applied angular
corrections on the tip/tilt mirror and the astronomical mount with amplitudes
of about 0.1 mrad. Taking into account the distance between the ground
terminal and the tether anchor-point of about 2 m (see fig. 2) this accounts
only partially (about 10%) for the 2 mm longitudinal motion of the payload. We
attribute the main part to a small coupling of the transverse motion of the
payload (about 3 cm amplitude) into the longitudinal axis along the line of
sight.

When comparing the in-flight uncompensated to ground uncompensated data (PSD
or MDEV) we notice that the in-flight measurements are significantly noisier
(by up to 3 orders of magnitude) at all frequencies up to 10 kHz. If the
uncompensated phase measurements were dominated by instrumental noise or
atmospheric turbulence we would expect the ground data to show similar or
larger noise (turbulence at 1.5 m above the ground is expected to be stronger
than at high elevation), but the contrary is the case. This, together with the
correlation between observed angular oscillations and distance oscillations
(see previous paragraph) suggests that the in-flight uncompensated phase noise
is largely dominated by the motion of our balloon and payload, leading to
corresponding distance changes, as one would expect.

At very high frequencies (above 10 kHz) all noise curves converge towards the
noise measured on our test fibre link. In that frequency region the results
are dominated by the measurement noise of our system and the phase
compensation loop (DPLL) noise. The locking bump corresponding to the
bandwidth of that loop is clearly visible around 30 kHz.

Turning now to the phase-noise compensated links (in flight and on ground) we
first notice that in most of the frequency range we do not reach the projected
performance from a purely distance limited calculation (see Appendix A). It is
only reached for the in flight measurement in the region between about 0.1 Hz
and 2 Hz. At lower frequencies we are limited by thermal effects on the
uncompensated fibres of the experiment, in particular the out of loop
measurement fibre (dashed yellow line in Fig. 2). At intermediate frequencies
we are limited by amplitude to phase noise (AM/PM) conversion as discussed
below. At high frequencies (above a few kHz) by the residual noise of our DPLL
as seen on the fibre link.

At the lowest frequencies (below 0.1 Hz in PSD, above 10 s in MDEV) thermal
effects from the uncompensated fibres are dominant. A linear fit to the phase
data yields a phase drift that corresponds to a fractional frequency offset of
$2\times 10^{-17}$ for the in flight data and $1\times 10^{-17}$ on ground. A
quadratic fit gives a fractional frequency drift of $1.6\times 10^{-20}$/s in
flight and $1.4\times 10^{-20}$/s on ground. With a thermal coefficient of the
fibres of $10^{-5}$/K and for 1 m uncompensated fibre this corresponds to
temperature variations of about 0.3 mK/s (=1.1 K/h), which is realistic given
that the equipment was outside on the airfield, without any thermal shielding
or temperature control. Part of this is expected to be due to the out of loop
measurement fibre i.e. not actually part of the link. In any case, some even
modest optimisation (reduce uncompensated fibre lengths, add thermal shielding
and control and/or correct in post-processing) should allow reducing long term
effects below $10^{-18}$ in fractional frequency offset and well below
$10^{-20}$/s in fractional frequency drift.

At intermediate frequencies (a few Hz to about a kHz) our noise limitation
comes from AM/PM conversion. The effect is located mainly in the mixer used to
down-convert the 80 MHz out of loop measurement to 1 MHz. As such it is not
part of the link itself, but only present in the verification measurement. We
have checked this effect by controlled modulation of the signal amplitude on
the ground link at CNES premises in Toulouse. The signal amplitude was
modulated by sinusoidally varying the tip/tilt mirror. The observed phase
modulation was proportional to the generated amplitude modulation with a
coefficient [29] of $K_{AP}\approx 0.09$ rad. Using that coefficient we
converted the amplitude fluctuations measured on our power monitoring diode
(see Fig. 2) to phase fluctuations, as plotted on figures 3 and 4. They fit
well to the measured phase noise. This measurement artefact can easily be
avoided in the future by redesigning our measurement chain. We think that it
is not a fundamental limitation of the link.

Finally, we discuss the possible extension to larger distances. Indeed,
applications like chronometric geodesy will require links to higher altitudes
(1-10 km) leading to line of sight distances of tens of km. Our current
configuration was designed for technology demonstration (tracking, phase noise
compensation) and is certainly not suited for such large distances. However,
the obtained results indicate that with appropriate design (10 cm size optics,
improved link efficiency, higher laser power, …) distances of several tens of
km should be possible. If necessary, the constraints on the power budget for
such long distances could be reduced by using an on-board bidirectional
amplifier (EDFA on Fig. 2) provided the mass and power budget of the airborne
carrier allows it. In the shorter term, we plan to study and optimize the one-
way (ground-balloon) power budget of our short link in order to better
understand the requirements for long distance links.

##  5 Outlook and conclusion

In conclusion, we presented the characterization of a folded free-space
optical link via a corner cube onboard a tethered balloon. The results,
obtained during a flight campaign in March 2023, demonstrate a fractional
frequency stability of $8\times 10^{-19}$ after 16 s averaging time, which is
the best result for airborne experiments reported so far. In addition, we
extensively identified and discussed the main limiting noise sources and how
they could be overcome in future implementations.

This first step paves the way to point-to-point ground links via an airborne
balloon relay (type (i) link mentioned in the introduction). This will be
achieved through a miniaturization of our active system to fit within the
balloon capability. Particular attention will be paid to the improvement of
the pointing and stabilization performance which appears to be limiting in
case of windy weather conditions. Such free-space comparisons will be useful
for many applications, like chronometric geodesy, using operational and
transportable in-field systems via airborne relays.

## Acknowledgements

This work was conducted in the framework of the Ph.D. thesis of N.M. co-funded
by the Labex First-TF and CNES, and the Ph.D. thesis of S.F. co-funded by AID
and CNES.

The authors would like thank the CNES balloon aeronauts (Brice Bellanger) for
their time and expertise, and colleagues from Météo-France (Axel Roy) for
feedback and data concerning their Helikite experience. We also thank Jean-
Daniel Deschênes for help with the DPLL software. Last but not least we
acknowledge fruitful collaboration and discussion over the last years with our
colleagues from the University of Western Australia (Sascha Schediwy, David
Gozzard, Benjamin Dix-Matthews among others), it is great and much easier to
advance together!

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available
at this time but may be obtained from the authors upon reasonable request.

## Appendix A: Calculation of the theoretical limit for the compensated noise

As discussed in section 4 the noise of the uncompensated link is dominated by
the balloon motion. We model it by a virtual frequency shift $\delta\nu(t)$
that occurs exactly half way between the Tx and Rx terminals. The received
frequency in the uncompensated link is

$\nu_{Rx}^{OFF}(t)=\nu_{L}+\delta\nu(t-T/2),$ (1)

where $\nu_{L}$ is the laser frequency and $T$ is the flight time between Tx
and Rx (in our case $T=2L/c$, where $L=300$ m). Thus we have for the PSD of
the out of loop diode (that measures $\nu_{Rx}^{OFF}(t)-\nu_{L}$):
$S_{OFF}(f)=S_{\delta\nu}(f)$.

For the compensated link the frequency at the Rx end is

$\nu_{Rx}^{ON}(t)=\nu_{L}+\Delta\nu(t-T)+\delta\nu(t-T/2),$ (2)

where $\Delta\nu(t)$ is the frequency correction applied by the AOM at the Tx
end. We neglect the Rx-AOM and the constant frequency of the Tx-AOM, which are
irrelevant for this calculation.

The error signal at the Tx end is set to zero by the DPLL hence

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\nu_{L}-\nu_{ret}(t)$ (3)
$\displaystyle=$ $\displaystyle\nu_{L}-\big{(}\nu_{L}+\Delta\nu(t-2T)$
$\displaystyle+$
$\displaystyle\delta\nu(t-3T/2)+\delta\nu(t-T/2)+\Delta\nu(t)\big{)}\,,$

where $\nu_{ret}(t)$ is the frequency of the signal after a full Tx-Rx-Tx
round trip. A first order expansion leads to

$\displaystyle 2\Delta\nu(t)-2T\Delta\dot{\nu}(t)$ $\displaystyle\simeq$
$\displaystyle-(2\delta\nu(t)-2T\delta\dot{\nu}(t))$
$\displaystyle\implies\,\,\Delta\nu(t)$ $\displaystyle\simeq$
$\displaystyle-\delta\nu(t)\,.$ (4)

Substituting (Appendix A: Calculation of the theoretical limit for the
compensated noise) into (2) gives

$\nu_{Rx}^{ON}(t)=\nu_{L}-\delta\nu(t-T)+\delta\nu(t-T/2)\,.$ (5)

Finally, the out of loop diode measures is $\nu_{Rx}^{ON}(t)-\nu_{L}$, and the
corresponding PSD is [30]

$\displaystyle S_{ON}(f)$ $\displaystyle=$ $\displaystyle 2(1-\cos(\pi
fT))S_{\delta\nu}(f)$ (6) $\displaystyle=$ $\displaystyle 2(1-\cos(\pi
fT))S_{OFF}(f)\,.$

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  * [27] All data acquisitions were intentionally stopped at durations sufficient for our purposes ($\sim 10^{2}$ s, beyond that we only see thermal effects, see sect. 4) in order to avoid gaps, although none were observed, and to leave time for different tests and variations of settings.
  * [28] On an modified Allan deviation a periodic effect of period $P$ gives rise to a bump at $\tau\simeq 0.31P$.
  * [29] We define the AM/PM coefficient by $\Delta\phi(t)=K_{AP}\frac{\Delta P(t)}{P_{0}}$, where $\Delta\phi(t)$ and $\Delta P(t)$ are the phase and power variations and $P_{0}$ is the average power.
  * [30] For a noise process $x(t)$ that is characterised by a PSD $S_{x}(f)$, the PSD of the linear combination $z(t)\equiv\sum{a_{i}x(t-T_{i})}$ (where $a_{i}$ are arbitrary real constants) can be derived from first principles (see e.g. Duchayne L., PhD-thesis, p.227, https://tel.archives-ouvertes.fr/tel-00349882) and is given by $S_{z}(f)=\left(\sum\limits_{i}a_{i}^{2}+\sum\limits_{i}\sum\limits_{j\neq i}a_{i}a_{j}{\rm cos}(2\pi f(T_{i}-T_{j}))\right)S_{x}(f)$.


# An elementary proof for dynamical scaling for certain fractional non-
homogeneous Poisson processes

Markus Kreer
Feldbergschule, Oberhöchstadter Strasse 20, D-61440 Oberursel, Germany
<EMAIL_ADDRESS>

###### Abstract

Dynamical scaling is an asymptotic property typical for the dynamics of first-
order phase transitions in physical systems and related to self-similarity.
Based on the integral-representation for the marginal probabilities of a
fractional non-homogeneous Poisson process introduced by Leonenko et al.
(2017) and generalising the standard fractional Poisson process, we prove the
dynamical scaling under fairly mild conditions. Our result also includes the
special case of the standard fractional Poisson process.

_Keywords_ method of steepest decent $\cdot$ M-Wright function $\cdot$
asymptotic analysis $\cdot$ self-similarity

##  1 Preliminaries and main results

Whereas the regular Poisson process can be generalized in a straight-forward
manner to a non-homogeneous Poisson process with a nicely behaved time-
dependent intensity function by a mere time change using the chain rule, this
trick does not work for fractional derivatives (e.g. [14]). The original
fractional Poisson process (fPp) as firstly introduced by [7] cannot be
generalized using his approach. A possible way to go forth is following [8]
who introduced the marginals for a fractional non-homogeneus Poisson process
(fnhPp) by equation Eq.(2.18) in their paper

$\displaystyle
P_{\beta}(n,t)=\int_{0}^{\infty}du~{}\frac{e^{-\Lambda(u)}\Lambda(u)^{n}}{n!}h_{\beta}(t,u)$
(1)

where $0<\beta<1$ and $\Lambda(t)=\int_{0}^{t}du~{}\lambda(u)$ with an
intensity function $\lambda(\cdot):[0,\infty)\rightarrow[0,\infty)$ and the
density of the inverse stable subordinator (e.g. ([11])

$\displaystyle h_{\beta}(t,x)=\frac{t}{\beta
x^{1+1/\beta}}~{}g_{\beta}\left(\frac{t}{x^{1/\beta}}\right)$ (2)

and

$\displaystyle
g_{\beta}(u)=\frac{1}{\pi}\sum_{j=1}^{\infty}(-1)^{j+1}\frac{\Gamma(\beta
j+1)}{\Gamma(j+1)}\frac{1}{u^{\beta j+1}}\sin(\beta\pi j)$ (3)

The authors in [8] show in their Corollary 1 that the marginals Eq.(1) satisfy
the following system of fractional differential-integral equations

$\displaystyle\frac{d^{\beta}}{dt^{\beta}}P_{\beta}(n,t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}du~{}\lambda(u)\left(-\frac{e^{-\Lambda(u)}\Lambda(u)^{n}}{n!}+\frac{e^{-\Lambda(u)}\Lambda(u)^{n-1}}{(n-1)!}\right)h_{\beta}(t,u),\hskip
28.45274ptn=1,2,3,...$
$\displaystyle\frac{d^{\beta}}{dt^{\beta}}P_{\beta}(0,t)~{}$ $\displaystyle=$
$\displaystyle~{}-\int_{0}^{t}du~{}\lambda(u)e^{-\Lambda(u)}h_{\beta}(t,u)\hskip
153.6447ptn=0$

with initial conditions $P_{\beta}(0,0)=1$ and $P_{\beta}(n,0)=0$ for
$n=1,2,...$. The fractional derivative is the fractional Caputo-Djrbashian
derivative which is defined as (e.g. [12])

$\displaystyle\frac{d^{\beta}}{dt^{\beta}}F(t)=\frac{1}{\Gamma(1-\beta)}\int_{0}^{t}\frac{ds}{(t-s)^{\beta}}~{}\frac{d}{ds}f(s)$

As expected these equations Eq.(LABEL:eq:nhfPp_eq) reduce to the usual
fractional Poisson process with Kolmogorov-Feller equations as discussed by
[1] (see also [10] for a general theory, and [6] with an emphasis of their
relation to Smoluchowsky-type coagulation-fragmentation equations) when
$\lambda(t)$ is chosen to be just a positive constant $\lambda$,

$\displaystyle\frac{d^{\beta}}{dt^{\beta}}P_{\beta}(n,t)=\lambda\left(P_{\beta}(n-1,t)-P_{\beta}(n,t)\right)$
(5)

where we have $0<\beta<1$ and $P_{\beta}(-1,t)=0$ for convenience and the same
initial conditions as above. This fact proven by [8] will be demonstrated
independently as we proceed with proving our main theorem.

###### Theorem 1

Let the function $\Lambda(\cdot)$ satisfy the following conditions

(i) $\lim_{x\rightarrow\infty}\Lambda(x)=\infty$ and

(ii) There exists some positive constant $c$ such that as $x\rightarrow\infty$

$\displaystyle\frac{\Lambda^{\prime}(x)}{\Lambda(x)}=\frac{c}{x}+\mathcal{O}\left(\frac{1}{x\log{x}}\right)$

Then for $0<\beta<1$ the marginal probabilities of the fnhPp as given in
Eq.(1) have the following limiting behaviour

$\displaystyle\lim_{n\rightarrow\infty,n=\Lambda(z_{0}t^{\beta})}nP_{\beta}(n,t)=\frac{z_{0}}{c}M_{\beta}(z_{0})$

where for any positive $z_{0}>0$ we relate the positive $n$ to any positive
$t$ by the equation $n=\Lambda(z_{0}t^{\beta})$, and $M_{\beta}(\cdot)$ is the
M-Wright function or Mainardi function defined by

$\displaystyle
M_{\beta}(z)=\frac{1}{\pi}\sum_{j=1}^{\infty}\frac{(-z)^{j-1}}{(j-1)!}\Gamma(\beta
j)\sin{(\pi\beta j)}$

From Theorem 1 we obtain immeadiately a useful

###### Corollary 2

For any positive number $r$ let $\Lambda(x)=x^{r}$. Then for $0<\beta<1$ the
marginal probabilities as given in Eq.(1) have the following limiting
behaviour

$\displaystyle\lim_{t\rightarrow\infty,nt^{-r\beta}=z_{0}^{r}}t^{r\beta}P_{\beta}(n,t)=\frac{z_{0}^{1-r}}{r}M_{\beta}(z_{0})$

for any $z_{0}>0$ relating $t$ and $n$ by the condition
$nt^{-r\beta}=z_{0}^{r}$. In particular for $r=1$ we obtain the dynamical
scaling result for the standard fPp satisfying the Kolmogorov-Feller equations
Eq. (5) with $\lambda=1$.

Note that for $\beta=1$ and $\lambda(t)=1$, when we recover the standard
Poisson process, the function $M_{\beta}(z)=0$ and our proof fails in this
case.

##  2 Proof of Theorem 1

We first show that the representation for the marginals as given in Eq.(1)
together with Eq.(2)-(3) by [8] can be expressed in terms of the M-Wright
function (Mainardi function). Setting in Eq.(1) for convenience $u=zt^{\beta}$
we get

$\displaystyle
P_{\beta}(n,t)=t^{\beta}\int_{0}^{\infty}dz~{}\frac{e^{-\Lambda(zt^{\beta})}\Lambda(zt^{\beta})^{n}}{n!}h_{\beta}(t,zt^{\beta})$

Because of Eq.(2)-(3) we obtain after some straight forward computations

$\displaystyle t^{\beta}h_{\beta}(t,zt^{\beta})$ $\displaystyle=$
$\displaystyle\frac{1}{\beta
z^{1+1/\beta}}~{}g_{\beta}\left(\frac{1}{z^{1/\beta}}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\beta
z^{1+1/\beta}}~{}\frac{1}{\pi}\sum_{j=1}^{\infty}(-1)^{j+1}\frac{\Gamma(\beta
j+1)}{\Gamma(j+1)}\frac{1}{\left(\frac{1}{z^{1/\beta}}\right)^{\beta
j+1}}\sin(\beta\pi j)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\sum_{j=1}^{\infty}(-1)^{j-1}\frac{\Gamma(\beta
j)}{\Gamma(j)}z^{j-1}\sin(\beta\pi j)=M_{\beta}(z)$

The last equality follows from the series expansion of the M-Wright function
(e.g. Eq. (10) in [9]) Therefore we have as a backbone of our investigation
the identity

$\displaystyle
P_{\beta}(n,t)=\int_{0}^{\infty}dz~{}\frac{e^{-\Lambda(zt^{\beta})}\Lambda(zt^{\beta})^{n}}{n!}M_{\beta}(z)$
(6)

Note that when $\Lambda(\cdot)$ is a linear function we obtain a useful Lemma
for fPp firstly stated by [4]:

###### Lemma 3

(Gorenflo and Mainardi) For the fractional Poisson probabilities
$P_{\beta}(n,t)$ we have the integral representation (the subordination
integral)

$\displaystyle P_{\beta}(n,t)$ $\displaystyle=$
$\displaystyle\frac{1}{n!}\int_{0}^{\infty}dt_{*}t_{*}^{n}~{}t^{-\beta}\exp{(-t_{*})}M_{\beta}(t_{*}t^{-\beta})$
(7) $\displaystyle=$
$\displaystyle\frac{1}{n!}t^{n\beta}\int_{0}^{\infty}dz~{}z^{n}e^{-zt^{\beta}}M_{\beta}(z)$

where we have set for convenience $z=t_{*}t^{-\beta}$ as new integration
variable.

Thanks to Lemma 3 and its Eq.(7), following as a special case from Eq.(6), we
see immediately that without loss of generality if $\lambda(t)=1$ and
consequently $\Lambda(t)=t$ is linear in $t$, the integration in the
fractional differential-integral equations Eq.(LABEL:eq:nhfPp_eq) on the
right-hand side can be carried out explicitly yielding the term
$\left(P_{\beta}(n-1,t)-P_{\beta}(n,t)\right)$ and we have thus recovered the
fractional Kolmogorov-Feller equations Eq.(5) from [1] as claimed. Therefore
our dynamical scaling result covers also the standard fPp. We finally want to
emphasize the fact that Lemma 3 is of highest practical relevance in computing
the marginal probabilities of the fPp as functions of time for applications in
statistics or physics.

Proof of Theorem 1

We write Eq.(6) as follows

$\displaystyle
P_{\beta}(n,t)=\frac{1}{n!}\int_{0}^{\infty}dz~{}e^{nf(z|t)}M_{\beta}(z)$

Here we have defined an auxilary function by

$\displaystyle
f(z|t)=\log{\Lambda(zt^{\beta})}-\frac{1}{n}\Lambda(zt^{\beta})$ (8)

and we treat $t$ as a parameter. We compute now the first and second
derivative of $f(\cdot|t)$ with respect to the first argument, namely

$\displaystyle
f^{\prime}(z|t)=\Lambda^{\prime}(zt^{\beta})t^{\beta}\left[\frac{1}{\Lambda(zt^{\beta})}-\frac{1}{n}\right]$
(9)

and

$\displaystyle
f^{\prime\prime}(z|t)=\Lambda^{\prime\prime}(zt^{\beta})t^{2\beta}\left[\frac{1}{\Lambda(zt^{\beta})}-\frac{1}{n}\right]-\left(\frac{\Lambda^{\prime}(zt^{\beta})t^{\beta}}{\Lambda(zt^{\beta})}\right)^{2}$
(10)

Due to the properties of $\Lambda(\cdot)$, condition (i), there exists for any
positive $n$ and for any $z_{0}>0$ a positive number $t$ solving ,
$f^{\prime}(z_{0}|t)=0$, yielding from Eq.(9)

$\displaystyle n=\Lambda(z_{0}t^{\beta})$ (11)

Note that from Eq.(11) we have with Eq. (8) that
$f(z_{0}|t)=\log{\Lambda(z_{0}t^{\beta})}-\Lambda(z_{0}t^{\Lambda})/n=\log{n}-1$.
And we see from Eq.(10) that $f^{\prime\prime}(z_{0}|t)<0$ and thus $z_{0}>0$
is a positive maximum of $f(\cdot|t)$, i.e. using condition (ii) on
$\Lambda(\cdot)$ we get

$\displaystyle\sqrt{-f^{\prime\prime}(z_{0}|t)}=\frac{\Lambda^{\prime}(z_{0}t^{\beta})t^{\beta}}{\Lambda(z_{0}t^{\beta})}=\frac{c}{z_{0}t^{\beta}}t^{\beta}+t^{\beta}\mathcal{O}\left(\frac{1}{t^{\beta}\log{t}}\right)$
(12)

Because the M-Wright function $M_{\beta}(\cdot)$ is analytic on the real line
(e.g. [3], [12]) we can evaluate the integral using the method of steepest
decent because the conditions for applying [13] Chapter 6 Proposition 2 and
Proposition 3 are satisfied:

$\displaystyle\int_{0}^{\infty}dz~{}e^{nf(z|t)}M_{\beta}(z)$ $\displaystyle=$
$\displaystyle\frac{\exp{(nf(z_{0}|t))}}{\sqrt{n}}\left(M_{\beta}(z_{0})~{}\sqrt{\frac{2\pi}{-f^{\prime\prime}(z_{0}|t)}}+\mathcal{O}\left(\frac{1}{n}\right)\right)$
$\displaystyle=$
$\displaystyle\frac{e^{nf(z_{0}|t)}}{\sqrt{n}}\sqrt{2\pi}\left(M_{\beta}(z_{0})~{}\frac{z_{0}}{c}+\mathcal{O}\left(\frac{1}{n}\right)+\mathcal{O}\left(\frac{1}{\log{t}}\right)\right)$

and with Stirling’s formula

$\displaystyle n!=\sqrt{2\pi
n}\left(\frac{n}{e}\right)^{n}\left(1+\mathcal{O}\left(\frac{1}{n}\right)\right)$

and Eq.(11) we obtain altogether

$\displaystyle\frac{1}{n!}\int_{0}^{\infty}dz~{}e^{nf(z|t)}M_{\beta}(z)$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2\pi
n}}\left(\frac{n}{e}\right)^{-n}\left(1+\mathcal{O}\left(\frac{1}{n}\right)\right)\cdot$
$\displaystyle\cdot
e^{nf(z_{0}|t)}\sqrt{\frac{2\pi}{n}}\left(M_{\beta}(z_{0})~{}\frac{z_{0}}{c}+\mathcal{O}\left(\frac{1}{n}\right)+\mathcal{O}\left(\frac{1}{\log{t}}\right)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{n}\left(M_{\beta}(z_{0})~{}\frac{z_{0}}{c}+\mathcal{O}\left(\frac{1}{n}\right)+\mathcal{O}\left(\frac{1}{\log{t}}\right)\right)$

where we have used $f(z_{0}|t)=\log{n}-1$ in the last line. Multiply both
sides with $n$ and note that for large increasing $n$ by the properties of
$\Lambda(\cdot)$ also $t$ increases and $z_{0}>0$ remains a maximum as long as
$n$ and $t$ are related by Eq. (11). This concludes the proof. $\blacksquare$

##  3 Outlook and application

Since [15] the dynamical scaling hypothesis became a “regular” feature in the
physical theory of phase transitions. It was originally discovered in
numerical studies and not long afterwards a mathematical proof for the simple
Smoluchowski coagulation equations with constant kernel was given by [5]. The
dynamical scaling hypothesis is expected to hold for generalized Smoluchowski
equations but not proven yet. The work of [6] related the fPp to a certain
class of Smoluchowski equations and thus a dynamical scaling of the marginal
probabilities would come to no surprise. Our current work has answered this in
the affirmative even for the more general class of fnhPp as introduced by [8].
This generalisation has the advantage that the dynamical scaling exponent of
time $t$ is not limited to positive numbers between 0 and 1 (which is the
range the order of the fractional derivative can take) but can be any positive
number due to the fact that the the choice of the intensity function
$\Lambda(\cdot)$ allows for some additional freedom.

How would one apply a fnhPp to the problem of phase transitons? In our case we
may imagine a solid crystal consisting of single monomers bound in a regular
lattice structure. When the crystal is heated the lattice takes up heat and
the monomeres start to oscillate and finally break free. At the end, the
lattice has been dissolved because all monomers have gone. The very simple
description of this “phase transition”, in which the ordered state (= crystall
lattice) has undergone a transition to an unordered state (free-moving
monomers corresponding to a liquid or a gas) is to count the number of
monomers breaking free from the remaining crystal structure, one at a time.
The reverse of this model would be monomers condensing to a newly formed
crystal. In fact, scientists have used this simple counting model to describe
the nucleation of lysozyme and paracetamol in aqueous droplets: [2] describe
the nucleation by a Poisson process and estimate its parameters.

A simple computation shows that a standard Poisson process does not possess a
regular dynamical scaling limit, namely with $\beta=1$ in Eq.(7) we have after
some straight-forward computations

$\displaystyle\lim_{t\rightarrow\infty,t^{-1}n=z_{0}}t^{1/2}P_{1}(n,t)=\left\\{\begin{array}[]{ll}\frac{1}{\sqrt{2\pi}}\hskip
14.22636pt\textrm{for}\hskip 5.69046ptz_{0}=1\\\ 0\hskip
28.45274pt\textrm{otherwise}\end{array}\right.$ (15)

With respect to the dynamical scaling hypothesis for phase transitions it
would be very interesting to analyse the nucleation data of [2] using a
fractional Poisson process as model.

## Acknowledgements

Stimulating discussions with Lukas Kreer and Theo Manoussos (both Johannes-
Gutenberg Universität, Mainz) and Dr. Ayşe Kızılersü (University of Adelaide)
are gratefully acknowledged.

This research did not receive any specific grant from funding agencies in the
public, commercial or not-for-profit sectors.

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