# Theoretical study of the process $D^{+}_{s}\to\pi^{+}K^{0}_{S}K^{0}_{S}$ and
the isovector partner of $f_{0}(1710)$

Xin Zhu  School of Physics and Microelectronics, Zhengzhou University,
Zhengzhou, Henan 450001, China  Institute of Modern Physics, Chinese Academy
of Sciences, Lanzhou 730000, China  De-Min Li<EMAIL_ADDRESS>School of
Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001,
China  En Wang<EMAIL_ADDRESS>School of Physics and Microelectronics,
Zhengzhou University, Zhengzhou, Henan 450001, China  Li-Sheng Geng
<EMAIL_ADDRESS>School of Physics, Beihang University, Beijing
102206, China  Beijing Key Laboratory of Advanced Nuclear Materials and
Physics, Beihang University, Beijing 102206, China  School of Physics and
Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China  Ju-Jun
Xie<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  School of Nuclear Sciences and Technology,
University of Chinese Academy of Sciences, Beijing 101408, China  School of
Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001,
China

###### Abstract

We present a theoretical study of $a_{0}(1710)$, the isovector partner of
$f_{0}(1710)$, in the process $D^{+}_{s}\to\pi^{+}K^{0}_{S}K^{0}_{S}$. The
weak interaction part proceeds through the charm quark decay process:
$c(\bar{s})\to(s+\bar{d}+u)(\bar{s})$, while the hadronization part takes
place in two mechanisms, differing in how the quarks from the weak decay
combines into $\pi K^{*}$ with a quark-antiquark pair $q\bar{q}$ with the
vacuum quantum numbers. In addition to the contribution from the tree diagram
of the $K^{*+}\to\pi^{+}K^{0}_{S}$, we have also considered the
$K^{*}\bar{K}^{*}$ final-state interactions within the chiral unitary approach
to generate the intermediate state $a_{0}(1710)$, then it decays into the
final states $K^{0}_{S}K^{0}_{S}$. We find that the recent experimental
measurements on the $K^{0}_{S}K^{0}_{S}$ and $\pi^{+}K^{0}_{S}$ invariant mass
distributions can be well reproduced, and the proposed mechanism can provide
valuable information on the nature of scalar $f_{0}(1710)$ and its isovector
partner $a_{0}(1710)$.

##  I Introduction

Though the scalar $f_{0}(1710)$ resonance with $I^{G}(J^{PC})=0^{+}(0^{++})$
is a well established state quoted in the Review of Particle Physics (RPP)
ParticleDataGroup:2020ssz , it has attracted a lot of discussions and debates
on its structure. The main decay channels of the $f_{0}(1710)$ resonance are
$K\bar{K}$ and $\eta\eta$, while the $\pi\pi$ decay branching ratio of the
$f_{0}(1710)$ resonance is very small ParticleDataGroup:2020ssz . This
indicates that $f_{0}(1710)$ resonance has a large $s\bar{s}$ component in its
wave function. This is indeed what was found in Ref. Close:2005vf . It has
also been suggested as a scalar glueball candidate Gui:2012gx ;
Janowski:2014ppa ; Fariborz:2015dou . Furthermore, the scalar mesons
$f_{0}(1370)$, $f_{0}(1500)$, and $f_{0}(1710)$ cannot be simultaneously
accommodated in the quark model, thus they were widely investigated by
different mixing schemes Achasov:1995nh ; Amsler:1995tu ; Amsler:1995td ;
Li:2000xy ; Li:2000yn ; Close:2000yk ; Li:2000cj .

On the other hand, the $f_{0}(1710)$ was proposed to be a state dynamically
generated from the vector meson-vector meson interactions Geng:2008gx ;
Geng:2009gb ; Du:2018gyn , which remains valid even after the vector meson-
pseudoscalar meson and pseudoscalar meson-pseudoscalar meson interactions are
included in the coupled channel approach Garcia-Recio:2013uva ; Wang:2019niy ;
Wang:2021jub . Within this picture, the $f_{0}(1710)$ couples mostly to the
$K^{*}\bar{K^{*}}$ channel and most properties of $f_{0}(1710)$ can be well
reproduced Wang:2021jub ; Nagahiro:2008bn ; Branz:2009cv ; Geng:2010kma ;
Wang:2011tm ; MartinezTorres:2012du ; Xie:2014gla ; Dai:2015cwa ; Dai:2018thd
; Molina:2019wjj .

In fact, in Refs. Geng:2008gx ; Du:2018gyn , an isospin one partner of the
$f_{0}(1710)$ state is also obtained, with its mass around $1780$ MeV and
negative $G$-parity. The $a_{0}(1710)$ also couples mostly to the
$K^{*}\bar{K}^{*}$ channel, but the $\rho\omega$ and $\rho\phi$ channels are
also important. Very similar conclusions are also found in Ref. Wang:2022pin ,
where these pseudoscalar-pseudoscalar coupled channels were taken into
account, while the obtained mass of $a_{0}(1710)$ of Ref. Wang:2022pin  is
smaller than those predicted in Refs. Geng:2008gx ; Du:2018gyn . The
properties of the $a_{0}(1710)$ of Refs. Geng:2008gx ; Geng:2009gb ;
Wang:2022pin  are collected in Table 1, where the results of Ref. Wang:2022pin
are obtained with a cutoff $q_{\rm max}=1000$ MeV. Within the ranges of the
model parameters of Ref. Du:2018gyn , the $a_{0}(1710)$ mass is predicted in
the range of $1750\sim 1790$ MeV. In addition, one isovector scalar resonance
with a mass of 1744 MeV is also predicted within the Regge trajectories
Wang:2017pxm .

Table 1:  Predicted properties of the $a_{0}(1710)$ state. $g_{K^{*}\bar{K}^{*}}$ stands for the coupling of $a_{0}(1710)$ to the $K^{*}\bar{K}^{*}$ channel. $\Gamma_{K\bar{K}}$ corresponds to the partial decay width of the $a_{0}(1710)\to K\bar{K}$. All are in units of MeV. Set | $M_{a_{0}(1710)}$ | $\Gamma_{a_{0}(1710)}$ | $g_{K^{*}\bar{K}^{*}}$ | $\Gamma_{K\bar{K}}$
---|---|---|---|---
I (Refs. Geng:2008gx ; Geng:2009gb ) | $1777$ | $148$ | $(7525,-i1529)$ | $36$
II (Ref. Wang:2022pin ) | $1720$ | $200$ | $(8731,-i2200)$ | $74$

Recently, the BESIII Collaboration has performed an amplitude analysis of the
process $D_{s}^{+}\to\pi^{+}K_{S}^{0}K_{S}^{0}$ BESIII:2021anf . It is found
that there is an enhancement in the $K^{0}_{S}K^{0}_{S}$ invariant mass
spectrum around $1.7$ GeV, which was not seen in the BESIII earlier
measurements of $D_{s}^{+}\to K^{+}K^{-}\pi^{+}$ BESIII:2020ctr . This
indicates the existence of the isospin one partner of the $f_{0}(1710)$
resonance, i.e., $a_{0}(1710)$. In addition, the $a_{0}(1710)$ state was also
observed in the $\pi\eta$ invariant mass spectrum of the
$\eta_{c}\to\eta\pi^{+}\pi^{-}$ decay by the BABAR Collaboration BaBar:2021fkz
. The Breit-Wigner mass and width of the $a_{0}(1710)$ state 111It should be
stressed that in Ref. BESIII:2021anf  BESIII does not distinguish between the
$a_{0}(1710)$ and $f_{0}(1710)$, and denotes the combined state as $S(1710)$.
are determined as,

$\displaystyle M_{a_{0}(1710)}$ $\displaystyle=$ $\displaystyle 1723\pm 11\pm
2~{}{\rm MeV},$ (1) $\displaystyle\Gamma_{a_{0}(1710)}$ $\displaystyle=$
$\displaystyle 140\pm 14\pm 4~{}{\rm MeV},$ (2)

by BESIII BESIII:2021anf , and

$\displaystyle M_{a_{0}(1710)}$ $\displaystyle=$ $\displaystyle 1704\pm 5\pm
2~{}{\rm MeV},$ (3) $\displaystyle\Gamma_{a_{0}(1710)}$ $\displaystyle=$
$\displaystyle 110\pm 15\pm 11~{}{\rm MeV},$ (4)

by BABAR BaBar:2021fkz .

Based on the new measurement of BESIII BESIII:2021anf , Ref. Dai:2021owu  has
investigated the process $D_{s}^{+}\to\pi^{+}K_{S}^{0}K_{S}^{0}$, where
$D^{+}_{s}\to\pi^{+}K^{*+}K^{*-}$, $\pi^{+}K^{*0}\bar{K}^{*0}$ firstly happen,
then undergo the $K^{*}\bar{K}^{*}$ final state interaction to give rise to
the final states $K\bar{K}$. Accordingly, the $a_{0}(1710)$ and $f_{0}(1710)$
resonances are dynamically generated from the $K^{*}\bar{K}^{*}$ final state
interaction. The production of $a_{0}(1710)$ and $f_{0}(1710)$ states in the
$D_{s}^{+}\to\pi^{+}K_{S}^{0}K_{S}^{0}$ and $D^{+}_{s}\to\pi^{+}K^{+}K^{-}$
reactions can be explained Dai:2021owu .

In this work, following Ref. Dai:2021owu , we will revisit the process
$D_{s}^{+}\to\pi^{+}K_{S}^{0}K_{S}^{0}$. In addition to the contributions of
the $a_{0}(1710)$ and $f_{0}(1710)$ states from the intermediate process
$D^{+}_{s}\to\pi^{+}K^{*}\bar{K}^{*}$, we will also study the contribution of
$K^{*}$, which could play a role in the intermediate process $D^{+}_{s}\to
K^{*+}\bar{K}^{0}\to\pi^{+}K^{0}\bar{K}^{0}$. We wish to go beyond the work of
Ref. Dai:2021owu  and study the whole $K^{0}_{S}K^{0}_{S}$ and
$\pi^{+}K^{0}_{S}$ invariant mass spectra, where we will focus on the roles
played by $a_{0}(1710)$ and $K^{*+}$ to describe the line shapes of
$K^{0}_{S}K^{0}_{S}$ and $\pi^{+}K^{0}_{S}$, rather than just the
$f_{0}(1710)$ and $a_{0}(1710)$ contributions extracted from the experimental
data, which was well described in Ref. Dai:2021owu .

The paper is organized as follows. In Sec. II, we present the theoretical
formalism of the $D_{s}^{+}\to\pi^{+}K_{S}^{0}K_{S}^{0}$ decay, and in Sec.
III, we show our numerical results and discussions, followed by a short
summary in Sec. IV.

##  II Formalism

The decay $D^{+}_{s}\to\pi^{+}K^{0}_{S}K^{0}_{S}$ can proceed via the $S$-wave
$K^{*}\bar{K}^{*}$ final state interaction of the intermediate
$D^{+}_{s}\to\pi^{+}K^{*}\bar{K}^{*}$ process, or through the intermediate
$K^{*}$ process of $D^{+}_{s}\to\bar{K}^{0}K^{*+}$ with $K^{*+}\to
K^{0}\pi^{+}$ decay in $P$-wave. In the following, we will present the
theoretical formalism of these two mechanisms respectively.

###  II.1 The mechanism of
$D^{+}_{s}\to\pi^{+}K^{*}\bar{K}^{*}\to\pi^{+}K^{0}_{S}K^{0}_{S}$ reaction

As shown in Refs. Dai:2021owu ; Molina:2019udw ; Wang:2021naf ; Duan:2020vye ,
a way for the $D^{+}_{s}\to\pi^{+}K^{0}_{S}K^{0}_{S}$ to proceed is the
following: 1) the charmed quark in $D^{+}_{s}$ turns into a strange quark with
a $u\bar{d}$ pair by the weak decay shown in Fig. 1; 2) the $s\bar{d}$ (Fig.
1(a)) or $u\bar{s}$ (Fig. 1(b)) pair, together with the $\bar{q}q$
($=\bar{u}u+\bar{d}d+\bar{s}s$) pair with the vacuum quantum numbers created
from vacuum, hadronizes into $(\pi K^{*})^{0}$ or $(\pi\bar{K}^{*})^{+}$, and
the other $u\bar{s}$ and $s\bar{d}$ will hadronize to $K^{*+}$ and
$\bar{K}^{*0}$, respectively; 3) the final-state interactions of the
$K^{*}\bar{K}^{*}$ will lead to dynamical generated $a_{0}(1710)$, and finally
it decays into $K^{0}_{S}K^{0}_{S}$. According to the topological
classification of weak decays in Refs. Chau:1982da ; Chau:1987tk  the above
processes proceed via the so-called internal $W$ emission mechanism.

Figure 1: The internal $W$ emission mechanisms for (a):
$D_{s}^{+}\to\pi^{+}K^{*-}K^{*+}$ and (b):
$D^{+}_{s}\to\pi^{+}K^{*0}\bar{K}^{*0}$.

The $D^{+}_{s}$ weak decay processes shown in Figs. 1 (a) and (b) can be
formulated as following,

$\displaystyle D^{+}_{s}$ $\displaystyle\to$ $\displaystyle V_{1}[s\bar{d}\to
s(\bar{u}u+\bar{d}d+\bar{s}s)\bar{d}](u\bar{s}\to K^{*+}),$ (5) $\displaystyle
D^{+}_{s}$ $\displaystyle\to$ $\displaystyle V_{2}[u\bar{s}\to
u(\bar{u}u+\bar{d}d+\bar{s}s)\bar{s}](\bar{d}s\to\bar{K}^{*0}),$ (6)

where $V_{1}$ and $V_{2}$ are the strength of the production vertices, and
contain all the dynamical factors. One can rewrite the two-quark two-antiquark
products in the following way

$\displaystyle\sum_{i=u,d,s}{s\bar{q}_{i}q_{i}\bar{d}}$ $\displaystyle=$
$\displaystyle M_{3i}M_{i2}=(M^{2})_{32},$ (7)
$\displaystyle\sum_{i=u,d,s}{u\bar{q}_{i}q_{i}\bar{s}}$ $\displaystyle=$
$\displaystyle M_{1i}M_{i3}=(M^{2})_{13},$ (8)

where $M$ is the $q_{i}\bar{q}_{j}$ matrix in the $SU(3)$ flavor space, which
is defined as

$\displaystyle M=\left(\begin{matrix}u\bar{u}&u\bar{d}&u\bar{s}\\\
d\bar{u}&d\bar{d}&d\bar{s}\\\ s\bar{u}&s\bar{d}&s\bar{s}\end{matrix}\right).$
(9)

The elements of matrix $M$ can be written in terms of the pseudoscalar ($P$)
or vector ($V$) mesons, which are given by Molina:2019udw ; Dai:2021owu ;
Wang:2021naf .

$\displaystyle
P=\left(\begin{matrix}\frac{\eta}{\sqrt{3}}+\frac{{\pi}^{0}}{\sqrt{2}}+\frac{{\eta}^{\prime}}{\sqrt{6}}&\pi^{+}&K^{+}\\\
\pi^{-}&\frac{\eta}{\sqrt{3}}-\frac{{\pi}^{0}}{\sqrt{2}}+\frac{{\eta}^{\prime}}{\sqrt{6}}&K^{0}\\\
K^{-}&\bar{K}^{0}&-\frac{\eta}{\sqrt{3}}+\frac{{\sqrt{6}\eta}^{\prime}}{3}\end{matrix}\right),$
(10)

and

$\displaystyle
V=\left(\begin{matrix}\frac{{\rho}^{0}}{\sqrt{2}}+\frac{{\omega}}{\sqrt{2}}&\rho^{+}&K^{*+}\\\
\rho^{-}&-\frac{{\rho}^{0}}{\sqrt{2}}+\frac{{\omega}}{\sqrt{2}}&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\end{matrix}\right).$ (11)

The hadronization processes at the quark level in Eqs. (7) and (8) can be
reexpressed at the hadronic level as,

$\displaystyle(M^{2})_{32}\to(V\cdot P)_{32}$ $\displaystyle=$
$\displaystyle\pi^{+}K^{\ast-}-\frac{1}{\sqrt{2}}\pi^{0}\bar{K}^{\ast 0},$
(12) $\displaystyle(M^{2})_{13}\to(P\cdot V)_{13}$ $\displaystyle=$
$\displaystyle\pi^{+}K^{\ast 0}+\frac{1}{\sqrt{2}}\pi^{0}K^{\ast+},$ (13)

where we have neglected those terms with no contribution to the intermediate
$\pi K^{*}\bar{K}^{*}$ state. Using Eqs. (12) and (13), we can rewrite Eqs.
(5) and (6) as

$\displaystyle D^{+}_{s}$ $\displaystyle\to$ $\displaystyle
V_{1}(\pi^{+}K^{\ast-}K^{\ast+}-\frac{1}{\sqrt{2}}\pi^{0}\bar{K}^{\ast
0}K^{\ast+}),$ (14) $\displaystyle D^{+}_{s}$ $\displaystyle\to$
$\displaystyle V_{2}(\pi^{+}K^{\ast 0}\bar{K}^{\ast
0}+\frac{1}{\sqrt{2}}\pi^{0}K^{\ast+}\bar{K}^{\ast 0}).$ (15)

To study the decay $D^{+}_{s}\to\pi^{+}a_{0}(1710)$ with $a_{0}(1710)$
dynamically generated from the final-state interaction of $K^{*}\bar{K}^{*}$,
we should sum Eqs. (14) and (15) and produce the combination of $K^{*+}K^{*-}$
and $K^{*0}\bar{K}^{*0}$ in isospin $I=1$. With the isospin doublet
$(K^{*+},K^{*0})$ and $(\bar{K}^{*0},-K^{*-})$ Close:1979bt , we obtain,

$\displaystyle|K^{\ast 0}\bar{K}^{\ast 0}\rangle$ $\displaystyle=$
$\displaystyle{\frac{1}{\sqrt{2}}}(|K^{\ast}\bar{K}^{\ast},I=1\rangle-|K^{\ast}\bar{K}^{\ast},I=0\rangle),$
$\displaystyle|K^{\ast+}K^{\ast-}\rangle$ $\displaystyle=$
$\displaystyle-{\frac{1}{\sqrt{2}}}(|K^{\ast}\bar{K}^{\ast},I=1\rangle+|K^{\ast}\bar{K^{\ast}},I=0\rangle).$

Thus, we have,

$\displaystyle\\!\\!V_{1}K^{\ast+}K^{\ast-}+V_{2}K^{\ast 0}\bar{K}^{\ast 0}$
(16) $\displaystyle=$
$\displaystyle\\!\\!-{\frac{V_{1}}{\sqrt{2}}}\left(|K^{\ast}\bar{K^{\ast}},I=1\rangle+|K^{\ast}\bar{K^{\ast}},I=0\rangle\right)$
$\displaystyle\\!\\!+\frac{V_{2}}{\sqrt{2}}\left(|K^{\ast}\bar{K^{\ast}},I=1\rangle-|K^{\ast}\bar{K^{\ast}},I=0\rangle\right)$
$\displaystyle=$
$\displaystyle\\!\\!{\frac{V_{2}-V_{1}}{\sqrt{2}}}|K^{\ast}\bar{K^{\ast}},I=1\rangle-{\frac{V_{2}+V_{1}}{\sqrt{2}}}|K^{\ast}\bar{K^{\ast}},I=0\rangle.$

We see that the phases of the above two terms have different signs. If one
term is dominant, the other one could be small and can be neglected. In this
work, we will focus on the contribution from $a_{0}(1710)$ and ignore the
$f_{0}(1710)$ contribution. This seems to be a reasonable choice given the
reasonable description of the invariant $K^{0}_{S}K^{0}_{S}$ and
$\pi^{+}K^{0}_{S}$ mass distributions as shown below.

After the production of the $K^{*}\bar{K}^{*}$ pair, the final-state
interaction in $S$-wave between $K^{*}$ and $\bar{K}^{*}$ takes place, in
which the $a_{0}(1710)$ is produced, and then it decays to
$K^{0}_{S}K^{0}_{S}$ in the final state. 222Note that the parameters $V_{1}$
and $V_{2}$ are assumed to be independent of the final-state interactions. In
Fig. 2 we show the rescattering diagram for the
$D^{+}_{s}\to\pi^{+}K^{*}\bar{K}^{*}\to\pi^{+}a_{0}(1710)\to\pi^{+}K^{0}_{S}K^{0}_{S}$
decay.

Figure 2: The diagram for the $K^{*}\bar{K}^{*}$ final state interaction for
the
$D^{+}_{s}\to\pi^{+}K^{*}\bar{K}^{*}\to\pi^{+}a_{0}(1710)\to\pi^{+}K^{0}_{S}K^{0}_{S}$
decay.

With the above formalism, the decay amplitude of the process shown in Fig. 2
can be written as 333We take
$|K^{0}\rangle=\frac{1}{\sqrt{2}}(|K^{0}_{S}\rangle+|K^{0}_{L}\rangle)$ and
$|\bar{K}^{0}\rangle=\frac{1}{\sqrt{2}}(|K^{0}_{S}\rangle-|K^{0}_{L}\rangle)$,
where we have ignored the effect of $CP$ violation.

$\displaystyle\mathcal{M}_{a}$ $\displaystyle=$
$\displaystyle{\frac{V_{2}-V_{1}}{4}}{\tilde{G}}_{K^{\ast}\bar{K}^{\ast}}(M_{K^{0}_{S}K^{0}_{S}})\times$
(17)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!{\frac{g_{K^{\ast}\bar{K^{\ast}}}g_{K\bar{K}}}{M_{K_{S}^{0}K_{S}^{0}}^{2}-M_{a_{0}(1710)}^{2}+iM_{a_{0}(1710)}\Gamma_{a_{0}(1710)}}},$

where $M_{K^{0}_{S}K^{0}_{S}}$ is the invariant mass of the
$K^{0}_{S}K^{0}_{S}$ system. We define $V_{P}=V_{2}-V_{1}$, which will be
determined from the branching fraction
Br$(D^{+}_{s}\to\pi^{+}K^{0}_{S}K^{0}_{S})$.

The $\tilde{G}_{K^{*}\bar{K}^{*}}$ is the loop function for the
$K^{*}\bar{K}^{*}$ pair, which depends on $M_{K^{0}_{S}K^{0}_{S}}$. Since the
$K^{*}$ and $\bar{K}^{*}$ have large total decay widths, they should be taken
into account. For that purpose, the $G_{K^{*}\bar{K}^{*}}$ is not $\tilde{G}$,
the loop function of two stable particles of masses $m_{1}$ and $m_{2}$, but
convoluted in the masses $m_{1}$ and $m_{2}$ with the mass distributions of
$K^{*}$ and $\bar{K}^{*}$ vector mesons, which can be done following Refs.
Geng:2008gx ; Molina:2008jw ; Xie:2013ula ,

$\displaystyle{G}_{K^{\ast}\bar{K}^{\ast}}(M_{K^{0}_{S}K^{0}_{S}})$
$\displaystyle=$
$\displaystyle\int_{m_{-}^{2}}^{m_{+}^{2}}\int_{m_{-}^{2}}^{m_{+}^{2}}d\tilde{m_{1}}^{2}d\tilde{m}_{2}^{2}\times$
(18)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\omega(\tilde{m}_{1}^{2})\omega(\tilde{m}_{2}^{2})\tilde{G}(M_{K^{0}_{S}K^{0}_{S}},\tilde{m}_{1}^{2},\tilde{m}_{2}^{2}),$

with

$\displaystyle\omega(\tilde{m}_{1}^{2})$ $\displaystyle=$
$\displaystyle{\frac{1}{N}}\text{Im}\left(\frac{1}{\tilde{m}_{1}^{2}-m_{K^{\ast}}^{2}+i\Gamma(\tilde{m}_{1}^{2})\tilde{m}_{1}}\right)$
(19) $\displaystyle N$ $\displaystyle=$
$\displaystyle\int_{\tilde{m}_{-}^{2}}^{\tilde{m}_{+}^{2}}d\tilde{m}_{1}^{2}\text{Im}\left(\frac{1}{\tilde{m}_{1}^{2}-m_{K^{\ast}}^{2}+i\Gamma(\tilde{m}_{1}^{2})\tilde{m}_{1}}\right),$
(20)

and

$\displaystyle\Gamma(\tilde{m}_{1}^{2})$ $\displaystyle=$
$\displaystyle\Gamma_{K^{\ast}}\frac{\tilde{k}^{3}}{k^{3}},$ (21)
$\displaystyle\tilde{k}$ $\displaystyle=$
$\displaystyle\frac{\lambda(\tilde{m}_{1}^{2},m_{\pi}^{2},m_{K}^{2})}{2\tilde{m}_{1}},$
(22)

where the K${\ddot{a}}$llen function
$\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2xz-2yz$. In this work, we take
$m_{+}^{2}=\left(m_{K^{\ast}}+2\Gamma_{K^{\ast}}\right)^{2}$,
$m_{-}^{2}=\left(m_{K^{\ast}}-2\Gamma_{K^{\ast}}\right)^{2}$, $m_{\pi}=138.04$
MeV and $m_{K}=495.644$ MeV. In addition, the masses, widths and spin-parities
of the involved particles are listed in Table 2.

Table 2: Masses, widths and spin-parities of the involved particles in this work. Particle | Mass (MeV) | Width (MeV) | Spin-parity ($J^{P}$)
---|---|---|---
$D^{+}_{s}$ | 1968.34 | 1.31 $\times 10^{-9}$ | $0^{-}$
$\pi^{+}$ | 139.5704 | — | $0^{-}$
$K^{0}_{S}$ | 497.611 | — | $0^{-}$
$K$ | 495.644 | — | $0^{-}$
$K^{*}$ | 893.605 | 49.05 | $1^{-}$
$K^{*+}$ | 891.66 | 50.8 | $1^{-}$
$K^{*0}$ | 895.55 | 47.3 | $1^{-}$

In the dimensional regularization scheme,
$\tilde{G}(s=M^{2}_{K^{0}_{S}K^{0}_{S}},m_{1}^{2},m_{2}^{2})$ can be written
as Molina:2008jw ; Xie:2013ula

$\displaystyle\tilde{G}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\Bigg{\\{}a_{\mu}+\text{In}\frac{m_{1}^{2}}{\mu^{2}}+\frac{m_{2}^{2}-m_{1}^{2}+s}{2s}\text{In}\frac{m_{2}^{2}}{m_{1}^{2}}$
(23)
$\displaystyle\frac{p}{\sqrt{s}}\bigg{[}\text{In}\left(s-\left(m_{2}^{2}-m_{1}^{2}\right)+2p\sqrt{s}\right)$
$\displaystyle+\text{In}\left(s+\left(m_{2}^{2}-m_{1}^{2}\right)+2p\sqrt{s}\right)$
$\displaystyle-\text{In}\left(-s+\left(m_{2}^{2}-m_{1}^{2}\right)+2p\sqrt{s}\right)$
$\displaystyle-\text{In}\left(-s-\left(m_{2}^{2}-m_{1}^{2}\right)+2p\sqrt{s}\right)\bigg{]}\Bigg{\\}}$

with

$\displaystyle p=\frac{\lambda^{1/2}(s,m_{1}^{2},m_{2}^{2})}{2\sqrt{s}},$ (24)

where $\mu$ is a scale of dimensional regularization, and $a_{\mu}$ is the
subtraction constant. We take $\mu=1000$ MeV and $a_{\mu}=-1.726$ as in Ref.
Geng:2008gx . It is worth mentioning that the only parameter dependent part of
$\tilde{G}$ is $a_{\mu}+{\rm ln}(m^{2}_{1}/\mu^{2})$. Any change in $\mu$ is
reabsorbed by a change in $a_{\mu}$ through $a_{\mu^{\prime}}-a_{\mu}={\rm
ln}(\mu^{\prime 2}/\mu^{2})$, so that the loop function $\tilde{G}$ is scale
independent. It should be noted that the loop function $\tilde{G}$ can be also
regularized with the cutoff method as in Refs. Wang:2021jub ; Wang:2022pin ;
Wang:2019niy ; Lu:2020ste ; Oller:1997ti ; Oset:1997it .

The so obtained real (solid curves) and imaginary (dashed curves) parts of the
loop function $G_{K^{*}\bar{K}^{*}}$ as a function of the $K^{0}_{S}K^{0}_{S}$
invariant mass are shown in Fig. 3. The results considering the $K^{*}$ width
are obtained with the dimensional regularization method as in Ref. Geng:2008gx
, while the results without considering the $K^{*}$ width are calculated with
the cutoff parameter of Ref. Wang:2022pin .

Figure 3: Real and imaginary parts of the loop function $G_{K^{*}\bar{K}^{*}}$
as a function of the invariant $K^{0}_{S}K^{0}_{S}$ mass computed in the
dimensional regularization method and the cutoff method.

In addition, the $g_{K\bar{K}}$ in Eq. (17) is the coupling constant of
$a_{0}(1710)$ to the $K\bar{K}$ channel, and it can be determined from the
partial decay width of $a_{0}(1710)\to K\bar{K}$, which is given by

$\displaystyle\Gamma_{K\bar{K}}=\frac{g^{2}_{K\bar{K}}}{8\pi}\frac{p_{K}}{M^{2}_{a_{0}(1710)}},$
(25)

where $p_{K}$ is the three momentum of the $K$ or $\bar{K}$ meson in the
$a_{0}(1710)$ rest frame. With these $\Gamma_{K\bar{K}}$ values of Refs.
Geng:2008gx ; Geng:2009gb  and Ref. Wang:2022pin  shown in Table 1, we obtain
$g_{K\bar{K}}=1966$ MeV and $2797$ MeV for Set I and Set II, respectively.
Note that from the partial decay width, one can only obtain the absolute value
of the coupling constant, but not the phase. In this work, we assume that
$g_{K\bar{K}}$ is real and positive.

###  II.2 The mechanism of
$D^{+}_{s}\to\bar{K}^{0}K^{*+}\to\pi^{+}K^{0}_{S}K^{0}_{S}$ reaction

In this section, we will present the formalism for the decay
$D_{s}^{+}\to\pi^{+}K_{S}^{0}K_{S}^{0}$ via the intermediate meson $K^{*+}$.
According to the RPP ParticleDataGroup:2020ssz , the absolute branching
fraction of the decay mode $D^{+}_{s}\to\bar{K}^{0}K^{*+}$ is $(5.4\pm
1.2)\%$, which is comparable to the absolute branching fraction of
$D^{+}_{s}\to\eta\rho^{+}$ that is $(8.9\pm 0.8)\%$. As a result, the
$D_{s}^{+}\to K^{0}_{S}K^{*+}$ is important to produce
$\pi^{+}K_{S}^{0}K_{S}^{0}$ in the final state through
$K^{*+}\to\pi^{+}K^{0}_{S}$ in $P$-wave, as shown in Fig. 4.

Figure 4: The decay $D_{s}^{+}\rightarrow\pi^{+}K_{S}^{0}K_{S}^{0}$ via the
intermediate vector $K^{\ast+}$. We also show the definition of the
kinematical ($p_{1}$, $p_{2}$, $p_{3}$, $p_{D^{+}_{s}}$) variables that we use
in the present calculation.

The decay amplitude for $D^{+}_{s}\to\pi^{+}K^{0}_{S}K^{0}_{S}$ from the
process shown in Fig. 4 can be obtained as

$\displaystyle\mathcal{M}_{b}$ $\displaystyle=$
$\displaystyle\frac{g_{D_{s}\bar{K}K^{*}}g_{K^{\ast}K\pi}}{2}\frac{1}{q^{2}-m_{K^{\ast+}}^{2}+im_{K^{\ast+}}\Gamma_{K^{\ast+}}}$
(26)
$\displaystyle\times\Bigg{[}\left(m_{K_{S}^{0}}^{2}-m_{\pi^{+}}^{2}\right)\left(1-\frac{q^{2}}{m_{K^{\ast+}}^{2}}\right)$
$\displaystyle+2p_{1}\cdot
p_{3}\frac{m_{\pi^{+}}^{2}-m_{K_{S}^{0}}^{2}-m_{K^{\ast+}}^{2}}{m_{K^{\ast+}}^{2}}$
$\displaystyle+2p_{2}\cdot
p_{3}\frac{m_{\pi^{+}}^{2}-m_{K_{S}^{0}}^{2}+m_{K^{\ast+}}^{2}}{m_{K^{\ast+}}^{2}}\Bigg{]}$
$\displaystyle+({\rm exchange~{}term~{}with}~{}p_{2}\leftrightarrow p_{3}),$

where $q^{2}=(p_{1}+p_{2})^{2}=M_{{\pi}K_{S}^{0}}^{2}$ is the invariant mass
squared of the $\pi^{+}K^{0}_{S}$ system. The $g_{D_{s}\bar{K}K^{*}}$ and
$g_{K^{*}K\pi}$ denote the coupling constants of
$D^{+}_{s}\to\bar{K}^{0}K^{*+}$ and $K^{*+}\to K^{0}\pi^{+}$, respectively.
With the masses of these particles given in Table 2, the branching fraction of
Br$(D^{+}_{s}\to\bar{K}^{0}K^{*+})=(5.4\pm 1.2)\%$ and the partial decay width
$K^{*+}\to K^{0}\pi^{+}$ quoted in the RPP ParticleDataGroup:2020ssz , we
obtain $g_{D_{s}\bar{K}K^{*}}=(1.05\pm 0.12)\times 10^{-6}$ and
$g_{K^{*}K\pi}=3.26$. Again, we assume that $g_{D_{s}\bar{K}K^{*}}$ and
$g_{K^{*}K\pi}$ are real and positive Ling:2021qzl . The uncertainty of
$g_{D_{s}\bar{K}K^{*}}$ originates from the uncertainty of the branching
fraction Br$(D^{+}_{s}\to\bar{K}^{0}K^{*+})$, while the uncertainty of
$g_{K^{*}K\pi}$ is ignored, since it is very small.

###  II.3 Invariant mass distributions

We can write the total decay amplitude of
$D^{+}_{s}\to\pi^{+}K^{0}_{S}K^{0}_{S}$ as follows,

$\displaystyle\mathcal{M}=\mathcal{M}_{a}+\mathcal{M}_{b},$ (27)

and the double differential width of the decay
$D^{+}_{s}\to\pi^{+}K^{0}_{S}K^{0}_{S}$ is

$\displaystyle\frac{d^{2}\Gamma}{dM_{K_{S}^{0}K_{S}^{0}}{dM_{{\pi}K_{S}^{0}}}}=\frac{M_{K_{S}^{0}K_{S}^{0}}M_{{\pi}K_{S}^{0}}}{128\pi^{3}m_{D_{s}^{+}}^{3}}(|\mathcal{M}_{a}|^{2}+|\mathcal{M}_{b}|^{2}),$
(28)

where the interference between ${\cal M}_{a}$ and ${\cal M}_{b}$ is neglected,
since these coupling constants are assumed to be real and positive, as
discussed above.

In Ref. BESIII:2021anf , by considering the interference term between
$a_{0}(1710)$ and $K^{*+}$, the extracted branching fraction
Br$(D^{+}_{s}\to\bar{K}^{0}K^{*+})$ is $(1.8\pm 0.2\pm 0.1)\%$, which deviates
from the CLEO result of Br$(D^{+}_{s}\to\bar{K}^{0}K^{*+})=(5.4\pm 1.2)\%$
CLEO:1989zcg . In this work, since the interference term is not included, we
use the CLEO result to determine the value of the coupling constant
$g_{D_{s}\bar{K}K^{*}}$.

Finally, one can easily obtain ${d\Gamma}/{dM_{K_{S}^{0}K_{S}^{0}}}$ and
${d\Gamma}/{dM_{{\pi}K_{S}^{0}}}$, by integrating Eq. (28) over each of the
invariant mass variables with the limits of the Dalitz Plot given in the RPP
ParticleDataGroup:2020ssz . For example, the upper and lower limits for
$M_{\pi^{+}K^{0}_{S}}$ are as follows:

$\displaystyle\left(M_{\pi^{+}K_{S}^{0}}^{2}\right)_{\text{max}}$
$\displaystyle=$
$\displaystyle\left(E_{\pi^{+}}^{\ast}+E_{K_{S}^{0}}^{\ast}\right)^{2}-$
$\displaystyle\left(\sqrt{E_{\pi^{+}}^{\ast
2}-m_{\pi^{+}}^{2}}-\sqrt{E_{K_{S}^{0}}^{\ast
2}-m_{K_{S}^{0}}^{2}}\right)^{2}$
$\displaystyle\left(M_{\pi^{+}K_{S}^{0}}^{2}\right)_{\text{min}}$
$\displaystyle=$
$\displaystyle\left(E_{\pi^{+}}^{\ast}+E_{K_{S}^{0}}^{\ast}\right)^{2}-$
$\displaystyle\left(\sqrt{E_{\pi^{+}}^{\ast
2}-m_{\pi^{+}}^{2}}+\sqrt{E_{K_{S}^{0}}^{\ast
2}-m_{K_{S}^{0}}^{2}}\right)^{2},$

where the $E_{\pi^{+}}^{\ast}$ and $E_{K_{S}^{0}}^{\ast}$ are the energies of
$\pi^{+}$ and $K_{S}^{0}$ in the $K_{S}^{0}K_{S}^{0}$ rest frame,
respectively,

$\displaystyle
E_{\pi^{+}}^{\ast}=\frac{m_{D_{s}^{+}}^{2}-M_{K_{S}^{0}K_{S}^{0}}^{2}-m_{\pi^{+}}^{2}}{2M_{K_{S}^{0}K_{S}^{0}}},$
$\displaystyle
E_{K_{S}^{0}}^{\ast}=\frac{M_{K_{S}^{0}K_{S}^{0}}^{2}-m_{K_{S}^{0}}^{2}+m_{K_{S}^{0}}^{2}}{2M_{K_{S}^{0}K_{S}^{0}}}.$
(29)

Similarly, one can obtain the upper and lower limits of
$M_{K^{0}_{S}K^{0}_{S}}$.

In Fig. 5 we show the Dalitz Plot of the
$D^{+}_{s}\to\pi^{+}K^{0}_{S}K^{0}_{S}$ reaction. The blue band stands for the
$K^{*+}$ region in the $\pi^{+}K^{0}_{S}$ channel. One can see that the
$K^{*+}$ energy region overlaps largely with the $a_{0}(1710)$ state in the
$K^{0}_{S}K^{0}_{S}$ channel.

Figure 5: Dalitz Plot for the decay
$D_{s}^{+}\rightarrow\pi^{+}K_{S}^{0}K_{S}^{0}$.

Because the factor $V_{P}$ is unknown, we determine it from the branching
fraction of $D^{+}_{s}\to\pi^{+}K^{0}_{S}K^{0}_{S}$, which is $(0.68\pm
0.04\pm 0.01)\%$ BESIII:2021anf . With the $a_{0}(1710)$ parameters given in
Table 1, we obtain

$\displaystyle V_{P}=(1.69\pm 0.55)\times 10^{-4}.$ (30)

for Set I, and

$\displaystyle V_{P}=(1.95\pm 0.64)\times 10^{-4}.$ (31)

for Set II.

##  III Results and Discussion

In this section, we present the numerical results for the invariant mass
distribution of $K^{0}_{S}K^{0}_{S}$ and $\pi^{+}K^{0}_{S}$ of the
$D_{s}^{+}\rightarrow\pi^{+}K_{S}^{0}K_{S}^{0}$ decay. To compare the
theoretical invariant mass distributions with the experimental measurements,
we introduce an extra global normalization factor $C$, which will be fitted to
the experimental data. In Fig. 6, we show our theoretical results for the
$K^{0}_{S}K^{0}_{S}$ invariant mass distribution. The red-solid curve stands
for the total contributions from the $a_{0}(1710)$ state and the vector
$K^{*+}$ meson, while the blue-dashed and green-dot-dashed curves correspond
to the contribution from only the $a_{0}(1710)$ and $K^{*+}$, respectively.
The red-solid curve has been adjusted to the strength of the experimental data
of BESIII BESIII:2021anf  at its peak by taking $C=2.3\times 10^{7}$ for both
Set I and Set II. One can see that the model results obtained with the
parameters of both Set I and Set II can reproduce the experimental data
reasonably well, and the $K^{*}$ plays an important role around the peak of
the $a_{0}(1710)$ state. It is clearly seen that the shape of $a_{0}(1710)$ in
Fig. 6 (b) is wider than that in Fig. 6 (a). One reason is that, as shown in
Table 1, the $a_{0}(1710)$ width of Set II is larger than the one of Set I.
The other reason is that the loop function $G_{K^{*}\bar{K}^{*}}$ obtained
with the cutoff regularization of Ref. Wang:2022pin  is smoother than the one
of Ref. Geng:2008gx , which can be seen in Fig. 3.

Figure 6: Invariant mass distribution of $K_{S}^{0}K_{S}^{0}$ for the
$D_{s}^{+}\rightarrow\pi^{+}K_{S}^{0}K_{S}^{0}$ decay, compared with the
experimental data taken from Fig. 2(a) of Ref. BESIII:2021anf . (a): results
of Set I; (b): results of Set II.

In Ref. Dai:2021owu , the vector-vector intermediate states were produced at
the first step with both the external and internal $W$-emission mechanisms,
and then the final-state interaction of vector-vector produces $f_{0}(1710)$
and $a_{0}(1710)$ and then they decay into $K^{0}_{S}K^{0}_{S}$ and
$K^{+}K^{-}$. By adjusting the effective parameters between these production
processes, the ratio of the branching fractions
Br$(D^{+}_{s}\to\pi^{+}K^{0}_{S}K^{0}_{S})$ and
Br$(D^{+}_{s}\to\pi^{+}K^{+}K^{-})$ from the $f_{0}(1710)$ and $a_{0}(1710)$
contribution can be reproduced Dai:2021owu . Clearly, this work and Ref.
Dai:2021owu  share the same mechanism for the final-state interactions. As a
result, both can describe the main feature of the $K^{0}_{S}K^{0}_{S}$ line
shapes. In principle, both the external and internal $W$-emission mechanisms
can play a role. However, a quantitative consideration of both mechanisms
inevitably introduces additional free parameters for the weak interaction
(more details can be found in Ref. Dai:2021owu ), which cannot yet be well
determined. Hence, we will leave a simultaneous consideration of both
mechanisms to a future study when more precise experimental data become
available.

It should be noted that the contribution of the $f_{0}(1710)$ state is not
considered in our calculation, while the data, on the other hand, contain the
contributions of both states. This implies that the peaks of $f_{0}(1710)$ and
$a_{0}(1710)$ overlap strongly. Otherwise, the sole contribution from
$a_{0}(1710)$ cannot describe the experimental data. The $f_{0}(1710)$ and
$a_{0}(1710)$ mixing can also be studied in the $J/\psi$ decays Molina:2019wjj
when more experimental data are available, just as the $a_{0}(980)$ and
$f_{0}(980)$ mixing BESIII:2018ozj ; Wu:2008hx ; Wu:2007jh  for the case of
$K\bar{K}$ molecules. In fact, the $a_{0}(980)$ and $f_{0}(980)$ mixing was
investigated in the decay $D^{+}_{s}\to\eta\pi^{0}\pi^{+}$ in Ref.
Achasov:2017edm  with the formalism built in a earlier work of Ref.
Achasov:1979xc , where the mixing of $a_{0}(980)$ and $f_{0}(980)$ resonances
that breaks the isospin invariance due to the $K^{+}$ and $K^{0}$ meson mass
difference.

In our model, the final $K^{0}_{S}K^{0}_{S}$ pair is produced from the
$K^{*}{\bar{K}}^{*}$ interaction, and the loop function $G_{K^{*}\bar{K}^{*}}$
is very small around the $a_{0}(980)$ [$f_{0}(980)$] pole region (see Fig. 3)
and the coupling of $a_{0}(980)$ [$f_{0}(980)$] to the $K^{*}\bar{K}^{*}$
channel Garcia-Recio:2013uva  is also small compared with the one of
$a_{0}(1710)$ to the $K^{*}\bar{K}^{*}$ channel. Hence, there are no
$a_{0}(980)$ and $f_{0}(980)$ signal in the $K^{0}_{S}K^{0}_{S}$ mass spectrum
of the $D_{s}^{+}\rightarrow\pi^{+}K_{S}^{0}K_{S}^{0}$ decay. On the other
hand, from Eq. (16), we find that the two phases in the $a_{0}(1710)$ and
$f_{0}(1710)$ productions have an opposite sign. If the production of
$a_{0}(1710)$ is constructive as shown in the new BESIII data BESIII:2021anf ,
one can expect no contribution from the mechanism shown in Fig. 1 to produce
the $f_{0}(1710)$ resonance in the $D^{+}_{s}\to\pi^{+}K^{+}K^{-}$ decay
BESIII:2020ctr ; Duan:2020vye ; Wang:2021ews . However, the $f_{0}(1710)$
could be produced via the external $W$ emission mechanism, and its signal is
expected in the process $D^{+}_{s}\to\pi^{+}K^{+}K^{-}$.

Next, we turn to the $\pi^{+}K^{0}_{S}$ invariant mass distributions. In Fig.
7 we show the theoretical results for the invariant $\pi^{+}K_{S}^{0}$ mass
distributions of the $D_{s}^{+}\rightarrow\pi^{+}K_{S}^{0}K_{S}^{0}$ decay. To
compare with the experimental results, we have multiplied a factor of two to
${d\Gamma}/{dM_{{\pi}K_{S}^{0}}}$, since the experimental distribution of
$\pi^{+}K^{0}_{S}$ contains two entries of events, one for each $K^{0}_{S}$
(see more details in Ref. BESIII:2021anf ). The peak of the $K^{*+}$ can be
well described. The contribution from $a_{0}(1710)$ is very small to the peak,
while its contribution to the threshold enhancement of the invariant
$\pi^{+}K^{0}_{S}$ mass distribution is significant.

Figure 7: Invariant mass distribution of $\pi^{+}K_{S}^{0}$ for the
$D_{s}^{+}\rightarrow\pi^{+}K_{S}^{0}K_{S}^{0}$ decay, compared with the
experimental data taken from Fig. 2(b) of Ref. BESIII:2021anf . (a): results
of Set I; (b): results of Set II.

In addition, with the model parameters as obtained above for the
$D_{s}^{+}\rightarrow\pi^{+}K_{S}^{0}K_{S}^{0}$ decay, we study the process of
$D_{s}^{+}\rightarrow\pi^{+}K^{+}K^{-}$, where the contributions from
$a_{0}(1710)$ and $K^{*}(892)$ are taken into account by assuming that the
mechanism of $D_{s}^{+}\rightarrow\pi^{+}K^{+}K^{-}$ is the same as the one of
the process of $D_{s}^{+}\rightarrow\pi^{+}K^{0}_{S}K^{0}_{S}$. The numerical
results for the $K^{+}K^{-}$ and $\pi^{+}K^{-}$ invariant mass distributions
are shown in Figs. 8 and 9, respectively. The experimental data are taken from
Ref. BESIII:2020ctr . If the $a_{0}(1710)$ plays the dominant role for the
structure around $M^{2}_{K^{+}K^{-}}=3$ GeV2, the $\pi^{+}K^{-}$ invariant
mass distribution in the low energy region can not be well described because
of the reflection effect of $a_{0}(1710)$. In Fig. 8, the $K^{*}(892)$
contribution is scaled by a factor of 4.3, as shown by the pink-dash-dashed
curve, and one can see that the $K^{*}(892)$ contribution is already enough to
reasonably describe both the $K^{+}K^{-}$ and $\pi^{+}K^{-}$ invariant mass
distributions in the energy region considered. which is also consistent with
the Dalitz Plot of $D_{s}^{+}\rightarrow\pi^{+}K^{+}K^{-}$, as shown in Fig. 6
of Ref. BESIII:2020ctr .

As discussed above, to describe well both
$D^{+}_{s}\to\pi^{+}K^{0}_{S}K^{0}_{S}$ and $D^{+}_{s}\to\pi^{+}K^{+}K^{-}$
reactions, one needs to consider other mechanisms, especially the contribution
from $f_{0}(1710)$. In this case, we will have more free parameters, and we
need more constraints from both theoretical and experimental sides. In the
present work, we focus on the role played by the $a_{0}(1710)$ in the
$D^{+}_{s}\to\pi^{+}K^{0}_{S}K^{0}_{S}$ decay, and it is found that the new
measurements of the $D^{+}_{s}\to\pi^{+}K^{0}_{S}K^{0}_{S}$ reaction can be
well reproduced, and the contribution of the $K^{*}(892)$ around the
$a_{0}(1710)/f_{0}(1710)$ peak is important.

Figure 8: Invariant mass distribution of $K^{+}K^{-}$ for the
$D_{s}^{+}\rightarrow\pi^{+}K^{+}K^{-}$ decay, compared with the experimental
data taken from Fig. 7(a) of Ref. BESIII:2020ctr . Figure 9: Invariant mass
distribution of $\pi^{+}K^{-}$ for the $D_{s}^{+}\rightarrow\pi^{+}K^{+}K^{-}$
decay, compared with the experimental data taken from Fig. 7(c) of Ref.
BESIII:2020ctr .

Finally, it is interesting to note that one can study the $a^{+}_{0}(1710)$
state in the $K^{+}K^{0}_{S}$ channel of the
$D^{+}_{s}\to\pi^{0}K^{+}K^{0}_{S}$ decay by including the contribution of the
final-state interaction of $K^{*+}\bar{K}^{*0}$, which can be easily obtained
by summing the second term of Eqs. (12) and (13). If the very small mass
difference of charged and neutral $K^{*}$ meson is neglected, it is expected
that the branching fraction of $D^{+}_{s}\to\pi^{0}K^{+}K^{0}_{S}$ should be
the same as the one of $D^{+}_{s}\to\pi^{+}K^{0}_{S}K^{0}_{S}$, and hence a
charged $a_{0}^{+}(1710)$ will be visible in the invariant $K^{+}K^{0}_{S}$
mass spectrum. Indeed, the $a_{0}^{+}(1710)$ was recently observed in the
decay of $D^{+}_{s}\to\pi^{0}K^{+}K^{0}_{S}$ by the BESIII Collaboration
BESIII:2022wkv .

##  IV Summary

In summary, we have studied the Cabibbo-favored process of
$D_{s}^{+}\rightarrow\pi^{+}K_{S}^{0}K_{S}^{0}$. By considering the decay
mechanism of internal $W^{+}$ emission, and hadronization of the $s\bar{d}$ or
$u\bar{s}$ with $q\bar{q}$ with the vacuum quantum numbers, we obtain
$\pi^{+}K^{*}\bar{K}^{*}$ in the first step, then the transition of
$K^{*}\bar{K}^{*}\to K^{0}_{S}K^{0}_{S}$ proceeds following final-state
interactions of the $K^{*}\bar{K}^{*}$ pair in the chiral unitary approach
where the $a_{0}(1710)$ state is dynamically generated. In addition, the tree
diagram of $K^{*+}\to\pi^{+}K^{0}_{S}$ is also taken into account.

We have calculated the $K^{0}_{S}K^{0}_{S}$ and $\pi^{+}K^{0}_{S}$ invariant
mass distributions, which are in good agreement with the experimental
measurements of BESIII BESIII:2021anf . We have found that the $K^{*}$ plays
an important role in the $a_{0}(1710)$ peak region. Our study shows that the
BESIII measurements support the $K^{*}\bar{K}^{*}$ molecular nature of the
$a_{0}(1710)$ and $f_{0}(1710)$ states and they overlap strongly in the data.

For the reproduction of the $a_{0}(1710)$ peak, it is found that the
contributions from both the tree diagram as shown in Fig. 4 and the
$K^{*}\bar{K}^{*}$ final-state interaction as shown in Fig. 2 are crucial. In
addition, within the proposed mechanism, it is expected that the charged
$a^{+}_{0}(1710)$ signal can show up in the $K^{+}K^{0}_{S}$ invariant mass
distribution of the $D^{+}_{s}\to\pi^{0}K^{+}K^{0}_{S}$ decay, which should be
checked by future experimental measurements.

## Acknowledgments

We would like to thank Prof. Eulogio Oset and Prof. Lian-Rong Dai for their
careful reading of the manuscript and useful discussions. This work is partly
supported by the National Natural Science Foundation of China under Grants
Nos. 12075288, 11735003, 11975041, 11961141004, 11961141012, and 12192263.
This work is supported by the Natural Science Foundation of Henan under Grand
No. 222300420554, the Key Research Projects of Henan Higher Education
Institutions under No. 20A140027, the Project of Youth Backbone Teachers of
Colleges and Universities of Henan Province (2020GGJS017), the Youth Talent
Support Project of Henan (2021HYTP002), the Fundamental Research Cultivation
Fund for Young Teachers of Zhengzhou University (JC202041042), and the Open
Project of Guangxi Key Laboratory of Nuclear Physics and Nuclear Technology,
No. NLK2021-08.

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Random multi-hooking networks

Kiran R. Bhutani111Department of Mathematics, The Catholic University of
America, Washington, D.C. 20064, U.S.A.; Email<EMAIL_ADDRESS>Ravi
Kalpathy222Department of Mathematics, The Catholic University of America,
Washington, D.C. 20064, U.S.A.; Email<EMAIL_ADDRESS>Hosam
Mahmoud333Department of Statistics, The George Washington University,
Washington, D.C. 20052, U.S.A.; Email<EMAIL_ADDRESS>

## Abstract

We introduce a broad class of multi-hooking networks, wherein multiple copies
of a seed are hooked at each step at random locations, and the number of
copies follows a predetermined building sequence of numbers.

We analyze the degree profile in random multi-hooking networks by tracking two
kinds of node degrees—the local average degree of a specific node over time
and the global overall average degree in the graph. The former experiences
phases and the latter is invariant with respect to the type of building
sequence and is somewhat similar to the average degree in the initial seed. We
also discuss the expected number of nodes of the smallest degree.

Additionally, we study distances in the network through the lens of the
average total path length, the average depth of a node, the eccentricity of a
node, and the diameter of the graph.

AMS subject classifications: Primary: 05C82, 90B15; Secondary: 60C05, 05C12.

Keywords: Hooking networks, random graph, degree profile, recurrence.

##  1 Introduction

Trees have long been in the focus of research on random graphs. The classic
types, such as those that appear in data structures [3, 10, 14, 15] and
digital processing [6, 12, 20], grow incrementally, one node at a time. In
more recent times, authors considered more complex types of random graphs
grown by adjoining entire graphs to a growing network [1, 2, 5, 7, 8, 9, 11,
13, 16, 17, 18, 21]. We consider a growing network model in which the number
of components attached at a stage follows a predetermined building sequence of
numbers.

Societies and social networks grow and change over time in multiple random
ways, which include growth patterns that add “components” at each step.
Networks grown by adding components reflect these dynamics better than
networks evolving on single node additions. One can embed a graph in a
predetermined growth structure leading to multiple scenarios of growing
networks.

In this paper, we develop a model where networks grow by hooking multiple
copies of the seed at multiple nodes of the growing network chosen in a random
fashion and study the theoretical and statistical properties of the networks
so generated.

##  2 The building sequence

We assume that a network grows by attaching a number of components at each
step to the existing structure, which starts with $\tau_{0}\geq 2$ vertices.
In the next subsection, we give a formal definition. Here, we only say a word
on the number of components added at each step. After $n$ steps of growth, the
number of components attached to obtain the next network is $k_{n}$, a
predetermined sequence of nonnegative numbers.

###  2.1 Regularity conditions

Let $\tau_{0}\geq 2$. This represents the number of nodes in a building block
(a seed). We grow the network by adding a number of copies of the seed at
places called latches. At each latch, a designated vertex in the seed (called
the hook) is fused with the latch. A formal definition of this process is
given in the sequel.

We shall consider adding $k_{n}\geq 1$ copies of the seed to construct the
$(n+1)$st network, under the following regularity conditions:

  * (R1)

$k_{n}\leq\tau_{0}+(\tau_{0}-1)\sum_{i=0}^{n-1}k_{i}$.

  * (R2)

$\lim_{n\to\infty}\frac{k_{n}}{\sum_{i=0}^{n}k_{i}}=a\in[0,1]$.

  * (R3)

$\lim_{n\to\infty}\frac{\tau_{0}}{k_{n}}=b\geq 0$.

A sequence of nonnegative integers $\\{k_{n}\\}_{n=0}^{\infty}$ satisfying
(R1)–(R3) is called a building sequence. Condition (R1) is to guarantee the
feasibility of choosing latches. At no point in time does the process require
more (distinct) latches than the number of nodes existing in the network.
Conditions (R2)–(R3) facilitate the existence of limits for properties of
interest and expedite finding their values. Note that $a=0$ and $b=0$ are both
allowed. For instance, for a constant sequence $k_{n}=k\in\mathbb{N}$, we have
$a=0$, and $b=\tau_{0}/k>0$, whereas when $k_{n}=n+1$, we have both $a=0$ and
$b=0$.

Regularity conditions (R1)–(R3) are not too restrictive and the class covered
by the investigation remains very broad. The examples that come up in practice
satisfy these regularity conditions. For example, at one extreme the building
sequence $k_{n}=1$ builds networks of linear growth, including trees. At the
other extreme, the case of equality in Condition (R1) builds a deterministic
network where the entire vertex set is chosen at each step (a take-all model);
such extremal case grows the network exponentially fast.

##  3 The multi-hooking network

A network grows as follows. We start with a connected seed graph $G_{0}$ with
vertex set of size $\tau_{0}$ and edge set of size $\eta$. One of the vertices
in the seed is designated as a hook (vertex $h$). When a copy of the seed is
adjoined to the network, it is the seed’s hook that latches into that larger
graph. The hooking is accomplished by fusing together the hook and a latch
(vertex) chosen from the network.

At step $n$, $k_{n-1}$ copies of the seed are hooked into the graph,
$G_{n-1}=(V_{n-1},{\cal E}_{n-1})$, with vertex set $V_{n-1}$ and edge set
${\cal E}_{n-1}$, that exists at time $n-1$. To complete the $n$th hooking
step, we sample $k_{n-1}$ latches from the graph $G_{n-1}$. The selection
mechanism can take a number of forms, such as choosing distinct hooks as
opposed to allowing repetitions.

We use the notation $|A|$ for the cardinality of a set $A$. We consider a
uniform model that selects $k_{n-1}$ distinct nodes in the network, with all
$|V_{n-1}|\choose k_{n-1}$ subsets being equally likely. In the language of
statistics, this boils down to sampling without replacement.

Figure 1 illustrates a seed and a network grown from it in three steps under
the building sequence $k_{n}=n+1$. So, $G_{1}$ grows by choosing a latch from
$G_{0}$ (the starred node in $G_{0}$), $G_{2}$ grows by choosing the two
starred nodes from $G_{1}$, and $G_{3}$ grows by choosing the three starred
nodes from $G_{2}$. The networks in Figure 1 have loops and multiple edges, as
we do not restrict the study to simple graphs.

$h$$\star$$G_{0}$$G_{1}$$G_{2}$$G_{3}$$\star$$\star$$\star$$\star$$\star$$h$$h$$h$
Figure 1: A seed (top) with a hook and three networks grown from it (second
row) under the building sequence $k_{n}=n+1$. The white vertices in the
network $G_{1}$, $G_{2}$, and $G_{3}$ represent the reference vertex.

###  3.1 Notation

The notation ${\rm Hypergeo}(t,r,s)$ stands for the hypergeometric random
variable associated with the random sampling of $s$ objects out of a total of
$t$ objects, of which $r$ objects are of a special type. So, the
hypergeometric random variable counts the number of special objects in the
sample.

It is customary to call the cardinality of the vertex set of a graph the order
of the graph and reserve the term size of the graph to the cardinality of the
set of edges in the graph. Let $V_{n}$ be the set of vertices of the graph
$G_{n}$, and ${\cal E}_{n}$ be the set of edges of that graph. Thus, the seed
$G_{0}=(V_{0},{\cal E}_{0})$ is a connected graph with the set $V_{0}$ of
vertices and the set ${\cal E}_{0}$ of edges.

Let $\tau_{n}$ be the order of the graph at age $n$. Hence, the cardinality of
the vertex set $V_{0}$ of the seed is $|V_{0}|=\tau_{0}$. The $n$th hooking
step adds $k_{n-1}$ copies of the seed at $k_{n-1}$ distinct latches chosen
uniformly at random from $G_{n-1}$. Each copy contributes $\tau_{0}-1$ new
vertices to the network. The reason for subtracting 1 is the absorption of the
hook. This gives the recurrence

$\tau_{n}=\tau_{n-1}+k_{n-1}(\tau_{0}-1).$ (1)

Unwinding this recurrence, we obtain

$\tau_{n}=(\tau_{0}-1)\sum_{i=1}^{n}k_{i-1}+\tau_{0}.$ (2)

We use the notation $\deg(v)$ to denote the degree of node $v$ in a given
graph, and we set $h^{*}=\deg(h)$.

###  3.2 Useful limits

By the regularity conditions, we can argue from (2) that

$\frac{\tau_{n}}{k_{n-1}}=(\tau_{0}-1)\frac{1}{k_{n-1}}\sum_{i=0}^{n-1}k_{i}+\frac{\tau_{0}}{k_{n-1}}\to\frac{\tau_{0}-1}{a}+b=:\gamma.$

Reorganize (1) as

$1=\frac{\tau_{n-1}}{\tau_{n}}+\frac{k_{n-1}}{\tau_{n}}(\tau_{0}-1).$

to find the limit

$\frac{\tau_{n-1}}{\tau_{n}}=1-\frac{k_{n-1}}{\tau_{n}}(\tau_{0}-1)\to
1-\frac{(\tau_{0}-1)}{\gamma}.$

##  4 A degree profile of the network

Various aspects of the degrees of nodes in a network are of interest in
different contexts. For example, in the language of epidemiology, the degree
of a node may be a useful representation of a highly infective person. From a
health policy point of view, having knowledge about the degrees in conjunction
with other graph parameters may help in identifying hot spots that trigger
outbreaks and may be useful in controlling and mitigating the contagion. In
the context of a social network, the degree of a node may represent the
popularity and social skills of the person represented by the node.

Equally interesting are the global overall average degree in the entire graph
(where we look at all the nodes), the local degree of a specific node during
its temporal evolution, and the number of nodes of the smallest degree. We
deal with the average behavior of each of these in a separate subsection. The
different aspects of the degree complete a profile of the graph.

###  4.1 Evolution of the degree of a specific node

Suppose a node appears for the first time at step $j$. What will become of its
degree at step $n$? At step $j$, several copies are added. To avoid a heavy
notation identifying the time of appearance $j$, the copy number, which node
within the copy to be tracked, and $n$, we use a simpler notation that needs
only $j$ and $n$, for after all nodes of the same degree in the seed have the
same distribution over time.

######  Theorem 4.1.

Suppose $\\{k_{n}\\}_{n=0}^{\infty}$ is a building sequence of the family of
graphs $\\{G_{n}\\}_{n=1}^{\infty}$. Let $X_{j:n}$ be the degree of a node at
time $n$ that had appeared for the first time at step $j$. If initially its
degree (in the seed) is $\delta$, then we have

${\mathbb{E}}[X_{j:n}]=\delta+h^{*}\sum_{i=j}^{n-1}\frac{k_{i}}{\tau_{i}}=\frac{h^{*}a}{(1-a)(\tau_{0}-1)+ab}(n-j)+o(n-j)+O(1).$

###### Proof.

Suppose a node $v$ appears at time $j$ for the first time. So, it belongs to
one of the copies adjoined to the graph at that time. As the graph evolves, in
any single step the degree of $v$ can increase, if it is one of the nodes
selected as latches in that step; otherwise its degree stays put, and when it
does increase, it goes up by $h^{*}=\deg(h)$, the degree of the hook in the
seed. This gives rise to a recurrence:

$X_{j:n}=X_{j:n-1}+h^{*}{\mathbb{I}}_{n-1}(v),$

where ${\mathbb{I}}_{n-1}(v)$ is an indicator of the event of choosing $v$
among the $k_{n-1}$ latches of that step of growth. On average, we have

${\mathbb{E}}[X_{j:n}]={\mathbb{E}}[X_{j:n-1}]+h^{*}\frac{{\tau_{n-1}-1\choose
k_{n-1}-1}}{{\tau_{n-1}\choose
k_{n-1}}}={\mathbb{E}}[X_{j:n-1}]+h^{*}\frac{k_{n-1}}{\tau_{n-1}}.$

Unwinding the recurrence, we obtain the exact average:

${\mathbb{E}}[X_{j:n}]=\delta+h^{*}\sum_{i=j}^{n-1}\frac{k_{i}}{\tau_{i}}.$

By the limits in Subsection 3.2, we obtain

${\mathbb{E}}[X_{j:n}]=O(1)+o(n-j)+\frac{h^{*}a}{(1-a)(\tau_{0}-1)+ab}(n-j).$

∎

######  Remark 4.1.

If $a=0$ the ${\mathbb{E}}[X_{j:n}]$ is only $o(n-j)$.

######  Remark 4.2.

Consider the case $a>0$. The average in Theorem 4.1 indicates that the degree
of a specific node experiences phases. The degree of a node in the early phase
with $j=j(n)=o(n)$ grows linearly with its age in the network. When
$j(n)\sim\rho n$, for $0<\rho<1$, we still get a linear growth, but the
coefficient of linearity is attenuated to
$(1-\rho)\,\frac{h^{*}a}{(1-a)(\tau_{0}-1)+ab}$. At $\rho=1$, we have
${\mathbb{E}}[X_{j:n}]=o(n)$.

######  Remark 4.3.

If $a=0$, we can only assert that ${\mathbb{E}}[X_{j:n}]=o(n-j)+\delta$. In
this case, a finer analysis is needed to identify the leading order of the
average degree of a node that appears at time $j$. For instance, in the case
of a tree grown from the complete graph $K_{2}$, we have
$\delta=1,h^{*}=1,k_{n}=1$, and $a=0$. The exact formula in this case yields

${\mathbb{E}}[X_{j:n}]=1+\sum_{i=j}^{n-1}\frac{1}{i+2}=H_{n+1}-H_{j+1}+1.$

Whence, we have the phases

${\mathbb{E}}[X_{j:n}]\sim\begin{cases}\ln n,&\mbox{if \ }j\mbox{\ is
fixed};\\\ \ln\frac{n}{j(n)},&\mbox{if \ }j(n)\to\infty\ \mbox{and \
}j(n)=o(n);\\\ \ln\frac{1}{\rho},&\mbox{if \ }j(n)\sim\rho
n,\leavevmode\nobreak\ \leavevmode\nobreak\ 0<\rho<1;\\\ 1,&\mbox{if \
}j(n)=n-o(n).\end{cases}$

###  4.2 The overall average degree

The main result about the overall average degree in the graph is developed in
this section. The result is expressed in terms of $\eta$, the number of edges
in the seed graph.

######  Theorem 4.2.

Suppose $\\{k_{n}\\}_{n=0}^{\infty}$ is a building sequence of the family of
graphs $\\{G_{n}\\}_{n=1}^{\infty}$. Let $Y_{n}$ be the degree of a randomly
chosen node in the graph $G_{n}$ at age $n$. We have

$\lim_{n\to\infty}{\mathbb{E}}[Y_{n}]=\frac{2\eta}{\tau_{0}-1}.$

###### Proof.

Upon hooking $k_{n-1}$ copies of the seed to $k_{n-1}$ distinct nodes of
$G_{n-1}=(V_{n-1},{\cal E}_{n-1})$, we add $\eta k_{n-1}$ edges to the graph.
Therefore, we have

$|{\cal E}_{n}|=|{\cal E}_{n-1}|+\eta k_{n-1}.$

This recurrence has the solution

$|{\cal E}_{n}|=\eta\Big{(}1+\sum_{i=0}^{n-1}k_{i}\Big{)}.$

Using the classical First Theorem of Graph Theory, we obtain

$\sum_{v\in V_{n}}\deg(v)=2{|\cal
E}_{n}|=2\eta\Big{(}1+\sum_{i=0}^{n-1}k_{i}\Big{)}.$

Scaling the equation by $\tau_{n}$, we get

$\frac{1}{\tau_{n}}\sum_{v\in
V_{n}}\deg(v)=\frac{2\eta}{\tau_{n}}\Big{(}1+\sum_{i=0}^{n-1}k_{i}\Big{)}.$

Taking limits, and using equation (2), we obtain

$\lim_{n\to\infty}{\mathbb{E}}[Y_{n}]=0+\lim_{n\to\infty}2\eta\,\frac{\sum_{i=0}^{n-1}k_{i}}{\tau_{n}}=\frac{2\eta}{\tau_{0}-1}.$

∎

######  Remark 4.4.

The average degree in the seed is $2\eta/\tau_{0}$. For any building sequence,
the asymptotic average degree in the graph is $2\eta/(\tau_{0}-1)$, only
slightly higher than the average degree in the initial seed. This should be
anticipated because the additions introduce a number of copies of the seed,
each of which has the degree properties of the seed with the hook eliminated.

###  4.3 Nodes of the smallest degree

We study only the nodes of the smallest degree. Let $d^{*}$ be the smallest
degree in the seed. Note that the smallest admissible degree in the graph is
$d^{*}$. After the network grows, the smallest degree in it may be $d^{*}$ or
higher. Let $X_{n}$ be the number of nodes of degree $d^{*}$ at time $n$.
Thus, $X_{0}$ is the number of nodes of degree $d^{*}$ in the seed. Later
graphs can have more nodes of degree $d^{*}$. The seed in Figure 1 has
$d^{*}=2$, and $X_{0}=1,X_{1}=2,X_{2}=3$, and $X_{3}=3$.

####  4.3.1 Stochastic recurrence

In the evolution at step $n$, we hook $k_{n-1}$ copies of the seed to the
graph $G_{n-1}$. Let $A_{0}$ be the event $\deg(h)=d^{*}$ and
$\mathbb{I}_{A_{0}}$ be an indicator that assumes value 1, if $\deg(h)=d^{*}$,
otherwise, it assumes the value 0. A latch of degree $d^{*}$ in the sample
will have a higher degree (namely, its degree goes up to $d^{*}+\deg(h)$) in
$G_{n}$. So, we lose such vertices in the count of $X_{n}$. If the hook degree
is $d^{*}$, every hooked copy contributes only $X_{0}-1$ vertices of degree
$d^{*}$.

For the case when $k_{n-1}=1$ and the latch is $\ell$, the change from
$X_{n-1}$ to $X_{n}$ for the four cases can be seen as shown in the table
below:

| ${\deg(\ell)}=d^{*}$ | ${\deg(\ell)}\neq d^{*}$
---|---|---
$\deg(h)=d^{*}$ | $(X_{n-1}-1)+(X_{0}-1)$ | $X_{n-1}+(X_{0}-1)$
$\deg(h)\neq d^{*}$ | $(X_{n-1}-1)+X_{0}$ | $X_{n-1}+X_{0}$

Thus, for any value of $k_{n-1}$, the count $X_{n}$ therefore satisfies a
(conditional) stochastic recurrence:

$\displaystyle X_{n}$
$\displaystyle=X_{n-1}+(X_{0}-1-\mathbb{I}_{A_{0}})\,{\rm
Hypergeo}(\tau_{n-1},X_{n-1},k_{n-1})$
$\displaystyle\qquad{}+(X_{0}-\mathbb{I}_{A_{0}})\big{(}k_{n-1}-{\rm
Hypergeo}(\tau_{n-1},X_{n-1},k_{n-1})\big{)}.$ (3)

####  4.3.2 The average proportion of nodes of degree $d^{*}$

Take (conditional) expectation of (3) to get

$\displaystyle{\mathbb{E}}\big{[}X_{n}\,|\,G_{n-1}\big{]}$
$\displaystyle=X_{n-1}+(X_{0}-1-\mathbb{I}_{A_{0}})\frac{X_{n-1}}{\tau_{n-1}}k_{n-1}$
$\displaystyle\qquad{}+(X_{0}-\mathbb{I}_{A_{0}})\Big{(}k_{n-1}-\frac{X_{n-1}}{\tau_{n-1}}k_{n-1}\Big{)}$
$\displaystyle=\Big{(}1-\frac{k_{n-1}}{\tau_{n-1}}\Big{)}X_{n-1}+(X_{0}-\mathbb{I}_{A_{0}})k_{n-1}.$
(4)

######  Theorem 4.3.

Suppose $\\{k_{n}\\}_{n=0}^{\infty}$ is a building sequence of the family of
graphs $\\{G_{n}\\}_{n=1}^{\infty}$, starting from a seed with $X_{0}$ nodes
of the smallest degree $d^{*}$. Let $X_{n}$ be the number of vertices of this
degree in the graph after $n$ steps of evolution according to the building
sequence. We have

${\mathbb{E}}[X_{n}]=(X_{0}-\mathbb{I}_{A_{0}})\sum_{i=1}^{n}k_{i-1}\prod_{j=i+1}^{n}\Big{(}1-\frac{k_{j-1}}{\tau_{j-1}}\Big{)}+X_{0}\prod_{j=1}^{n}\Big{(}1-\frac{k_{j-1}}{\tau_{j-1}}\Big{)}.$

Subsequently, the average proportion converges to a limit independent of the
limits $a$ and $b$; namely we have the convergence

${\mathbb{E}}\Big{[}\frac{X_{n}}{\tau_{n}}\Bigr{]}\to\frac{X_{0}-\mathbb{I}_{A_{0}}}{\tau_{0}}.$

###### Proof.

Taking a double expectation of (4) yields

${\mathbb{E}}[X_{n}]=\Big{(}1-\frac{k_{n-1}}{\tau_{n-1}}\Big{)}\,{\mathbb{E}}[X_{n-1}]+(X_{0}-\mathbb{I}_{A_{0}})\,k_{n-1}.$
(5)

This recurrence equation is of the standard linear form

$y_{n}=g_{n}y_{n-1}+h_{n},$ (6)

with solution

$y_{n}=\sum_{i=1}^{n}h_{i}\prod_{j=i+1}^{n}g_{j}+y_{0}\prod_{j=1}^{n}g_{j}.$
(7)

So, the sought solution for the average of the number of nodes of degree
$d^{*}$ (for $n\geq 1$) is

${\mathbb{E}}[X_{n}]=(X_{0}-\mathbb{I}_{A_{0}})\sum_{i=1}^{n}k_{i-1}\prod_{j=i+1}^{n}\Big{(}1-\frac{k_{j-1}}{\tau_{j-1}}\Big{)}+X_{0}\prod_{j=1}^{n}\Big{(}1-\frac{k_{j-1}}{\tau_{j-1}}\Big{)}.$

The strategy for the asymptotic part of the statement is two-fold: We prove
the existence of a limit (under any building sequence) for the proportion from
the exact solution. We then find the value of the limit from the recurrence
under the mild regularity conditions imposed on the building sequence.

First, express the expected proportion as

${\mathbb{E}}\Big{[}\frac{X_{n}}{\tau_{n}}\Big{]}=(X_{0}-\mathbb{I}_{A_{0}})\sum_{i=1}^{n}\frac{k_{i-1}}{\tau_{n}}\prod_{j=i+1}^{n}\Big{(}1-\frac{k_{j-1}}{\tau_{j-1}}\Big{)}+\frac{X_{0}}{\tau_{n}}\prod_{j=1}^{n}\Big{(}1-\frac{k_{j-1}}{\tau_{j-1}}\Big{)};$
(8)

at $i=n$, the first product does not exist, and is taken to be 1, as usual.
Let

$c_{n}=\sum_{i=1}^{n}\frac{k_{i-1}}{\tau_{n}}\prod_{j=i+1}^{n}\Bigl{(}1-\frac{k_{j-1}}{\tau_{j-1}}\Bigr{)}=\sum_{i=1}^{n-1}\frac{k_{i-1}}{\tau_{n}}\prod_{j=i+1}^{n}\Bigl{(}1-\frac{k_{j-1}}{\tau_{j-1}}\Bigr{)}+\frac{k_{n-1}}{\tau_{n}};$

We manipulate this to turn it into a recurrence as follows:

$\displaystyle c_{n+1}$
$\displaystyle=\sum_{i=1}^{n}\frac{k_{i-1}}{\tau_{n+1}}\prod_{j=i+1}^{n+1}\Bigl{(}1-\frac{k_{j-1}}{\tau_{j-1}}\Bigr{)}+\frac{k_{n}}{\tau_{n+1}}$
$\displaystyle=\frac{\tau_{n}}{\tau_{n+1}}\Big{(}1-\frac{k_{n}}{\tau_{n}}\Big{)}\sum_{i=1}^{n}\frac{k_{i-1}}{\tau_{n}}\prod_{j=i+1}^{n}\Bigl{(}1-\frac{k_{j-1}}{\tau_{j-1}}\Bigr{)}+\frac{k_{n}}{\tau_{n+1}}$
$\displaystyle=\Big{(}\frac{\tau_{n}-k_{n}}{\tau_{n+1}}\Big{)}c_{n}+\frac{k_{n}}{\tau_{n+1}}$
$\displaystyle=\Big{(}\frac{\tau_{n+1}-(\tau_{0}-1)k_{n}-k_{n}}{\tau_{n+1}}\Big{)}c_{n}+\frac{k_{n}}{\tau_{n+1}}$
$\displaystyle=\Big{(}\frac{\tau_{n+1}-\tau_{0}k_{n}}{\tau_{n+1}}\Big{)}c_{n}+\frac{k_{n}}{\tau_{n+1}}$

Rearrange the recurrence in the form

$c_{n+1}-\frac{1}{\tau_{0}}=c_{n}-\frac{\tau_{0}k_{n}}{\tau_{n+1}}\,c_{n}+\frac{k_{n}}{\tau_{n+1}}-\frac{1}{\tau_{0}}=\Big{(}c_{n}-\frac{1}{\tau_{0}}\Big{)}\Big{(}\frac{\tau_{n+1}-\tau_{0}k_{n}}{\tau_{n+1}}\Big{)},$

leading to the inequality

$\displaystyle\Big{|}c_{n+1}-\frac{1}{\tau_{0}}\Big{|}$
$\displaystyle\leq\Big{|}c_{n}-\frac{1}{\tau_{0}}\Big{|}\,\Big{|}\frac{\tau_{n+1}-\tau_{0}k_{n}+k_{n}}{\tau_{n+1}}\Big{|}$
$\displaystyle=\Big{|}c_{n}-\frac{1}{\tau_{0}}\Big{|}\,\frac{\tau_{n}}{\tau_{n+1}}$
$\displaystyle\leq\Big{|}c_{n-1}-\frac{1}{\tau_{0}}\Big{|}\,\frac{\tau_{n}}{\tau_{n+1}}\times\,\frac{\tau_{n-1}}{\tau_{n}}$
$\displaystyle\ \ \vdots$
$\displaystyle\leq\Big{|}c_{0}-\frac{1}{\tau_{0}}\Big{|}\,\frac{\tau_{n}}{\tau_{n+1}}\times\,\frac{\tau_{n-1}}{\tau_{n}}\times\cdots\times\frac{\tau_{0}}{\tau_{1}}.$

Noting that the sum in $c_{n}$ is empty at $n=0$, we have $c_{0}=0$ and the
bounds simplify to $0\leq|c_{n}-1/\tau_{0}|\leq 1/\tau_{n}$. So, both inferior
and superior limits of $|c_{n}-1/\tau_{0}|$ are equal to $0$, which furnishes
the existence of a limit for $c_{n}$ equal to $1/\tau_{0}$, too.

As for the remainder part

$r_{n}:=\frac{X_{0}}{\tau_{n}}\prod_{j=1}^{n}\Bigl{(}1-\frac{k_{j-1}}{\tau_{j-1}}\Bigr{)},$

in (8), it clearly converges to 0, as $\tau_{n}$ is increasing, and the
product is bounded from above by 1. Plugging in the limits
$\lim_{n\to\infty}c_{n}=1/\tau_{0}$ and $\lim_{n\to\infty}r_{n}=0$ in (8), we
reach the conclusion that

$\lim_{n\to\infty}{\mathbb{E}}\Bigl{[}\frac{X_{n}}{\tau_{n}}\Bigr{]}=\frac{X_{0}-\mathbb{I}_{A_{0}}}{\tau_{0}}.$

∎

######  Remark 4.5.

In the case when the hook is not of the smallest degree $d^{*}$, we have
${\mathbb{E}}[X_{n}/\tau_{n}]\to X_{0}/\tau_{0}$. The initial proportion of
nodes of the smallest degree in the seed is preserved on average in larger
subsequent graphs.

######  Remark 4.6.

In the case when the hook is of the smallest degree $d^{*}$, we have
${\mathbb{E}}[X_{n}/\tau_{n}]\to(X_{0}-1)/\tau_{0}$. The long-term proportion
of nodes of the smallest degree is less than the proportion of nodes of degree
$d^{*}$ in the seed.

######  Remark 4.7.

In the case when the hook is the only node of the smallest degree $d^{*}$ in
the seed, we have $X_{0}=1$, and ${\mathbb{E}}[X_{n}/\tau_{n}]\to 0$, for all
$n\geq 1$. Indeed, the degree $d^{*}$ disappears after the first latching at
the initial hook and never reappears.

######  Remark 4.8.

The limit in Theorem 4.3 is more than just an ultimate value in the take-all
case. In this case, it is the actual value for each $n\geq 0$, which can be
seen from the recurrence. The only term that does not vanish is the last term
in sum, yielding
$(X_{0}-I_{A_{0}})k_{n-1}/\tau_{n}=(X_{0}-I_{A_{0}})/\tau_{0}$.

##  5 Distances in the network

We measure node distances in $G_{n}$ relative to a reference point (vertex).
We take the reference to be the hook of $G_{0}$. We look at two (related)
kinds of distances: The total path length and the average distance in the
graph. Let the nodes of the $n$th graph be labeled with the numbers
$1,2,\ldots,\tau_{n}$, with 1 being reserved for the reference vertex and the
rest of the nodes are arbitrarily assigned distinct numbers from the set
$\\{2,\ldots,\tau_{n}\\}$. The depth of a node in the network is its distance
from the reference vertex (i.e., the length of the shortest path from the node
to the reference vertex measured in the number of edges). We denote the depth
of the $i$th node in the $n$th network by $\Delta_{n,i}$ The total path length
is the sum of all depths; namely it is

$T_{n}=\sum_{i=1}^{\tau_{n}}\Delta_{n,i}.$

For instance, the networks $G_{0},G_{1},G_{2}$, and $G_{3}$ in Figure 1 have
total path lengths $T_{0}=3,T_{1}=6$, $T_{2}=18$, and $T_{3}=45$,
respectively.

###  5.1 Average total path length

As the network grows, at step $n$, a sample of size $k_{n-1}$ latches is
chosen from $G_{n-1}$ to grow $G_{n-1}$ into $G_{n}$. Suppose these latches
are $\ell_{1},\ldots,\ell_{k_{n-1}}$ at depths $d_{1},\ldots,d_{k_{n-1}}$. In
view of the absorption of the hooks of the added graphs, a copy’s hook fused
at the $j$th latch adds $\tau_{0}-1$ nodes, which appear in $G_{n}$ at depths
equal to their distance from the hook of the copy translated by an additional
distance from the latch to the reference vertex. So, collectively, the
vertices of the copy hooked to $\ell_{j}$ increase the total path length by
$T_{0}+(\tau_{0}-1)d_{j}$. We have a conditional recurrence:

${\mathbb{E}}\big{[}T_{n}\,|\,G_{n-1},d_{1},\ldots,d_{k_{n-1}}\big{]}=T_{n-1}+\sum_{j=1}^{k_{n-1}}(T_{0}+(\tau_{0}-1)d_{j}).$

Averaging over the graphs and the choices of the latches within, we get

${\mathbb{E}}[T_{n}]={\mathbb{E}}[T_{n-1}]+T_{0}k_{n-1}+(\tau_{0}-1)\sum_{j=1}^{k_{n-1}}{\mathbb{E}}[\Delta_{n-1,\ell_{j}}].$
(9)

######  Lemma 5.1.

$\displaystyle\sum_{j=1}^{k_{n-1}}{\mathbb{E}}[\Delta_{n-1,\ell_{j}}]=\frac{k_{n-1}}{\tau_{n-1}}\,{\mathbb{E}}[T_{n-1}].$

###### Proof.

Condition on the event $\ell_{1}=i_{1},\ldots,\ell_{k_{n-1}}=i_{k_{n-1}}$, to
get

$\displaystyle\sum_{j=1}^{k_{n-1}}{\mathbb{E}}[\Delta_{n-1,\ell_{j}}]$
$\displaystyle=\sum_{j=1}^{k_{n-1}}\sum_{1\leq
i_{1}<i_{2}<\cdots<i_{k_{n-1}}\leq\tau_{n-1}}{\mathbb{E}}\big{[}\Delta_{n-1,\ell_{j}}\,|\,d_{1}=i_{1},\ldots
d_{k_{n-1}}=i_{k_{n-1}}\big{]}$
$\displaystyle\qquad\qquad\qquad\qquad{}\times{\mathbb{P}}(d_{1}=i_{1},\ldots,d_{k_{n-1}}=i_{k_{n-1}}).$

The subsets of size $k_{n-1}$ latches that appear in a sample of vertices from
$G_{n-1}$ are all equally likely, and we get

$\displaystyle\sum_{j=1}^{k_{n-1}}{\mathbb{E}}[\Delta_{n-1,\ell_{j}}]$
$\displaystyle\qquad=\frac{1}{{{\tau_{n-1}}\choose{k_{n-1}}}}\sum_{j=1}^{k_{n-1}}{\mathbb{E}}\Big{[}\Big{(}\sum_{1\leq
i_{1}<i_{2}<\cdots<i_{k_{n-1}}\leq\tau_{n-1}}\Delta_{n-1,\ell_{j}}\Big{)}\,\big{|}\,\ell_{1}=i_{1},\ldots\ell_{k_{n-1}}=i_{k_{n-1}}\Big{]}.$

Let us write out the inner sum in expanded form:

$\displaystyle\Delta_{n-1,1}+\Delta_{n-1,2}+\cdots+\Delta_{n-1,k_{n-1}}$
$\displaystyle\qquad{}+\Delta_{n-1,1}+\Delta_{n-1,2}+\cdots+\Delta_{n-1,k_{n-1}-1}+\Delta_{n-1,k_{n-1}+1}$
$\displaystyle\qquad\ \,{}\vdots$
$\displaystyle\qquad{}+\Delta_{n-1,\tau_{n-1}-k_{n-1}+1}+\Delta_{n-1,\tau_{n-1}-k_{n-1}+2}+\cdots+\Delta_{n-1,\tau_{n-1}}.$

Upon a reorganization collecting similar terms, we get

${\tau_{n-1-1}\choose
k_{n-1}-1}(\Delta_{n-1,1}+\Delta_{n-1,2}+\cdots+\Delta_{n-1,\tau_{n-1}})={\tau_{n-1-1}\choose
k_{n-1}-1}T_{n-1}.$

Plugging this expression in the expectation, we proceed to

$\sum_{j=1}^{k_{n-1}}{\mathbb{E}}[\Delta_{n-1,\ell_{j}}]=\frac{{\tau_{n-1-1}\choose
k_{n-1}-1}}{{\tau_{n-1}\choose
k_{n-1}}}\,{\mathbb{E}}[T_{n-1}]=\frac{k_{n-1}}{{\tau_{n-1}}}\,{\mathbb{E}}[T_{n-1}].$

∎

######  Theorem 5.1.

Suppose $\\{k_{n}\\}_{n=0}^{\infty}$ is a building sequence of the family of
graphs $\\{G_{n}\\}_{n=1}^{\infty}$, starting from a seed of total path length
$T_{0}$. Let $T_{n}$ be the total path length after $n$ steps of evolution
according to the building sequence. We have

${\mathbb{E}}[T_{n}]=T_{0}\tau_{n}\Big{(}\sum_{i=1}^{n}\frac{k_{i-1}}{\tau_{i}}+\frac{1}{\tau_{0}}\Big{)}.$

###### Proof.

By Lemma 5.1 and the recurrence (9), we have a recurrence for the average
total path length:

$\displaystyle{\mathbb{E}}[T_{n}]$
$\displaystyle={\mathbb{E}}[T_{n-1}]+T_{0}k_{n-1}+\frac{(\tau_{0}-1)k_{n-1}}{\tau_{n-1}}\,{\mathbb{E}}[T_{n-1}]$
$\displaystyle=\Big{(}1+\frac{(\tau_{0}-1)k_{n-1}}{\tau_{n-1}}\Big{)}{\mathbb{E}}[T_{n-1}]+T_{0}k_{n-1}.$

Again, the recurrence is of the standard form (6) with the solution (7). In
the specific case at hand, this solution is

${\mathbb{E}}[T_{n}]=T_{0}\sum_{i=1}^{n}k_{i-1}\prod_{j=i+1}^{n}\Big{(}1+\frac{(\tau_{0}-1){k_{j-1}}}{\tau_{j-1}}\Big{)}+T_{0}\prod_{j=1}^{n}\Big{(}1+\frac{(\tau_{0}-1)k_{j-1}}{\tau_{j-1}}\Big{)}.$

The recurrence (1) on the order of the graph simplifies the solution into
telescopic products

${\mathbb{E}}[T_{n}]=T_{0}\sum_{i=1}^{n}k_{i-1}\prod_{j=i+1}^{n}\frac{\tau_{j}}{\tau_{j-1}}+T_{0}\prod_{j=1}^{n}\frac{\tau_{j}}{\tau_{j-1}}=T_{0}\tau_{n}\Bigl{(}\sum_{i=1}^{n}\frac{k_{i-1}}{\tau_{i}}+\frac{1}{\tau_{0}}\Bigr{)}.$

∎

###  5.2 Average depth

Theorem 5.1 provides a benchmark for the calculation of the average depth. Let
the depth of a randomly selected node in the $n$th network be $D_{n}$.

######  Corollary 5.1.

${\mathbb{E}}[D_{n}]=T_{0}\sum_{i=1}^{n}\frac{k_{i-1}}{\tau_{i}}+D_{0}.$

###### Proof.

Given a specific development leading to $G_{n}$, the average depth in that
graph is

${\mathbb{E}}\big{[}D_{n}\,|\,G_{n}\big{]}=\frac{\Delta_{n,1}+\cdots+\Delta_{n,\tau_{n}}}{\tau_{n}}=\frac{T_{n}}{\tau_{n}}.$

Upon taking expectation, it follows that
${\mathbb{E}}\big{[}D_{n}]={\mathbb{E}}[T_{n}]/\tau_{n}$. The form given in
the statement ensues from Theorem 5.1. ∎

######  Corollary 5.2.

Under the regularity conditions (R1)–(R3), we have the asymptotic equivalent

${\mathbb{E}}[D_{n}]=\frac{aT_{0}}{\tau_{0}-1+ab}\,n+o(n),\qquad\mbox{as \
}n\to\infty.$

######  Remark 5.1.

Corollary 5.2 is more useful when

$\lim_{n\to\infty}\frac{k_{n}}{\sum_{i=0}^{n}k_{i}}=a\not=0.$

When $a=0$, as in the case of trees for example, one needs to sharpen the
argument to find the leading asymptotic term, as we do in some specific cases
below.

###  5.3 Distances under specific building sequences

At one extreme, there is the sequence of least possible growth ($k_{n}=1$). At
the other extreme, we have a take-all model ($k_{n}=\tau_{n}$) in which all
the nodes of $G_{n}$ are taken as latches for $\tau_{n}$ copies of the seed.

In the case of $k_{n}=k$, for fixed $k\in\mathbb{N}$, of nearly the least
growth, the average depth is

${\mathbb{E}}[D_{n}]=T_{0}\sum_{i=1}^{n}\frac{k}{\tau_{i}}+D_{0}.$

The limit $a$ in regularity condition (R2) is 0, and Corollary 5.2 only tells
us that ${\mathbb{E}}[D_{n}]=o(n)$. However, we can sharpen the asymptotic
equivalence from the specific construction of the case.

Here, we have $\tau_{i}=(\tau_{0}-1)i+\tau_{0}$, which gives

${\mathbb{E}}[D_{n}]=\frac{T_{0}k}{\tau_{0}-1}\sum_{i=1}^{n}\frac{1}{i+\frac{\tau_{0}}{\tau_{0}-1}}+D_{0}.$

In terms of the generalized harmonic numbers444Customarily, $H_{n}(0)$ is
denoted by $H_{n}$.

$H_{n}(x)=\frac{1}{1+x}+\frac{1}{2+x}+\cdots+\frac{1}{n+x},$

the depth in the near-least-growth is compactly expressed as

${\mathbb{E}}[D_{n}]=\frac{T_{0}k}{\tau_{0}-1}H_{n}\Big{(}\frac{\tau_{0}}{\tau_{0}-1}\Big{)}+D_{0}\sim\frac{T_{0}k}{\tau_{0}-1}\ln
n,\qquad\mbox{as \ }n\to\infty.$

######  Remark 5.2.

The case $k=1$ and $\tau_{0}=2$ grows a recursive tree. The seed is a rooted
tree on two vertices, in which $T_{0}=1$ and $D_{0}=\frac{1}{2}$. In this
case, the average depth becomes

${\mathbb{E}}[D_{n}]=H_{n}(2)+\frac{1}{2}=H_{n+2}-1\sim\ln n,\qquad\mbox{as \
}n\to\infty,$

which recovers a known result [19].

######  Remark 5.3.

At the other end of the spectrum, there is the take-all model, in which
$k_{i}=\tau_{i}$, leading at step $n$ to a graph of order
$\tau_{n}=\tau_{0}^{n+1}$. Here, the limit $a$ is $(\tau_{0}-1)/\tau_{0}$ and
the limit $b$ is 0. According to Corollary 5.2, we have
${\mathbb{E}}[D_{n}]\sim D_{0}n$, as $n\to\infty$. This asymptotic estimate
can be sharpened as the case is amenable to exact calculation:

${\mathbb{E}}[D_{n}]=T_{0}\sum_{i=1}^{n}\frac{\tau_{i-1}}{\tau_{i}}+D_{0}=T_{0}\sum_{i=1}^{n}\frac{1}{\tau_{0}}+D_{0}=D_{0}(n+1).$

##  6 Eccentricity

The eccentricity $C(v)$ of a node $v$ in a graph $\cal G$ is the distance
between $v$ and a vertex farthest from $v$ in $\cal G$. The eccentricity is
instrumental in constructing a notion of the diameter of a graph (extreme
distances). We use the eccentricity of the hook and the various latches
selected in $G_{n-1}$ to determine the diameter of the graph $G_{n}$.

The eccentricity is technically defined as follows. If $\cal Q$ is a path in a
graph $\cal G$, we denote its length by $|\cal Q|$ (the number of edges in
it). There can be several paths joining two vertices $u$ and $v$ in $\cal G$,
and the distance between $u$ and $v$, denoted by $d(u,v)$, is the length of
the shortest such path. That is, with ${\cal P}(u,v)$ denoting the collection
of paths between $u$ and $v$, the distance between these two nodes is given by

$d(u,v)=\min_{{\cal Q}\in{\cal P}(u,v)}|{\cal Q}|.$

The eccentricity $C(v)$ of a vertex $v$ in a graph with vertex set $\cal V$
is:

$C(v)=\max_{u\in\cal V}d(v,u)=\max_{u\in\cal V}\min_{{\cal Q}\in{\cal
P}(v,u)}|{\cal Q}|.$

For instance, the eccentricity in Figure 1 of the reference vertex of $G_{0}$
is 2, of the reference vertex in $G_{1}$ is 2 as well, but of the reference
vertex in $G_{2}$ is 4 and becomes 6 in $G_{3}$.

###  6.1 Eccentricity of a node in $G_{n}$

The $k_{n-1}$ nodes selected as latches from the graph $G_{n-1}=(V_{n-1},{\cal
E}_{n-1})$ are vertices that play a key role in designing the network at stage
$n$ and onward and contribute significantly in determining the diameter of the
graph at the next stage.

As a node’s eccentricity changes over time, its value at step $n$ in $G_{n}$
may be different from its value at step $n-1$ in $G_{n-1}$. We need an
eccentricity notation reflecting the possible change over time. For that we
use $C_{n}(v)$ to speak of the eccentricity of a vertex $v$ in $G_{n}$.

If $v\in V_{n}$ is a vertex in a copy of $G_{0}$ latched at a vertex
$\ell_{i}\in V_{n-1}$, we express that by saying $v\in V_{0}^{co_{i}}$,
otherwise we say $v\in V_{n-1}$. We now introduce some notation:

  1. 1.

$\mathbb{L}_{n}=\\{\ell_{1},\ell_{2},\cdots\ell_{k_{n}}\\}$ is the set of
latches selected in the graph $G_{n}$ to produce the graph $G_{n+1}$.

  2. 2.

$\widetilde{C}_{n}(v)=C_{n}(v)\,|\,G_{n-1},\mathbb{L}_{n-1}$. This is the
conditional eccentricity $C_{n}(v)$ of the node $v$ in the graph $G_{n}$,
given $G_{n-1}$ and the $k_{n-1}$ latches in it.

  3. 3.

For any $v\in V_{n-1}$, we define
$d_{v}^{\\#}=\max_{\ell_{j}\in\mathbb{L}_{n-1}}d(v,\ell_{j})$. So,
$d_{v}^{\\#}$ is the maximum distance from $v$ to the nodes in
$\mathbb{L}_{n-1}$.

Also, in what follows we use the notation ${\mathbb{I}}_{\cal C}$ to indicate
a predicate (condition) $\cal C$. So, it is 1, when $\cal C$ holds, and is 0,
otherwise.

######  Theorem 6.1.

Suppose $\\{k_{n}\\}_{n=0}^{\infty}$ is a building sequence of the family of
graphs $\\{G_{n}\\}_{n=1}^{\infty}$. Let $v$ be a node in the graph $G_{n}$.
Conditional upon the choice of the latches $\ell_{1},\ldots,\ell_{k_{n-1}}$ in
$G_{n-1}$, the eccentricity $C_{n}(v)$ is given by

$\widetilde{C}_{n}(v)={\small\begin{cases}\max\Big{\\{}C_{n-1}(v),d_{v}^{\\#}+C_{0}(h)\Big{\\}},&\mbox{if
\ }v\in V_{n-1};\\\ \\\ \\\ \max\Big{\\{}d(v,h)+C_{n-1}(\ell_{i}),\\\
\qquad{}\max\big{\\{}C_{0}(v),\big{(}d(v,h)+d_{\ell_{i}}^{\\#}+C_{0}(h)\big{)}{\mathbb{I}}_{\\{k_{n-1}>1\\}}\big{\\}},&\mbox{if
\ }v\in V_{0}^{co_{i}}.\end{cases}}$

###### Proof.

The graph $G_{n}$ is obtained by attaching a copy of the seed $G_{0}$ at each
of the latches $\ell_{1},\ell_{2},\cdots,\ell_{k_{n-1}}$ selected in the graph
$G_{n-1}$.

We denote the vertex set of the $r$th copy of the seed, for
$r=1,\ldots,k_{n-1}$, by $V_{0}^{{\rm co}_{r}}$. We now compute the distance
from a node $v$ to a vertex $u$ in $G_{n}$ by considering the four cases:

| $u\in V_{n-1}$ | $u\in V_{0}^{{\rm co}_{j}}$
---|---|---
$v\in V_{n-1}$ | $d(v,u)$ | $d(v,\ell_{j})+d(h,u)$
$v\in V_{0}^{{\rm co}_{i}}$ | $d(v,h)+d(\ell_{i},u)$ | $d(u,v){\mathbb{I}}_{i=j}+(d(v,h)+d(\ell_{i},\ell_{j})+d(h,u)){\mathbb{I}}_{i\not=j}$

For a given $v\in V_{n}$, the maximum (over the range of $u$ and $j$) in each
block is

| $u\in V_{n-1}$ |  $u$ in a copy
---|---|---
${v\in V_{n-1}}$ | $C_{n-1}(v)$ | $d_{v}^{\\#}+C_{0}(h)$
$v\in V_{0}^{{\rm co}_{i}}$ | $d(v,h)+C_{n-1}(\ell_{i})$ | $\max\big{\\{}C_{0}(v),\big{(}d(v,h)+d_{\ell_{i}}^{\\#}+C_{0}(h)\big{)}{\mathbb{I}}_{\\{k_{n-1}>1\\}}\big{\\}}$

The result now follows. ∎

######  Remark 6.1.

Suppose a vertex $\ell$ is chosen as a latch from $G_{n-1}$. From Theorem 6.1,
we pick up the top line and write

$\widetilde{C}_{n}(\ell)=\max\big{\\{}C_{n-1}(\ell),\displaystyle
d_{\ell}^{\\#}+C_{0}(h)\big{\\}}.$

If $k_{n-1}=1$, then $d_{\ell}^{\\#}=0$, in which case we have
$\widetilde{C}_{n}(\ell)=\max\big{\\{}C_{n-1}(\ell),C_{0}(h)\big{\\}}.$

##  7 Diameter of the graph $G_{n}$

The diameter of a connected graph with vertex set $\cal V$ is the longest
distance between any two nodes in it [4]. That is, the diameter is the maximum
eccentricity, $\max_{v\in\cal V}C(v)$. For example, the diameters of the
graphs $G_{0},G_{1},G_{2}$ and $G_{3}$ in Figure 1 are respectively $2,4,6$
and $10$.

We now introduce some additional notation:

  1. 1.

$\widetilde{\mbox{ D}_{n}}=\mbox{D}_{n}\,|\,G_{n-1},\mathbb{L}_{n-1}$. This is
the conditional diameter $\mbox{D}_{n}$ of the graph $G_{n}$ given $G_{n-1}$
and the $k_{n-1}$ latches in $\mathbb{L}_{n-1}$ (see Subsection 6.1 for the
definition of $\mathbb{L}_{n-1}$).

  2. 2.

Only for $k_{n}>1$, we define
$q_{n}=\max_{\ell,\tilde{\ell}\in\mathbb{L}_{n}}d(\ell,\tilde{\ell})=\max_{\ell\in\mathbb{L}_{n}}\ell^{\\#}$.
Thus, $q_{n}$ computes the maximum distance between any two latches in
$G_{n}$.

  3. 3.

$\alpha_{n}=\max_{\ell\in\mathbb{L}_{n}}C_{n}(\ell)$. Thus, $\alpha_{n}$ is
the maximum eccentricity of a latch in $G_{n}$.

######  Theorem 7.1.

Suppose $\\{k_{n}\\}_{n=0}^{\infty}$ is a building sequence of the family of
graphs $\\{G_{n}\\}_{n=1}^{\infty}$. The conditional diameter
$\widetilde{\mbox{D}_{n}}=\mbox{D}_{n}\,|\,G_{n-1},\mathbb{L}_{n-1}$ of a
graph of age $n$ is given by

$\widetilde{\mbox{D}_{n}}=\max\Big{\\{}\mbox{D}_{n-1},\leavevmode\nobreak\
\leavevmode\nobreak\
\big{(}2C_{0}(h)+q_{n-1}\big{)}{\mathbb{I}}_{\\{k_{n-1}>1\\}},\leavevmode\nobreak\
\leavevmode\nobreak\ C_{0}(h)+\alpha_{n-1}\Big{\\}}.$

###### Proof.

The (conditional) diameter $\widetilde{\mbox{D}_{n}}$ of $G_{n}$ may remain
the same as the diameter $\mbox{D}_{n-1}$ of $G_{n-1}$,555In the graph $G_{2}$
in Figure 1, if we pick the three latches at distances 2,3,4 from the top
vertex, the diameter of the graph so obtained in step 3 will be equal to
$\mbox{D}_{2}$, the diameter of $G_{2}$. unless we can find longer paths in
$G_{n}$. The latter case arises, if

  * (a)

There is a pair $x$ and $y$ of latches in $G_{n-1}$, and a pair of vertices
(say $u$ in the copy latched at $x$ and $v$ in the copy latched at $y$), such
that $d(u,x)+d(x,y)+d(y,v)>\mbox{D}_{n-1}$. The case can be, only if
$k_{n-1}>1$. The longest such distance is obtained by maximizing over $x,y,u$
and $v$ to obtain $2C_{0}(h)+q_{n-1}$.666This situation occurs in the graph
$G_{3}$ in Figure 1.

  * (b)

Or, we can find a vertex $u$ far enough from a latch $\ell$ in $G_{n-1}$ and
another vertex $v$ in the copy latched at $\ell$ such that
$d(u,\ell)+d(\ell,v)>\mbox{D}_{n-1}$. The longest such distance is obtained by
maximizing over $\ell,u$ and $v$ to obtain $\alpha_{n-1}+C_{0}(h)$.

The longest distance in the graph is the maximum of the three possibilities
discussed. ∎

######  Remark 7.1.

Consider the case where, at stage $n$ (for each $n\geq 1$), we pick among the
$k_{n-1}$ latches two, say $\ell,\tilde{\ell}$ in $G_{n-1}$, such that
$d(\ell,\tilde{\ell})$ is the diameter of $G_{n-1}$. Note that this selection
mechanism is no longer random in the sense discussed in all the preceding
sections. Let us call the diameter of the graph so constructed
$\mbox{D}_{n-1}^{*}$. This is only possible if $k_{n-1}>1$, for each $n$. By
arguments similar to what we used in the proof of Theorem 7.1, we get
$\mbox{D}_{n}^{*}=2C_{0}(h)+\mbox{D}_{n-1}^{*}$. Unwinding we get
$\mbox{D}_{n}^{*}=2nC_{0}(h)+\mbox{D}_{0}$.

Remark 7.1 shows that, under this special hooking mechanism, the diameter
$\mbox{D}_{n}$ at step $n$ only requires the knowledge of the seed graph and
$n$. It does not take into consideration how many latches were picked at
stages $1$ through $n-1$ as long as there are two latches $\ell,\tilde{\ell}$
picked at each stage such that $d(\ell,\tilde{\ell})$ is maximum.

## Acknowledgment

The authors would like to sincerely thank Dr. Shaimaa M. Abd-Elaal, Egyptian
Ministry of Health and Population, for many fruitful discussions that gave
them a perspective on applications of the model proposed.

## Competing interests

The authors declare none.

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