# Non-homogeneous exoplanets in metric-affine gravity

ALEKSANDER KOZAK Institute of Theoretical Physics, University of Wrocław, pl.
Maxa Borna 9,
50-204 Wrocław, Poland
<EMAIL_ADDRESS>ANETA WOJNAR Laboratory of Theoretical Physics,
Institute of Physics, University of Tartu, W. Ostwaldi 1,
50411 Tartu, Estonia
<EMAIL_ADDRESS>
###### Abstract

The improved description of the planets’ interior is provided. We examine the
modified gravity effects on the Earth-like planets composed of the iron core
and silicate mantle. We confirm that the mass-radius relations as well as
density profiles differ with respect to the commonly adopted Newtonian models.

(exo)-planets, modified gravity.

##  I Introduction

General Relativity (GR) is one of the most thoroughly tested physical theories
Will:2014kxa . It was not only able to explain various phenomena that could
not be accounted for on the ground of previous theory, but also successfully
predicted existence of some exotic objects, such as black holes. Evidence for
their existence was provided by a recent detection of gravitational waves
TheLIGOScientific:2017qsa , as well as a direct observation of a shadow of a
black hole at the centre of M87 Akiyama:2019cqa ; aki2 ; aki3 ; god . However,
not all cosmological and astrophysical data fits into the scenario predicted
by GR. For example, it is unclear how the currently observed accelerated
expansion of the Universe should be explained, as well as the problem of dark
matter. These problems were addressed by modified gravity theories
Copeland:2006wr ; Nojiri:2006ri ; nojiri2 ; nojiri3 ; Capozziello:2007ec ;
Carroll:2004de , introducing some deviations from Einstein’s theory. Within
the context of modified gravity, the space-time singularities and high-energy
unification scenarios were also investigated Senovilla:2014gza ; ParTom ;
BirDav . Another major problem that cannot be satisfactorily solved by GR is
the excess of maximum mass of compact objects and binary black hole mergers as
compared to theoretical predictions lina ; as ; craw ; NSBH ; abotHBH ; sak3 .

In this paper, we focus on a narrow class of modified theories of gravity,
called $f(R)$ theories, in order to investigate how their non-relativistic
limit affects internal composition of low-mass objects. The problems sketched
above concern mostly high-energy regime or large, cosmological scales, but it
might be possible that also weak-field limit will be altered, allowing one to
detect deviation from GR on much smaller scales. For example, it has been
shown that non-relativistic description of objects such as main-sequence and
white dwarf stars is changed in modified gravity Saito:2015fza ; olek ;
olmo_ricci ; review ; cantata ; Chandra ; Saltas:2018mxc ; Jain:2015edg ;
Banerjee:2017uwz ; Wojnar:2020wd ; Belfaqih:2021jvu ; Sakstein:2015zoa ;
Sakstein:2015aac ; Crisostomi:2019yfo ; Olmo:2019qsj ; Rosyadi:2019hdb ; early
; chow ; Benito2021 ; Wojnar:2020frr . On the other hand, planets in the
context of alternative theories of gravity were not studied too much in the
literature. The evolution of Jupiter-like planets and fragmentation process
were investigated in an_jowisz , while mass-radius relation in olek2 . Apart
from these works, they have been mainly used to test and constrain theories of
gravity in the case of precessions of planetary perihelion in the Solar
System, see e.g. BD ; melinda ; iorio ; harko ; hatice ; bag ; seba ; schmidt
; harko2 ; iorio2 ; fatibene ; bhat , as well as modifications to the third
Kepler’s law kepler1 ; kepler2 .

We adopt so-called ’Palatini formalism’, in which one considers the metric
tensor to be a priori independent of the affine connection. This is motivated
by the fact that logically these two structures - affine and metric, the
former being responsible for parallel transport and the latter for measurement
of distances, time, volumes, and also for setting up causal structure of
spacetime - are unrelated. The connection is assumed to be symmetric, it also
does not enter the matter part of the action. Thanks to this assumptions, the
theory is metric, i.e. particles follow geodesics determined by the dynamical
metric. The Palatini formalism, when applied to $f(R)$ theories, exhibits some
advantages over the metric approach. For example, one does not need any
screening mechanism to hide the effect of a scalar degree of freedom present
in the metric theory, since the field is not dynamical olmo2011 . Also, they
provide a framework in which the inflation appears in a natural way
borowiec2016 .

In the previous work olek2  we have revealed that in the mass-radius relations
for a cold homogeneous sphere one deals with the additional degeneracy caused
by metric-affine gravity, and that particular models of gravity modify the
interior structure of terrestrial planets. Since that finding carries a
possibility to test theories of gravity with the use of earth- and marsquakes,
as detailed discussed in olek2 , it is crucial to improve our planets’
modelling by taking into account a more complex interior structure and
realistic geometry. Therefore, in what follows, we will improve our analysis
by adding a second layer - our toy planet will now consist of an iron core and
silicate mantle, in the proportions adequate to the description of the Earth-
like exoplanets. Before doing so, in the next section we will briefly recall
the theoretical background and structural equations with well-tested equations
of state for the iron and silicates. In the section (III) we present the
boundary conditions and numerical solutions of the modified equations in the
form of the mass-radius relations and density profiles for the Palatini
quadratic model. The last section is dedicated to the short conclusions.

We use $(-+++)$ signature convention and $\kappa^{2}=8\pi G/c^{4}$.

##  II Theoretical background

###  II.1 Conformal invariants approach to Palatini $f(\mathcal{R})$ gravity

General action for metric scalar-tensor theories of gravity reads as follows:

$\begin{split}S[g,\Phi,\psi_{m}]=&\frac{1}{2\kappa^{2}}\int\sqrt{-g}\Big{[}\mathcal{A}(\Phi)R(g)-\mathcal{B}(\Phi)(\partial\Phi)^{2}\\\
&-\mathcal{V}(\Phi)\Big{]}d^{4}x+S_{\text{matter}}[e^{2\alpha(\Phi)}g,\psi_{m}].\end{split}$
(1)

with $\psi_{m}$ denoting matter fields. There are four functions of $\Phi$
entering the action: $\mathcal{A},\mathcal{B}$ and $\alpha$ denoting different
types of coupling: to the curvature, to the kinetic term and anomalous to
matter, respectively, and also the self-interaction potential of the scalar
field, represented by the function $\mathcal{V}$. This action will provide
mathematical framework for analysis of Palatini $f(\mathcal{R})$ gravity given
by the action:

$S[g,\Gamma,\psi_{m}]=\frac{1}{2\kappa^{2}}\int\sqrt{-g}f(\mathcal{R})d^{4}x+S_{\text{matter}}[g,\psi_{m}],$
(2)

where $\mathcal{R}=g^{\mu\nu}\mathcal{R}_{\mu\nu}(\Gamma)$111In what follows
we will denote the Palatini curvature by a calligraphic $\mathcal{R}$, and the
metric curvature by $R$.. Here, $\Gamma$ is treated as an object independent
of the metric tensor and scalar field.

The theory has a scalar-tensor representation in the metric formulation:

$\begin{split}S[g,\Phi,\psi_{m}]=&\frac{1}{2\kappa^{2}}\int\sqrt{-g}\Big{[}\Phi
R(g)+\frac{3}{2\Phi}(\partial\Phi)^{2}\\\
&-\mathcal{V}(\Phi)\Big{]}d^{4}x+S_{\text{matter}}[g,\psi_{m}],\end{split}$
(3)

that can be achieved by means of a Legendre transformation:
$\Phi=df/d\mathcal{R}$ and
$V(\Phi)=f^{\prime}(\mathcal{R}(\Phi))\mathcal{R}(\Phi)-f(\mathcal{R}(\Phi))$
and integrating out the auxiliary connection field. The four functions of the
scalar field from (1) can be now identified with:

$\begin{split}&\mathcal{A}(\Phi)=\Phi,\quad\mathcal{B}(\Phi)=-\frac{3}{2\Phi},\\\
&\mathcal{V}(\Phi)=V(\Phi),\quad\alpha(\Phi)=0.\end{split}$ (4)

Variation w.r.t. dynamical variables entering the action (3) leads to the
following field equations:

$\displaystyle\begin{split}&\Phi
G_{\mu\nu}-\frac{3}{4\Phi}g_{\mu\nu}g^{\alpha\beta}\partial_{\alpha}\Phi\partial_{\beta}\Phi+\frac{3}{2\Phi}\partial_{\mu}\Phi\partial_{\nu}\Phi\\\
&-(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu})\Phi+\frac{1}{2}V(\Phi)g_{\mu\nu}=\kappa^{2}T_{\mu\nu},\end{split}$
(5a) $\displaystyle V(\Phi)-\frac{1}{2}\Phi
V^{\prime}(\Phi)=\frac{\kappa^{2}}{2}T\,.$ (5b)

One can introduce Weyl transformations of the metric tensor in order to change
the parametrization of (1) in such a way that the field equations involving
new functions $\\{\mathcal{A},\mathcal{B},\mathcal{V},\alpha\\}$ will be more
approachable. The transformation is defined as:

$\displaystyle\begin{cases}\bar{g}_{\mu\nu}=e^{2\gamma(\Phi)}g_{\mu\nu}\\\
\bar{\Phi}=\bar{f}(\Phi),\end{cases}$ (6)

where one allows the scalar field to change as well. Under these
transformations, the four functions defining so-called conformal frame also
change. It is possible, however, to construct quantities preserving their
functional form when (6) is applied to dynamical variables. Such quantities
will be called (conformal) invariants; the ones used in this paper are the
following:

$\displaystyle\mathcal{I}_{1}(\Phi)$ $\displaystyle=$
$\displaystyle\frac{\mathcal{A}(\Phi)}{e^{2\alpha(\Phi)}},$ (7)
$\displaystyle\mathcal{I}_{2}(\Phi)$ $\displaystyle=$
$\displaystyle\frac{\mathcal{V}(\Phi)}{\mathcal{A}^{2}(\Phi)},$ (8)
$\displaystyle\frac{d\mathcal{I}}{d\Phi}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\mathcal{B}}{\mathcal{A}}+\left(\frac{3\mathcal{A}^{\prime}}{2\mathcal{A}}\right)^{2}},$
(9)

where prime denotes differentiating w.r.t. the scalar field. For Palatini
$f(R)$ theory, the invariants are:

$\displaystyle\mathcal{I}_{1}(\Phi)$ $\displaystyle=$ $\displaystyle\Phi,$
(10) $\displaystyle\mathcal{I}_{2}(\Phi)$ $\displaystyle=$
$\displaystyle\frac{V(\Phi)}{\Phi^{2}},$ (11)
$\displaystyle\frac{d\mathcal{I}}{d\Phi}$ $\displaystyle=$ $\displaystyle 0,$
(12)

with the potential function left unspecified (it is determined by the choice
of the $f$ function). Each of the invariants introduced above has a meaning
within the context of scalar-tensor theories of gravity. The first one
represents non-minimal coupling between the scalar field and curvature, and if
it is constant then it will be always possible to find a conformal frame, in
which $\Phi$ can be viewed as just another matter field. $\mathcal{I}_{2}$
generalizes the self-interaction potential, while $\mathcal{I}$ plays the role
of invariant scalar field. Its vanishing means that the field is non-dynamical
for a certain class of theories; conformal transformations do not change this
fundamental property of such theories. As one can see, the invariant vanishes
for Palatini $f(\mathcal{R})$ gravity, which is a well-known fact. Since the
field is non-dynamical for this theory, it can be related in an algebraic way
to the trace of stress-energy tensor, as suggested by the field equations (5).

###  II.2 Planets’ structure equations in Palatini gravity

The hydrostatic equilibrium equations for the Palatini $f(\mathcal{R})$
gravity in the case of the quadratic functional

$f(\mathcal{R})=\mathcal{R}+\beta\mathcal{R}^{2}$

and the perfect fluid energy-momentum tensor
$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}$ were presented in various
works (see aneta_stab ; olek ). In what follows, we will need their simplified
forms since the terms proportional to $p/c^{2}$ can be neglected in comparison
to the energy density. Therefore, the structural equations are olek2

$\begin{split}&p^{\prime}=-\frac{G\mathcal{M}\rho}{r^{2}\mathcal{I}^{1/2}_{1}}\left(1-\frac{2G\mathcal{M}}{c^{2}r\mathcal{I}^{1/2}_{1}}\right)^{-1}\left(1+\frac{4\pi\mathcal{I}_{1}^{\frac{3}{2}}r^{3}}{c^{2}\mathcal{M}}\frac{\mathcal{I}_{2}}{2\kappa^{2}}\right)\left(\frac{r}{2}\partial_{r}\ln\mathcal{I}_{1}+1\right)\\\
&-c^{2}\rho\>\partial_{r}\ln\mathcal{I}_{1}\end{split}$ (13)

with the invariants taking the following forms for the given model of gravity:

$\displaystyle\mathcal{I}_{1}=1+2\alpha\rho,\;\;\;\;\mathcal{I}_{2}=\frac{2\alpha\kappa^{2}c^{2}\rho^{2}}{\left(1+2\alpha\rho\right)^{2}}.$

We have introduced a redefined Starobisnki parameter
$\alpha:=2c^{2}\kappa^{2}\beta$. Moreover, the non-relativistic mass was
reduced to:

$\mathcal{M}(r)=\int^{r}_{0}4\pi\tilde{r}^{2}\frac{\rho-\alpha\rho^{2}}{\left(1+2\alpha\rho\right)^{1/2}}\left[1+\frac{\tilde{r}}{2}\partial_{\tilde{r}}\ln\left(1+2\alpha\rho\right)\right]d\tilde{r}.$
(14)

Apart from the structural equations, one needs to have an equation of state
(EoS) which describes the iron core and silicate mantle in our 2-layer model
of a terrestrial planet, and boundary conditions, discussed in the section
(III). Since we are also dealing with the pressure ranges $p\gtrsim
10^{4}\,\text{GPa}$, the electron degeneracy must be taken into account. To do
so, one matches the Birch EoS birch ; poi  with the Thomas-Fermi-Dirac EoS
(TFD) thomas ; fermi ; dirac ; feynmann . The last one also takes into account
a density-dependent correlation energy terms sal  being a result of
interactions between electrons themselves which obey the Pauli exclusion
principle and move in the Coulomb field of the nuclei. Such a merger can be
approximated by a modified polytropic equation of state (SKHM EoS) seager

$\rho(p)=\rho_{0}+cp^{n},$ (15)

whose best-fit parameters $\rho_{0}$, $c$, and $n$ for iron and silicate (Mg,
Fe)SiO3 are given in the table 1. The additional term $\rho_{0}$ includes the
incompressibility of solids and liquids at low pressures. The validity of the
equation (15) with the given fits reaches $p=10^{7}\,\text{GPa}$, and hence it
will be the maximal value of the central pressure considered in our numerical
analysis.

Table 1: Best-fit parameters for the SKHM EoS (15) obtained in the referenceseager . Material |  $\rho_{0}$ (kg m-3) |  $c$ (kg m-3 Pa${}^{-n})$ | $n$
---|---|---|---
Fe($\alpha$) | 8300 | 0.00349 | 0.528
(Mg, Fe)SiO3 | 4260 | 0.00127 | 0.549

##  III Numerical solutions

Since we are interested in the non-homogeneous planets, we need to use
separate equations of state for different layers. In this paper, we limit our
attention to spherical objects composed of two layers: the inner iron core and
the mantle made of silicate perovskite (Mg, Fe)SiO3. Such a chemical
composition was chosen to render more realistically structure of rocky planets
such as Earth or Earth-like exoplanets.

Before going further, let us briefly discuss matching the core and the mantle.
Junction conditions in the case of Palatini were studied in olmojunction  \-
let us however notice that our case differs from it, since in that work
interior solution was matched with the exterior one, providing additional
conditions on the matter fields. In our case, our main concern is related
rather to the so-called "singular parameter", discussed in olek2 , since it
was evident that for some particular combination of the parameter’s value and
equation of state the hydrostatic equilibrium equation could not have any
physical solutions. Therefore, one needs to be careful when performing the
calculations, especially for negative values of the parameter $\alpha$.

In order to obtain density profiles of planets, one needs to make use of Eqs
(13), (14) and (15) to write down the considered system of equations in the
following form:

$\displaystyle\begin{cases}\rho^{\prime}(r)=f_{1}(\rho,\mathcal{M},r),\\\
\mathcal{M}^{\prime}(r)=f_{2}(\rho,\mathcal{M},r),\end{cases}$ (16)

with the boundary conditions $\mathcal{M}(0)=0$ and $\rho(0)=\rho_{c}$,
integrated until the density drops to the ambient one. If one wishes to find
the density profile for a planet of a given mass $m$ (i.e. solve the above
system for $\mathcal{M}(0)=0$ and $\mathcal{M}(R_{p})=m$, where $R_{p}$ is the
planet radius), then it is necessary to find the internal density first; in
our case, it was achieved by a simple shooting method. We also set the core
mass fraction to be 0.3, based on the most common Earth and Earth-like models
of terrestrial planets icarus . At the core-mantle boundary we demand that the
core pressure and the mantle one are equal, which gives a condition for the
initial density in the second layer. The numerical calculations were performed
using a Python script.

Figure 1: [color online] Mass-radius relation for small planets composed of
two layers: iron core and perovskite silicate mantle. The results were
obtained for different values of the Starobinsky parameter $\beta$. The solar-
system planets were included, as well as some TRAPPIST-1 exoplanets, denoted
by letters trappist . Figure 2: [color online] Mass-radius relation for large
planets composed of two layers: iron core and perovskite silicate mantle. The
results were obtained for different values of the Starobinsky parameter
$\beta$. We added a point representing mass and radius of Kepler-145b
exoplanet xie  to illustrate that the effect of modified gravity can be large
for bigger planets.

We present the results of numerical integration in the figures above and
below. Figures 1 and 2 show the mass-radius relation for small and bigger
planets, respectively, for various values of the Starobinsky parameter
$\beta$. Here, we use the parameter $\alpha=2c^{2}\kappa^{2}\beta$ to quantify
deviation from GR. We see that the bigger values of the parameter correspond
to bigger (and less dense) planets of the same mass. This can be also seen
when density profile, shown in Figure 4, is investigated. Here, we plotted the
profiles for planets of mass equal to the Earth’s one. It means that, for
different values of $\beta$, they will have various values of the central
density. It can be seen that there is a tendency toward smaller densities (and
hence pressures) in the interior of planets for positive values of $\beta$.
Also, it might seem surprising that the mass is the same in all three cases,
since the area under the curve corresponding to larger value of $\beta$ is
smaller than the other three. However, we must remember that the mass is not
simply obtained by integrating $\int_{0}^{R_{p}}4\pi r^{2}\rho dr$, but there
are also contributions from modifications of the theory (see the equation (14)
and detailed discussion in olek2  and delacruz ). One must notice that in the
Palatini gravity, the effects coming from modifications of geometry do not
contribute to the overall mass outside the sphere, as the density of matter
vanishes there and hence, the integrand in (14).

One can also ask why the best-fit curve for the planet Earth in the Figure 1
is the one for $\alpha=10^{-13}$, which seems to be in conflict with well-
established data about the planet. However, one must remember that our model
is greatly simplified, as it is usually assumed that Earth consists of six
layers at least, not of only two. On the other hand, it is possible to choose
such a core-mass fraction, that the Earth lies exactly on a curve representing
two-layer planets made of iron and perovskite silicate when no modification of
gravity is introduced (i.e. $\alpha=0$). We repeated the calculations for
different mass fractions and discovered that the best-fit curve is the one for
a planet whose iron core contributes only $10\%$ of its total mass (see Fig.
3). This finding is in agreement with the results obtained by Seager et al.
(see seager ).

It is also reasonable to ask whether omitting terms containing pressure in the
structural TOV equation (13) was justified, since the corrections coming from
modifications of gravity might be less important that the pressure. In order
to check that, we included the pressure in (13) and computed the density
profiles for a planet of mass equal to one Earth’s mass and compared the
results. As it turned out, there were no differences exceeding the
computational accuracy of our program.

Figure 3: [color online] Mass-radius relation for small planets with different
core-mass fractions when there is no modification of gravity taken into
account. The blue dot represents planet Earth. The calculations were carried
out assuming two-layer composition of the planets (iron core and perovskite
silicate mantle). Figure 4: [color online] Density profile for planets
composed of two layers. Different curves correspond to different values of the
Starobinsky parameter $\beta$.

##  IV Conclusions

In this paper, we have presented numerical solutions to modified Tolman-
Oppenheimer-Volkoff equations for non-homogeneous, Earth-like planets in the
Palatini $f(\mathcal{R})=\mathcal{R}+\beta\mathcal{R}^{2}$ gravity. As it
turns out, modifications of the Lagrangian influencing the high-energy limit
of the theory, have an impact on non-relativistic regime as well, leading to
alteration of interior structure of small, dense objects. This was shown by
writing the relevant equations - TOV and mass relation - in terms of conformal
invariants, allowing one to analyze the same equations for various theories
within one framework. Since the matter is non-relativistic, we neglected
pressure in the equations. The last missing piece was the equation of state
for the matter, which was used to relate pressure derivative with derivative
of the density. We chose a modified polytropic equation of state, following
seager , for iron and perovskite silicate, which are the main two compounds
building the Earth’s core and mantle. Numerical integration was performed for
a planet whose core mass fraction is 0.3, i.e. after reaching 30$\%$ of total
mass, the equation of state was changed from the one for iron to the one for
the silicate. As it turned out, the final result is very sensitive to the
choice of internal density. A small deviation from the physical density
resulted in an incorrect mass-to-radius ratio, therefore some datapoints were
excluded from our analysis. That fact did not influence the final result,
showing that for different values of the parameters $\beta$, planets of the
same mass will have different radii.

Figures 1 and 2 show that there exists some degeneracy due to the choice of
the theory of gravity one is analyzing. The same planet made of certain
compounds in the Newtonian theory of gravity might have a different
composition in a modified version of the theory, i.e. curves representing
different proportions of chemical compounds (or even different substances
building the planet in general) in different gravity theories might overlap,
so that it might be impossible to say what exactly their properties are and
whether they are, for example, capable of supporting life. This effect, while
minuscule for small planets, should be significant for bigger ones. For
example, for a rocky planet of a mass equal to approximately 37 Earth’s
masses, which is the case of a Kepler-145b exoplanet xie , this effect should
have an influence on our prediction of its internal structure, and also on
jovian-like planets, as the one discover recently in an external galaxy
stefano . We also remark that, comparing these new results to the ones
obtained in our previous paper olek2 , we see that due to addition of iron
core, the average density of planets increased and now the Earth is
significantly above the curve representing the unmodified theory. Now, the
best-fit curve would be for $\alpha=10^{-13}$, but this might be misleading
since the realistic model would assume more layers, made of lighter
substances.

Figure 4 tells us something about the measured structure itself. While we will
probably never be able to perform seismic measurements on distant exoplanets,
discovering to a bigger accuracy its internal constitution, we have at our
disposal some experimental data acquired here on Earth prem . We know, to a
satisfying accuracy, the internal structure of the Earth, i.e. what layers it
is composed of, what their thickness is, chemical composition of each of them,
and so on. On the other hand, modifications of gravity influence the thickness
of each layer, as it can be seen in Figure 4. Although our analysis was
carried out for a much simpler model than Earth, which is usually modelled
with at least six different layers, the point is clear: having accurate data
obtained from seismic experiments, we can constrain possible values of the
parameter $\beta$ (or other parameters for different theories of gravity) by
comparing the reference model of Earth to the one predicted by a new gravity
theory. One may argue that such corrections introduced by modified gravity
will be too small to be detected here on Earth. However this fact can allow to
put a constraint on the maximum absolute value of $\beta$, and as the
measurements get more accurate, the theory will be confirmed or rejected.

Acknowledgement. This work was supported by the EU through the European
Regional Development Fund CoE program TK133 "The Dark Side of the Universe".
AK is a beneficiary of the Dora Plus Program, organized by the University of
Tartu.

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# DV-Hop localization based on Distance Estimation using Multinode and Hop
Loss in WSNs

Penghong Wang, Xingtao Wang, Wenrui Li, Xiaopeng Fan, and Debin Zhao, This
work was supported in part by the National Key Research and Development
Program of China (2021YFF0900500), and the National Science Foundation of
China (NSFC) under grant 61972115.Penghong Wang, Xingtao Wang and Wenrui Li
are with the School of Computer Science, Harbin Institute of Technology,
Harbin 150001, China. e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>21B903007@stu.hit.edu.cn). Xiaopeng Fan and Debin Zhao are with the School of
Computer Science, Harbin Institute of Technology, Harbin 150001, China, and
also with the PengCheng Lab, Shenzhen 518055, China. e-mail<EMAIL_ADDRESS>dbzhao@hit.edu.cn). Corresponding author: Xiaopeng Fan.

###### Abstract

Location awareness is a critical issue in wireless sensor network
applications. For more accurate location estimation, the two issues should be
considered extensively: 1) how to sufficiently utilize the connection
information between multiple nodes and 2) how to select a suitable solution
from multiple solutions obtained by the Euclidean distance loss. In this
paper, a DV-Hop localization based on the distance estimation using multinode
(DEMN) and the hop loss in WSNs is proposed to address the two issues. In
DEMN, when multiple anchor nodes can detect an unknown node, the distance
expectation between the unknown node and an anchor node is calculated using
the cross domain information and is considered as the expected distance
between them, which narrows the search space. When minimizing the traditional
Euclidean distance loss, multiple solutions may exist. To select a suitable
solution, the hop loss is proposed, which minimizes the difference between the
real and its predicted hops. Finally, the Euclidean distance loss calculated
by the DEMN and the hop loss are embedded into the multi-objective
optimization algorithm. The experimental results show that the proposed method
gains 86.11% location accuracy in the randomly distributed network, which is
6.05% better than the DEM-DV-Hop, while DEMN and the hop loss can contribute
2.46% and 3.41%, respectively.

###### Index Terms:

Wireless sensor networks, DV-Hop, distance estimation using multinode, hop
loss, multi-objective optimization.

##  I Introduction

Wireless sensor networks (WSNs) have attracted much attention in many
fields[1, 2, 3], e. g. industry, agriculture, medical treatment, and military.
In such applications, the localization of sensor nodes is a crucial research
issue. Especially, WSNs are composed of a group of tiny sensor nodes equipped
with small batteries, which have the characteristics of low energy consumption
and low cost. This low-cost requirement limits many range-based sensor node
location methods [4, 5, 6, 7], as the range-based localization method relies
on the physical characteristics (such as signal strength or arrival time) of
the received signal to measure the distance between sensor nodes. This
localization methods increase energy consumption and localization cost of
sensors. On the contrary, the range-free localization methods [8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40] can obtain the locations of unknown
nodes without additional hardware.

Range-free localization schemes generally predict the location of unknown
nodes based on the connectivity among sensor nodes. Fig. 1 is an illustration
of WSN, which contains seven sensor nodes, of which three are anchor nodes
(i.e., nodes with a known location, recorded as $a_{1}$, $a_{2}$ and $a_{3}$)
and four are unknown nodes (i.e., nodes with an undisclosed location, recorded
as $u_{1},u_{2},u_{3}$ and $u_{4}$). The edge between two nodes indicates that
these two nodes are neighbors, i.e., they can communicate with each other and
vice versa. Based on the locations of $a_{1}$, $a_{2}$ and $a_{3}$ and the
hops to each unknown node ($u_{1},u_{2},u_{3}$ or $u_{4}$), the locations of
these unknown nodes can be predicted.

Figure 1: An illustration of WSNs. Anchor nodes ($a_{1},a_{2}$ and $a_{3}$)
and unknown nodes ($u_{1},u_{2},u_{3},$ and $u_{4}$).

Distance vector hop (DV-Hop) is a typical range-free localization algorithm
proposed by Niculescu and Nath [8]. Its positioning process includes three
stages. 1) Minimum hops statistics: record the minimum connection hops from an
anchor node to other nodes; 2) Distance estimation: calculate the distances
per hop of anchor nodes based on these geographical locations and the minimum
connection hops between them, and calculate the distance between an unknown
node and an anchor node based on the minimum connection hops and distance per
hop of anchor node; 3) Location prediction: predict the geographical location
of each unknown node based on the obtained information.

Most of the existing range-free localization works draw on the experience of
the DV-Hop scheme, such as [10, 11, 12, 13, 15, 17, 18, 19, 20, 16, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 41, 14, 36, 37, 40]
proposing various improvement strategies in the stage of distance calculation
and position prediction. Specifically, [12, 23, 24, 25, 26, 27, 31] use a
single-objective intelligent optimization algorithm for location prediction.
[20, 29, 32, 33, 34, 35, 40] develop a multi-objective optimization model to
predict the locations of unknown nodes. [15, 16, 28, 40, 29, 42] develop
distance estimation strategies between an unknown node and an anchor node.

In these works, the following two issues should be considered extensively: 1)
how to sufficiently utilize the connection information between multiple nodes
and 2) how to select a suitable solution from multiple solutions obtained by
Euclidean distance loss. To address the two issues, a DV-Hop localization
based on distance estimation using multinode (DEMN) and the hop loss in the
WSNs is proposed. Specifically, the DEMN strategy aims to leverage the
information constraints provided by multiple anchor nodes to narrow the
potential search space of an unknown node and improve the accuracy of
estimated distance. The hop loss can help the algorithm select a suitable
solution from the set of potential solutions obtained in iterative
calculations. These schemes offer a promising solution to enhance the
localization accuracy in WSNs. The main contributions can be summarized as
follows:

  1. 1)

We propose DEMN, which predicts the distance from an unknown node to an anchor
node using multinode information. Specifically, when multiple anchor nodes can
detect an unknown node, the distance expectation between the unknown node and
an anchor node is calculated using the cross domain information and is
considered as the expected distance between them, which narrows the search
space.

  2. 2)

We propose the hop loss, which minimizes the difference between the real and
its predicted hops. Hop loss enables us to choose the most suitable solution
as the output solution when there are multiple potential solutions.

  3. 3)

We embed the Euclidean distance loss and the hop loss into a multi-objective
genetic algorithm. The simulation results show that the proposed method gains
86.11% location acuracary in the randomly distributed network, which is 6.05%
better than the DEM-DV-Hop, while the DEMN and the hop loss can contribute
2.46% and 3.41%, respectively.

The rest of this paper is organized as follows: Section II reviews the related
works on the DV-Hop algorithm. Section III details the distance estimation
using multinode (DEMN). Section IV presents the hop loss. Section V introduces
the multi-objective optimization embedding the Euclidean distance loss
calculated by the DEMN and the hop loss. Section VI provides the simulation
results. Finally, the conclusion of this paper is summarized.

##  II Related Work

Most range-free localization algorithms need to obtain the necessary
information to predict the geographical locations of unknown nodes, such as
the locations of anchor nodes and the connectivity between nodes. The DV-Hop
algorithm is proposed by Niculescu et al. [8] provides a simple calculation
scheme, and its process is as follows. Firstly, beacon nodes disseminate data
packets and flood the entire network. Thereinto, the structure of the package
is {$ID,L_{i},Hop_{i}$}, including the ID, locations of anchor nodes, and
initial hop ($Hop_{i}=0$) information. When other nodes receive the packet,
they will update the $Hop_{i}$ in the package and save the copy to their
table. When the flooding process converges, each node has a table containing
the minimum hop count to other nodes. Then, based on the geographical location
of anchor nodes and the minimum number of hops between nodes, the average
distance ($Avg\\_DisHo{p_{i}}$) per hop of an anchor node is calculated by

$Avg\\_DisHo{p_{i}}=\frac{{\sum\limits_{j}{\sqrt{{{({x_{i}}-{x_{j}})}^{2}}+{{({y_{i}}-{y_{j}})}^{2}}}}}}{{\sum\limits_{j}{Ho{p_{i,j}}}}},$
(1)

where ${({x_{i}},{y_{i}})}$ and ${({x_{j}},{y_{j}})}$ represent the locations
of node ${a}_{i}$ and node ${a}_{j}$ respectively, ${Hop}_{i,j}$ denotes the
minimum hop count between ${a}_{j}$ and ${a}_{i}$. The estimated distance
between ${u}_{k}$ and ${a}_{i}$ is calculated by

${Dis}_{i,k}=Avg\\_DisHo{p_{i}}\cdot{Hop}_{i,k},$ (2)

where ${Hop}_{i,k}$ denotes the minimum hop count between ${u}_{k}$ and
${a}_{i}$. The least square method is used to predict the location of
${u}_{k}$.

Figure 2: Distance estimation using multinode. ${a}_{i}$ and ${a}_{j}$: two
distinct anchor nodes, ${u}_{k}$: an unknown node, ${o}_{l}$: an ordinary
node, $D_{1}$ and $D_{2}$: a part of the cross domain, $m$: the hop count,
$R$: communication radius, $d_{i,j}$: the distance between node ${a}_{i}$ and
node ${a}_{j}$.

To improve the localization accuracy of DV-Hop, scholars have developed
different improvement strategies. Specially, [12] models the location problem
as a minimum optimization problem for the estimation distance and embeds the
model into the intelligent optimization algorithm. [23, 24, 25, 26, 27, 31]
focused on improving the location prediction algorithm for unknown nodes. [26]
improves the global search ability of the algorithm by introducing
optimization of inertia weights and selection strategies of adaptive
functions. [31] proposes an adaptive strategy for crossover probability and
mutation probability to improve the convergence performance of genetic
algorithms. [15, 16, 28, 40, 29, 25] improve the accuracy of the estimated
distance by improving the calculation method of the distance between an
unknown node and an anchor node. For example, Cui et al.[25] believe that
estimating the distance between nodes based on discrete hops in DV-Hop can
result in significant errors. Therefore, when nodes follow a uniform
distribution, they propose to transform discrete hop into continuous hop
count. It can obtain more accurate hop count values among nodes, thereby
improving the accuracy of distance estimation. Wang et al. [40] proposed a
distance estimation model based on the distribution characteristics of nodes.
This model mathematically derives the farthest and average distance detected
by an anchor node under different hops, improving the predicted distance’s
accuracy.

In recent years, scholars have also developed some new localization schemes
for different localization scenarios, such as positioning sensors in irregular
areas [20, 34, 40] or sensors with obstacles [33, 41, 14]. For example, [41]
judges whether there is a severe detour in the connectivity of nodes through
the theoretical analysis of the minimum hops of nodes and geometric
constraints. Then, the geometric constraints of anchor pairs are used to
estimate the unknown node position. In addition, [33] introduces the
theoretical distance per hop of anchor nodes and propose a multi-objective
localization loss models, then embed these loss models into the multi-
objective optimization algorithm to predict the geographical location of each
unknown node. Specially, [22] takes the discrepancy between the predicted
connectivity of the nearest neighbor node and its basic connectivity as a
constraint to optimize the expected location.

For achieving more accurate location estimation in wireless sensor networks,
two critical issues need to be extensively considered: 1) how to effectively
leverage the connection information between multiple nodes and 2) how to
select the most suitable solution from multiple solutions obtained by the
Euclidean distance loss. To address these issues, this paper proposes a novel
approach called ”DV-Hop localization based on the DEMN and the hop loss” for
WSNs. The proposed method differs from the approach presented in [22] in the
following ways: 1) [22] constructs a constraint optimization problem based on
the real connectivity between any two adjacent nodes, while the hop loss
considers multi-hop information between nodes. 2) The DEMN technique narrows
the search space of unknown nodes and helps to obtain accurate estimated
distances. In summary, the proposed DV-Hop localization scheme, incorporating
DEMN and hop loss, addresses the key challenges of leveraging connection
information and selecting appropriate solutions. These differences from [22]
enable the proposed method to achieve more accurate and reliable location
estimation in WSNs.

##  III Distance Estimation using Multinode

In this section, we provide a detailed introduction to the design of DEMN. The
primary objective of the proposed DEMN strategy is to enhance the accuracy of
the estimated distance between an unknown node and an anchor node in WSNs. It
is achieved by utilizing the information constraints obtained from multiple
anchor nodes, which helps to reduce the potential search space for each
unknown node. As a result, the accuracy of distance estimation is
significantly improved. Fig. 2(a) illustrates the sensor nodes deployed in a
2D plane following a uniform distribution. In this figure, $a_{i}$ and $a_{j}$
represent two anchor nodes, while $u_{k}$ denotes an unknown node located in a
cross domain that can be detected by both node $a_{i}$ and node $a_{j}$. The
parameter $UB_{i,m}$ represents the upper bound detected by node $a_{i}$ when
the hop count is $m$. The calculation of $UB_{i,m}$ is based on the approach
described in [40]. Fig. 2(b) further illustrates an example analysis of
modeling these sensor nodes in a Cartesian coordinate system. This
representation allows for a better understanding and visualization of the
sensor node distribution and their relationships. By combining the insights
from Fig. 2(a) and (b), the DEMN strategy optimizes the distance estimation
process, taking into account the constraints imposed by multiple anchor nodes
and effectively narrowing down the search space for unknown nodes. The
relevant information is defined as follows:

Definition 1. Let $Circle(a_{0},r_{0})$ define a circle with the center of
$a_{0}$ and the radius of $r_{0}$, its function in Cartesian coordinate system
is expressed as $\Psi_{0}:x^{2}+y^{2}=r_{0}^{2}$.

Definition 2. Let $a_{i}$ define the origin and has $(0,0)$ as coordinates.
The coordinates of $a_{i}$ are $(d_{i,j},0)$.

Based on the above definition, $Circle(a_{i},{UB}_{i,m})$ can be represented
as $\Psi_{1}:x^{2}+y^{2}={{UB}_{i,m}}^{2}$, and $Circle(a_{j},R)$ can be
represented as $\Psi_{2}:{(x-d_{i,j})}^{2}+y^{2}={R}^{2}$, where $x$ and $y$
are two variables. The coordinate
$(\frac{{UB_{i,m}}^{2}{+d}^{2}-R^{2}}{2d_{i,j}},0)$ of $o_{l}$ is obtained by
combining $\Psi_{1}$ and $\Psi_{2}$. The expected distance between $u_{k}$ in
the cross domain and $a_{i}$ is considered as the expected (estimated)
distance between them, and it is calculated by

$\begin{aligned}
{E\\_dis}_{i,k}&=\frac{\int_{d_{i,j}-R}^{\frac{{UB_{i,m}}^{2}{+d}^{2}-R^{2}}{2d_{i,j}}}{\int_{0}^{\sqrt{R^{2}-{(x-d_{i,j})}^{2}}}\sqrt{x^{2}+y^{2}}dydx}}{{D_{1}}}\\\
&+\frac{\int_{\frac{{UB_{i,m}}^{2}{+d_{i,j}}^{2}-R^{2}}{2d_{i,j}}}^{UB_{i,m}}{\int_{0}^{\sqrt{{UB_{i,m}}^{2}-x^{2}}}\sqrt{x^{2}+y^{2}}dydx}}{{D_{2}}}\end{aligned},$
(3)

where $d_{i,j}$ denotes the distance between node $a_{i}$ and $a_{j}$.
${D_{1}}$ and ${D_{2}}$ respectively refer to the area of the corresponding
regin, and they are calculated by

$\begin{aligned}
{D_{1}}=\int_{d_{i,j}-R}^{\frac{{UB_{i,m}}^{2}{+d}^{2}-R^{2}}{2d_{i,j}}}{\sqrt{R^{2}-{(x-d_{i,j})}^{2}}dx}\\\
\end{aligned},$ (4)

$\begin{aligned}
{{D_{2}}}=\int_{\frac{{UB_{i,m}}^{2}{+d_{i,j}}^{2}-R^{2}}{2d_{i,j}}}^{UB_{i,m}}{\sqrt{{UB_{i,m}}^{2}-x^{2}}dx}\\\
\end{aligned}.$ (5)

In Eq. (3), we set $m$ to be less than 3. When $m\geq 3$, the expected
distance follows the calculation method proposed by [40].

Algorithm 1 Calculation of the ${E\\_dis}_{i,k}$.

Input: $d_{i,j}$: the distance between node $a_{i}$ and node $a_{j}$

$hop_{i,k}$ and $hop_{i,k}$

Output: ${E\\_dis}_{i,k}$

1:Calculate $d_{i,j}$ between node $a_{i}$ and $a_{j}$.

2:for $k=1\to N_{u}$ do

3: for $i=1\to N_{a},j=1\to N_{a}$ do

4: if $hop_{i,k}=m$ & $hop_{j,k}=1$ then

5: ${{D_{1}}}\leftarrow Eq.(\ref{SD1})$

6: ${{D_{2}}}\leftarrow Eq.(\ref{SD2})$

7: ${E\\_dis}_{i,k}\leftarrow Eq.(\ref{Expdis})$

8: end

9: end

10:end

The proposed DEMN strategy exhibits significant differences and advantages
compared to the distance estimation model introduced in [40]. The DEMN
strategy is based on multiple anchor nodes to narrow the potential search
space of an unknown node and enhance the accuracy of estimated distance. As
depicted in Fig. 2(b), assuming $m=1$, the previous work [40] defines the
potential search space of an unknown node as the semicircular region $D$
covered by $Circle(a_{i},UB_{i,1})$. In contrast, the proposed DEMN defines it
as the region $D_{1}+D_{2}$. This is a crucial improvement that significantly
narrows the potential search space for an unknown node.

Algorithm 1 provides the implementation details of the ${E\\_dis}_{i,k}$.
After obtaining the ${E\\_dis}_{i,k}$, we used it to construct a Euclidean
distance loss. The traditional Euclidean distance loss is expressed as

$\min\;{f_{1}}{({x_{k}},{y_{k}})}=\sum\limits_{i=1}^{N_{a}}{(\sqrt{{{({x_{i}}-{x_{k}})}^{2}}+{{({y_{i}}-{y_{k}})}^{2}}}-Dis{{}_{i,k}})^{2}},$
(6)

where ${({x_{i}},{y_{i}})}$ and ${({x_{k}},{y_{k}})}$ denote the locations of
node ${a}_{i}$ and node ${u}_{k}$ respectively, ${Dis}_{i,k}$ represents the
estimated distance between node ${u}_{k}$ and node ${a}_{i}$. In this paper,
${Dis}_{i,k}$ refers to ${E\\_dis}_{i,k}$ and it is calculated by Eq.(3). To
reduce computational costs, the DEMN only considers the scenario where an
unknown node is detected by two anchor nodes and the hop counts are
$hop_{i,k}<3$ and $hop_{j,k}=1$ respectively.

##  IV Hop Loss

Minimizing the Euclidean distance loss may obtain multiple equivalent
solutions. Assuming there are two anchor nodes ${a}_{i}$ and ${a}_{j}$, as
well as an unknown node ${u}_{k}$, as shown in Fig. 3. Where ${u}^{gt}_{k}$
denotes the ground truth locations of node ${u}_{k}$, $E\\_dis_{i,k}$ and
$E\\_dis_{j,k}$ are the predicted distances from node ${u}_{k}$ to two anchor
nodes, respectively. When the Euclidean distance loss is minimal, i.e.
$f_{1}({x_{k}},{y_{k}})=0$, we obtain two predicted locations
${u}^{{pred}_{1}}_{k}$ and ${u}^{{pred}_{2}}_{k}$. As shown in Fig. 3,
${u}^{{pred}_{2}}_{k}$ is more suitable than ${u}^{{pred}_{1}}_{k}$. In this
case, Euclidean distance loss is ineffective in selecting a suitable solution,
however, randomly selecting one of the solutions may cause a serious error. To
address this issue, this section proposes the hop loss function, which can
choose a more suitable solution by minimizing the difference between the real
hop and its predicted hop.

Figure 3: Design principle of hop loss. ${u}^{{pred}_{1}}_{k}$ and
${u}^{{pred}_{2}}_{k}$: two predicted locations of node ${u}_{k}$,
${u}^{gt}_{l}$: the ground truth location of node ${u}_{l}$, ${a}_{i}$ and
${a}_{j}$: two distinct anchor nodes.

Figure 4: An example of the real and its predicted hops. ${Hop}^{real}$: the
real hops, ${Hop}^{pred}$: the predicted hops.

Fig. 4 illustrates an example of the real and its predicted hops. In the
figure, the left part represents the real hop (${Hop}_{i,k}^{real}$) between
nodes, which is obtained by the flooding broadcast of packets[8]. The right
part is the predicted hop (${Hop}_{i,k}^{pred}$) between nodes, which is
calculated based on the predicted locations of nodes, as shown in Algorithm 2.
The smaller the difference between ${Hop}_{i,k}^{pred}$ and
${Hop}_{i,k}^{real}$, the more closer ${Hop}_{i,k}^{pred}$ to
${Hop}_{i,k}^{real}$, which means the chosen solution is more accurate. Based
on the above analysis, the hop loss is proposed as

$\begin{aligned}
\min\;f_{2}({Hop}_{i,k}^{pred})=&\sum_{k=i}^{N}{\sum_{i=1}^{N}{({Hop}_{i,k}^{real}-{Hop}_{i,k}^{pred})}^{2}}\\\
&,\forall{Hop}_{i,k}^{real}<3\\\ \end{aligned}.$ (7)

Algorithm 2 Calculation of the ${Hop}_{i,k}^{pred}$ .

Input: Anchor node locations, $R$ and the expected locations of unknown nodes

Output: ${Hop}_{i,k}^{pred}$

1:for $i=1\to N,k=i\to N$ do

2: Calculate the $Euc\\_Dis_{i,k}^{pred}$ between node $o_{i}$ and $o_{k}$

3: if $Euc\\_Dis_{i,k}^{pred}<R$ then

4: Node $o_{i}$ and $o_{k}$ are adjacent nodes

5: Record their indexes [$row,col$]

6: end

7:end

8:$G\leftarrow graph(col,row)$

9:${Hop}_{i,k}^{pred}\leftarrow BFsearch(G)$

Algorithm 2 provides the implementation details of the ${Hop}_{i,k}^{pred}$,
where $G$ denotes the undirected graph, $o_{i}$ and $o_{k}$ denote the
ordinary sensor nodes, $Euc\\_Dis_{i,k}^{pred}$ denotes the predicted
Euclidean distance between node $o_{k}$ and node $o_{i}$. Firstly, we combine
the predicted locations of unknown nodes obtained from each iteration of the
algorithm with the anchor nodes locations to calculate
$Euc\\_Dis_{i,k}^{pred}$. Secondly, record adjacent node pairs by comparing
$Euc\\_Dis_{i,k}^{pred}$ and communication radius $R$. Thirdly, these adjacent
nodes are constructed into an undirected graph. Finally, the
${Hop}_{i,k}^{pred}$ in the graph is obtained by the breadth-first search
function, i.e., $BFsearch(G)$.

##  V Multi-Objective Optimization

In this section, we embed the Euclidean distance loss calculated by DEMN and
the hop loss into the multi-objective genetic algorithm (NSGI-II [43]) for
iterative optimization. It is emphasized that the main contributions of this
paper are as follows: 1) The DEMN strategy. The proposed approach effectively
utilizes the constraint information provided by multiple anchor nodes to
narrow down the potential search space of an unknown node. This constraint
information enhances the accuracy of predicting the distance between an
unknown node and an anchor node, leading to more reliable localization. 2) The
hop loss. The incorporation of hop loss in the optimization process enhances
the ability to select suitable solutions and make the localization results
more accurate and robust. The multi-objective genetic algorithm serves as a
necessary tool to solve the locations of unknown nodes. Algorithm 3 outlines
the pipeline of a multi-objective genetic algorithm. Where more details about
crossover, mutation and selection operation, non-dominated sorting, and
crowding distance calculation are following [43].

Algorithm 3 Multi-objective genetic algorithm.

Input: Node set: includes anchor nodes and unknown nodes

Output: The locations of unknown nodes

1:Initialize parameters with Table I and initial population.

2:${E\\_dis}_{i,k}$ $\leftarrow$ Algorithm 1

3:while $iter<MIter$ do

4: Perform crossover and mutation operations based on $Pc$ and $Pm$.

5: ${Hop}_{i,\ j}^{pred}$ $\leftarrow$ Algorithm 2

6: $f_{1}$ $\leftarrow$ Eq.(6)

7: $f_{2}$ $\leftarrow$ Eq.(7).

8: Perform non-dominated sorting and crowding distance calculation based on
the values of $f_{1}$ and $f_{2}$.

9: Select population individuals based on sorting results.

10: $iter=iter+1$

11:end

After finishing the iterations, we obtain a set of solutions {$Indl_{1}$, …,
$Indl_{i}$, …, $Indl_{Ps}$}. $Indl_{i}$ denotes the $i^{th}$ individual in the
population, it contains the predicted locations of all unknown nodes. Based on
the predicted locations of the unknown nodes represented by $Indl_{i}$, we can
obtain the values of two Loss function $f_{1}(Indl_{i})$ and
$f_{2}(Indl_{i})$. Especially, we set the individual with the minimum hop loss
value (i.e., min $f_{2}(Indl_{i})$) as the final output solution of the
algorithm.

In summary, the integration of the Euclidean distance loss, hop loss, and the
multi-objective genetic algorithm enables the proposed approach to iteratively
optimize the localization of unknown nodes in the WSNs. The contributions of
this paper lie in the effective utilization of constraint information provided
by multiple anchor nodes, the introduction of hop loss, and the utilization of
the multi-objective genetic algorithm to achieve more accurate and reliable
localization results.

##  VI Simulation And Analysis

In this section, we provide the experimental results and the comparison with
the state-of-the-art results, including DV-Hop [8], CC-DV-Hop[22], OCS-DV-
Hop[23], CCS-DV-Hop[30], IAGA-DV-Hop[31], NSGAII-DV-Hop[33] and DEM-DV-
Hop[40].

###  VI-A Experimental Setup

Datasets. We choose four types of node distribution data for localization
performance evaluation. As shown in Fig. 6, the four kinds of sensor test
networks are random networks and C-, O- and X-shaped networks. Each test
network contains the true locations of 100 sensor nodes distributed in a
100m×100m region. These nodes’ first $5\%-30\%$ are the anchor node locations,
and the rest are unknown nodes’ ground truth (gt) locations.

TABLE I: Simulation parameters.

Parameters |  Value |  Parameters |  Value
---|---|---|---
$Pc$ |  0.9 |  $Pm$ |  0.1
$N_{u}$ |  70-95 |  $R$ (m) |  25-40
Population size |  20 |  $MIter$ |  500
Independent repeat test |  50 |  – |  –

*$Pc$: crossover probability, $Pm$: mutation probability, $N_{u}$: the number of unknown nodes, $MIter$: maximum iterations, independent repeat test: the number of independent experiments; population size: the number of individuals in the population.

Parameters. The detailed experimental parameters of the algorithm are listed
in Table I. The parameter settings employed in this study are consistent with
those utilized in the algorithms proposed by Cai et al.[33] and Wang et
al.[40].

Figure 5: Data sets of different shape test networks. “$\star$”: an anchor node, “$\bullet$”: an unknown node.  TABLE II: The $ALEs$ of the randomly distributed network in the experiment. ${N}_{a}$ | 5 | 10 | 15
---|---|---|---
$R$ (m) | 25 | 30 | 35 | 40 | 25 | 30 | 35 | 40 | 25 | 30 | 35 | 40
DV-Hop[8] | 80.12 | 195.27 | 142.59 | 122.09 | 38.41 | 32.75 | 33.08 | 30.55 | 28.65 | 29.06 | 32.86 | 27.09
CC-DV-Hop[22] | 40.61 | 122.80 | 83.35 | 85.22 | 34.31 | 30.53 | 29.14 | 27.07 | 23.67 | 27.09 | 28.89 | 23.47
OCS-DV-Hop[23] | 50.02 | 71.04 | 76.08 | 76.52 | 31.65 | 26.47 | 30.96 | 26.82 | 24.36 | 21.70 | 27.04 | 23.16
CCS-DV-Hop[30] | 50.17 | 69.12 | 73.54 | 76.49 | 31.65 | 25.51 | 29.34 | 25.60 | 24.35 | 21.28 | 26.40 | 22.82
IAGA-DV-Hop[31] | 102.37 | 92.81 | 82.74 | 75.33 | 30.34 | 27.35 | 29.90 | 26.01 | 24.72 | 21.41 | 25.83 | 21.06
NSGAII-DV-Hop[33] | 91.05 | 86.90 | 71.29 | 63.36 | 30.14 | 26.85 | 29.27 | 25.60 | 23.68 | 22.10 | 25.32 | 20.99
DEM-DV-Hop[40] | 85.00 | 82.20 | 70.03 | 59.64 | 27.54 | 23.72 | 26.56 | 24.12 | 20.77 | 19.92 | 22.30 | 19.30
Ours (DEMN+Hop loss) | 80.82 | 76.98 | 59.77 | 43.95 | 19.59 | 16.60 | 17.77 | 14.53 | 16.49 | 15.16 | 15.00 | 12.31
Ours ($MIter=300$) | 87.67 | 82.23 | 62.67 | 46.60 | 22.24 | 18.42 | 18.81 | 15.46 | 17.76 | 16.43 | 15.87 | 13.03
Ours ($MIter=400$) | 83.03 | 78.67 | 60.89 | 44.76 | 20.36 | 17.18 | 18.14 | 14.91 | 16.84 | 15.49 | 15.22 | 12.55
DEMN | 84.18 | 80.50 | 68.10 | 58.24 | 24.57 | 20.82 | 24.18 | 22.03 | 18.15 | 17.61 | 19.61 | 16.49
Hop loss | 86.05 | 80.59 | 58.65 | 45.49 | 23.78 | 18.79 | 20.35 | 17.00 | 18.52 | 17.09 | 19.53 | 16.05
${N}_{a}$ | 20 | 25 | 30
$R$ (m) | 25 | 30 | 35 | 40 | 25 | 30 | 35 | 40 | 25 | 30 | 35 | 40
DV-Hop[8] | 31.98 | 29.12 | 28.19 | 26.16 | 27.77 | 27.36 | 26.38 | 26.10 | 33.78 | 29.63 | 30.87 | 26.42
CC-DV-Hop[22] | 26.71 | 24.37 | 23.73 | 22.48 | 22.41 | 23.61 | 21.45 | 22.98 | 27.65 | 26.78 | 25.33 | 20.83
OCS-DV-Hop[23] | 24.40 | 21.91 | 20.94 | 22.62 | 20.42 | 18.75 | 19.85 | 19.02 | 21.89 | 18.65 | 22.27 | 19.26
CCS-DV-Hop[30] | 24.25 | 22.10 | 20.99 | 22.13 | 21.23 | 19.10 | 20.16 | 19.13 | 22.29 | 19.09 | 21.85 | 19.36
IAGA-DV-Hop[31] | 23.29 | 20.22 | 21.35 | 20.39 | 20.38 | 18.60 | 19.47 | 18.77 | 21.34 | 18.27 | 19.64 | 18.50
NSGAII-DV-Hop[33] | 22.70 | 21.39 | 21.06 | 20.10 | 19.92 | 18.94 | 19.40 | 18.34 | 20.48 | 18.66 | 19.63 | 18.30
DEM-DV-Hop[40] | 18.59 | 19.49 | 19.26 | 17.78 | 18.24 | 17.01 | 18.32 | 16.87 | 18.21 | 16.78 | 17.59 | 16.41
Ours (DEMN+Hop loss) | 16.03 | 13.84 | 13.03 | 11.13 | 14.01 | 12.29 | 11.83 | 10.36 | 14.02 | 12.45 | 11.45 | 9.95
Ours ($MIter=300$) | 16.95 | 14.56 | 13.60 | 11.92 | 14.52 | 12.95 | 12.37 | 11.16 | 14.51 | 12.86 | 11.99 | 10.62
Ours ($MIter=400$) | 16.25 | 14.07 | 13.20 | 11.42 | 14.13 | 12.44 | 11.99 | 10.64 | 14.15 | 12.62 | 11.63 | 10.21
DEMN | 17.66 | 16.81 | 16.61 | 15.41 | 15.73 | 14.90 | 15.26 | 14.57 | 15.85 | 14.48 | 14.87 | 14.05
Hop loss | 17.59 | 15.26 | 15.91 | 15.36 | 15.45 | 13.44 | 14.78 | 14.01 | 15.48 | 13.58 | 14.73 | 13.80
TABLE III: Overall performance on randomly distributed networks.

Algorithms |  $ALA$ (%) |  $APG$ (%) | TC (s)
---|---|---|---
DV-Hop[8] | 70.19 | 15.92 $\uparrow$ | 0.015
OCS-DV-Hop[23] | 76.89 | 9.22 $\uparrow$ | 13.76
NSGAII-DV-Hop[33] | 77.86 | 8.25 $\uparrow$ | 10.95
CC-DV-Hop[22] | 74.37 | 11.75 $\uparrow$ | 197.58
CCS-DV-Hop[30] | 77.36 | 8.75 $\uparrow$ | 15.09
IAGA-DV-Hop[31] | 77.66 | 8.45 $\uparrow$ | 12.82
DEM-DV-Hop[40] | 80.06 | 6.05 $\uparrow$ | 11.23
Ours | 86.11 | 0.00 | 19
Ours ($MIter=300$) | 85.20 | 0.91 $\uparrow$ | 11.4
Ours ($MIter=400$) | 85.83 | 0.28 $\uparrow$ | 15.2
DEMN | 82.52 | 3.59 $\uparrow$ | 11.2
Hop loss | 83.47 | 2.64 $\uparrow$ | 19

*$ALA$: average localization accuracy; TC: time consumption.

TABLE IV: The $ALEs$ of the C-shaped network in the experiments ${N}_{a}$ | 5 | 10 | 15
---|---|---|---
$R$ (m) | 25 | 30 | 35 | 40 | 25 | 30 | 35 | 40 | 25 | 30 | 35 | 40
DV-Hop[8] | 88.45 | 120.14 | 104.47 | 90.64 | 66.83 | 93.20 | 81.80 | 66.76 | 67.79 | 91.93 | 76.57 | 70.52
CC-DV-Hop[22] | 83.98 | 73.33 | 64.06 | 63.87 | 52.25 | 47.20 | 46.30 | 38.77 | 52.76 | 50.12 | 43.18 | 45.59
OCS-DV-Hop[23] | 47.77 | 40.02 | 37.57 | 36.71 | 42.26 | 35.14 | 35.29 | 35.48 | 45.67 | 36.35 | 34.05 | 33.46
CCS-DV-Hop[30] | 50.09 | 41.14 | 38.41 | 34.29 | 42.97 | 32.88 | 33.34 | 33.02 | 44.90 | 36.29 | 33.24 | 32.96
IAGA-DV-Hop[31] | 116.33 | 103.63 | 86.13 | 78.20 | 41.76 | 32.55 | 32.97 | 31.71 | 42.97 | 34.98 | 32.95 | 33.11
NSGAII-DV-Hop[33] | 69.82 | 58.85 | 58.03 | 54.38 | 39.33 | 31.76 | 28.24 | 28.91 | 40.23 | 31.26 | 27.23 | 28.72
DEM-DV-Hop[40] | 62.34 | 53.11 | 56.83 | 51.81 | 40.21 | 31.55 | 27.69 | 27.37 | 38.44 | 30.04 | 26.35 | 26.99
Ours | 57.69 | 43.35 | 35.78 | 33.37 | 34.55 | 27.32 | 21.63 | 21.21 | 33.99 | 27.64 | 22.86 | 21.95
${N}_{a}$ | 20 | 25 | 30
$R$ (m) | 25 | 30 | 35 | 40 | 25 | 30 | 35 | 40 | 25 | 30 | 35 | 40
DV-Hop[8] | 61.31 | 68.12 | 59.57 | 52.60 | 60.91 | 59.27 | 53.28 | 46.08 | 63.28 | 53.51 | 49.10 | 42.31
CC-DV-Hop[22] | 43.91 | 38.03 | 41.07 | 37.84 | 44.02 | 39.50 | 40.02 | 34.51 | 44.88 | 40.87 | 42.16 | 35.14
OCS-DV-Hop[23] | 42.61 | 35.09 | 35.85 | 32.45 | 54.28 | 39.17 | 40.97 | 36.02 | 53.55 | 40.38 | 42.64 | 37.98
CCS-DV-Hop[30] | 43.65 | 34.97 | 35.51 | 32.39 | 50.81 | 37.29 | 39.86 | 34.16 | 53.97 | 38.72 | 42.37 | 37.79
IAGA-DV-Hop[31] | 35.59 | 29.70 | 29.84 | 28.84 | 39.66 | 31.05 | 31.81 | 29.17 | 38.63 | 30.95 | 31.99 | 28.93
NSGAII-DV-Hop[33] | 36.04 | 30.44 | 27.23 | 27.17 | 38.23 | 30.83 | 28.11 | 27.21 | 37.17 | 31.52 | 27.87 | 26.32
DEM-DV-Hop[40] | 35.44 | 29.32 | 25.34 | 26.04 | 37.23 | 29.52 | 25.96 | 25.89 | 35.98 | 29.60 | 25.84 | 25.23
Ours | 31.00 | 25.54 | 21.31 | 20.60 | 30.96 | 24.98 | 21.09 | 20.04 | 29.38 | 25.02 | 20.87 | 19.21
TABLE V: The $ALEs$ of the O-shaped network in the experiments ${N}_{a}$ | 5 | 10 | 15
---|---|---|---
$R$ (m) | 25 | 30 | 35 | 40 | 25 | 30 | 35 | 40 | 25 | 30 | 35 | 40
DV-Hop[8] | 84.04 | 152.28 | 121.71 | 100.22 | 50.68 | 136.31 | 109.29 | 92.96 | 53.49 | 101.91 | 80.13 | 68.32
CC-DV-Hop[22] | 41.56 | 35.11 | 46.24 | 36.64 | 42.70 | 27.78 | 24.31 | 34.78 | 38.49 | 28.15 | 21.56 | 26.27
OCS-DV-Hop[23] | 43.29 | 46.62 | 43.39 | 42.82 | 37.64 | 34.66 | 31.86 | 28.95 | 38.10 | 30.04 | 26.93 | 24.01
CCS-DV-Hop[30] | 41.80 | 45.16 | 43.57 | 42.69 | 37.74 | 33.09 | 30.36 | 27.84 | 38.07 | 30.02 | 27.32 | 23.97
IAGA-DV-Hop[31] | 62.26 | 61.53 | 56.40 | 51.91 | 34.49 | 30.54 | 26.55 | 29.53 | 30.79 | 24.50 | 21.08 | 21.77
NSGAII-DV-Hop[33] | 55.11 | 59.35 | 46.04 | 47.87 | 35.77 | 27.84 | 24.62 | 27.98 | 32.53 | 22.33 | 20.59 | 20.24
DEM-DV-Hop[40] | 50.72 | 52.84 | 42.33 | 44.42 | 33.02 | 24.85 | 21.96 | 22.57 | 32.57 | 21.42 | 18.70 | 17.49
Ours | 75.45 | 56.14 | 40.80 | 37.33 | 32.43 | 21.29 | 18.34 | 17.14 | 30.67 | 19.81 | 16.93 | 14.28
${N}_{a}$ | 20 | 25 | 30
$R$ (m) | 25 | 30 | 35 | 40 | 25 | 30 | 35 | 40 | 25 | 30 | 35 | 40
DV-Hop[8] | 47.14 | 92.79 | 73.85 | 64.74 | 48.10 | 83.41 | 67.19 | 57.83 | 48.06 | 47.54 | 44.59 | 37.29
CC-DV-Hop[22] | 37.39 | 27.11 | 24.26 | 26.05 | 38.29 | 26.88 | 21.37 | 23.35 | 38.77 | 24.99 | 27.43 | 24.28
OCS-DV-Hop[23] | 40.07 | 34.53 | 28.52 | 24.77 | 46.46 | 35.16 | 30.84 | 24.87 | 48.43 | 38.93 | 32.48 | 28.96
CCS-DV-Hop[30] | 40.23 | 33.09 | 27.53 | 23.95 | 46.32 | 34.70 | 30.14 | 24.97 | 50.08 | 36.90 | 32.00 | 26.80
IAGA-DV-Hop[31] | 29.90 | 23.26 | 20.35 | 19.15 | 32.32 | 22.45 | 20.43 | 18.94 | 31.53 | 22.25 | 21.01 | 19.00
NSGAII-DV-Hop[33] | 30.11 | 21.33 | 18.92 | 17.37 | 31.92 | 21.22 | 19.10 | 17.83 | 30.26 | 20.57 | 19.76 | 17.94
DEM-DV-Hop[40] | 29.25 | 20.87 | 17.24 | 14.92 | 30.73 | 21.03 | 16.83 | 14.51 | 28.90 | 20.09 | 17.06 | 15.16
Ours | 28.16 | 18.71 | 15.60 | 12.93 | 27.63 | 17.91 | 14.98 | 12.34 | 25.24 | 17.18 | 14.95 | 12.41
TABLE VI: The $ALEs$ of the X-shaped network in the experiments ${N}_{a}$ | 5 | 10 | 15
---|---|---|---
$R$ (m) | 25 | 30 | 35 | 40 | 25 | 30 | 35 | 40 | 25 | 30 | 35 | 40
DV-Hop[8] | 58.46 | 108.88 | 95.24 | 91.84 | 60.38 | 96.73 | 86.75 | 83.64 | 51.12 | 81.10 | 70.13 | 72.57
CC-DV-Hop[22] | 52.30 | 49.44 | 49.42 | 45.36 | 55.27 | 46.77 | 45.76 | 40.71 | 43.97 | 38.37 | 35.97 | 36.14
OCS-DV-Hop[23] | 42.85 | 49.13 | 47.49 | 38.15 | 39.75 | 40.24 | 35.80 | 32.52 | 42.18 | 41.91 | 37.48 | 35.86
CCS-DV-Hop[30] | 42.69 | 48.68 | 45.04 | 34.45 | 40.18 | 40.08 | 34.36 | 30.64 | 38.37 | 38.90 | 33.59 | 32.61
IAGA-DV-Hop[31] | 47.46 | 49.47 | 43.92 | 40.17 | 33.99 | 35.08 | 31.10 | 30.07 | 30.83 | 29.51 | 26.14 | 27.25
NSGAII-DV-Hop[33] | 42.37 | 44.45 | 38.87 | 34.70 | 33.27 | 31.89 | 27.84 | 27.18 | 29.96 | 28.18 | 23.68 | 24.20
DEM-DV-Hop[40] | 40.43 | 42.36 | 37.50 | 33.24 | 32.95 | 30.90 | 26.03 | 25.14 | 29.39 | 27.26 | 21.81 | 21.27
Ours | 32.46 | 33.69 | 30.59 | 25.42 | 28.13 | 23.95 | 19.30 | 18.38 | 26.36 | 23.07 | 17.76 | 17.24
${N}_{a}$ | 20 | 25 | 30
$R$ (m) | 25 | 30 | 35 | 40 | 25 | 30 | 35 | 40 | 25 | 30 | 35 | 40
DV-Hop[8] | 49.07 | 78.38 | 68.07 | 67.90 | 52.52 | 69.58 | 61.29 | 56.98 | 54.51 | 56.82 | 49.58 | 47.62
CC-DV-Hop[22] | 42.36 | 40.13 | 36.76 | 38.06 | 45.50 | 44.27 | 40.72 | 39.52 | 45.83 | 44.09 | 39.54 | 39.61
OCS-DV-Hop[23] | 43.51 | 45.41 | 37.52 | 36.84 | 45.50 | 45.87 | 39.09 | 42.20 | 45.51 | 49.90 | 42.38 | 45.61
CCS-DV-Hop[30] | 42.11 | 43.44 | 35.62 | 36.04 | 46.39 | 46.22 | 38.32 | 40.86 | 44.27 | 48.19 | 41.28 | 44.55
IAGA-DV-Hop[31] | 30.42 | 29.60 | 27.35 | 27.01 | 29.89 | 28.34 | 26.35 | 26.90 | 26.34 | 26.89 | 25.70 | 25.96
NSGAII-DV-Hop[33] | 27.94 | 26.55 | 23.04 | 23.18 | 28.38 | 26.49 | 22.24 | 23.18 | 25.18 | 24.32 | 20.91 | 20.91
DEM-DV-Hop[40] | 27.70 | 25.03 | 21.65 | 20.96 | 28.10 | 25.47 | 20.23 | 20.52 | 25.05 | 23.51 | 18.81 | 18.02
Ours | 23.85 | 20.91 | 17.38 | 17.06 | 24.65 | 21.15 | 16.51 | 16.91 | 22.82 | 19.68 | 14.47 | 15.07
TABLE VII: A comprehensive result comparison.  Node distribution type | C-shaped networks | O-shaped networks | X-shaped networks |
---|---|---|---|---
Algorithms |  $ALA$ (%) |  $APG$ (%) |  $ALA$ (%) |  $APG$ (%) |  $ALA$ (%) |  $APG$ (%) |
OCS-DV-Hop[23] | 57.10 | 17.84 $\uparrow$ | 70.79 | 9.76 $\uparrow$ | 58.03 | 21.74 $\uparrow$ |
NSGAII-DV-Hop[33] | 60.57 | 14.37 $\uparrow$ | 66.69 | 13.86 $\uparrow$ | 58.75 | 21.02 $\uparrow$ |
CC-DV-Hop[22] | 61.45 | 13.49 $\uparrow$ | 67.24 | 13.31 $\uparrow$ | 60.20 | 19.57 $\uparrow$ |
CCS-DV-Hop[30] | 66.54 | 8.40 $\uparrow$ | 75.01 | 5.54 $\uparrow$ | 71.26 | 8.51 $\uparrow$ |
IAGA-DV-Hop[31] | 68.81 | 6.13 $\uparrow$ | 76.09 | 4.46 $\uparrow$ | 74.07 | 5.70 $\uparrow$ |
DEM-DV-Hop[40] | 70.00 | 4.94 $\uparrow$ | 78.04 | 2.51 $\uparrow$ | 75.51 | 4.26 $\uparrow$ |
Ours | 74.94 | 0.00 | 80.55 | 0.00 | 79.77 | 0.00 |

Figure 6: The $ALA$ of different algorithms. (a): randomly distributed
network; (b): C-shaped network; (c): O-shaped network; (d): X-shaped network.
TS: Telecommunication Systems; JPDC: Journal of Parallel and Distributed
Computing; JSEN: IEEE Sensors Journal; TVT: IEEE Transactions on Vehicular
Technology; TMC: IEEE Transactions on Mobile Computing; NC: Neurocomputing.

Figure 7: The $95\%$ confidence interval for $ALEs$ samples. (a): randomly
distributed network; (b): C-shaped network; (c): O-shaped network; (d):
X-shaped network.

Evaluation Metrics. The normalized average localization errors ($ALEs$)
represent the most commonly used metric for evaluating localization, while the
average performance gain ($APG$) indicates the improvement of the proposed
model in comparison to other algorithms. In this paper, we employ both to
assess our model’s reliability and robustness. The $ALEs$ are calculated by

$ALEs=\frac{100\%}{{N_{u}\cdot
R}}\sum\limits_{k=1}^{N_{u}}{\sqrt{{{(x_{k}^{gt}-{x_{k}})}^{2}}+{{(y_{k}^{gt}-{y_{k}})}^{2}}}},$
(8)

where $(x_{k},y_{k})$ and $(x_{k}^{gt},y_{k}^{gt})$ represent the expected
coordinates and ground truth coordinates of an $u_{k}$ respectively. And the
$APG$ is calculated by

$APG={\rm mean}\sum{({ALEs}_{other}-{ALEs}_{our})},$ (9)

where ${ALEs}_{other}$ represent the $ALEs$ of other algorithms used for
comparison in this paper. In addition, we provide confidence interval (where,
the confidence probability is $95\%$) analysis for $ALEs$. Assuming $ALEs$
follow a Gaussian distribution, the confidence interval is calculated by

$[\bar{X}-\frac{S}{\sqrt{n}}t_{{\frac{\alpha}{2}}}(n-1),\bar{X}+\frac{S}{\sqrt{n}}t_{{\frac{\alpha}{2}}}(n-1)],$
(10)

where $\bar{X}$ denotes the sample mean, $S^{2}$ denotes the sample variance,
$n$ denoites the number of samples, and $\alpha$ denotes the significance
level. In this work, $\bar{X}={\frac{1}{n}}\sum_{i}^{n}{ALE}_{i}$,
$S^{2}=\frac{\sum_{i}^{n}{({ALE}_{i}-\bar{X})}^{2}}{n-1}$, $n=50$, and
$\alpha=0.05$.

###  VI-B Experimental Results

Table II compares the performance with the existing excellent work under a
randomly distributed network. When $N_{a}>5$, the model proposed in this paper
obtains the minimum $ALEs$. When $N_{a}=5$, our algorithm also shows strong
competitiveness. When $R=25$ or $30$, the $ALEs$ of the proposed model are
greater than CC-DV-Hop[22], OCS-DV-Hop[23] and CCS-DV-Hop[30], and the minimum
$ALEs$ are maintained in other cases. When $N_{a}=30$ and $R=40$, the $ALEs$
of the proposed model are less than $10\%$. In addition, Table II also
provides $ALEs$ of the proposed model when $MaxIter=300$ or $400$, and the
results show that the proposed model still achieves the best localization
accuracy. Additionally, we conduct an ablation study to demonstrate the
isolated impacts of each design component within our proposed approach. We
illustrate how proposed DEMN in section III and the hop loss in section IV
enhance the baselines (i.e., DEM-DV-Hop[40]). We utilize the identical solving
algorithm as the baselines[40], which can more convincingly demonstrate the
advantages of the proposed model. The results suggest that replacing the
distance estimation model with DEMN in the baselines results in a reduction of
$0.82\%-3.06\%$ in $ALEs$ compared to the baselines, while increasing APG by
$2.46\%$. Correspondingly, the localization performance achieved solely
through the hop loss surpasses that of the baselines and only falls short when
${N}_{a}=5$ and $R=25$ are considered. Its $APG$ and maximum performance gain
are $3.41\%$ and $14.15\%$, respectively. When combining DEMN with the hop
loss, we observed a decrease of $2.56\%-15.69\%$ in $ALEs$ compared to
baselines, with an $APG$ of $6.05\%$. It is worth noting that both proposed
models contribute to improving localization performance, thus validating their
effectiveness.

Table III shows the overall performance comparison on the randomly distributed
network. The results show that when $MIter=500$, the average localization
accuracy ($ALA=1-mean(ALEs)$) of the proposed model is up to $86.11\%$, which
is $15.92\%$ higher than the classical DV-Hop[8] algorithm. Compared with the
current best performance DEM-DV-Hop[40], the $APG$ of the proposed model is up
to $6.05\%$. In addition, Table III also provides the time consumption of each
algorithm. When $MIter=500$, the time consumption of the proposed model is
$3.91-8.05s$ more than those of the excellent current algorithm [23, 33, 30,
40, 31]. When $MIter=300$, the time consumption of the proposed model is
equivalent to or even smaller than those of these comparison algorithms. Our
model still maintains significant performance advantages in this case, with an
$APG$ of $5.14\%-10.84\%$.

Table IV compares the performance with the existing excellent work under the
C-shaped network. The localization accuracy of the proposed model is better
than that of the comparison algorithm in most cases. Only when $N_{a}=5$ and
$R=25$ or $30$, the localization accuracy is inferior to those of OCS-DV-
Hop[23] and CCS-DV-Hop[30]. Compared with the NSGAII-DV-Hop[33], the $ALEs$
are reduced by $3.62\%-22.26\%$, and compared with the DEM-DV-Hop[40], the
$ALEs$ are reduced by $2.4\%-21.05\%$.

Table V compares the performance with the existing excellent work under the
O-shaped network. When $N_{a}>5$, the $ALEs$ of the proposed model are
significantly smaller than those of all comparison algorithms. When $N_{a}=5$
and $R=25$ or $30$, the localization performance of the proposed model is
inferior to those of CC-DV-Hop[22], OCS-DV-Hop[23], CCS-DV-Hop[30] and DEM-DV-
Hop[40]. When $N_{a}=5$ and $R=40$, the $ALEs$ of the proposed model are more
significant than that of CC-DV-Hop[22], but less than those of other
comparison algorithms. When $N_{a}=5$ and $R=35$, the proposed model has the
minimum $ALEs$ compared with these state-of-the-art algorithms.

Table VI compares the performance with the existing excellent work under the
X-shaped network. The results show that the $ALEs$ of the proposed model are
significantly lower than other comparison algorithms in all cases. Compared
with NSGAII-DV-Hop[33] and DEM-DV-Hop[40], the localization accuracy of the
proposed model is improved by $2.35\%-10.77\%$ and $2.23\%-8.67\%$
respectively.

Table VII shows the $ALA$ of the proposed model on C-, O- and X-shaped
networks and its $APG$ compared with other algorithms. In this comparison, we
have opted not to include the DV-Hop[8] algorithm and the test data when
$N_{a}=5$ due to some algorithms having $ALEs$ exceeding $100\%$, rendering
the results meaningless. The results indicate that the proposed model achieves
$ALAs$ of $74.94\%$, $80.55\%$, and $79.77\%$ on C-, O- and X-shaped datasets,
respectively. Compared to the comparison algorithm, our model obtains the
$APGs$ ranging from $4.94\%-17.84\%$, $2.51\%-13.86\%$, and $4.26\%-21.74\%$,
respectively.

Fig. 6 visually shows the $ALAs$ of different algorithms on four test
datasets. As can be seen in Fig.6, under the randomly distributed network, the
$ALA$ of NSGAII-DV-Hop[33] outperforms that of OCS-DV-Hop[23] and CC-DV-
Hop[22]. However, under the C-, O- and X-shaped networks, the $ALA$ of NSGAII-
DV-Hop[33] is inferior to that of CC-DV-Hop[22]. In contrast, our model
outperforms other algorithms with the highest $ALA$ in four data sets,
demonstrating its robustness and generalization ability.

Fig. 7 provides the $95\%$ confidence intervals (CI) for the samples of $ALEs$
obtained under the conditions of $N_{a}=20$ and $R=25$. Factors influencing
the width of the CI consist of the confidence level, the dispersion in the
sample, and the sample size. For each algorithm investigated in this paper,
the sample size and confidence level remain consistent throughout the
analysis. Therefore, the greater the dispersion, the wider the confidence
interval. The findings demonstrate that across the four distinct test
networks, the overall distribution of CI is smaller than comparison
algorithms. The width of the CI is slightly wider than the comparison
algorithm, i.e. the dispersion of $ALEs$ samples is slightly higher than
comparison algorithms.

##  VII Conclusions

In this paper, to address the following two issues: 1) how to sufficiently
utilize the connection information between multiple nodes and 2) how to select
a suitable solution from multiple solutions obtained by Euclidean distance
loss, we propose a DV-Hop localization based on the distance estimation using
multinode (DEMN) and the hop loss in wireless sensor networks. In DEMN, when
multiple anchor nodes can detect an unknown node, the distance expectation
between the unknown node and an anchor node is calculated using the cross
domain information and is considered as the expected distance between them.
Compared to the distance estimation based on a single anchor node, DEMN
narrows the search space, resulting in more accurate distance estimation. When
minimizing the traditional Euclidean distance loss, multiple solutions may
exist. To select a suitable solution, the hop loss is proposed, which
minimizes the difference between the real and its predicted hops. Finally, the
Euclidean distance loss calculated by DEMN and the hop loss are embedded into
the multi-objective optimization algorithm. The experimental results show that
the proposed method gains 86.11% location accuracy in the randomly distributed
network, which is 6.05% better than the DEM-DV-Hop, while DEMN and the hop
loss can contribute 2.46% and 3.41%, respectively. Moreover, experimental
results on C-, O-, and X-shaped networks show that the proposed method
significantly improves the localization performance.

In future work, we will study DEMN when an unknown node is detected by more
than two anchor nodes, while saving computational costs. In addition, we will
also consider the calculation method of the expected distance when the nodes
in the cross domain follow different distributions.

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