††thanks: These authors contributed equally to this work.

# Magnetic Feshbach resonances in collisions of 23Na40K with 40K

Xin-Yao Wang1,2,4,5,6 Matthew D. Frye3,∗<EMAIL_ADDRESS>Zhen
Su2,4,5 Jin Cao2,4,5 Lan Liu2,4,5 De-Chao Zhang2,4,5 Huan Yang2,4,5 Jeremy M.
Hutson3<EMAIL_ADDRESS>Bo Zhao2,4,5<EMAIL_ADDRESS>Chun-Li
Bai1,2,6<EMAIL_ADDRESS>Jian-Wei Pan2,4,5<EMAIL_ADDRESS>1Beijing National
Laboratory for Molecular Sciences, Key Laboratory of Molecular Nanostructure
and Nanotechnology, CAS Research/Education Center for Excellence in Molecular
Sciences, Institute of Chemistry, Chinese Academy of Sciences, Beijing 100190,
China  2Hefei National Laboratory for Physical Sciences at the Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China  3Joint Quantum Centre (JQC) Durham-Newcastle,
Department of Chemistry, Durham University, South Road, Durham DH1 3LE, United
Kingdom  4Shanghai Branch, CAS Center for Excellence and Synergetic Innovation
Center in Quantum Information and Quantum Physics, University of Science and
Technology of China, Shanghai 201315, China  5Shanghai Research Center for
Quantum Sciences, Shanghai 201315, China  6University of Chinese Academy of
Sciences, Beijing 100049, China

###### Abstract

We present measurements of more than 80 magnetic Feshbach resonances in
collisions of ultracold 23Na40K with 40K. We assign quantum numbers to a group
of low-field resonances and show that they are probably due to long-range
states of the triatomic complex in which the quantum numbers of the separated
atom and molecule are approximately preserved. The resonant states are not
members of chaotic bath of short-range states. Similar resonances are expected
to be a common feature of alkali-metal diatom + atom systems.

##  I Introduction

Ultracold molecules offer great opportunities to study many areas of physics,
including fundamental chemical reaction dynamics Balakrishnan:FH2:2001 ;
Tscherbul:JCP:2006 ; Ospelkaus:react:2010 ; Rui2017 ; Hu:2019 , quantum
simulation and computing DeMille:2002 ; Ni:Swap:2018 ; Blackmore:2019 ;
Sawant:qudit:2019 , and precision measurements of fundamental constants
Safronova:2018 ; Kobayashi:mep-ratio:2019 ; Augenbraun:2020 ; Fitch:2020 .
There has been considerable success in producing ultracold ground-state
molecules, either by associating pairs of ultracold atoms Ni:KRb:2008 ;
Lang:ground:2008 ; Danzl:ground:2010 ; Takekoshi:RbCs:2014 ; Molony:RbCs:2014
; Wang:2015 ; Park:NaK:2015 ; Guo:NaRb:2016 ; Seesselberg:2018 ;
Yang:K_NaK_resonances:2019 ; Voges:NaK:2020  or by direct laser cooling of
molecules Barry:2014 ; Truppe:MOT:2017 ; Anderegg:2018 ; Caldwell:2019 ;
Ding:2020 ; Augenbraun:2020 . Understanding collisions of ultracold molecules
with themselves and ultracold atoms is now essential to allow control and
further cooling of these molecules Son:2020 ; Valtolina:2020 ; Matsuda:2020 ;
Jurgilas:2021  to form the dense stable gases needed for many applications.
Further, at such low temperatures, the molecular collisions are highly quantum
mechanical in nature, and are thus of fundamental interest in their own right
and a powerful tool to unravel the nature of molecular interactions.

Magnetic Feshbach resonances are one of the most important features of
ultracold collisions Chin:RMP:2010 . They occur when the energy of a
(quasi-)bound state is tuned across the incoming collision threshold by an
external magnetic field. They have been studied extensively in atomic
scattering, where they are used to tune interaction strengths to control
collisional cooling, study degenerate quantum gases, and investigate strongly
interacting systems, amongst other applications. They are also used to
associate pairs of atoms into weakly bound molecules through
magnetoassociation Kohler:RMP:2006  and provide highly sensitive probes of
atomic interaction potentials.

Magnetic Feshbach resonances in collisions involving molecules are far more
challenging to understand. Full scattering calculations are hampered by the
need for huge basis sets and pathological dependence on uncertainties in the
interaction potential Wallis:LiNH:2011 . Mayle _et al._ Mayle:2012 ;
Mayle:2013  suggested that the combination of deep anisotropic potentials and
large masses produce a dense forest of chaotic short-range states that
dominate the scattering properties and cause large collisional sticking times.
This idea has been influential, and there is significant experimental evidence
for such effects in molecule+molecule scattering Gregory:RbCs-collisions:2019
; Gregory:RbCs-complex-lifetime:2020 ; Liu:2020 . However, the densities of
states are now expected to be much lower than the original predictions
Christianen:density:2019 , and are also substantially lower for atom+molecule
than for molecule+molecule complexes. There is great interest Quemener2012  in
whether atom+molecule scattering is dominated by such chaotic states, or
whether the important features are caused by long-range bound states akin to
those involved in atom+atom magnetic Feshbach resonances. Recently, magnetic
Feshbach resonances have been observed in collisions of 23Na40K with 40K
Yang:K_NaK_resonances:2019 ; however, only a few resonances were observed, so
it was not possible to determine the nature of the resonant states or assign
quantum numbers to them.

In this paper, we extend the experimental work of Yang _et al._
Yang:K_NaK_resonances:2019  and carry out a theoretical study of the
resonances. We perform extensive measurements of the Feshbach resonances in 25
collision channels in the magnetic field range between 16 G and 120 G; more
than 80 new atom-molecule Feshbach resonances are observed. We use this large
set of results to show that the resonances cluster into groups with clear
patterns. We assign both spin and rovibrational character to the bound states
responsible for the resonances, and conclude that they are relatively simple
long-range states and not chaotic short-range states.

##  II observation of the atom-molecule Feshbach resonances

The atom-molecule Feshbach resonances are detected by observing the loss of
molecules from an ultracold atom-molecule mixture as a function of magnetic
field. The experimental procedures are similar to those in
Yang:K_NaK_resonances:2019 . We first prepare a two-species dark-spot magneto-
optical trap of 23Na and 40K atoms and load the atoms into a clover-leaf
magnetic trap to perform forced evaporative cooling. The atoms are then loaded
into a crossed-beam optical dipole trap ($\lambda$=1064 nm) to perform further
evaporative cooling. We typically create about $3\times 10^{5}$ Na atoms and
$1.6\times 10^{5}$ K atoms at a temperature of about 500 nK.

We create weakly bound Feshbach molecules from this atomic mixture by two-
photon Raman photoassociation. The technical details are given in our previous
work Yang:K_NaK_resonances:2019 . We use 3 different target Feshbach states
and magnetic fields, each associated with an atomic Feshbach resonance
Park:2012 , depending on the desired hyperfine level of the molecule. The
atoms are prepared in an initial state different from the one which shows the
Feshbach resonance and coupled to the Feshbach state by blue-detuned Raman
light fields. For transitions involving changing the 40K (23Na) state, the
single-photon detuning is about $2\pi\times 250$ GHz ($2\pi\times 252$ GHz)
relative to the D2 transition of 40K (23Na) atoms, and the Rabi frequency for
the corresponding $m_{f}$-changing atomic transitions are about $2\pi\times
50$ kHz ($2\pi\times 33$ kHz). Details of the states and resonances involved
are given in Table 1.

We transfer the molecules from the Feshbach state to a desired hyperfine level
of the rovibronic ground state by stimulated Raman adiabatic passage (STIRAP).
The 23Na40K molecule in its rovibrational ground state has 36 hyperfine states
due to the nuclear spins $i_{\textrm{Na}}=3/2$ and $i_{\textrm{K}}=4$. In the
free molecule these are coupled only weakly, so at the magnetic field of the
experiment they are decoupled with well conserved projections
$m_{\textrm{Na}}$ and $m_{\textrm{K}}$, respectively; we therefore label
molecular states $(m_{\textrm{Na}},m_{\textrm{K}})_{\textrm{mol}}$. In this
work, 9 different hyperfine levels are prepared. The hyperfine states
$(3/2,-4)_{\textrm{mol}}$, $(1/2,-4)_{\textrm{mol}}$,
$(-3/2,-3)_{\textrm{mol}}$, $(-1/2,-4)_{\textrm{mol}}$ are created at about 85
G; the first of these is the absolute ground state. Molecule loss in the
atom+molecule mixture is measured in the magnetic field range between 16 G and
106 G in steps of about 0.5 G. The hyperfine states
$(-1/2,-3)_{\textrm{mol}}$, $(-1/2,-2)_{\textrm{mol}}$,
$(-3/2,-2)_{\textrm{mol}}$, $(-3/2,-1)_{\textrm{mol}}$, and
$(-3/2,-4)_{\textrm{mol}}$ are prepared at about 102 G. The magnetic field
range studied is between 16 G and 120 G. The intermediate states used in the
STIRAP are the hyperfine levels of a mixed rovibronic state formed from
$B^{1}\Pi$ $|v=12,J=1\rangle$ and $c^{3}\Sigma$ $|v=35,J=1\rangle$
Park:2_photon:2015 . These hyperfine levels can be labelled by the quantum
numbers $F_{1}=i_{\rm{Na}}+J$ and $F=F_{1}+i_{\rm{K}}$. The hyperfine
interaction between the nuclear spin of K atoms and the electron spin is
small, and the quantum numbers $F_{1}$, $m_{F_{1}}$, and
$m_{F}=m_{F_{1}}+m_{i_{\rm{K}}}$ are approximately good quantum numbers. The
intermediate states used to prepare different hyperfine levels of the ground
states are listed in Table 1.

Table 1: The atomic Feshbach resonances used, the hyperfine levels of the intermediate states, and the polarization of the STIRAP lasers required to prepare the molecule in the desired hyperfine level of the rovibronic ground state. The quantum numbers are defined in the main text. The atom-atom resonance at 104 G for $(1,0)_{\textrm{Na}}+(9/2,-9/2)_{\textrm{K}}$ has not been reported before, and its location has been determined by measuring the binding energy of the Feshbach molecules.  Initial state | Feshbach resonance |  Hyperfine level of ground state | Intermediate state | Laser polarization
---|---|---|---|---
$F_{1}$ | $m_{F_{1}}$ | $m_{i_{\textrm{K}}}$ | $m_{F}$ | Stokes | pump
$(1,1)_{\textrm{Na}}+(9/2,-5/2)_{\textrm{K}}$ |  $(1,1)_{\textrm{Na}}+(9/2,-7/2)_{\textrm{K}}$ at 110 G | $(-1/2,-2)_{\textrm{mol}}$ | $1/2$ | $-1/2$ | $-2$ | $-5/2$ | $\pi$ | $\pi$
$(-1/2,-3)_{\textrm{mol}}$ | $1/2$ | $-1/2$ | $-3$ | $-7/2$ | $\pi$ | $\sigma^{-}$
$(-3/2,-1)_{\textrm{mol}}$ | $3/2$ | $-3/2$ | $-1$ | $-5/2$ | $\pi$ | $\pi$
$(-3/2,-2)_{\textrm{mol}}$ | $3/2$ | $-3/2$ | $-2$ | $-7/2$ | $\pi$ | $\sigma^{-}$
$(1,1)_{\textrm{Na}}+(9/2,-7/2)_{\textrm{K}}$ |  $(1,1)_{\textrm{Na}}+(9/2,-9/2)_{\textrm{K}}$ at 85 G | $(3/2,-4)_{\textrm{mol}}$ | $5/2$ | $3/2$ | $-4$ | $-5/2$ | $\pi$ | $\sigma^{+}$
$(1/2,-4)_{\textrm{mol}}$ | $5/2$ | $1/2$ | $-4$ | $-7/2$ | $\pi$ | $\pi$
$(-1/2,-4)_{\textrm{mol}}$ | $1/2$ | $-1/2$ | $-4$ | $-9/2$ | $\pi$ | $\sigma^{-}$
$(-3/2,-3)_{\textrm{mol}}$ | $3/2$ | $-3/2$ | $-3$ | $-9/2$ | $\pi$ | $\sigma^{-}$
$(1,1)_{\textrm{Na}}+(9/2,-9/2)_{\textrm{K}}$ |  $(1,0)_{\textrm{Na}}+(9/2,-9/2)_{\textrm{K}}$ at 104 G | $(-3/2,-4)_{\textrm{mol}}$ | $3/2$ | $-3/2$ | $-4$ | $-11/2$ | $\pi$ | $\sigma^{-}$

In the experiment, the pump laser coupling the Feshbach state and the
intermediate state is a 805 nm TA-Pro laser, and the Stokes laser coupling the
ground state and the intermediate state is a 567 nm TA-SHG laser. The two
lasers are locked to a ULE cavity to suppress the noises. The Stokes light is
$\pi$ polarized and the required polarization of the pump light is $\pi$,
$\sigma^{+}$ or $\sigma^{-}$. Due to the limited optical access, the two
lasers propagate perpendicular to the magnetic field. The $\pi$ polarization
is achieved by setting the polarization of the lasers parallel to the magnetic
field. The $\sigma$ polarization is applied by setting the polarization of the
pump laser to $\sigma^{+}+\sigma^{-}$ linear polarization. A detailed
theoretical analysis of the coupling strength between the Feshbach states and
the hyperfine levels of the intermediate states is given in Ref. LiuY2020, .
There are many possible pathways for STIRAP and in principle many hyperfine
levels of the rovibronic ground state can be populated. We have chosen
hyperfine levels of the excited states so that the desired pathway dominates
over the others and the target hyperfine level is prepared. The purity of the
hyperfine states of the ground-state molecule is checked by observing that the
molecule number after a round-trip STIRAP does not oscillate as a function of
the hold time Liu:2019 .

The 40K atom has electron spin $s=1/2$ in addition to nuclear spin
$i_{\textrm{K}}=4$; these couple strongly to form $f=9/2$ and $7/2$ at zero
field, with projection $m_{f}$. At the fields of the experiment, $f$ is a
nearly good quantum number and we label the atomic states of 40K by
$(f,m_{f})_{\textrm{at}}$. The atoms are prepared in the states
$(9/2,-9/2)_{\textrm{at}}$, $(9/2,-7/2)_{\textrm{at}}$, and
$(9/2,-5/2)_{\textrm{at}}$.

After the atom-molecule mixture is prepared, we ramp the magnetic field to the
desired value in a few milliseconds and then hold the atom-molecule mixture
for about 7 ms. We program the profile of current of the coil to compensate
the residual magnetic field caused by the eddy currents, so that the magnetic
field can reach the desired value in about 4 to 6 ms. During the hold time,
the magnetic field drifts and fluctuates, and the magnitude of the uncertainty
of the magnetic field is proportional to the magnetic field range we ramp.
When the magnetic field is ramped from about 102 G to about 16 G, the
uncertainty of the magnetic field during the hold time is about 800 mG. The
hold time of the atom-molecule mixture is controlled by removing the 40K atoms
using resonant light pulses. After that, the magnetic field is ramped back to
the initial value in 3 ms. After about 20 ms when the magnetic field is
stable, we transfer the ground-state molecules to the Feshbach state by
reverse STIRAP. The Feshbach molecules are either directly detected by the
standard time-of-flight absorption imaging or are dissociated into free atoms,
which are then detected by absorption imaging. During the hold time, molecules
are lost from the trap due to inelastic 2-body atom-molecule collisions or
3-body recombination. At resonance, both these loss processes are enhanced and
we see characteristic dips in the remaining molecule number as a function of
field. We identify these loss features as the atom-molecule Feshbach
resonances. Narrow resonances in the low-field region are challenging to
detect in the current experiment, both because the long field ramp to the
target field creates uncertainties in the final field, and because the signal-
to-noise ratio is reduced if molecules are lost in traversing higher-field
resonances on the way.

We have searched for Feshbach resonances in 25 of the 27 possible combinations
of these 9 molecular states and 3 atomic states. For the combinations
$(-3/2,-4)_{\textrm{mol}}+(9/2,-5/2)_{\textrm{at}}$ and
$(-1/2,-3)_{\textrm{mol}}+(9/2,-5/2)_{\textrm{at}}$, the background loss rates
are very large and our method is not applicable. The resonances for
$(-3/2,-2)_{\textrm{mol}}$ at magnetic fields between 43 G and 120 G were
studied in our previous work Yang:K_NaK_resonances:2019 , but here we extend
the magnetic field range down to 16 G. In total, we observe 84 new atom-
molecule Feshbach resonances. Example loss features are shown in Fig. S6. The
other 80 loss features are displayed in the supplementary material. We fit the
loss features with Gaussian functions to extract a position $B_{0}$ and width
$\delta$. Note that the observed width of the loss feature $\delta$ is not the
same quantity as the theoretical width $\Delta$ that describes the pole in
scattering length associated with the resonance. We give results for all
resonances observed in Tables 2 and 3, for the low- and high-field groups of
resonances, respectively.

Figure 1:  Example loss features for different collision channels. The
remaining molecule numbers are shown as a function of the magnetic field. The
solid lines are the fitted Gaussian functions. Error bars represent the
standard deviation.

##  III Assignment of the low-field atom-molecule Feshbach resonances

In order to interpret these resonances, we need to describe the resonant bound
states that cause them and assign their quantum numbers
$X{{}^{\textrm{res}}}$. We look for explanations in terms of resonances of as
few different characters as possible; here the character is defined by the
rovibrational state of the triatomic complex and the changes $\Delta
X=X{{}^{\textrm{res}}}-X{{}^{\textrm{in}}}$ between incoming and resonant
states for the remaining quantum numbers. If a resonance of a particular
character exists at one threshold, a resonance of that character is expected
to exist at every threshold where it is allowed.

Table 2: Atom-molecule Feshbach resonances in the low-field group, along with the assigned quantum numbers for the bound states that cause them as described in the main text.  Initial State | Measured | Assigned quantum numbers
---|---|---
$m_{\textrm{Na}}$ | $m_{\textrm{K}}$ |  $B_{0}$ (G) |  $\delta$ (G) | $m_{f}$ | $m_{\textrm{Na}}$ | $m_{\textrm{K}}$ | $N$ | $M_{N}$
Initial state of the K atom $(f,m_{f})=(9/2,-9/2)$
$-3/2$ | $-3$ | 20.9 | 3.2 | $-7/2$ | $-3/2$ | $-4$ | 1 or 2 | 0
$-3/2$ | $-2$ | 19.4 | 2.6 | $-7/2$ | $-3/2$ | $-3$ | 1 or 2 | 0
$-3/2$ | $-1$ | 19.0 | 1.0 | $-7/2$ | $-3/2$ | $-2$ | 1 or 2 | 0
$-1/2$ | $-4$ | 21.8 | 3.9 | $-7/2$ | $-3/2$ | $-4$ | 1 or 2 | 0
$-1/2$ | $-3$ | 19.3 | 5.4 | $-7/2$ |  $-3/2$ or $-1/2$ |  $-3$ or $-4$ | 1 or 2 | 0
$-1/2$ | $-2$ | 19.3 | 2.9 | $-7/2$ |  $-3/2$ or $-1/2$ |  $-2$ or $-3$ | 1 or 2 | 0
$1/2$ | $-4$ | 16.6 | 0.7 | $-7/2$ | $1/2$ | $-4$ | 1 or 2 | $-1$
$1/2$ | $-4$ | 20.9 | 3.9 | $-7/2$ | $-1/2$ | $-4$ | 1 or 2 | 0
$3/2$ | $-4$ | 17.1 | 0.9 | $-7/2$ | $3/2$ | $-4$ | 1 or 2 | $-1$
$3/2$ | $-4$ | 20.1 | 1.7 | $-7/2$ | $1/2$ | $-4$ | 1 or 2 | 0
Initial state of the K atom $(f,m_{f})=(9/2,-7/2)$
$-3/2$ | $-4$ | 19.6 | 2.0 | $-5/2$ | $-3/2$ | $-4$ | 1 or 2 | $-1$
$-3/2$ | $-3$ | 22.4 | 2.8 | $-5/2$ | $-3/2$ | $-4$ | 1 or 2 | 0
$-3/2$ | $-2$ | 23.6 | 3.6 | $-5/2$ | $-3/2$ | $-3$ | 1 or 2 | 0
$-3/2$ | $-1$ | 24.3 | 2.5 | $-5/2$ | $-3/2$ | $-2$ | 1 or 2 | 0
$-1/2$ | $-4$ | 23.5 | 5.2 | $-5/2$ | $-3/2$ | $-4$ | 1 or 2 | 0
$-1/2$ | $-3$ | 22.9 | 5.9 | $-5/2$ |  $-3/2$ or $-1/2$ |  $-3$ or $-4$ | 1 or 2 | 0
$-1/2$ | $-2$ | 23.1 | 4.0 | $-5/2$ |  $-3/2$ or $-1/2$ |  $-2$ or $-3$ | 1 or 2 | 0
$1/2$ | $-4$ | 24.8 | 5.7 | $-5/2$ | $-1/2$ | $-4$ | 1 or 2 | 0
$3/2$ | $-4$ | 19.8 | 0.6 | $-5/2$ | $3/2$ | $-4$ | 1 or 2 | $-1$
$3/2$ | $-4$ | 24.9 | 1.5 | $-5/2$ | $1/2$ | $-4$ | 1 or 2 | 0
Initial state of the K atom $(f,m_{f})=(9/2,-5/2)$
$-3/2$ | $-3$ | 26.8 | 3.9 | $-3/2$ | $-3/2$ | $-4$ | 1 or 2 | 0
$-3/2$ | $-2$ | 29.7 | 5.2 | $-3/2$ | $-3/2$ | $-3$ | 1 or 2 | 0
$-3/2$ | $-1$ | 31.7 | 1.9 | $-3/2$ | $-3/2$ | $-2$ | 1 or 2 | 0
$-1/2$ | $-4$ | 26.1 | 4.9 | $-3/2$ | $-3/2$ | $-4$ | 1 or 2 | 0
$-1/2$ | $-2$ | 27.2 | 1.0 | $-3/2$ |  $-3/2$ or $-1/2$ |  $-2$ or $-3$ | 1 or 2 | 0
$1/2$ | $-4$ | 25.1 | 0.4 | $-3/2$ | $1/2$ | $-4$ | 1 or 2 | $-1$
$1/2$ | $-4$ | 29.7 | 3.0 | $-3/2$ | $-1/2$ | $-4$ | 1 or 2 | 0
$3/2$ | $-4$ | 31.2 | 3.1 | $-3/2$ | $1/2$ | $-4$ | 1 or 2 | 0

In the three-atom complex, the diatom rotation $n$ couples to the angular
momentum $L$ of the atom and diatom around one another to form $N$, which
represents the total spin-free angular momentum. The quantum numbers
$(v,n,L,N)$, with $v$ the diatom vibrational quantum number, provide a good
representation in the long-range region where the interaction potential is
weak, except for coupling to spins and the magnetic field. The incoming states
relevant to the experiment all have
$v{{}^{\textrm{in}}}=n{{}^{\textrm{in}}}=0$; since $L{{}^{\textrm{in}}}=0$
dominates at the temperature of the experiment, this implies
$N{{}^{\textrm{in}}}=0$. The electronic interaction potential is strong and
highly anisotropic at short range, which results in complicated states with
strongly mixed $n$, $L$, and $v$, but it does not mix $N$.

Table 3: Atom-molecule Feshbach resonances in the high-field group, along with the assigned quantum numbers for the bound states that cause them as described in the main text.  Initial State | Measured | Assigned quantum number
---|---|---
$m_{\textrm{Na}}$ | $m_{\textrm{K}}$ |  $B_{0}$ (G) |  $\delta$ (G) | $m_{f}$
Initial state of the K atom $(f,m_{f})=(9/2,-9/2)$
$-3/2$ | $-4$ | 48.4 | 4.8 | $-7/2$
$-3/2$ | $-3$ | 47.8 | 3.0 | $-7/2$
$-3/2$ | $-2$ | 48.5 | 2.1 | $-7/2$
$-3/2$ | $-2$ | 90.3 | 2.2 | $-7/2$
$-3/2$ | $-1$ | 50.6 | 1.1 | $-7/2$
$-3/2$ | $-1$ | 59.6 | 1.3 | $-7/2$
$-3/2$ | $-1$ | 95.3 | 1.9 | $-7/2$
$-1/2$ | $-4$ | 43.4 | 0.6 | $-7/2$
$-1/2$ | $-4$ | 47.7 | 1.9 | $-7/2$
$-1/2$ | $-3$ | 45.1 | 0.5 | $-7/2$
$-1/2$ | $-3$ | 48.8 | 1.7 | $-7/2$
$-1/2$ | $-2$ | 50.5 | 1.3 | $-7/2$
$-1/2$ | $-2$ | 59.7 | 3.4 | $-7/2$
$-1/2$ | $-2$ | 94.9 | 0.6 | $-7/2$
$1/2$ | $-4$ | 43.4 | 0.4 | $-7/2$
$1/2$ | $-4$ | 45.4 | 0.4 | $-7/2$
$1/2$ | $-4$ | 48.8 | 1.1 | $-7/2$
$3/2$ | $-4$ | 81.5 | 0.3 | $-7/2$
$3/2$ | $-4$ | 83.4 | 0.3 | $-7/2$
Initial state of the K atom $(f,m_{f})=(9/2,-7/2)$
$-3/2$ | $-4$ | 63.2 | 12.3 | $-5/2$
$-3/2$ | $-3$ | 52.0 | 0.5 | $-7/2$
$-3/2$ | $-3$ | 57.6 | 5.3 | $-7/2$
$-3/2$ | $-3$ | 102.9 | 3.0 | $-7/2$
$-3/2$ | $-2$ | 54.6 | 0.6 | $-7/2$
$-3/2$ | $-2$ | 59.4 | 3.1 | $-5/2$
$-3/2$ | $-2$ | 101.1 | 0.6 | $-7/2$
$-3/2$ | $-2$ | 106.3 | 1.7 | $-7/2$
$-3/2$ | $-1$ | 57.7 | 0.7 | $-5/2$
$-3/2$ | $-1$ | 62.2 | 2.2 | $-5/2$
$-3/2$ | $-1$ | 106.0 | 0.7 | $-7/2$
$-1/2$ | $-4$ | 57.5 | 2.5 | $-5/2$
$-1/2$ | $-3$ | 59.4 | 2.7 | $-5/2$
$-1/2$ | $-3$ | 68.7 | 3.7 | $-5/2$
$-1/2$ | $-3$ | 101.0 | 0.3 | $-7/2$
$-1/2$ | $-3$ | 106.5 | 1.1 | $-7/2$
$-1/2$ | $-2$ | 62.3 | 2.2 | $-5/2$
$-1/2$ | $-2$ | 72.1 | 1.8 | $-5/2$
$1/2$ | $-4$ | 51.8 | 0.3 | $-7/2$
$1/2$ | $-4$ | 59.2 | 0.9 | $-5/2$
Initial state of the K atom $(f,m_{f})=(9/2,-5/2)$
$-3/2$ | $-3$ | 57.8 | 3.6 | $-7/2$
$-3/2$ | $-3$ | 63.1 | 1.2 | $-5/2$
$-3/2$ | $-3$ | 70.0 | 2.4 | $-5/2$
$-3/2$ | $-3$ | 102.4 | 2.0 | $-7/2$
$-3/2$ | $-2$ | 68.2 | 1.0 | $-5/2$
$-3/2$ | $-2$ | 74.5 | 3.7 | $-5/2$
$-3/2$ | $-2$ | 83.5 | 1.6 | $-3/2$
$-3/2$ | $-1$ | 72.9 | 1.2 | $-5/2$
$-3/2$ | $-1$ | 79.3 | 2.3 | $-3/2$
$-3/2$ | $-1$ | 87.2 | 0.7 | $-3/2$
$-3/2$ | $-1$ | 90.1 | 1.2 | $-3/2$
$-1/2$ | $-4$ | 56.8 | 4.0 | $-7/2$
$-1/2$ | $-4$ | 62.6 | 1.0 | $-5/2$
$-1/2$ | $-4$ | 69.5 | 1.3 | $-5/2$
$-1/2$ | $-4$ | 78.8 | 2.0 | $-3/2$
$-1/2$ | $-4$ | 100.9 | 2.1 | $-7/2$
$-1/2$ | $-2$ | 55.9 | 2.1 | $-7/2$
$-1/2$ | $-2$ | 79.5 | 2.3 | $-3/2$
$-1/2$ | $-2$ | 83.3 | 0.8 | $-3/2$
$-1/2$ | $-2$ | 90.4 | 1.9 | $-3/2$
$1/2$ | $-4$ | 51.7 | 3.2 | $-7/2$
$1/2$ | $-4$ | 63.7 | 0.3 | $-5/2$
$1/2$ | $-4$ | 73.7 | 0.8 | $-5/2$
$1/2$ | $-4$ | 77.7 | 0.4 | $-3/2$
$1/2$ | $-4$ | 83.7 | 1.2 | $-3/2$
$3/2$ | $-4$ | 50.5 | 0.3 | $-7/2$
$3/2$ | $-4$ | 56.2 | 1.4 | $-7/2$

The interaction potential between ground-state NaK and K is strongly
anisotropic, but there is only a single Born-Oppenheimer potential-energy
surface that correlates with the incoming states. There is thus no strong
coupling due to differences between two potentials, as there is in collisions
of two alkali-metal atoms, or as there would be in collisions of an electron-
spin-doublet molecule with an alkali-metal atom. The important couplings are
expected to include various Zeeman, spin-rotation, and hyperfine couplings
Frye:triatomic-complexes:2021 ; Frye:triatomic-threshold-complexes:2021 . A
detailed description and estimation of the strength of these couplings is
beyond the scope of this paper, but they are expected to be fairly weak and to
act essentially perturbatively. This means that the bound states can be
usefully labeled with the same spin quantum numbers as the incoming states,
even if there is significant coupling of $s$ to the molecular nuclear spins at
short range. This situation has some similarities with that in alkali+1S
systems Zuchowski:RbSr:2010 ; Brue:LiYb:2012 ; Brue:AlkYb:2013 ;
Barbe:RbSr:2018 ; Yang:CsYb:2019 .

We consider first the group of resonances between 16 and 32 G, which we refer
to as the low-field group. 28 such resonances are currently observed. The
general layout of the resonances in field must come from the atomic quantum
numbers $(f,m_{f})$, since the associated magnetic moments are far larger than
the magnetic moments of the molecule; one or both of $f$ and $m_{f}$ must
change for the resonant state to cross the incoming threshold. The lowest
state of 40K is $(9/2,-9/2)_{\textrm{at}}$ and is fully spin-stretched, so the
resonances observed for this incoming state must have $\Delta m_{f}=+1$. This
selection rule can explain resonances for all three atomic states
investigated, so it seems likely that this character applies to all resonances
in the low-field group.

Figure 2:  Observed positions of resonances in the low-field group (black
dots), with projections back to zero field assuming
$f{{}^{\textrm{res}}}=9/2,\Delta m_{f}=+1$ (red dashed lines) and
$f{{}^{\textrm{res}}}=7/2,\Delta m_{f}=+1$ (blue dashed lines). Black solid
lines indicate thresholds. The weak Zeeman effect of the molecular spins is
neglected.

In order to determine $f{{}^{\textrm{res}}}$ for these resonances, we consider
what each possibility would imply about the levels. We expect that, towards
zero field, sublevels with different $m_{f}{{}^{\textrm{res}}}$ will approach
one another to form a single clear group, which may have internal structure
due to hyperfine couplings in the triatomic complex. We therefore calculate
the implied energy of the resonant state, determining
$m_{f}{{}^{\textrm{res}}}$ from $\Delta m_{f}=+1$ and considering
$f{{}^{\textrm{res}}}=9/2$ or $7/2$. We project each bound state back to zero
field, assuming it remains at a constant binding energy below its
corresponding threshold; this neglects mixing between different levels in the
group, but should be a good first approximation. The results of this are shown
in Fig. 2.

The red lines in Fig. 2 show that the pattern of resonances is consistent with
a closely spaced group of zero-field bound states with
$f{{}^{\textrm{res}}}=9/2$. These have binding energies centred near 8 MHz and
spread across about 5 MHz. Such a closely spaced group is unlikely to occur by
coincidence. They produce a tightly bunched group of resonances at each atomic
threshold, spread over less than 6 G for each $m_{f}$. Conversely, the blue
lines for an assignment to $f{{}^{\textrm{res}}}=7/2$ diverge towards zero
field and would imply zero-field binding energies spread across at least 20
MHz. However, in this case the states with a given value of
$m_{f}{{}^{\textrm{res}}}$ would be expected to cause additional resonances
with $\Delta m_{f}=0$ and $-1$ when they cross the higher thresholds. In fact
the number of resonances at higher thresholds is similar to that at the
$m_{f}=-9/2$ threshold. It is thus probable that the low-field resonances
arise from states with $f{{}^{\textrm{res}}}=9/2$.

To assign any further quantum numbers, we must look at the detailed patterns
of resonances. The low-field resonances are sometimes observed in pairs at the
same threshold, so there must be (at least) two characters of resonances in
this group. For the thresholds where both resonances are observed, there is a
consistent pattern that the one at higher field is wider than the one at lower
field, and we label these as having characters A and B respectively. For
thresholds where there is only one resonance observed, it is possible that one
of the resonances is systematically missing due to selection rules, but it is
also possible that the second resonance may be present but not observed.

There is one resonance observed for the incoming combination
$(-3/2,-4)_{\textrm{mol}}+(9/2,-7/2)_{\textrm{at}}$, for which the molecule is
in its spin-stretched state. This resonance must have $\Delta
m_{\textrm{mol}}=\Delta m_{\textrm{Na}}+\Delta m_{\textrm{K}}\geq 0$. Combined
with $\Delta m_{f}=+1$, as discussed above, and conservation of total
projection, $\Delta M_{\textrm{tot}}=\Delta m_{\textrm{mol}}+\Delta
m_{f}+\Delta M_{N}=0$, this implies that $\Delta M_{N}\leq-1$ and so
$N{{}^{\textrm{res}}}\neq 0$. States with different $N{{}^{\textrm{res}}}$ are
expected to be far apart in energy, so all states in this group must have the
same value of $N{{}^{\textrm{res}}}$, but may have different
$M_{N}{{}^{\textrm{res}}}$. There are no terms in the fine, hyperfine or
Zeeman Hamiltonian that can directly change $N$ by more than 2, and only the
small nuclear electric quadrupole interaction has $\Delta M_{N}>1$
Frye:triatomic-threshold-complexes:2021 ; the most likely interpretation is
therefore that $N{{}^{\textrm{res}}}=1$ or 2, and that the two characters of
resonances in the low-field group are $(\Delta M_{N},\Delta
m_{\textrm{mol}})=(-1,0)$ and $(0,-1)$. The latter is missing at the threshold
$(-3/2,-4)_{\textrm{mol}}+(9/2,-7/2)_{\textrm{at}}$ because the molecule is
spin-stretched, so the observed resonance must have character $(-1,0)$.
However, it is not easily assigned as either A or B from its position and
width.

For the incoming combination
$(-3/2,-4)_{\textrm{mol}}+(9/2,-9/2)_{\textrm{at}}$, no resonance is observed
at all, but we expect one with character $(-1,0)$. This resonance might be
missed either because it is outside the region of observation or because it is
too narrow. Both of these are more likely if it is of character B, so we
tentatively assign character A to be $(0,-1)$ and character B to be $(-1,0)$.

This assignment determines the spin character of the resonant bound states
almost completely. The resulting quantum numbers are included in Tables II and
III. The only remaining ambiguities are in cases where different values of
$\Delta m_{\textrm{Na}}$ and $\Delta m_{\textrm{K}}$ can sum to give the same
$\Delta m_{\textrm{mol}}$. This might, for example, lead to additional
resonances for collisions involving $(-1/2,-3)_{\textrm{mol}}$, but the
present experiments have not resolved any such features.

Our assignment implies that there is a rovibrational state of the triatomic
complex with $N{{}^{\textrm{res}}}\geq 1$ that lies very close to threshold.
If this has $f{{}^{\textrm{res}}}=9/2$, it is bound by less than 10 MHz. For a
system with unknown scattering length, an argument based on single-channel
quantum defect theory Gao:2000 ; Frye:triatomic-threshold-complexes:2021
indicates there is a 9% (4%) prior probability of finding a p (d)-wave state
within 10 MHz of threshold. The real system is more complex than a single
channel, but this estimate is appropriate for a state so close to threshold.
Conversely, if the state has $f{{}^{\textrm{res}}}=7/2$, its binding energy
must be in a narrow range within 50 MHz of the atomic hyperfine splitting; the
corresponding probability for this is 2% for either a p-wave or a d-wave
state. The final possibility is a state with $n>0$, for which $N>0$ is allowed
even if $L=0$. The probability of such a short-range bound state existing in
the top 10 MHz of the well is low for any individual value of $n$ and $v$, but
much larger when the full phase space is considered Christianen:density:2019 .
We estimate the density of states near threshold to be 0.3 GHz-1 for NaK+K
Frye:triatomic-complexes:2021 , so there is only about a 0.3% prior
probability of finding such a state within 10 MHz of threshold.

The bound state that causes the low-field resonances is therefore most likely
a very weakly bound long-range state. Such a bound state will have the
character predominantly of the separated atom and molecule in their respective
states. This presents a physical picture where the complicated short-range
rovibrational couplings are relatively unimportant and the relevant states of
the complex are remarkably simple. This is in sharp contrast to the picture of
resonances caused by chaotic short-range states, as suggested by Mayle _et
al._ Mayle:2012 .

The long-range nature of the states responsible for the low-field resonances
suggests that similar resonances will be present in other alkali-metal
atom+diatom systems. For each value of $L{{}^{\textrm{res}}}$, there will
always be one long-range rovibrational state of the triatomic complex within a
certain “bin” of energy below threshold Gao:2000 . For low
$L{{}^{\textrm{res}}}$, such a state is likely to couple to the incoming
states to produce a group of magnetically tunable Feshbach resonances below a
certain characteristic magnetic field. The depth of the top bin and the
corresponding characteristic field depend on the masses of the atom and
molecule and on the long-range dispersion coefficient. If
$f{{}^{\textrm{res}}}=9/2$, the least-bound state for 23Na40K+40K lies in the
upper part of its bin, but for other systems this may not be the case;
nevertheless, analogous resonances are typically expected at fields below 400
G for heavier systems Frye:triatomic-threshold-complexes:2021 .

The remaining 66 resonances span from 43 G to 107 G. We will refer to these as
the high-field group. They do not form so coherent a pattern as the low-field
group, and may arise from sets of resonances with several different
characters. In the same way as for the low-field group, the resonances at
thresholds with $m_{f}{{}^{\textrm{in}}}=-9/2$ must have $\Delta m_{f}=+1$.
However, at thresholds with $m_{f}{{}^{\textrm{in}}}=-7/2$ there are a larger
number of resonances than for $m_{f}{{}^{\textrm{in}}}=-9/2$. This increase is
probably caused by resonances with $\Delta m_{f}=0$, which are allowed at
thresholds with $m_{f}{{}^{\textrm{in}}}\geq-7/2$. The further increase in the
number of resonances at thresholds with $m_{f}{{}^{\textrm{in}}}=-5/2$ can
similarly be attributed to $\Delta m_{f}=-1$. As discussed above, at least one
of $f$ or $m_{f}$ must change between the incoming and bound states, so
$\Delta m_{f}=0$ implies $\Delta f\neq 0$. We thus conclude that the high-
field resonances probably arise from resonant states with
$f{{}^{\textrm{res}}}=7/2$.

Figure 3:  Observed positions of resonances in the high-field group (black
dots), with projections back to zero field assuming
$f{{}^{\textrm{res}}}=7/2,\Delta m_{f}=+1$ (blue dashed lines);
$f{{}^{\textrm{res}}}=7/2,\Delta m_{f}=0$ (green dashed lines); or
$f{{}^{\textrm{res}}}=7/2,\Delta m_{f}=-1$ (magenta dashed lines). Each
resonance is tentatively assigned to one of these options. Black solid lines
indicate thresholds. The weak Zeeman effect of the molecular spins is
neglected.

We project the bound states for this group of resonances back to zero field as
described above. We take $f{{}^{\textrm{res}}}=7/2$ for all resonances in this
group, but $m_{f}{{}^{\textrm{res}}}$ is not unambiguously assigned for
$m_{f}{{}^{\textrm{in}}}\geq-7/2$, so we pick values that produce as simple a
pattern as possible. The implied bound-state pattern is shown in Fig. 3. Many
of the states appear to converge towards a zero-field grouping between $-105$
and $-130$ MHz, relative to the $f{{}^{\textrm{in}}}=9/2$ thresholds, with a
few surrounding outliers. This is not intended to provide a definitive
assignment for individual states, but to indicate that a plausible
interpretation along these lines may be possible. The zero-field states
between $-200$ and $-250$ MHz seem to form a separate group, but there are too
few of them to identify clear patterns.

The character of the states responsible for the high-field resonances remains
uncertain, but there are a few likely possibilities. They might arise from
states with $v{{}^{\textrm{res}}}=n{{}^{\textrm{res}}}=0$, bound by just the
1.3 GHz of the atomic hyperfine splitting; they might arise from a low-lying
excited rotational state with $n{{}^{\textrm{res}}}>0$, bound by the
rotational excitation energy $bn(n+1)$ where $b/h=2.9$ GHz, either with or
without atomic hyperfine excitation; or they might arise from a complicated
short-range state of the kind suggested by Mayle _et al._ Mayle:2012 . The
first of these possibilities would still have substantial long-range
character, with decreasing amounts for the subsequent possibilities.

##  IV conclusion

In summary, we have performed a detailed experimental and theoretical study of
the ultracold atom-molecule Feshbach resonances between 23Na40K and 40K. More
than 80 new Feshbach resonances have been observed for a range of initial
states. We have presented a detailed theoretical interpretation of the low-
field group of resonances and assigned quantum numbers to the bound states
that cause them. This allows us to determine the nature of the resonant
states: they are likely to be remarkably simple long-range states with the
character of the separated atom and molecule. Such states are very different
from the chaotic short-range states proposed by Mayle _et al._ Mayle:2012 .
The simplicity of the long-range states allows understanding and offers the
possibility of controlling collisions using these resonances. We expect
similar resonances to be observable in other alkali-metal atom + diatom
systems.

###### Acknowledgements.

This work was supported by the National Key R&D Program of China (under Grant
No. 2018YFA0306502), the National Natural Science Foundation of China (under
Grant No. 11521063, 11904355), the Chinese Academy of Sciences, the Anhui
Initiative in Quantum Information Technologies, Shanghai Municipal Science and
Technology Major Project (Grant No.2019SHZDZX01), Shanghai Rising-Star Program
(Grant No. 20QA1410000). This work was supported by the U.K. Engineering and
Physical Sciences Research Council (EPSRC) Grant No. EP/P01058X/1. X.-Y. Wang
led the experimental work, and Matthew D. Frye led the theoretical work.

## Data availability statement

All data that support the findings of this study are included within the
article (and any supplementary information files).

These authors contributed equally to this work.

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## supplementary materials

In the supplementary material, the loss spectrum for the other 80 resonances
between 23Na40K and 40K are shown in the Figs. S1-S6.

Figure S1:  The low-field resonances with the initial state of the 40K atom in
$(9/2,-9/2)$. The remaining molecule numbers are shown as a function of the
magnetic field. The solid lines are the fitted Gaussian functions. Error bars
represent the standard deviation. Figure S2:  The low-field resonances with
the initial state of the 40K atom in $(9/2,-7/2)$. The remaining molecule
numbers are shown as a function of the magnetic field. The solid lines are the
fitted Gaussian functions. Error bars represent the standard deviation. Figure
S3:  The low-field resonances with the initial state of the 40K atom in
$(9/2,-5/2)$. The remaining molecule numbers are shown as a function of the
magnetic field. The solid lines are the fitted Gaussian functions. Error bars
represent the standard deviation. Figure S4:  The high-field resonances with
the initial state of the 40K atom in $(9/2,-9/2)$. The remaining molecule
numbers are shown as a function of the magnetic field. The solid lines are the
fitted Gaussian functions. Error bars represent the standard deviation. Figure
S5:  The high-field resonances with the initial state of the 40K atom in
$(9/2,-7/2)$. The remaining molecule numbers are shown as a function of the
magnetic field. The solid lines are the fitted Gaussian functions. Error bars
represent the standard deviation. Figure S6:  The high-field resonances with
the initial state of the 40K atom in $(9/2,-5/2)$. The remaining molecule
numbers are shown as a function of the magnetic field. The solid lines are the
fitted Gaussian functions. Error bars represent the standard deviation.


Soumita Das, Anupam Biswas 11institutetext: Department of Computer Science and
Engineering,
National Institute of Technology, Silchar-788010, Assam, India
22institutetext: Université Paris-Saclay, INRAE,
UMR 518 MIA, Palaiseau, France
Email:22email: wingsoffire72<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Direct Comparative Analysis of Nature-inspired Optimization Algorithms on
Community Detection Problem in Social Networks

Soumita Das 11 Bijita Singha 11 Alberto Tonda 22 Anupam Biswas 11

###### Abstract

Nature-inspired optimization Algorithms (NIOAs) are nowadays a popular choice
for community detection in social networks. Community detection problem in
social network is treated as optimization problem, where the objective is to
either maximize the connection within the community or minimize connections
between the communities. To apply NIOAs, either of the two, or both objectives
are explored. Since NIOAs mostly exploit randomness in their strategies, it is
necessary to analyze their performance for specific applications. In this
paper, NIOAs are analyzed on the community detection problem. A direct
comparison approach is followed to perform pairwise comparison of NIOAs. The
performance is measured in terms of five scores designed based on prasatul
matrix and also with average isolability. Three widely used real-world social
networks and four NIOAs are considered for analyzing the quality of
communities generated by NIOAs.

###### keywords:

Nature Inspired Optimization Algorithms, Community Detection, Fitness
Function, Direct Comparison

##  1 Introduction

In today’s world, majority of the problems are complex in nature and requires
optimization of diverse objectives such as minimization of costs, energy
consumption and/ or maximization of efficiency, sustainability and
performance. Specifically, optimization problems are often subject to a set of
complex, non-linear constraints. To solve optimization problems in an
effective and time efficient manner, numerous Nature-inspired Optimization
Algorithms (NIOAs) are developed [1, 2, 3]. NIOAs are typically based on
randomization concept and are used for both continuous and discrete
optimization problems. An extensive comparative study of several NIOAs
algorithms for continuous and discrete optimization has been performed in [4,
5]. In another work [6], a comparative analysis of NIOAs on ten continuous and
discrete optimization problems has been carried out. In addition to this,
numerous methods have been introduced which developed the discrete version of
a continuous optimization problem [7, 8]. An example of a discrete
optimization problem is community detection. It is discrete in the sense that
each of the solution element in a solution vector with N-dimensions can take
only discrete values. Several NIOAs algorithms on community detection have
been proposed [9]. Comparative study of few NIOA based community detection has
also been carried out [10, 11]

The general principle to solve the community detection problem is to maximize
intra-community connectivity (vertices/ entities of the same community are
strongly connected) and minimize inter-community connectivity (vertices/
entities belonging to different communities are loosely connected). However,
the measure of cohesiveness may vary depending on the type of network
(unweighted, weighted, directed, undirected, multiple edges, dynamic etc.). In
this paper, we have considered only undirected and unweighted networks for
carrying out our experiments and analyze the performance of NIOAs algorithms
on community detection. The contributions of this paper are listed as follows:

  * •

A considerable variety of NIOAs algorithms such as Grey Wolf Optimizer (GWO),
Moth-Flame Optimization (MFO), Sine-Cosine Algorithm (SCA) and Whale
Optimization Algorithm (WOA) have been used to detect communities in a
network.

  * •

A comparative performance analysis based on AVerage Isolability (AVI) has been
carried to determine the quality of communities identified by the
corresponding baselines.

  * •

Communities obtained from the respective baseline algorithms are directly
compared with each other based on D-scores (direct comparison) and K-scores
(overall comparison).

The organization of the rest of the paper is as follows: Section 2 emphasizes
on the baseline NIOAs algorithms, Section 3 briefs about the community
detection problem, Section 4 discusses about the direct comparative analysis
measure, Section 5 is dedicated to experimental analysis and Section 6
concludes the paper.

Figure 1: Generic Flow diagram of NIOAs.

##  2 Nature Inspired Optimization Algorithms

NIOAs share a set of steps that is portrayed by the generic workflow of the
algorithm in Figure 1. In the first step, the algorithm generates a set of
candidate solutions. This candidate solution generation is called population
initialization (X) which requires setting of three parameters such as
population size, number of dimensions and setting the range of value of
solution element. The second step deals with evaluation of the goodness of
each of the candidate solution using fitness function. Following this, the
termination criteria or the maximum number of iterations (MaxIt) is assigned
in third step. Until the termination criteria is satisfied, a set of
procedures are repeated as enumerated in the given figure by a), b), c) and
d). Firstly, position of each solution vector is updated. Next, the fitness of
the updated position vector is computed and compared with the previous
fitness. Subsequently, the best solution vector is selected and current
iteration counter is incremented by one. Then, after the termination condition
is satisfied, the algorithm returns the best solution vector. In this section,
we have discussed about some of the best performing algorithms in NIOAs realm
which are as follows.

###  2.1 Grey Wolf Optimizer (GWO)

This is a population based optimization algorithm inspired by the hunting
mechanism of grey wolves found in nature [12]. The wolves are categorized in
descending order of leadership hierarchy as $\alpha$, $\beta$, $\delta$ and
$\omega$ such that $\alpha$, $\omega$ lies at the top and bottom of hierarchy
respectively. GWO algorithm starts with population initialization followed by
computation of fitness of wolves where the best three wolves are designated as
$\alpha$, $\beta$ and $\delta$. Next, the distance between each wolf and prey
is computed by,

$\overrightarrow{D}=\mid\overrightarrow{C}.\overrightarrow{X_{p}}(t)-\overrightarrow{X}(t)\mid$
(1)

where $t$ represents number of iterations, $\overrightarrow{C}$ indicates
coefficient vector, $\overrightarrow{X_{p}}$, $\overrightarrow{X}$ is location
vector of prey and grey wolf respectively. Thereafter, position of grey wolf
is updated using the following formula,

$\overrightarrow{X}(t+1)=\overrightarrow{X_{p}}(t)-\overrightarrow{A}.\overrightarrow{D}$
(2)

where $A$ is a vector coefficient in [0,2]. Then, position of prey is updated
according to the following formula,

$\overrightarrow{X_{p}}(t+1)=\frac{(\overrightarrow{X_{1}}+\overrightarrow{X_{2}}+\overrightarrow{X_{3}})}{3},$
(3)

where $\overrightarrow{X_{1}},\overrightarrow{X_{2}},\overrightarrow{X_{3}}$
represents position vector of $\alpha,\beta,\delta$ wolves respectively. These
set of steps are repeated until termination criteria is satisfied. Ultimately,
GWO algorithm returns the best position vector for $\alpha$ which indicates
the best solution of the problem under consideration.

###  2.2 Sine-Cosine Algorithm (SCA)

It is also a population based optimization algorithm where the search for
optimal solution is inspired by the sine and cosine trigonometric functions
[13]. Initially, SCA algorithm starts with population initialization where
each individual is represented by $X_{i}=(x_{i1},...,x{ij},...,x_{iD})$ in the
D-dimensional search space. Next, the optimal solution is obtained using sine
and cosine functions depicted by the following formula,

$X_{i}^{t+1}=X_{i}^{t}+r_{1}\times\sin(r_{2})\times\mid
r_{3}X_{best}^{t}-X_{i}^{t}\mid,~{}~{}r_{4}<0.5$ (4)

$X_{i}^{t+1}=X_{i}^{t}+r_{1}\times\cos(r_{2})\times\mid
r_{3}X_{best}^{t}-X_{i}^{t}\mid,~{}~{}r_{4}\geq 0.5,$ (5)

where $X_{i}^{t}$ indicates the position of search space at $t^{th}$
iteration, $X_{best}^{t}$ refers to the best position in $t^{th}$ iteration.
Equation 4 and 5 indicates that SCA comprises of four key parameters such as
$r_{1}$, $r_{2}$, $r_{3}$ and $r_{4}$ where $r_{1}$ represents the search
region. This region lies either between the search agent and target or
outside, $r_{2}$ refers to the extent the movement is done towards or outside
the target, $r_{3}$ is used to emphasize ($r_{3}>1$) or de-emphasize
($r_{3}<1$) the current optimal solution in order to compute the distance to
be covered by search agents and $r_{4}$ is used to explore the search space
deterministically by switching between sine and cosine functions.

###  2.3 Moth-Flame Optimization (MFO)

It is a population based optimization algorithm inspired by the transverse
orientation of moths around light sources [14]. Moths travel long distances in
a straight line by maintaining a fixed angle with the moon. MFO algorithm
basically comprises of three primary steps. The first step is population
initialization of moths using a matrix $M(t)$ in a D-dimensional search space.
Next, fitness of individual moths are stored in an array.

This is followed by storing the flames which are the best positions obtained
by moths when searching the search space and is similarly represented in
matrix $F(t)$ and it’s corresponding fitness values are stored in array
$OF(t)$. Next, as the moths come across flames/ artificial light, they try to
maintain a similar fixed angle with the flames resulting into a deadly spiral
path towards the flames. Therefore, the second step is associated with
updating the position of moths using the following formula,

$M_{i}(t)=Dis_{i}(t)\times e^{bk}\times\cos(2\pi k)+F_{j}(t),$ (6)

$Dis_{i}(t)=\mid F_{j}(t)-M_{i}(t)\mid,$ (7)

where $M_{i}(t)$ refers to the moth’s position in $i^{th}$ iteration,
$Dis_{i}(t)$ represents the distance moth $M_{i}(t)$ and corresponding flame
$F_{j}(t)$, $k$ is a random number that lies in the range [-1,1], $b$ depicts
shape of logarithmic spiral.

###  2.4 Whale Optimization Algorithm (WOA)

It is a population based optimization algorithm inspired by the hunting
mechanism of humpback whales [15]. Firstly, population of search agents is
initialized and fitness of individual search agents is computed. Considering
the fitness values, the current best search agent is assumed to be the target
prey. Secondly, the position of other search agents are updated near the
target prey based on parameters $p$ and $A$. These parameters controls
position updating by incorporation of these parameters into three different
rules such as encircling prey where $p<0.5$ and $\mid A\mid<1$, search for
prey where $p<0.5$ and $\mid A\mid\geq 1$ and spiral updating position where
$p\geq 0.5$. The position of search agent $\overrightarrow{X}(t+1)$ using is
updated by encircling prey at iteration $t+1$ using Equation 8 and Equation 9.

$\overrightarrow{D}=\mid\overrightarrow{C}.\overrightarrow{X}^{*}-\overrightarrow{X}(t)\mid$
(8)

$\overrightarrow{X}(t+1)=\mid\overrightarrow{X}^{*}(t)-\overrightarrow{A}.\overrightarrow{D}\mid,$
(9)

$\overrightarrow{A}=2\overrightarrow{a}.\overrightarrow{r}-\overrightarrow{a}$
(10)

$\overrightarrow{C}=2.\overrightarrow{r}$ (11)

$\overrightarrow{X}^{*}$ represents the best search agent in the current
iteration $t$, $\overrightarrow{X}(t)$ represents position of a search agent
at iteration $t$, the value of $a$ decreases from 2 to 0 over the iterations,
$r$ is a random number in range [0,1]. Next, searching for prey is similar to
encircling prey. However, the only difference is that $\overrightarrow{X}^{*}$
is replaced with a randomly selected search agent $\overrightarrow{X}_{rand}$.
In spiral position update, the positions of individual search agents are
updated using the following equation,

$\overrightarrow{X}(t+1)=\overrightarrow{\acute{D}}.e^{bl}.\cos{(2\pi
l)}+\overrightarrow{X^{*}}(t),$ (12)

where
$\overrightarrow{\acute{D}}=\mid\overrightarrow{X}^{*}-\overrightarrow{X}(t)\mid$
which indicates the difference of the distance between the target prey and the
search agent at the current iteration, $b$ is constant, $l\in[-1,1]$. The
position of search agents are updated until the termination criteria and
finally WOA algorithm returns the best search agent.

##  3 Community Detection Problem

The problem of community detection in networks belong to the class of graph
partitioning problem, and it is thus a NP-hard problem [16]. Therefore, it has
received a lot of attention in recent years and several community detection
methods have been introduced for identifying communities in networks. A
network comprises of a set of entities and relationships/ connections shared
by the entities. Networks are represented in the form of a graph indicated by
$G(V,E)$ comprising of nodes ($V$) referring to entities and edges ($E$)
specifying connections. The problem is to divide the network into several
communities $C=\\{C_{1},C_{2},C_{3},..,C_{k}\\}$ where each community say
$C_{i},~{}~{}~{}\forall i={1,2,..,k}$ consists of a set of nodes belonging to
$V$ such that the number of connections within $C_{i}$ should be maximized and
number of connections between $C_{i}$ and other communities should be
minimized. These maximization or minimization requires the use of fitness
function in order to obtain the best solution.

Suppose, $G(V,E)$ is divided into $l$ feasible partitions
$P=\\{P_{1},P_{2},P_{3},..,P_{l}\\}$. Then, community detection problem is
formulated as an optimization problem using the following equation,

$f(P^{best})=maxf(P),$ (13)

where $P^{best}$ is the desired partition of the network obtained by
incorporating a fitness function $f$ which evaluates the goodness of the
network.

Fitness function: It is required to find the best solution in an optimization
problem. Here, as we are considering community detection as an optimization
problem, so for fitness computation, community evaluation metrics such as are
modularity, Normalized Mutual Information (NMI), purity, Adjusted Random Index
(ARI) etc. are used [17, 18]. Modularity is used to measure the quality of
community, whereas NMI, purity, ARI is used to measure accuracy of community.
Depending on the cardinality of fitness function used, community detection
problem is classified as single-objective optimization problem and multi-
objective optimization problem [19].

##  4 Direct comparative analysis

The rapid growth of NIOAs have necessitated the performance evaluation of the
respective algorithms. Though several statistical measures such as mean,
standard deviation and median are used for performance comparison purpose, but
these measures do not directly compare the solutions given by two separate
algorithms say primary algorithm ($A_{p}$) and alternative algorithms
($A_{q}$), where $A_{p}$ refers to those algorithms whose performance is to be
evaluated and $A_{q}$ refers to the set of algorithms with which $A_{p}$ is to
be compared. In this paper, we have used D-scores and K-scores for direct
comparison and overall comparison respectively to evaluate the quality of
communities [20].

Direct Optimality (DO:) $A_{p}$ is compared with $A_{q}$ in terms of
optimality by combining the comparative performance considering best
performance, average performance and worst performance of $A_{p}$ with respect
to $A_{q}$ denoted by $O_{1},O_{2},O_{3}$ respectively and is defined by,

$DO=O_{1}+0.5*O_{2}-O_{3}$ (14)

Direct Comparability (DC): $A_{p}$ is compared with algorithm $A_{q}$ in terms
of three levels of abstractions such as win, tie and loose denoted by $C_{1}$,
$C_{2}$ and $C_{3}$ respectively and is defined by,

$DC=C_{1}+0.5*C_{2}-C_{3}$ (15)

Overall Optimality (KO): The overall optimality of $A_{p}$ is computed based
on three levels of abstraction such as best, average and worst irrespective of
win or loose indicated by $K_{1}^{0}$, $K_{2}^{0}$ and $K_{3}^{0}$
respectively and is defined by,

$KO=K_{1}^{0}+0.5*K_{2}^{0}-K_{3}^{0}$ (16)

Overall Comparability (KC): $A_{p}$ is compared with $A_{q}$ by considering
overall comparability in all three levels of abstraction such as win, tie and
loose indicated by $K_{1}^{c}$, $K_{2}^{c}$ and $K_{3}^{c}$ respectively and
is defined by,

$KC=K_{1}^{c}+0.5*K_{2}^{c}-K_{3}^{c}$ (17)

Overall Together (KT): It is used to interpret that $A_{p}$ performs better
than $A_{q}$ considering that abstraction levels such as best $\&$ average and
win $\&$ tie are overlapping and is defined by,

$KT=\frac{a+b+d+e}{n}$ (18)

where a,b, c and d represents the overlapping abstraction levels, $n$
indicates total number of possible combinations of abstraction levels.

Table 1: Dataset Statistics. First column contains dataset details, $\\#~{}Nodes$ refers to number of nodes, $\\#~{}Edges$ refers to number of edges, Avg. degree indicates average degree of the graph. Dataset |  # Nodes |  # Edges |  Avg. degree
---|---|---|---
Karate [21] |  34 |  78 |  4.58
Dolphin [22] |  62 |  159 |  5.12
Football [23] |  115 |  613 |  10.66

##  5 Experimental Analysis

In this work, experiments are conducted on several widely used real-world
datasets such as karate network [21], dolphin network [22] and football
network [23] summarized in Table 1. Several state-of-the-art NIOAs algorithms
such as GWO, MFO, SCA and WOA have been used on community detection to perform
a comparative analysis of these algorithms using average isolability and five
different performance measures based on optimality and comparability [24].
Also, the performance of NIOAs algorithms for community detection is highly
dependent on parameter settings. Therefore, in this section, we discuss about
algorithm parameter settings, average isolability and result analysis.

###  5.1 Algorithm parameter settings

There are two types of parameters in NIOAs algorithms namely, common
parameters and algorithm specific parameters. Parameters that are common in
all NIOAs algorithms are called common parameters and parameters specific to a
particular NIOAs algorithm are the algorithm specific parameters. There are
particularly three common parameters namely population size, number of
dimensions and number of iterations which are described below.

Number of dimensions: In community detection context, number of dimensions is
equal to the total number of nodes present in a network. The size of candidate
solution is equal to the number of dimensions. Total number of such candidate
solutions indicates population size.

Population size: The population size needs to be carefully initialized because
the best solution might be dependent on population size. Setting a high
population size improves the search capability but leads to increase in time
complexity of the algorithm. In our experiments, we have set the population
size as 30.

Number of iterations: It is also a key parameter to find the optimal solution.
Initially, current iteration is set to 1. For specification of number of
iterations, two aspects are to be considered. Firstly, if the number of
iterations is small, then the optimal solution might not be found. Whereas,
large number of iterations increases time complexity of optimization
algorithms and may lead to redundancy i.e. iterations may continue even after
attaining the best solution. Therefore, number of iterations must be carefully
set.

###  5.2 Average Isolability

It is required to compare and improve the candidate solutions to obtain a near
optimal solution. In our experiment, we have considered individual cluster
specific fitness function namely, AVerage Isolability (AVI) [24] where the
objective is to examine the ability of a cluster to isolate itself from rest
of the network by examining the nodes based on the strength of connections.
Therefore, to find the optimal solution, we have maximized AVI. For an
undirected graph, Isolability of a cluster $C_{i}$ is defined by,

$\text{Isolability }(C_{i})=\frac{\\{(u,v)\mid
u\in_{C_{i}}v\\}}{\\{\\{(u,v);(u,w)\\}\mid u\in_{C_{i}}v~{}\&w\notin
C_{i}\\}},$ (19)

where, the numerator term indicates connections within the community $C_{i}$
and denominator is the total number of connections. Next, AVI is defined by,

$Q_{AVI}(G,C)=\frac{1}{K}\sum Isolability(C_{i}),$ (20)

where $k$ indicates total number of clusters in $G(V,E)$.

Table 2: Comparative performance of MFO algorithm with alternative algorithms based on D-scores and K-scores. Dataset | GWO | SCA | WOA
---|---|---|---
DO | DC | KO | KC | KT | DO | DC | KO | KC | KT | DO | DC | KO | KC | KT
Karate | 0.98 | 0.98 | 1.00 | 1.00 | 1.00 | 0.98 | 0.98 | 1.00 | 1.00 | 1.00 | 0.06 | 0.75 | 0.53 | 1.0 | 1.0
Dolphin | 1.44 | 0.76 | 1.00 | 0.78 | 1.00 | 1.44 | 0.76 | 1.00 | 0.78 | 1.00 | 0.17 | 0.17 | 0.31 | 0.30 | 0.64
Football | 1.44 | 0.75 | 1.00 | 0.76 | 1.00 | 1.44 | 0.75 | 1.00 | 0.76 | 1.00 | 1.44 | 0.75 | 1.00 | 0.76 | 1.00

###  5.3 Result Analysis

Quality of communities given by GWO, MFO, SCA and WOA have been analyzed on
three widely used real-world datasets. The analysis has been carried based on
the emphasizing on the quality of the community given by each baseline
algorithm and performing comparative evaluation. AVI value is used for quality
evaluation. In addition to this, performance analysis based on one-to-one
comparison (D-scores) and one-to-many comparison (K-scores) is performed.

KarateDolphinFootball$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.24$$0.15$$2.88\cdot
10^{-2}$$0.31$$0.16$$3.29\cdot 10^{-2}$$0.25$$0.16$$2.89\cdot
10^{-2}$$0.17$$0.13$$2.94\cdot 10^{-2}$Average IsolabilityGWOMFOSCAWOA Figure
2: Comparative analysis of GWO, MFO, SCA and WOA based on Average Isolability.

####  5.3.1 Result analysis with Average Isolability:

The AVI scores of the communities given by GWO, MFO, SCA and WOA on real-world
datasets are shown in Figure 2. Let us try to analyze the performance of these
algorithms with the help of this figure. Here, the X-axis represent real-world
datasets namely karate, dolphin and football; Y-axis represents AVI score. The
performance of GWO, MFO, SCA and WOA is shown using teal, lime, yellow and
green colored bars respectively. The values corresponding to each bar
indicates AVI score of the respective algorithms on a given dataset. Higher
AVI score indicates good performance of corresponding algorithm and the
performance deteriorates with decrease of AVI score. Therefore, the results
shown in Figure 2 indicates that MFO algorithm gives the best performance on
all the datasets and WOA algorithm shows the worst performance on karate and
dolphin dataset. Whereas, GWO algorithm shows the worst performance on
football dataset.

KarateDolphinFootball$0$$0.5$$1$$1.5$$2$$2.5$$9.06\cdot
10^{-2}$$0.25$$0.47$$0.67$$1.02$$1.44$$0.22$$-3.21\cdot
10^{-2}$$0.22$$0.9$$1.08$$5\cdot 10^{-2}$ADOGWOMFOSCAWOA (a) Comparative
analysis based on ADO.

KarateDolphinFootball$-0.2$$0$$0.2$$0.4$$0.6$$0.8$$1$$1.2$$8.33\cdot
10^{-2}$$0.1$$0.16$$0.9$$0.57$$0.75$$5.13\cdot 10^{-2}$$-2.87\cdot
10^{-2}$$6.45\cdot 10^{-2}$$0.63$$0.85$$-1.9\cdot 10^{-2}$ADC (b) Comparative
analysis based on ADC.

KarateDolphinFootball$-0.5$$0$$0.5$$1$$1.5$$2$$-0.67$$-0.71$$-0.36$$0.84$$0.77$$1$$-0.67$$-0.63$$-0.32$$0.8$$0.7$$-0.32$AKO
(c) Comparative analysis based on AKO.

KarateDolphinFootball$-1$$-0.5$$0$$0.5$$1$$1.5$$2$$-0.76$$-0.75$$-0.5$$1$$0.62$$0.76$$-0.76$$-0.75$$-0.49$$0.94$$0.66$$-0.58$AKC
(d) Comparative analysis based on AKC.

KarateDolphinFootball$0$$0.2$$0.4$$0.6$$0.8$$1$$1.2$$1.4$$1.6$$1.8$$0.14$$0.13$$0.26$$1$$0.88$$1$$0.13$$0.11$$0.25$$0.98$$0.89$$0.22$AKT
(e) Comparative analysis based on AKT.

Figure 3: Comparative analysis based on average D-score and K-scores i.e. ADO,
ADC, AKO, AKC and AKT values for the of communities identified with GWO, MFO,
SCA and WOA on real-world datasets.

####  5.3.2 Result analysis based on D-scores and K-scores:

D-scores and K-scores are used to evaluate the performance of all possible
combinations of the baseline algorithms in terms of the quality of communities
given by the respective algorithms. All such combinations of baseline
algorithms indicated by $(A_{p},A_{q})$ is considered as a comparable
algorithm pair. The results of comparable algorithm pairs such as
$(A_{p}=MFO,A_{q}=GWO)$, $(A_{p}=MFO,A_{q}=SCA)$ and $(A_{p}=MFO,A_{q}=WOA)$
in terms of D-scores and K-scores on karate, dolphin and football dataset are
summarized in Table 2. Then, average DO (ADO), average DC (ADC) score, average
KO (AKO), average KC (AKC) and average KT (AKT) is obtained by summation of
corresponding DO, DC, KO, KC and KT scores of all comparable algorithm pairs
with $A_{p}=MFO$ divided by the total number of such pairs and the results are
shown in Figure 3. High ADO, ADC, AKO, AKC, AKT scores indicate that $A_{p}$
performs better than $A_{q}$ in terms of optimality and comparability. For
each dataset and corresponding performance measure, highest positive score
obtained by the respective algorithm is ranked as 1, second highest is ranked
as 2 and so on. Following this ranking procedure, MFO algorithm is ranked as 1
and hence, it is the best performing algorithm in terms of D-scores, K-scores.
Following this strategy, SCA gives the worst performance.

##  6 Conclusion

A quality measure based on connection strength associated with a cluster
called average isolability and a direct comparison approach based on five
scores designed based on prasatul matrix is used to evaluate the quality of
communities considering optimality and comparability. Four NIOAs and three
widely used real-world datasets are used to perform comparative analysis.
Results based on average isolability indicate that the MFO algorithm gives the
best performance on all datasets. Whereas, WOA algorithm has the worst
performance on karate and dolphin datasets, GWO algorithm has the worst
performance on football datasets. Following this, the performance analysis
based on the five scores derived from prasatul matrix suggests that the MFO
algorithm achieves the best performance and the SCA algorithm gives the worst
performance.

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