# Fragility of the antichiral edge states under disorder

Marwa Mannaï1,2 Eduardo V. Castro3,4 Sonia Haddad2,5,6 1Center for Quantum and
Topological Systems, NYUAD Research Institute, New York University Abu Dhabi,
UAE
2Laboratoire de Physique de la Matière Condensée, Faculté des Sciences de
Tunis, Université Tunis El Manar, Campus Universitaire 1060 Tunis, Tunisia
3 Centro de Física das Universidades do Minho e Porto, Departamento de Física
e Astronomia, Faculdade de Ciências, Universidade do Porto, 4169-007 Porto,
Portugal
4 Beijing Computational Science Research Center, Beijing 100084, China
5 Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse
38, Dresden 01187, Germany
6 Institute for Theoretical Solid State Physics, IFW Dresden, Helmholtzstr.
20, 01069 Dresden, Germany

(August 31, 2024)

###### Abstract

Chiral edge states are the fingerprint of the bulk-edge correspondence in a
Chern insulator. Co-propagating edge modes, known as antichiral edge states,
have been predicted to occur in the so-called modified Haldane model
describing a two-dimensional semi-metal with broken time reversal symmetry.
These counterintuitive edge modes are argued to be immune to backscattering
and extremely robust against disorder. Here, we investigate the robustness of
the antichiral edge states in the presence of Anderson disorder. By analysing
different localization parameters, we show that, contrary to the general
belief, these edge modes are fragile against disorder, and can be easily
localized. Our work provides insights to improve the transport efficiency in
the burgeoning fields of antichiral topological photonics and acoustics.

##  I Introduction

The hallmark property of topological materials is the the occurrence of
topologically protected edge channels propagating at the boundaries of the
bulk structure Hasan-Rev ; Qi-Rev ; Bansil . Regarding their robustness
against disorder, these states can support dissipationless current flow. The
flagship edge states are the chiral modes of the Haldane Chern insulator
Haldane  describing a spinless electronic system on a honeycomb lattice, where
time reversal symmetry (TRS) is broken by complex hopping integrals between
next nearest-neighboring (NNN) atoms. The latter are characterized by a
complex phase $\Phi$, which results into staggered magnetic fluxes having the
same configuration in the two honeycomb sublattices. Different systems AQHE-
rev ; Haldane2 ; Haldane3 ; photonic ; Niu ; Wang ; Yang ; Khan ; Ni ; kagome
; Kee ; Chern-gr ; Chern-weyl  have been proposed to host chiral edge states
which have been observed in photonic crystals Wang09 , optical lattices cold ,
acoustic systems acoustic , thin film of magnetic topological insulators Rosen
, magnetic Weyl semimetals Chern-weyl  and in nanomechanical graphene gr .

By flipping the sign of the complex phase of the NNN integrals in one
sublattice, Colomés and Franz Franz  obtained the so-called modified Haldane
model (mHM) describing a semi-metal with broken TRS resulting from a valley-
dependent pseudo-scalar potential that offsets the Dirac point energies. In a
zizgag ribbon geometry, the mHM gives rise to co-propagating edge modes, known
as the antichiral (AC) edge states, connecting the oppositely shifted Dirac
points. Regarding the ungapped spectrum of the model, the AC modes are
counterbalanced by an equal number of gapless bulk states which propagate in
the opposite direction.

AC edge states are expected to be implemented in transition-metal
dichalcogenides Franz , exciton-polariton systems Mandal ; Mandal22 ,
gyromagnetic photonic crystals Chen20 , acoustic resonators JAP21 , twisted
van der Waals multilayers Denner , combined systems of Haldane model Cheng ,
Heisenberg ferromagnets on honeycomb lattice Bhowmick20 , and Floquet lattices
Floquet .

Recently, there has been a growing interest in the topological properties of
the antichiral edge states and their possible applications Mannai ; Roche ;
supra ; Chen ; AC-mag ; helical ; coexist . It has been shown that a Bernal
stacked bilayer of the mHM gives rise to a Chern insulator with a Chen number
$C=0,\,\pm 1,\,\mathrm{or}\,\pm 2$ Marwa23 . One-way bulk states were
predicted in a strip of alternately stacked modified Haldane lattices with
opposite complex phases heteroHM .

The experimental realization of the AC edge states has been reported in a
microwave-scale gyromagnetic photonic crystal Zhou ; photonic22  and in
electrical circuits Yang21 . AC surface states, a 2D extension of the one-
dimensional AC edge states of the mHM, have also been observed in photonic
crystal Weyl ; surface-AC .
The outcomes of these studies point towards the promising applications of AC
edge states in topological photonics. But, are AC edge states topologically
robust?
As the mMH is gapless, the AC edge states cannot be protected by a bulk
topological invariant. However, they were predicted to be robust against
disorder in the same way as the zero energy edge modes of a graphene zigzag
ribbon Franz .

In this work, we address the robustness of the AC edge states against Anderson
disorder. We first compute, based on the coupling matrix approach, the winding
number of a one-dimensional (1D)-reduced mHM and analyze its behavior under an
on-site disorder. Then, we focus on strips described by the Haldane model (HM)
and the mHM. We compute, using the transfer matrix method, the localization
lengths of, respectively, the chiral and the AC edge states. Our results show
that, contrary to the chiral states, the AC edge modes are not robust and can
be easily localized by defects which mix them with their counterbalancing bulk
modes.
The paper is organized as follow. In section II, we derive the 1D description
of the mHM which is mapped to an extended Su-Schrieffer-Heeger (SSH) model
with momentum dependent hopping integrals. We then study the behavior of the
corresponding winding number and the inverse participation ratio (IPR) in the
presence of Anderson type disorder. In section III, we compute the
localization lengths of zigzag ribbons of the HM and the mHM with different
widths. We benchmark the behaviors of the associated localization lengths
under different disorder amplitudes and concentrations. The concluding section
IV summarizes our results.

##  II 1D-reduced modified Haldane model

###  II.1 Modified Haldane model: a brief review

Figure 1: (a) Pattern of the NNN hopping processes of the mHM. The blue and
red arrows indicate the directions along which the electron gains a phase
$\Phi$. $t$ and $t_{2}$ denote, respectively, the NN and NNN hopping
integrals. $\left(\mathbf{a}_{1},\,\mathbf{a}_{2}\right)$ is the lattice basis
and $a$ is the lattice parameter. (b) The mHM on a zigzag nanoribbon structure
of width $W$. A lattice site is expressed as
$\mathbf{R}_{m,n}=m\mathbf{a}_{1}+n\mathbf{a}_{2}$. The dashed rectangle
delimits the ribbon unit cell used to define the reduced 1D-mHM.

We consider the mHM where the NNN hopping integrals have a complex phase with
opposite signs in the two honeycomb sublattices (Fig. 1 (a)). The
corresponding Hamiltonian is

$\displaystyle H=t\sum_{\langle
i,j\rangle}c_{i}^{\dagger}c_{j}+t_{2}\sum_{\langle\langle
i,j\rangle\rangle}e^{i\Phi_{i,j}}c_{i}^{\dagger}c_{j},$ (1)

where $c_{i}$ is the annihilation operator of a spinless electron on an atom
of the honeycomb lattice, $t$ and $t_{2}$ are the hopping integrals between,
respectively, the nearest neighboring (NN) and the next nearest neighboring
(NNN) atoms, and $\Phi_{i,j}=\Phi$ ($-\Phi$) for NNN hopping processes along
(in the opposite direction to) the pattern shown in Fig. 1 (a). Without loss
of generality, we hereafter take $t_{2}=0.1t$ and $\Phi=\frac{\pi}{2}$ Franz .

The corresponding band structure, in a zigzag nanoribbon, shows a semi-
metallic behavior with co-propagating gapless AC edge states which connect the
oppositely shifted Dirac points (Fig. 2).

Figure 2: Energy spectrum of a mHM on a ribbon of a width $W=100$. The red
lines indicate the positions of the Dirac points connected by the co-
propagating AC edge states. Calculations are done for $t_{2}=0.1t$, and
$\Phi=\frac{\pi}{2}$.

Based on the conductance behavior of the mHM in a nanoribbon, Colomés and
Franz Franz  argued that the AC edge modes are robust against on-site
disorder. By increasing the ribbon length, the conductance reaches a value of
2 (in units of $e^{2}/h$), which was ascribed to the two AC edge modes crossed
by the Fermi level. This behavior is in contrast with that observed in
graphene zigzag ribbon Gr-cond , where the conductance is found to collapse.
According to the authors Franz , the AC edge states remain delocalized even at
large impurity concentrations, which was attributed to their large
localization length compared to that of the bulk modes Franz . However, the
authors pointed out that the behavior of the counter-propagating modes,
accompanying the AC edge states, is unclear. These states are found to be
concentrated along the sample boundaries instead of being fully extended over
the width. To get insights on the topological character of the AC edge states,
the authors Franz  analyzed the topology of the zero energy modes of the
zigzag graphene ribbon which are at the origin of the AC modes of the mHM.
They defined a 1D-reduced graphene Bloch Hamiltonian by fixing the momentum
component along the zigzag direction of the ribbon, and derived the
corresponding winding number Franz . The latter shows a quantized nonzero
value, indicating that the pristine zigzag edge modes of graphene are
topologically protected in the same way as the chiral symmetry class AIII of
topological insulators AZ . Since the Bloch Hamiltonian of the mHM, (see next
section), has the same eigenstates as graphene, Colomés and Franz Franz
concluded that the AC edge states have the same topological properties as the
zigzag edge modes of graphene. As a consequence they are expected to be robust
against disorder.
Such conclusion does not take into account the presence of the counter-
propagating bulk modes which can couple to the AC edge states in the presence
of defects Mandal22 .

To address the topology of the AC edge states, we derive, in the following,
the effective 1D-mHM and the corresponding winding number based on the
approach of Ref. [SSH-HM, ], where the authors showed that the 2D HM can be
mapped into an extended SSH model SSH . We then analyse the behavior of the
winding number of the mHM in the presence of Anderson type disorder Phil .

###  II.2 1D-modified Haldane model: winding number

To obtain the reduced 1D-Hamiltonian of the mHM, we start by performing a
partial Fourier transformation of the real space Hamiltonian (Eq. 1), with the
respect to the $x$ component, along the zigzag ends, of the atomic positions.
The annihilation operator $c_{i}\equiv
c_{\alpha}\left(\mathbf{R}_{m,n}\right)$ of an atom $\alpha$ ($\alpha=A,\,B$)
belonging to the unit cell $\mathbf{R}_{m,n}=m\mathbf{a_{1}}+n\mathbf{a_{2}}$,
can be written as

$\displaystyle
c_{i}=\frac{1}{\sqrt{N_{1}}}\sum_{k_{1}}e^{ik_{1}x_{m,n}^{\alpha}}c_{\alpha,n}(k_{1}),$
(2)

where ($\mathbf{a_{1}},\mathbf{a_{2}}$) is the graphene basis (Fig. 1),
$x_{m,n}^{\alpha}$ is the position of the atom $\alpha$ in the unit cell
$\mathbf{R}_{m,n}$, $k_{1}=\mathbf{k}\cdot\frac{\mathbf{a_{1}}}{a}$ is the
momentum component along $\mathbf{a_{1}}$ axis, and $a$ is the graphene
lattice parameter. This Fourier transform turns out to write the Hamiltonian
in the so-called basis II Bena ; JN , or within the convention II Vanderbilt ,
which takes into account the atom positions in the unit cells. This basis
should be used to derive the physical properties of the system JN , which are
the hopping integrals of the $k_{1}$-dependent Hamiltonian $H(k_{1})$ in terms
of which is expressed the Hamiltonian of Eq. 1 as

$\displaystyle H$ $\displaystyle=$ $\displaystyle\sum_{k_{1}}H(k_{1}),$ (3)
$\displaystyle H(k_{1})$ $\displaystyle=$ $\displaystyle
J_{1}\sum_{n}c_{A,n+1}^{\dagger}c_{B,n}+J_{1}^{\prime}\sum_{n}c_{A,n}^{\dagger}c_{B,n}$
$\displaystyle+$ $\displaystyle
J_{2}\sum_{n,\alpha}c_{\alpha,n+1}^{\dagger}c_{\alpha,n}+\mu(k_{1})\sum_{n,\alpha}c_{\alpha,n}^{\dagger}c_{\alpha,n}+h.c.,$

where $J_{1}=t$, $J_{1}^{\prime}=2t\cos\left(\frac{a}{2}k_{1}\right)$,
$J_{2}=2t_{2}\cos\left(\frac{a}{2}k_{1}-\Phi\right)$ and
$\mu(k_{1})=2t_{2}\cos\left(ak_{1}+\Phi\right)$.

$H(k_{1})$ is reminiscent of the Hamiltonian of the extended SSH model SSH-HM
with an equal onsite chemical potential $\mu_{A}=\mu_{B}=\mu(k_{1})$.

To derive the winding number of the reduced 1D Hamiltonian (Eq. LABEL:mHM2),
we perform a second Fourier transform with respect to the atomic position
along the $\mathbf{a_{2}}$ direction as

$\displaystyle
c_{\alpha,n}(k_{1})=\frac{1}{\sqrt{N_{2}}}\sum_{k_{2}}e^{ik_{2}X_{n}}c_{\alpha,k_{1}}(k_{2}),$
(5)

where $X_{n}=na$ is the position of the unit cell, $N_{2}$ is the number of
unit cells along $\mathbf{a_{2}}$, and
$k_{2}=\mathbf{k}\cdot\frac{\mathbf{a_{2}}}{a}$. Here, we adopted the so-
called basis I description Bena ; JN ; Vanderbilt  which gives rise, unlike
the basis II, to a periodic Bloch Hamiltonian which will be used to define the
winding number Cayssol .
Carrying out the Fourier transform, we obtain the Hamiltonian
$H_{k_{1}}(k_{2})$ of the reduced 1D mHM as

$\displaystyle
H_{k_{1}}(k_{2})=d_{0,k_{1}}(k_{2})\sigma_{0}+\mathbf{d}_{k_{1}}(k_{2})\cdot\mathbf{\sigma},$
(6)

where $\mathbf{\sigma}$ are the sublattice Pauli matrices, $\sigma_{0}$ is the
$2\times 2$ identity matrix,
$d_{0,k_{1}}(k_{2})=J_{2}\cos\left(k_{2}a\right)+\mu(k_{1})$,
$d_{x,k_{1}}(k_{2})=J_{1}^{\prime}+J_{1}\cos\left(k_{2}a\right)$,
$d_{y,k_{1}}(k_{2})=J_{1}\sin\left(k_{2}a\right)$, and $d_{z,k_{1}}(k_{2})=0$.
The corresponding dispersion relation is
$E_{k_{1}}(k_{2})=d_{0,k_{1}}(k_{2})\pm|\mathbf{d}_{k_{1}}(k_{2})|$, which
describes two bands separated by a gap that closes at the Dirac points
$K_{\xi}=\left(k_{1}=\xi\frac{2\pi}{3a},k_{2}=\frac{\pi}{a}\right)$ if
$|J_{1}^{\prime}|=J_{1}$. Here $\xi$ is the valley index on which depends the
scalar term $d_{0,k_{1}}(k_{2})$ responsible of the offset of the Dirac point
energies.

The reduced 1D-mHM given by Eq. 6 breaks TRS due to the complex phase of the
NNN hopping integrals (Eq. LABEL:mHM2). Chiral and particle-hole symmetries
are also broken since $\sigma_{z}H_{k_{1}}(k_{2})\sigma_{z}\neq-
H_{k_{1}}(k_{2})$ and $\sigma_{z}H_{k_{1}}^{\ast}(k_{2})\sigma_{z}\neq-
H_{k_{1}}(-k_{2})$, respectively.

In the absence of NNN hopping processes, $H_{k_{1}}(k_{2})$ reduces to the
standard SSH model SSH , with effective NN hopping integrals $J_{1}$ and
$J^{\prime}_{1}$ (Eq. LABEL:mHM2), characterized by its winding number

$\displaystyle\nu_{k_{1}}$ $\displaystyle=$ $\displaystyle\frac{i}{\pi}\int
dk_{2}\langle u_{k_{1}}(k_{2})|\partial_{k_{2}}u_{k_{1}}(k_{2})\rangle$ (7)
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}dk_{2}\frac{d\Phi_{k_{1}}(k_{2})}{dk_{2}},$

where
$|u_{k_{1}}(k_{2})\rangle=\frac{1}{\sqrt{2}}\left(e^{i\Phi_{k_{1}}(k_{2})},\pm
1\right)^{\dagger}$ and
$\Phi_{k_{1}}(k_{2})=\mathrm{Im}\left[\ln\left(J_{1}^{\prime}+J_{1}e^{ik_{2}a}\right)\right]$.

The winding number, given by Eq. 7, can be also ascribed to the reduced 1D-mHM
Hamiltonian (Eq. 6) which differs from the SSH Hamiltonian by a diagonal term.
The latter does not affect the eigenstates on which depends the winding
number. It comes out that the 1D-mHM has a winding number satisfying SSH

$\displaystyle|\nu_{k_{1}}|=\left\\{\begin{array}[]{cc}1,&\;\mathrm{if}\;|J_{1}^{\prime}|<J_{1},\\\
0,&\;\mathrm{if}\;|J_{1}^{\prime}|>J_{1},\\\
\mathrm{undefined}&\;\mathrm{if}|J_{1}^{\prime}|=J_{1}.\end{array}\right.$
(11)

The pristine 1D-mHM is then topologically non trivial if
$\frac{2\pi}{3a}<k_{1}<\frac{4\pi}{3a}$, where the limiting values correspond
to the Dirac point positions along the zigzag ribbon direction. This result is
in agreement with that found in Ref. [Franz, ] where the authors ascribed the
topology of the AC edge states to the non-vanishing winding number of the
zigzag graphene ribbon in the absence of NNN term.

To address the topological protection of the AC edge states, one needs to
study the behavior of the winding number $\nu_{k_{1}}$ of the 1D-mHM under
disorder.

###  II.3 Disordered 1D-modified Haldane model: winding number

We consider the 1D-mHM in the presence of an onsite Anderson disorder which
modifies the last term in the Hamiltonian $H(k_{1})$ (Eq. LABEL:mHM2) as

$\displaystyle\mu(k_{1})\sum_{n,\alpha}c_{\alpha,n}^{\dagger}c_{\alpha,n}\longrightarrow\sum_{n,\alpha}\mu_{n}(k_{1})c_{\alpha,n}^{\dagger}c_{\alpha,n},$
(12)

where $\mu_{n}(k_{1})=\mu(k_{1})+Uw_{n}$, $U$ is the disorder amplitude, and
$w_{n}$ is a uniform random number
$w_{n}\in\left[-\frac{1}{2},\frac{1}{2}\right]$. In the presence of disorder,
the winding number cannot be derived analytically since the integration over
the Brillouin zone is no more possible regarding the broken translational
symmetry. The winding number can be computed numerically using the complex
matrix method CM , where the momentum component $k_{2}$, in Eq. 7, is replaced
by a phase twist $\theta$ of the real space single-particle wavefunction.
For a long chain ($L\gg W$) (Fig. 1), one can use the numerical approach of
Ref. [CM, ] which significantly reduces the computational time by carrying out
the calculations in the momentum space with twisted boundary conditions. Based
on this approach, we computed the averaged winding number
$\langle\nu_{k_{1}}\rangle$ of the disordered 1D-mHM described by the
Hamiltonian $H(k_{1})$ (Eq. LABEL:mHM2) with the disorder potential given by
Eq. 12. The results are depicted in Fig. 3.

Figure 3: Averaged winding number of the reduced 1D-mHM as a function of the
normalized disorder amplitude $U/t$ for different disorder concentrations
$n_{I}$. Calculations are done for $t_{2}=0.1t$, $\Phi=\frac{\pi}{2}$,
$k_{1}=\frac{7\pi}{6a}$ and a ribbon width $W=75$. The average is taken over
$100$ disorder configurations.

In the pristine system ($U=0$), $\langle\nu_{k_{1}}\rangle$ is quantized to 1,
but it drastically vanishes under disorder. The strong suppression of the
quantization occurs at relatively weak disorder amplitude ($U<0.2t$) and is
particularly independent of the impurity concentration. This behavior shows
that the winding number cannot be taken as a probe for the topological
properties of the disordered 1D-mHM since the chiral symmetry, defining the 1D
topological insulator class, is broken under Anderson disorder.
It is worth to stress that the Chern number quantization of the HM is found to
be a robust feature CM . The corresponding plateau at $C=1$ persists up to a
strong disorder amplitude ($U\sim 4t$), which expresses the extreme robustness
of the chiral edge states compared to the AC edge channels.

###  II.4 Disordered 1D-modified Haldane model: IPR

Besides the winding number, the disorder-induced localization of the AC edge
states can be characterized by the inverse participation ratio (IPR) defined
for an eigenstate $|\varphi_{n}\rangle$ of the Hamiltonian by
$\text{IPR}\left(|\varphi_{n}\rangle\right)=\sum_{i}|\varphi\left(\mathbf{r}_{i}\right)|^{4}$,
where $\varphi\left(\mathbf{r}_{i}\right)$ is the amplitude of the eigenstate
in the site $|\mathbf{r}_{i}\rangle$.
In the thermodynamic limit, the IPR of a delocalized state vanishes, while it
remains finite for a localized state and reaches 1 for a state completely
localized on one lattice site.

In the SSH model, the IPR of the bulk states increases with increasing
disorder amplitude, which indicates a disorder induced localization SSH-IPR .
However, the SSH edge states are found to have a decreasing IPR reflecting a
defect-induced broken chiral symmetry, which delocalize the edge modes SSH-IPR
.

Figure 4 represents the IPR as a function of the disorder amplitude of the
mid-gap states of the reduced 1D-mHM and 1D-reduced HM at different impurity
concentrations. The IPR are averaged over 100 disorder configurations. At low
concentrations ($n_{I}\sim 0.1$), Fig. 4 (a) shows that the IPR of the AC edge
states decreases slowly with increasing disorder amplitude, which corresponds
to the disorder-induced delocalization resulting, as in the SSH model, from
the chiral symmetry breaking. At higher concentrations, the AC edge states are
delocalized by defects up to a critical value of the disorder amplitude above
which they undergo an Anderson localization. The localization regime is
rapidly reached by increasing $n_{I}$ since the number of defected sites
increases Franz . It comes out that the decrease of the IPR of the AC edge
modes does not reflect the robustness of these states.

In the case of the HM (Fig. 4 (b)), the chiral symmetry is already broken in
the pristine system and the drop of the IPR indicates the delocalization of
the chiral edge states which expand on the sites closer to the boundaries
before mixing with the bulk edge states above a critical disorder amplitude.
The latter is shifted towards smaller values as the impurity concentration
increases. The chiral edge states remain, then, localized around the system
ends at relatively weak disorder amplitudes, which is an indicator of a
protected bulk topology.

Figure 4: Averaged IPR of the mid-gap edge states as a function of the
normalized disorder amplitude $U/t$ at different impurity concentrations
$n_{I}$ for the 1D reduced (a) mHM and (b) HM. The IPR is averaged over $100$
random configurations. Calculations are done for $t_{2}=0.1t$, $\Phi=\pi/2$,
$k_{1}=\frac{7\pi}{6a}$ and a chain of $n=150$ sites (Fig. 1(b)).

Actually, the IPR behavior should be taken with a grain of salt since non-
topological states can have IPR exhibiting similar behavior as topological
modes IPR . To avoid confusing conclusions, we compute, in the following, the
localization lengths of the edge states of the 2D mHM and the 2D HM.

##  III  2D-modified Haldane model: Localization lengths

$\begin{array}[]{ccc}\includegraphics[width=151.76633pt]{HM-W05t-ni03t.eps}\includegraphics[width=151.76633pt]{HM-W05t-ni06.eps}\includegraphics[width=151.76633pt]{HM-W05t-ni1.eps}\\\
\includegraphics[width=151.76633pt]{HM-W2t-ni03.eps}\includegraphics[width=151.76633pt]{HM-W2t-ni06.eps}\includegraphics[width=151.76633pt]{HM-W2t-ni1.eps}\\\
\includegraphics[width=151.76633pt]{HM-W5t-ni03.eps}\includegraphics[width=151.76633pt]{HM-W5t-ni06.eps}\includegraphics[width=151.76633pt]{HM-W5t-ni1.eps}\end{array}$

Figure 5: Normalized localization length $\Lambda_{M}=\frac{\lambda_{M}}{M}$
of the 2D HM as a function of the energy $E$ expressed in unit of the NN
integral $t$. The results are shown for different ribbon sizes $M$.
Calculations are done for $t_{2}=0.2t$, and $\Phi=\frac{\pi}{2}$. Each column
(row) is for a given disorder concentration $n_{I}$ (disorder amplitude $U$).
The First, second and third columns correspond respectively to an impurity
concentration $n_{I}=0.3$, $n_{I}=0.6$, and $n_{I}=1$ while the first, middle
and last rows correspond to, respectively, $U=0.5t$, $U=2t$, and $U=5t$.

$\begin{array}[]{ccc}\includegraphics[width=151.76633pt]{mHM-W05t-ni03.eps}\includegraphics[width=151.76633pt]{mHM-W05t-ni06.eps}\includegraphics[width=151.76633pt]{mHM-W05t-ni1.eps}\\\
\includegraphics[width=151.76633pt]{mHM-W2t-ni03.eps}\includegraphics[width=151.76633pt]{mHM-W2t-ni06.eps}\includegraphics[width=151.76633pt]{mHM-W2t-ni1.eps}\\\
\includegraphics[width=151.76633pt]{mHM-W5t-ni03.eps}\includegraphics[width=151.76633pt]{mHM-W5t-ni06.eps}\includegraphics[width=151.76633pt]{mHM-W5t-ni1.eps}\end{array}$

Figure 6: Normalized localization length $\Lambda_{M}=\frac{\lambda_{M}}{M}$
of the 2D mHM as a function of the energy $E$ expressed in unit of the NN
integral $t$. The data are the same as in Fig. 5.

We consider a graphene zigzag ribbon with $N_{L}$ unit cells along the zigzag
periodic boundary and $M$ unit cells in the transverse direction, with $M\ll
N_{L}$. The system can be descried by $N_{L}$ chains coupled by NN and NNN
hopping processes, where each chain contains in total 2M atomic sites (Fig.
1).

Based on the transfer matrix method (TMM) TMM1 ; TMM2 , we computed the
normalized localization length $\Lambda_{M}=\frac{\lambda_{M}}{M}$ of the HM
and the mHM which are plot, respectively, in Figs. 5 and 6. $\lambda_{M}$ is
the localization length defined as $\lambda_{M}=\gamma^{-1}$ and $\gamma$ is
the Lyapunov exponent of the system.

The behavior of $\Lambda_{M}$ with the system width $M$ gives insights into
the localization properties of the wavefunction in the presence of disorder: a
decrease of $\Lambda_{M}$ with increasing $M$ is the signature of a disorder
induced localization of the wavefunction. However, if $\Lambda_{M}$ increases
or remains unchanged as $M$ increases, then the wavefunction is extended. Fig.
5 shows that, as the ribbon width ($M$) increases, $\Lambda_{M}$ of the HM
decreases, except at two energies where it remains unchanged. These energies
correspond to the chiral edge states which keep their extended character along
the sample terminations. These edge states survive even at strong disorder
amplitude $U=5t$ and high impurity concentration $n_{I}=1$, where all the
sites are subject to the random distributed impurities (Fig. 5 (i)). In this
disorder configuration, $\Lambda_{M}$ decreases by increasing the system width
$M$, for all the energies, except at two values corresponding to the chiral
modes. This indicate that all the bulk states undergo an Anderson localization
while the chiral states preserve their extended character.

In the case of the mHM (Fig. 6), one cannot distinguish between the AC edge
states and the bulk modes. Due to defect, the AC edge states can be coupled to
the accompanying bulk states with which they propagate at the bulk boundaries
Franz ; Mandal22 . In particular, at strong disorder amplitude (Fig. 6 (c),
(f) and (i)), $\Lambda_{M}$ decreases with increasing $M$ over all the energy
spectrum, which means that the AC edge and the bulk states are localized by
the disorder.

##  IV Concluding remarks

We addressed the robustness of the antichiral edge states (AC) occurring in
the so-called modified Haldane model (mHM) Franz , describing a semi-metal
with a broken TRS induced by a valley dependent scalar potential. We analyzed
the behavior of the AC in the presence of Anderson disorder. We first
discussed the reduced 1D mHM, which was reported Franz  to have a topological
character described by a momentum-dependent winding number. Our results show
that this topological invariant is drastically suppressed even at small
disorder amplitude. To avoid any misleading or incomplete conclusions drawn up
from the reduced 1D model, we studied the localization behavior of the edge
states of the HM and mHM in a ribbon geometry with a finite size. Using the
transfer matrix method, we computed the localization lengths and showed that
the AC edge states of the mHM are easily localized by disorder while the
chiral ones remain robust even at relatively strong disorder amplitude. To sum
up, we showed that the AC edge states are not robust against disorder
regarding their coupling to the counter-propagating pseudo-bulk states. The
non-vanishing conductance of the mHM ribbon, reported in Ref. [Franz, ], is
not necessarily a signature of the robustness of the AC edge states. It can be
due to the pseudo-bulk states which are found to be peculiarly localized at
the ribbon boundaries Franz  The fragile character of the AC edge states can
be checked in photonic crystals Mandal , electrical circuits Yang21  and
acoustic systems acoustic . Our results can be used in the rapidly growing
fields of topological photonics Ozawa  and acoustics topo-acous , with
antichiral propagating modes JAP21 ; Chen20 ; Zhou ; photonic22 ; surface-AC ,
where lattice defects can be controlled to improve the robustness of the edge
state transport.

##  V Acknowledgment

We thank A. Cook and G. Lange for fruitful discussions and a critical reading
of the manuscript. This work was supported by the Tunisian ”Ministère de
l’Enseignement Supérieur et de la Recherche Scientifique”. M. M. is grateful
for the financial support from the University of Porto. E. V. C. acknowledges
partial support from FCT-Portugal through Grant No. UIDB/04650/2020. S. H.
acknowledges the Max Planck Institute for the Physics of Complex Systems and
the Institute for Theoretical Solid State Physics at IFW for kind hospitality
and financial support.

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# Federated Learning and Blockchain-enabled Fog-IoT Platform for Wearables in
Predictive Healthcare

Marc Jayson Baucas, , Petros Spachos, , and Konstantinos N. Plataniotis This
work was supported in part by the Natural Sciences and Engineering Research
Council (NSERC) of Canada. M. Baucas and P. Spachos are with the School of
Engineering, University of Guelph, Guelph, ON, N1G2W1, Canada. (e-mail:
<EMAIL_ADDRESS>petros@uoguelph.ca). K. Plataniotis is with the Department
of Electrical and Computer Engineering, University of Toronto, Toronto, ON,
M5S3G4, Canada. (e-mail<EMAIL_ADDRESS>
###### Abstract

Over the years, the popularity and usage of wearable Internet of Things (IoT)
devices in several healthcare services are increased. Among the services that
benefit from the usage of such devices is predictive analysis, which can
improve early diagnosis in e-health. However, due to the limitations of
wearable IoT devices, challenges in data privacy, service integrity, and
network structure adaptability arose. To address these concerns, we propose a
platform using federated learning and private blockchain technology within a
fog-IoT network. These technologies have privacy-preserving features securing
data within the network. We utilized the fog-IoT network’s distributive
structure to create an adaptive network for wearable IoT devices. We designed
a testbed to examine the proposed platform’s ability to preserve the integrity
of a classifier. According to experimental results, the introduced
implementation can effectively preserve a patient’s privacy and a predictive
service’s integrity. We further investigated the contributions of other
technologies to the security and adaptability of the IoT network. Overall, we
proved the feasibility of our platform in addressing significant security and
privacy challenges of wearable IoT devices in predictive healthcare through
analysis, simulation, and experimentation.

###### Index Terms:

Machine learning, Data privacy, Predictive models, Distributed systems, Health
care services, Platforms, Testbed, Health informatics, Fog network, Security,
Private Blockchain, Privacy, Scalability, Internet of Things.

##  I Introduction

The usage of wearable Internet of Things (IoT) devices in healthcare is
rising. Due to their availability and sensing capability, these devices
collect physiological data from patients and provide real-time diagnosis [1].
Wearable IoT devices have caused remote healthcare to make a paradigm shift
into predictive diagnosis and reliable early detection. The data collected by
these devices partnered with different learning techniques aid in predictive
healthcare services. Doctors can analyze data such as their patient’s
activities and accurately predict anomalies and threats against their health
[2]. They can also prescribe treatments for preventing and addressing these
detected concerns. However, this breakthrough has limitations in its
technology. Challenge within a network that employs wearable IoT devices cause
impasses in predictive healthcare.

An open issue for wearable IoT devices in predictive healthcare is the amount
of data it needs to be effective. A large amount of personal data is
collected, resulting in security and privacy concerns due to the nature of the
data used for analysis [3]. The wearable devices’ limitations on processing
capabilities lead to vulnerabilities and potential leakages in sensitive
patient information [4]. Another issue is the integrity and reliability of the
service[5]. Structuring healthcare to prioritize certain aspects can cause
trade-offs in others. Service integrity is crucial for this field in remote
healthcare that relies on wearable data accuracy and predictive model
precision. One more issue is the adaptability of the network that deploys and
serves these predictive healthcare services [6]. Wearable IoT device
standardization is a significant concern to IoT networks due to the
heterogeneity it introduces. This diversity results in demands for continuous
maintenance and updates to the medical server to ensure that it is up to date
with every newly introduced wearable IoT device. As a result, concerns about
adaptability limit the healthcare network from fully remaining relevant and
sustainable over long stretches of its service.

In this work, we propose a fog-IoT platform to address these issues. We use
federated learning to preserve patient data privacy and the integrity of the
network’s predictive services [7]. Also, we incorporate blockchain technology
to address the security issues in wearable IoT devices through its access
control and cryptographic structure [8, 9]. Finally, we combine these
technologies within a fog-based IoT architecture to enforce decentralized
servers and resource reallocation to improve the adaptability and
sustainability of the overall network [10]. The main contributions of this
work are:

  * •

We present a fog-based IoT platform using federated learning and blockchain
technology to preserve patient data privacy and improve the security of data
within the network.

  * •

We designed a testbed that simulates and evaluates the proposed
implementation. We used model accuracy to observe the platform’s ability to
preserve the integrity of a predictive service.

The rest of this paper is organized as follows. A discussion on the background
of our study and a brief literature review are in Section II. Our proposed
design and the methodology followed are in Section III. The presentation of a
developed testbed and a discussion of the results are in Section IV. Finally,
our conclusions are in Section V.

##  II Background and Related Works

We provide a brief discussion on the relevant works in blockchain technology
and federated learning implementations for wearable IoT devices in healthcare.

###  II-A Benefits of Using Wearable IoT Devices in Healthcare

Wearable IoT devices can form a network of sensing devices that collect data
from points of interest for predictive analysis. In predictive healthcare,
these devices collect physiological information from patients for better
clinical decisions and to provide a prediction. Each obtained statistic
reflects the status of their health, which can identify future issues and
potential risks. A benefit of using wearable IoT devices in healthcare is to
reduce hospital congestion and medical examination costs via remote diagnoses
and long-distance patient monitoring. In [11], they introduce a wearable Tele-
ECG system for heart rate monitoring. The proposed design merges the latest
technologies in textile electrodes (TE), Bluetooth low energy (BLE), and
smartphones to create a portable means of monitoring and evaluating the
condition of a patient’s heart for potential anomalies. With features such as
geographical tracking, medical history storage, and remote patient monitoring,
the system establishes a light and cost-effective alternative for patients
suffering from heart issues that need constant observation and evaluation.

Another advantage of using wearable IoT devices in predictive healthcare is
improving the quality of early disease and fault detection in medical centres.
In [12], they present a review of various biomedical IoT devices that raise
the quality of diagnosis in healthcare services. The survey includes wearable
sensors as one of the leading technologies that enable advancements in
predictive healthcare IoT. Services are now made remote and analytic. Patients
can use technologies and wearable sensors that can help give doctors early
information on potential causes and triggers of health complications. As a
result, there is improvement in predictive anomaly detection without
constraints in geographical location and data resource availability. Also,
these devices widen the scope of medical centres toward their patients.
Doctors can use trend and behaviour analysis to pinpoint anomalies in a
patient’s health. At the same time, medical professionals can evaluate
diseases with improved precision due to leveraged real-time and early
detection [13].

###  II-B Challenges of Using Wearable IoT Devices in Healthcare

Although the usage of wearable IoT devices shows potential to raise the
quality of services in healthcare, they also pose the following concerns.

####  II-B1 Network Security and Data Privacy

Introducing wearable devices to collect patient data adds more endpoints to
the server of the healthcare service. Data collection and monitoring are
convenient due to these remote services. However, introducing more devices
introduces more vulnerabilities to the network. Increasing the endpoints
increases the potential areas where malicious users can attack and steal data.
As a result, there is a greater demand for better security as the network
grows. Also, privacy becomes a concern due to the sensitive information
transmitted from the wearable device to the server. Therefore, network
security and data privacy concerns grow as the network expands with more
wearable devices.

Different strategies and technologies have been proposed to address this issue
[14, 15]. In [14], they present a multi-keyword search mechanism to preserve
data privacy within the IoT network. They aim to strengthen the encryption of
the information transmitted from the patient to the medical centre. This
strategy achieves forward privacy preservation, which results in a more
guaranteed security system for users. Another example that aims for privacy
preservation is in [15]. They use deep learning to secure the network by
separating private from raw data. This strategy results in minimizing the
chances of leaking sensitive information.

Our approach takes key strengths from these two strategies. We take the
ability of cryptographic techniques to reinforce data transmissions and
combine them with the adaptability of machine learning techniques to create a
secure means of preserving patient data. As a result, we chose blockchain
technology for its strong encryption and federated learning for its adaptive
ability to improve data privacy while keeping its trained models optimized.

####  II-B2 Data Integrity and Precision

An advantage of using wearable IoT devices in healthcare is the real-time
diagnosis and early detection of illness and medical anomalies within a
patient. However, this can be affected by the quality of the sensor and the
data processing scheme behind the service. Potential misdiagnosis is high if
the sensor or the evaluation method is not up to standard. Another source of
error could come from how the data is collected and stored on the server.
Preserving physiological information integrity obtained from the wearable
devices is needed as it travels from patient to server. It ensures that the
data is recent and precise to the current status of the source. Also, the data
handling during aggregation should be carried out with great precision so that
the evaluation of the patient is accurate. A service that can not collect,
transmit, and process the data correctly and efficiently will result in
inaccurate results. Therefore, the integrity and precision of information a
concern that needs addressing for a healthcare service that uses wearable IoT
devices to be effective.

Approaches that improve the integrity of sensed data from IoT devices in
healthcare services are proposed in [16, 17]. In [16], a novel data aggregator
for efficient and secure analysis in IoT monitoring is presented. Its scheme
involves a cryptographic accumulator that securely collects data from wearable
IoT devices. Another example is in [17]. This work highlights the advantages
of integrating blockchains with IoT. They aim to enhance the performance of
healthcare services that use wearable IoT devices by using the security
advantages of blockchain technology. Similar to the approach in [16], both
strategies focus on the ability of cryptographic technologies to ensure the
accuracy of sensed data within the IoT network.

Our proposed approach differs as it incorporates federated learning for a more
secure data aggregator. We combine it with decentralized organizations
provided by blockchain and create a platform that ensures a secure medium for
accurate data analysis in healthcare.

####  II-B3 Network Structure Adaptability and Flexibility

Wearable IoT devices introduce different sensors and technologies that collect
and process data. Due to their multiple advantages, healthcare services should
include them in their systems. However, innovations and improvements are
iterative as changes arise often. Although this is a sign of technological
growth and advancements in the healthcare sector, it dictates the pace at
which current infrastructures need to keep up. Device diversity has always
been a concern for IoT networks due to the need for ongoing standardization.
As this new technology is still in development, services that use it should be
able to follow the development process. It is the same for wearable IoT
devices used in healthcare. The demand for technologies that detect each one
is high as new diseases and virus strains often appear. However, every new
technology introduced a need for the service to adapt. Networks that are less
flexible result in losing their resources. Some even end up losing their
service and infrastructure altogether. Therefore, there is a need for
adaptability and flexibility in the structure of healthcare networks for the
continuously evolving nature of wearable IoT devices.

Some approaches focus on improving the flexibility of wearable devices to
tackle this issue. In [18], they present an innovative wrist-worn prototype
for patient monitoring, which also functions as an IoT gateway. Some sensors
in remote healthcare focus on sensing the environment around the patient to
ensure that the condition of their surrounding is beneficial to their health.
Through it, ambient sensing devices can connect to the healthcare network. As
a result, it sets up a platform that can easily integrate and synchronize
other devices under it. In [19], they present a wearable patch that functions
similarly to the prototype in [18] by creating a flexible IoT gateway for
connecting wearable devices. Instead of ambient sensors, their portal focuses
on ECGs and PPGs.

Our approach is different as we aim to standardize devices through the fog by
moving analysis closer to the edge. As a result, we focus on data management
standardization instead of device standardization due to the diversity of
wearable devices. Instead of specialization through prototypes, we plan to use
fog-IoT paired with decentralized technologies such as blockchain and
federated learning to develop a modular IoT network for wearable devices in
healthcare.

###  II-C Federated Learning for Wearable IoT Devices

Federated learning is a machine learning technique that takes a distributed
approach in training its models [20]. It uses its decentralized strategy to
utilize global knowledge collected from its clients. In IoT, federated
learning has two components; the client and the server [21]. Its flow of
operations for federated learning starts with each client training their model
using their raw data. Then, the server aggregates and compiles the resulting
models into a global model that it redistributes to each client’s use. As a
result, each client receives the most optimal model given global knowledge
gathered through local training. This procedure is ongoing, and the global
model is updated every time the client provides new knowledge.

Federated learning is well-known for its capability of effectively preserving
the privacy of data [22]. Requiring clients to transmit raw data directly to
the server before it is analyzed can cause vulnerabilities. Without a strongly
reinforced network, malicious attackers aiming to steal or tamper can target
the data. Federated learning protects raw data by creating a strategy that
moves the analysis locally. There are lesser avenues for privacy leakages
since the server only cares about the resulting trained model from each client
[23]. As a result, this structure has an improved trusted architecture
compared to standard network arrangements. Aside from its security advantages,
federated learning also improves the sustainability of IoT networks. It
provides a dynamic learning strategy that keeps global knowledge for services
up to date [24]. Commonly, the server aggregates all the data first before it
uses it to train the global model. Federated learning allows new correlations
and behaviour changes from collected data to be detected by the server
earlier. A fully centralized scheme is slowed down and saturated due to the
volume of the data it uses to train its models. Federated learning can reduce
training time by running it locally and in parallel. Also, there is a
significant reduction in the data size as each client trains its model.

Healthcare services that involve wearable IoT devices revolve around
continuous data sensing, learning, and analysis. Federated learning can
provide a sustainable and decentralized scheme to optimize these services in
healthcare [25]. We chose to use this technology in our platform due to all
the improvements it can contribute to IoT services, specifically healthcare.
Our proposed design is different because we aim to reinforce federated
learning with the security provided by blockchains. Also, the foundation of
our network will be a fog IoT network. The diversity of wearable IoT devices
and their limited processing capabilities is a constraint. However, we can
overcome this by moving the training to the fog. This arrangement still allows
training locally while considering the processing limits of wearable IoT
devices. The result is a decentralized IoT network that can maximize the
potential of federated learning in keeping data analysis in healthcare
services secure and sustainable.

###  II-D Blockchain for Wearable IoT Devices

A blockchain is a cryptographic ledger that provides a distributed service for
information storage and security [26]. This decentralized control establishes
a trusted system that will only grant access to users acknowledged by the
blockchain ledger [27]. This feature protects the data from external or
unwanted modifications and makes it immutable. For wearable IoT devices in
healthcare, the network can use blockchains as a tamper-proof database for
patient data storage.

Its unique structure addresses different security issues in IoT networks. In
[28], they use the unique consent mechanism of blockchains to secure user data
privacy in wearable fitness devices. They aim to prove the feasibility of the
trustworthiness of blockchains when fortifying the data privacy of wearable
IoT device data. Another proposed use for the access control of blockchains is
in [29]. They developed a lightweight authentication scheme that classifies
mobile devices and differentiates them from fake data injections or
illegitimate users. Their designs show the potential of blockchain technology
in securing the healthcare server and the monitoring devices used by its
services. In [30], they use a different feature of blockchains in reinforcing
their healthcare service that uses wearable IoT devices. They integrate the
encryption model of blockchains to improve the security of the IoT network.
Their design creates a searchable encryption technique that assists in
securing collected COVID-19 data. It shows the effectiveness of blockchain
architecture in helping protect the healthcare IoT network from several
security threats like malicious data injection and hijacking sessions.

Our design differs from the previously presented implementations because we
aim to use a private blockchain [31]. Public blockchains use a reward-based
protocol called “proof-of-work” (PoW) in granting access to their devices
[32]. Although this protocol is secure, its processing requirement is a caveat
for wearable IoT devices in healthcare. In [33], we highlighted how some
wearable IoT devices are incapable of adapting to this authentication
structure due to their low-cost and low-end designs. Therefore, we need a
different blockchain architecture that can cater to the design nature of
wearable devices. Private blockchains function differently. They incorporate a
more trust-based protocol. This approach turns the blockchain into a
hyperledger that can regulate its devices based on a list of trusted devices
defined by the network owner. Healthcare IoT networks can benefit from this
approach more since the need for high processing power for their wearable
devices is reduced. As a result, the types of IoT-based wearable devices they
can incorporate into their services do not limit the healthcare network [34].

##  III Design and Methodology

The following is a discussion of our proposed platform, including a
description of the design and the different components.

###  III-A Architecture Overview

We propose a fog-IoT-based approach to secure data exchange from wearable IoT
devices used for predictive healthcare services. We aim to improve the overall
structure’s service integrity and flexibility. To further enhance the
distributive organization of our network, we chose to use federated learning.
With its ability to aggregate learning models, we can sort our data and keep
private information secure. Also, it can reduce the sources of leakage since
each client is only required to send their locally trained model to the
server. Integrating federated learning improves the sustainability of the
predictive healthcare service by providing a system that can organize the
model data and provide a global model that represents the overall knowledge
obtained through all the local training. The result is a scalable design that
introduces flexibility by keeping the network distributed while ensuring that
it does not impede the learning and analysis of the service.

Next, we combine federated learning with private blockchain technology for
more secure client authorization. Also, it is treated as a hyperledger to
store the IDs of each client device and locally trained model. Due to its
immutability, tracking and logging each trained model for version control and
debugging improves the data analysis and learning aspect of healthcare
networks and their services.

Commercial wearable devices are diverse. Some are low-end by design. To
address this, we moved the process of locally training the model from the edge
of the network to the fog. We can reallocate the training process by providing
an intermediary fog device. Usually, training takes a large portion of
processor capacity. Unfortunately, not every wearable IoT device can do it
effectively. For our platform to be able and handle a diverse pool of wearable
devices, we offloaded the processing required to a fog device.

The proposed network arrangement is shown in Fig. 1. First, the fog device
collects all sensed data from each wearable IoT device and trains the model
locally. Each one sends its local model to the server for storage and
tracking. Next, the server aggregates all the knowledge from the local models
and generates a global model. Then, the server sends this model to each
wearable IoT device through the fog nodes for further data analysis. We
investigated the feasibility of our platform by simulating the intended data
flow of our design and the effectiveness of our federated learning system.
Lastly, our implementation aims to provide a low-cost setup that simulates our
platform under a resource-constrained environment. We presented a top-down
view of our architecture and its data flow. The following details will
encompass the other design choices we made for further establishing the
predictive healthcare service that our design models.

Figure 1: Network arrangement for the proposed fog-based platform for wearable
IoT devices.

###  III-B Dataset and Neural Network

To establish the predictive healthcare service our platform will use to test
its feasibility, we looked for a dataset and neural network configuration that
fits our remote monitoring data flow. We chose a standard classifier design
and organized the dataset to minimize its impact on our platform experiments.
Our focus is on the effectiveness of our platform in securing wearable IoT
device data and ensuring the integrity and flexibility of the healthcare
service, and not maximizing the accuracy of the classifier. Therefore, the
classifier we used in testing our platform is for human activity recognition
(HAR). It uses accelerometer data from various commercial wearable IoT devices
such as smartphones and smartwatches to determine the posture and condition of
each user [35]. Healthcare services use these to monitor the safety of their
patients. For example, it can detect the falling pattern of a person and
enable real-time response. Also, they can monitor people’s daily exercise and
regimen for further consultation and improvements for their health [36].
Similarly, we will use it to simulate patient data for training HAR models to
better reactive analysis and accident detection in wearable IoT device-based
remote monitoring.

The dataset we selected is the Human Activity Recognition Using Smartphones
Data Set from the UCI Machine Learning Repository[37]. It is from an
experiment with 30 volunteers within 19-48 years. Each individual performed
six activities: walking, walking upstairs, walking downstairs, sitting,
standing, and lying down. Each volunteer did each action wearing a Samsung
Galaxy SII smartphone on their waist. The result is a dataset that contains
10299 instances of 3-axial linear acceleration and 3-axial angular velocity
captured from the accelerometer and gyroscope of the mobile device. It
encompasses 561 labelled features with time and frequency domain variables. We
chose this classifier and dataset combination because the data is from
smartphones, which is more common to the public. This selection provides a
better dataset scope that represents a larger population. Our focus is on the
federated learning aspect of the design. We can minimize any impacts of
extraneous variables that could interfere with our experiments and the
precision of its results by using an already tested and documented classifier.
Using it provides convenience and efficiency as we simulate the HAR-based
predictive service testbed for our experiments.

We implement a one-dimension convolutional neural network (CNN) because the
dataset is a sequence with only a single-dimensional source. Also, it is a
design that minimizes the overall processing requirement when training and
testing the model. Since we aim for a low-cost approach, we intend to limit
the impact of resource requirements. We coded it using a combination of the
Keras and Tensorflow libraries in Python. The dataset was already pre-grouped
to have a 70-30 split for training and testing. We use the same split when
training and testing our CNN to avoid generating any significant impact from
the nature of the classifier or the data. Also, we refrained from potentially
over-tuning the CNN by only calibrating through the epoch number. We wanted to
ensure that the experiments focused on the performance of the federated
learning aspect of our design.

###  III-C Federated Learning Implementation

We implemented the federated learning aspect to incorporate distributive
learning with the HAR predictive model. We used the previously presented CNN
as the learning model for our cloud and fog device. The target of the adopted
CNN is to learn from various human activities using accelerometer and
gyroscope data. The result is a HAR model for potential anomaly detection in
patient movement and activeness. Federated learning aims to solve for an
optimal global model $w_{G}$ that minimizes the weighted average loss of all
clients. It takes the global loss function $\mathcal{L}$ and simplifies it
into a summation of losses obtained from the local models $w_{L}$ of each
client shown in the following:

$w_{G}=\operatorname*{arg\,min}_{w_{L}}\mathcal{L}(w_{L})=\operatorname*{arg\,min}_{w_{L}}\left[\frac{1}{K}\sum_{k=1}^{K}\mathcal{L}_{k}(w_{L})\right],$
(1)

where $\mathcal{L}_{i}$ is the local loss function by client $k$. We
translated the solution for this federated learning problem as a python script
executed by the central server. It starts by iterating through each local
model we obtained through the fog devices.

Next, we extract the weights of each model and optimize them by scaling them
according to the total number of clients and their accuracy. We optimized our
weights by giving the model with the highest accuracy the best scaling factor
and gave the rest lower scale values relative to the number of total fog
clients in the network. This scaling scheme ensures that the best-performing
local model is the base, and the rest are additional references for further
learning reinforcement. While scaling, the server generates a CNN with the
same layer design. Then, the script aggregates the optimized weights via
averaging, takes the resulting solution, and sets it into the prepared model,
which becomes the global model. This model encompasses all the knowledge
obtained from each locally trained CNN. A graphical representation that shows
the flow of our federated learning system is in Fig. 2.

Figure 2: Flow of operations for our implemented federated learning system
within our platform.

###  III-D Private Blockchain Implementation

We incorporated a private blockchain to manage the training information shared
by the devices that will train the HAR-based predictive model. The blockchain
protocols create an access level ensuring that only trusted devices can access
the predictive model’s local training and global knowledge. We coded it in
Python. Each block is a class definition that contains an ID, a list of
training model results, and a hashed string of the previous block. The private
blockchain authorizes communication between the server and the client based on
their assigned ID. Also, it collects the locally trained models and stores
them periodically within a block. These blocks are chained cryptographically
by generating a hash of the previous block and storing it within the new one.
The resulting data structure is a log of every training model sent to the
server and the other actions the network carried out. The blockchain becomes a
hyperledger that stores the history of every operation done by the network.
Also, since it is tamper-proof, these records are protected. We also
initialize copies of the blockchain and distribute them to each client to keep
the data decentralized and free from manipulation outside the authorized
devices. This distribution allows the service to check the validity of each
client and the local models they are training and sending to the server.

With the private blockchain, the information and operation within the network
remain secure. Tampering becomes more difficult with the hyperledger keeping
track of all the changes to the global model. It can record the results of
each local training through the obtained models from the fog clients. Also,
private and raw data from the clients are not sent to the server, keeping them
secure from any form of leakage or malicious attacks. Overall, the blockchain
sets up the network by securing its endpoints and reinforcing its
decentralization.

###  III-E Hardware and Software Specifications

To simulate the predictive service based on the HAR model, we selected
hardware that fits our aim to have a low-cost design. Our setup makes use of
Raspberry Pi 3 Model Bs. We chose Pis due to their portability,
programmability, and modularity for rapid programming. Also, we selected a
low-cost and low-end device to simulate the capabilities of most wearable IoT
devices. Each Pi has a Quad-core ARM Cortex A53 processor that runs on a
benchmark of 1.2 GHz. Our focus is on model accuracy while processing speed is
currently not a concern. We also used Pis for both the fog devices and the
central server of our design. Each Pi is pre-loaded with a Raspbian-Jesse
operating system. We installed Python 3.6 on every Pi as our programming
language due to its flexibility and diverse selection of open-source labels.
The selected version is the most recent version that works with all the Python
libraries needed for our design. As for the wearable IoT device that
represents the edge of our network, we based it on the same smartphones from
the HAR dataset we chose to use. It is a Samsung Galaxy SII that uses a Dual-
core 1.5 GHz Scorpion processor.

##  IV Results and Discussions

We tested our proposed platform to evaluate its feasibility by addressing data
analysis security, integrity, and flexibility.

###  IV-A Testbed

We structured our testbed to simulate data flow from the fog to the cloud
server. It is to observe the behaviour of our platform under the standard
network interactions of local and central servers of a fog-IoT network. In
designing our testbed, we use low-cost and low-end devices to evaluate our
design under minimal processing capabilities. We chose this approach to show
the feasibility of our proposal when using simple IoT devices. It gives an
estimate for a benchmark on the minimum system requirements for our
implementation. We decided to use Raspberry Pi 3 Bs to model the fog devices
because it is modular, portable, and low-cost. During the early stages of
implementation, we discovered that the Pi could run the classifier training
and the blockchain scripts without any issues. Therefore, we continued to use
Pis as both the fog and the cloud servers in our network. Another reason why
we chose to stay with just one type of technology was to limit the potential
impact of any processing or data capacity advantage if we used a device with a
better processor as a server. Since Pi’s are modular, we replicated each node
by flashing an identical copy of the operating system to each device. As a
result, we arranged the network to have a Pi as the cloud server with 10 Pis
that serve as the fog nodes.

First, we assume that each fog server already contains the wearable IoT device
data, so the training and testing dataset is pre-loaded in each Pi. Since our
focus is on the federated learning aspect carried out by the fog-IoT servers,
we can skip the simulations of data transmissions between the wearable and the
fog devices. Next, each fog device will train its models locally and send them
to the central cloud server via wireless file transfer. The server will wait
until it receives all the local models before the next step. Next, it will
aggregate the weights of each model and scale it according to its accuracy. We
implemented a scaling scheme that gives the highest possible scaling factor to
the local model with the highest accuracy. At the same time, the rest of the
models will receive the same scaling factor but are significantly lower. Then,
the server will take the resulting scaled weights and set them on a
structurally identical CNN, resulting in the global model. For our test, we
plan to compare the accuracy of the global model with the average of the
locally trained models from each fog device. We elected to use this metric to
evaluate the behaviour of our platform. Also, we aim to ensure that our
implemented federated learning system does not disrupt the integrity of the
classifier. Finally, to investigate the security and adaptability of our
design, we later discuss the strengths of our platform that we observed as we
tested and simulated the platform.

###  IV-B Integrity Evaluation via Training Model Accuracy

Our test investigated the ability of our platform to maintain the integrity of
the predictive data analysis of the classifier within the healthcare network.
We used a 1D CNN to classify HAR using the HAR dataset from the UCI Machine
Learning Repository. We chose to evaluate the feasibility of our design by
comparing the accuracy between the global and the local models.

First, we trained the model locally using 10 Pis pre-loaded with the
classifier and the dataset. We arranged the models by assigning a number to
each corresponding client. The fog devices trained each local model using the
same 70% split of the dataset. Also, we tested each model with the remaining
30% of the dataset. We tried to minimize the complexity of our classifier and
focused on the performance of our federated learning system by doing the
minimal tuning. The only parameters we tinkered with were the epoch number and
the batch size, which we set to 10 and 8, respectively. The resulting model
yielded a 90.43% testing accuracy. We generated its validation-to-training
loss and accuracy plots as shown in Fig. 3 and Fig. 4, respectively. These
results are a sample of 1 out of 10 fog clients. Each presents the reasonable
validity of the classifier at minimal tuning in correctly identifying the
different labelled human actions based on accelerometer and gyroscope data.

Figure 3: Sample training and validation loss of one local model yielding a
testing accuracy of 90.43%. Figure 4: Sample training and validation accuracy
of one local model yielding a testing accuracy of 90.43%.

Upon further analyzing these graphs, we observed a spike in the validation
loss on epoch 6 of Fig. 3. This behaviour is due to the learning rate causing
the trained model to be volatile at this point. Since the graph stabilizes, it
does not significantly impact the overall model. Also, there is a similar
spike on epoch 5 of Fig. 4. This behaviour could reflect potential volatility
within the dataset. However, since the amplitude of the fluctuations is within
a reasonable range of +/- 2% from the median, it shows that the data’s
volatility is not significant enough to render the model inaccurate.

Figure 5: Confusion matrix of testing the global model with aggregated weights
from 10 local models.

Overall, each trained model yielded similar behavioural trends but had varying
final accuracy values within a range of +/- 1.28% from the median. This range
shows the impact of the data on the accuracy of the model. However, it is not
significant enough to invalidate the training results. Therefore, we took the
average of their performances and used it to represent the benchmark of the
classifier for each collective number of clients. After getting the average
accuracy of the local models, we compared it with the global model to evaluate
our implemented federated learning system.

First, we generated the global model for each N number of clients. We have a
visualization of our global model’s performance after aggregating the weights
of local models from 10 clients through the confusion matrix in Fig. 5. Next,
we compared the average accuracy of the N local models to their corresponding
global models. This method allows us to see the model’s ability to keep up
with the average performance of a collection of locally trained models. We
obtained the average accuracy of the local models while increasing the number
of clients. Then, we generated their corresponding global model and recorded
its accuracy.

Finally, we plotted the comparison between the accuracy of the global model
and the averaged local model shown in Fig. 6. Based on the results, we can
observe that as the number of clients increases, the accuracy of the global
model keeps up and even eventually performs better than the average accuracy
of its local models. Through this observation, we can see the ability of our
implemented federated learning system to minimize any loss from aggregating
weights. We can attribute this improvement in accuracy to our optimization
scheme that assigns a higher scaling factor for the local model with the best
performance. Also, we can see how the accuracy of the global model peaks at
91.75%.

Figure 6: Difference in testing accuracy between global model and average
local model as the number of clients increases.

We can conclude that our scaling scheme can maximize the classifier’s
performance by prioritizing the local models that yield the best results.
Also, the platform performed effectively even with the low-cost devices
functioning as servers. Therefore, we can further reinforce the argument that
our design can be scalable and efficient even under resource constraints.
Since the Pis are modular and portable, our platform can cover remote
monitoring applications constrained by distance and mobility. Overall, this
shows the potential of our design as an effective and scalable approach that
can benefit wearable IoT device-based predictive healthcare even with resource
constraints.

###  IV-C Security and Adaptability Analysis

The following is an analysis of the security and adaptability of our design.
Each evaluation is from observations from our simulations and testing:

####  IV-C1 Security

Our design choices aim to preserve the data privacy of wearable IoT devices.
Federated learning allows us to keep raw data from leaving the local network.
Since only the models are transmitted, limiting potential leakages protects
the data of wearable IoT devices. With blockchains, our platform can create a
more exclusive network where only trusted devices can send data to the cloud
server. As a result, the hyperledger can reduce the endpoints that expose the
server to malicious attacks such as spoofing and impersonation. Also, this can
reduce the threat of Denial of Services (DoS) attacks. Since the blockchain
can regulate which devices can actively communicate with the server, it can
serve as a rate limiter for any targeted attack. Blockchains provide an extra
level of security due to their built-in access control and cryptographic
structure. Overall, each technology integrated into our platform can preserve
the data privacy of wearable IoT devices.

####  IV-C2 Adaptability

Based on our observations, our fog-based IoT architecture was able to aid in
implementing both federated learning and blockchain technology with ease. We
can attribute it to the decentralized structure of fog-IoT. Due to the fog
architecture, both technologies can perform optimally under this network
arrangement and utilize their strengths. Device diversity causes network
strains due to the constant demand for updates and maintenance. However,
federated learning reduces this strain by standardizing transmitted
information into a uniform package. This package is the locally trained model.
Lessening the demand for the cloud server to keep up with the latest wearable
device allows it to focus on deploying services efficiently while keeping up
with the most recent global knowledge. As long as the classifier structure
remains uniform, the server only needs to regenerate the same model with
different weights. As a result, the regeneration and aggregation become
modular to the service. This modularity contributes to the adaptability of the
network. It enables the use of other classifiers and wearable IoT device
groups.

Keeping the network decentralized through the fog also allows better
clustering. This design reduces the time required to update all its servers
and devices. For instance, take a heterogeneity equation $H$ based on the
distribution of worker update time $\phi_{w}$ presented in the following:

$H=1-\frac{1}{W-1}\sum_{w=1}^{W-1}\frac{\phi_{W}}{\phi_{w}},$ (2)

According to Eq. 2, a higher worker count $W$ lowers the overall heterogeneity
$H$ of the network. Also, we assume that $\phi_{W}=Min(\phi_{1}...\phi_{w})$.
This assumption means that the heterogeneity value of the network is
proportional to how spread out is the update times of each worker. The higher
this value $H$, the lesser the network is impacted by the updates times of its
devices. In a standard cloud IoT network, the number of workers equals the
number of wearable IoT devices it holds. As a result, the heterogeneity value
will be high due to the diversity of the wearable devices connecting to the
server. With varying update times, the
$\sum_{w=1}^{W-1}\frac{\phi_{W}}{\phi_{w}}$ term will be asymptotic to 0. When
plugged in the rest of the equation, the heterogeneity value is maximized and
asymptotic to +1. The closer this value is to 1, the more heterogeneous the
network. Our fog-based IoT approach allows us to reduce the impacts of the
update time of each worker by standardizing them. Instead of having each
wearable IoT device as a separate worker, they are clustered by the fog device
and become the new worker. With standardized devices, the
$\sum_{w=1}^{W-1}\frac{\phi_{W}}{\phi_{w}}$ term will be asymptotic to $W$.
When plugged in the rest of the equation, the heterogeneity value is minimized
and asymptotic to -1. The closer this value is to -1, the less heterogeneous
the network. Ideally, we aim to minimize the heterogeneity value to indicate a
network less affected by the diversity of its worker’s update times.

With our experiments, the update time is each fog device’s training and
communication time. We can observe in the testbed design that having a
distributive arrangement can reduce the impact of the diversity of wearable
IoT devices on the overall update time by offloading the training and
processing to the fog. Instead of many wearable IoT devices that can yield
varying update times, sending training data to the cloud server can be
standardized. So theoretically, we can infer that our platform will result in
a lower heterogeneity value. This analysis highlights the benefits of the
distributive approach that we proposed.

Also, the resulting accuracy of the global model presents how the precision of
the predictive healthcare service is not affected by the shift in the location
of the learning process. Since the edge device is only required to send data
securely, it does not need standardization as long as the fog device is aware
of the data it receives. Also, having one less complex process within a
sensing device makes its design more robust. Removing this constraint helps
the server keep up with the constant introduction of new wearable IoT devices
by simplifying the data flow within the network. This change creates a more
adaptable network that can cater to a diverse pool of wearable IoT devices.

###  IV-D Future Work and Recommendations

Further improvements to the design can be using other metrics for testing.
Another performance metric we can include for future iterations can be the
propagation delay between the fog and the cloud when transporting the training
model results. This addition introduces potential dynamic scenarios that
further test the mobility and reach of our platform. Another metric could be
the power consumption of the servers. The coverage of servers must reach
remote areas in extreme cases of predictive healthcare. As a result, more
portable and power-efficient designs are in demand for sustainability and
longer service uptime. This design addition introduces the potential design
and device optimizations that make the platform more cost-efficient while
managing the predictive healthcare service. Another improvement is to test the
platform with more than one neural network design and dataset. This addition
can further emphasize our platform’s modularity and performance under
different configurations.

##  V Conclusion

We propose a platform that addresses the issues of data privacy, service
integrity, and network structure adaptability of wearable IoT devices in
predictive healthcare. We used federated learning for its ability to
effectively aggregate local models into a global entity to ensure the
integrity of the predictive service. We further incorporated private
blockchain technology to reinforce the overall network security. Lastly, we
have fog-IoT as the base for offloading and process redistribution. By
evaluating the implemented federated learning system in terms of model
accuracy, we observed its feasibility in maintaining the integrity of the HAR
classifier. Next, we discussed the effectiveness of our platform even when
using a low-cost and low-end device as its fog and cloud server. Then, we
analyzed our design choices and highlighted its strengths in terms of privacy
preservation, security, and network adaptability. Overall, through our testing
and evaluation, we saw the feasibility and potential of our proposed platform
in addressing the security, integrity, and adaptive issues of wearable IoT
devices in predictive healthcare.

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