# Inducing Skyrmion Flop Transitions in Co8Zn8Mn4 at Room Temperature

Simon A. Meynell  These authors contributed equally.  Department of Physics,
University of California, Santa Barbara 93106 CA, USA  Yolita M. Eggeler
These authors contributed equally.  Materials Department and Materials
Research Laboratory, University of California, Santa Barbara CA 93106 USA
Laboratory for Electron Microscopy, Karlsruhe Institute of Technology, D-76131
Karlsruhe, Germany  Joshua D. Bocarsly  Materials Department and Materials
Research Laboratory, University of California, Santa Barbara CA 93106 USA
Daniil A. Kitchaev  Materials Department and Materials Research Laboratory,
University of California, Santa Barbara CA 93106 USA  Bailey E. Rhodes
Materials Department and Materials Research Laboratory, University of
California, Santa Barbara CA 93106 USA  Tresa M. Pollock  Materials Department
and Materials Research Laboratory, University of California, Santa Barbara CA
93106 USA  Stephen D. Wilson  Materials Department and Materials Research
Laboratory, University of California, Santa Barbara CA 93106 USA  Anton Van
der Ven  Materials Department and Materials Research Laboratory, University of
California, Santa Barbara CA 93106 USA  Ram Seshadri  Materials Department and
Materials Research Laboratory, University of California, Santa Barbara CA
93106 USA  Department of Chemistry and Biochemistry, University of California,
Santa Barbara CA -93106 USA  Marc De Graef  Department of Materials Science
and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA  Ania
Bleszynski Jayich<EMAIL_ADDRESS>Department of Physics, University of
California, Santa Barbara 93106 CA, USA  Daniel S. Gianola<EMAIL_ADDRESS>Materials Department and Materials Research Laboratory, University of
California, Santa Barbara CA 93106 USA

###### Abstract

Magnetic skyrmions are topologically-protected spin textures that manifest in
certain non-centrosymmetric ferromagnets under the right conditions of
temperature and field. In thin film skyrmion hosts, demagnetization effects
combined with geometric confinement can result in two distinct types of spin
textures: those with their axis of symmetry in the plane of the film (IP) and
those with their axis pointing out-of-plane (OOP). Here we present Lorentz
transmission electron microscopy evidence in conjunction with numerical
modeling showing a flop transition between IP and OOP skyrmions in Co8Zn8Mn4
at room temperature. We show that this skyrmion flop transition is
controllable via the angle of the external field relative to the film normal
and we illustrate how this transition depends on thickness. Finally, we
propose a skyrmion-writing device that utilizes the details of this
transition.

††preprint: APS/123-QED

##  I Introduction

Since their postulation in 1994 [1], the vortex-like quasiparticles known as
magnetic skyrmions, have been the focus of efforts to create high-information-
density, nanoscale digital storage devices [2]. The skyrmion is a nanoscale
[3] cylindrical magnetic field twist whose spins wind in a topologically
stable vortex structure [4]. The ability to manipulate skyrmions even at very
low current density [5, 6] has made them attractive for digital memory storage
applications due to the consequent low-power requirements [7, 8, 9].
Furthermore, topological protection can result in an extremely long lifetime,
which is one of the prerequisites for a bit in a digital memory element [10].
Another essential requirement for a useful digital memory element is the
ability to controllably write and delete bits on demand. The challenge of
writing and deleting skyrmions is a major obstacle to developing skyrmion-
based devices and is an ongoing scientific and technological area of
investigation [11, 12, 13].

Many of the proposed methods for writing, or nucleating, skyrmions involve
large current pulses that ease the local nucleation of a skyrmion [14, 15, 12,
16]. Despite many successful demonstrations of skyrmion injection by these
methods, developing defect-free nucleation mechanisms that do not rely on
driving large current pulses through the microstructure would broaden the
applicability of skyrmion-based devices and enable new device architectures.

Central to the physics of skyrmion nucleation and their stability is a chiral
term arising in the spin-spin Hamiltonian known as the Dzyaloshinskii-Moriya
interaction (DMI) that arises in the presence of broken inversion symmetry.
This term can stabilize a wide variety of long-range chiral magnetic orders
such as magnetic helices, conical structures, and skyrmions [17, 18, 1, 19,
20]. Among skyrmions in thin films stabilized by bulk DMI, there exist two
primary structure types [21] — skyrmions with their axis of symmetry within
the thin-film plane [22, 23, 24], which we refer to as in-plane (IP), and the
more conventional type whose axis of symmetry lies parallel to the film
normal, which we refer to as out-of-plane (OOP) [25]. While there is no
fundamental difference between these skyrmion types in bulk materials, the
large shape anisotropy of thin films causes these two skyrmion types to differ
in their energy densities, regions of phase stability, and behaviors with
respect to external stimuli.

Figure 1:  (a) Schematic of the experimental geometry. The angle, $\alpha$, is
defined as the angle between the film normal and the applied field. (b)
Schematic of an in-plane skyrmion in a semi-infinite film of thickness $t$.
The z direction of the coordinate system is normal to the film; x and y lie in
the film plane. (c) Magnetic phase diagram for bulk Co8Zn8Mn4 as a function of
applied magnetic field, $H$, and temperature, $T$. The helical, conical, and
skyrmion magnetic orderings dominate at low temperature but terminate in the
fluctuation-disordered (fd) regime as temperature increases. At high field and
high temperatures, the material saturates into a ferromagnetic ordering.

The relative stabilities of OOP and IP skyrmions, as well as the transitions
between the two types, have been minimally explored. The field direction plays
a large role in determining the stable orientation of the skyrmion; a large
field tends to destabilize skyrmions with their axis of symmetry orthogonal to
the field direction. For example, in [Pt/Co/Ta] multilayers, it was
demonstrated that tilting the applied field away from the thin film normal
destabilizes out-of-plane skyrmions and preferentially stabilizes a helical
state [26]. Separately, it has been shown that in non-centrosymmetric magnets,
the field angle controls a transition between a helical state and OOP
skyrmions [27].

We posit that in addition to a transition between helical and OOP skyrmions,
the field angle can be used to control the skyrmion axis of symmetry. By
sweeping through the field angle, skyrmions may be reoriented between IP and
OOP via a flop-transition, similar to the one predicted in Ref. [28]. Most
proposed skyrmion nucleation methods to date involve the injection of OOP
skyrmions because they have the preferred small size and well-established
methods of current manipulation necessary for racetrack memory storage.
However, if skyrmion flop transitions could be controllably induced, then
devices that nucleate IP skyrmions could be used as a source for OOP
skyrmions.

In this contribution, we report direct observations of flop transitions
between in-plane and out-of-plane skyrmions by systematically varying the
magnetic field _in situ_ within a transmission electron microscope (TEM). Our
specimen geometry enables us first to nucleate in-plane skyrmions from a bulk
topological charge reservoir and then exploit a flop transition to switch
between in-plane and out-of-plane skyrmions controllably. We use a combination
of a tilted magnetic field and a variable film thickness to elucidate the
stability limits of the two skyrmion geometries and, consequently, identify
the conditions for the flop transition. Informed by these observations and
corroborating magnetic simulations, we propose a novel and potentially
scalable racetrack memory component that benefits from both orientation types
of skyrmions.

##  II Methods

The bulk skyrmion host CoxZnyMnz ($x+y+z=20$), crystallizing in the
$\beta$-Mn-type structure, generally supports skyrmions across a wide range of
stoichiometries. Previous investigations have revealed that room temperature
skyrmion formation is limited to a narrow composition window centered around
Co8Zn8Mn4 [29, 30, 25]. A polycrystalline ingot of this composition was
prepared from elemental starting materials of Co powder, Zn shot, and Mn
pieces, which were weighed with an 8:8:4 stoichiometry totaling a mass of 5 g
following a procedure reported in previous investigations [29, 30]. The metals
were sealed in an evacuated silica ampule and heated to 1,000∘C for 25 hr and
then cooled to 925∘C at a rate of 1∘C$\cdot$hr-1. Finally, the sample was
homogenized at 925∘C for 48 hr. before quenching to room temperature in a
water bath. The magnetic phase diagram near room temperature of the bulk
sample was inferred using AC magnetic susceptibility measurements, as has
previously been performed in $\beta$-Mn structure skyrmion hosts [29, 31]. AC
susceptibility as a function of applied DC magnetic field (0 $<H<$ 50 mT) was
measured using a Quantum Design MPMS3 Magnetometer at fixed temperatures
ranging from 310 K to 274 K. The phase boundaries between helical, conical,
skyrmion, conical, and ferromagnetic states were determined from alternating
local maxima and minima in plots of $d\chi$/$dH$ _vs._ $H$ as previously
described [31].

The Co8Zn8Mn4 TEM specimen was prepared by micromachining from the
polycrystalline ingot, using a focused ion beam incorporated within an FEI
Helios Dual-beam Nanolab 600. A 15 $\mu$m x 7 $\mu$m rectangular specimen was
extracted from the Co8Zn8Mn4 bulk sample with an initial thickness of 400-500
nm and attached to a TEM Cu support grid for further thinning down to electron
transparency at an average thickness of $90\text{\,}\mathrm{nm}$. The thinning
procedure resulted in a wedge-shaped specimen geometry attached to a bulk
reservoir as schematically demonstrated in (Fig. 1(a)), with the thickness of
the thin film ranging from $160\text{\,}\mathrm{nm}$ to
$90\text{\,}\mathrm{nm}$, measured from cross-sectional imaging using
secondary electrons at 10 keV.

Lorentz TEM (LTEM) was used to observe the magnetic structures as a function
of the magnetic field using two approaches. Fig. 1(a) shows the sample and
field geometry for this experiment. The first experiment used a fixed specimen
tilt angle $\alpha$ and varied the applied magnetic field strength, $H$, by
changing the objective lens strength on a FEI ThemIS 60-300 TEM operating in
Lorentz mode at 200 kV. The magnetic field as a function of the objective lens
excitation was calibrated using a Hall probe TEM holder. The second set of
experiments varies $\alpha$ at a fixed $H=$120\text{\,}\mathrm{mT}$$ generated
by a weakly excited objective lens operating at low magnification in a Thermo
Fisher Talos 200FX TEM operating at 200 keV. The behavior of the magnetic
structures was observed by obtaining a series of Fresnel images at large
defocus conditions in the range of several mm. A through-focus image series
was reconstructed to generate magnetization maps using an electron wave phase
retrieval method that employs the transport-of-intensity equation [32]. This
approach uses the defocused image series at under- and overfocused conditions
(centered at zero defocus) to recover a phase map, which in turns enables the
calculation of phase gradients, which are related to the in-plane components
of the integrated magnetic induction [33].

We use MuMax3 [34] to model the impact of thin-film anisotropy on the
energetics of IP and OOP skyrmion states. The material parameters used in the
calculations are
$D_{\textrm{bulk}}=$5.3\text{\times}{10}^{-4}\text{\,}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$$,
$M_{\textrm{sat}}=$410\text{\,}\mathrm{k}\mathrm{A}\mathrm{/}\mathrm{m}$$, and
$A_{\textrm{exch}}=$4.6\text{\times}{10}^{-12}\text{\,}\mathrm{J}\mathrm{/}\mathrm{m}$$.
The size of the numerical system is $32\times 32\times 32$ cells with a cell
size of $7\text{\,}\mathrm{nm}$. Periodic boundary conditions are enforced in
the XY-plane.

To calculate the competing energetics of OOP and IP skyrmions, we initialize
their magnetization using an idealized Bloch skyrmion, where the magnetization
points along z (x) for OOP (IP) skyrmions. Figure 1(b) shows an example of a
simulated IP skyrmion. For OOP skyrmions, $H$ is set along the z-direction,
and for IP skyrmions, $H$ is tilted at an angle $\alpha=15\degree$ from the
z-axis towards the x-axis. The magnetic structure is then relaxed using the
conjugate gradient method to arrive at a stable configuration of spins. Then
$\alpha$ is gradually varied in steps of $0.25\degree$, with a conjugate
gradient minimization of the spin structure under each condition. We repeat
this process for different field magnitudes
($$50\text{\,}\mathrm{mT}$-$150\text{\,}\mathrm{mT}$)$ and film thicknesses
($$60\text{\,}\mathrm{nm}$-$200\text{\,}\mathrm{nm}$$).

##  III Results and Discussion

As has previously been observed with bulk magnetic measurements and LTEM
measurements, Co8Zn8Mn4 can host skyrmions at room temperature under a
moderate applied magnetic field. Figure 1(c) depicts the bulk magnetic phase
diagram of our sample of Co8Zn8Mn4 inferred using AC magnetic susceptibility,
which is consistent with previous reports [29]. The skyrmion magnetic phase
competes with other chiral magnetic orderings: at very low applied fields, a
helical magnetic phase is observed. The helical phase transitions to a conical
phase upon application of larger fields in which the helical moments cant in
the direction of the applied field before eventually reaching a fully field-
polarized ferromagnetic state. Skyrmions exist as the stable phase in a small
pocket at moderate magnetic fields ($\approx$$10\text{\,}\mathrm{mT}$) within
the conical phase space. Here we directly confirm the existence of skyrmions
in thin-films of Co8Zn8Mn4 via LTEM imaging.

To describe the geometry of skyrmions within a thin film of thickness $t$ in
Co8Zn8Mn4, we adopt the coordinate system shown in Figure 1(b), where the
$z$-axis is parallel to the film normal and the $x$-axis points along the in-
plane component of the external field. For example, an in-plane skyrmion (IP)
has its axis within the $x-y$ plane of the film as depicted in Fig. 1(b). In
the bulk limit, $t\rightarrow\infty$, the skyrmion axis of symmetry is
parallel to $H$. We note that magnetocrystalline anisotropy is not considered
because of the polycrystalline nature of our films: in our images, we see
multiple grain boundaries, and the magnetic textures are uninterrupted across
them, indicating shape anisotropy dominates.

In a thin sample, the shape anisotropy leads to a large demagnetizing field
that changes the relative stability of IP and OOP skyrmions as a function of
$\alpha=\arctan(H_{z}/H_{x})$. Generally, the demagnetizing field in a thin
film suppresses the out-of-plane component of the magnetization [35]. Since
skyrmions have a net magnetization along their high-symmetry axis, this
phenomenon is expected to decrease the critical angle $\alpha^{*}$ of the
skyrmion flop transition. This trend persists as long as the film thickness
$t$ is larger than the skyrmion diameter (given by the material’s helical
pitch $\lambda$). For $t<\lambda$, the IP phase is suppressed by confinement.
We leverage this confinement-induced shape anisotropy effect to study the
skyrmion flop transition using the variable-thickness sample geometry
illustrated in Figure 1(a). By varying $\alpha$ and measuring over different
regions of the sample, we can explore the effects of both thickness and
angular dependence for the skyrmions observed in the Co-Zn-Mn sample.

###  III.1 Constant angle skyrmion nucleation

For novel data storage technologies, the OOP skyrmion has been the preferred
focus of investigations because of its small diameter, allowing for high
information density, and the existence of established techniques for current-
driven spatial manipulation. Most observations of skyrmion nucleation are of
the OOP variety. Here, we inject IP skyrmions from a bulk reservoir into a
thin film and subsequently nucleate OOP skyrmions using the OOP-IP flop
transition. This technique forms the basis for our proposed hybrid device
discussed in section III.3.

Figure 2:  (a) LTEM sequence of images of a wedge of CoZnMn with the applied
field held at a constant angle ($\alpha=9\degree$) while increasing magnetic
field strength B. For small fields, the in-plane skyrmions only extend part
way into the film. As the field strength increases, the skyrmions contract in
size and thus can reach into the thinner regions of the film. (b) A proposed
mechanism for in-plane skyrmion nucleation. The large shape anisotropy in the
wedge section suppresses the out-of-plane component of the tilted external
field, leaving only the in-plane component, stabilizing the in-plane
skyrmions.

Figure 2 shows a series of defocused LTEM images of the wedge sample at
varying field strengths with a fixed tilt angle $\alpha=9\degree$. As will be
shown later, this $\alpha$ is large enough such that IP skyrmions will always
be favored in the geometry of our sample. The left side of the images is near
to the bulk material with the foil thinning towards the right. At a low field
of $H=$30\text{\,}\mathrm{mT}$$, we observe stripes with a mean spacing of
$\sim$140\text{\,}\mathrm{nm}$$ emerging from the bulk reservoir on the left
and extending toward thinner regions of the film as the field increases. The
stripes have two possible interpretations: this texture is either a 1D lattice
of IP skyrmions, or it is a helical/conical state with $Q\cdot B_{x}=0$. We
infer that these are IP skyrmions because the in-plane component of the film
is pointing along the skyrmion axis. Were this a Bloch-type helical/conical
phase, it would be energetically favorable for the wave vector of the helix to
point along the in-plane field component (_i.e._ the stripes would appear
perpendicular to the direction in which we observe them). We further compare
the reconstructed magnetization to our data and find good agreement between a
simulated in-plane skyrmion and the LTEM image.

For low fields, IP skyrmions extend through the thicker regions of the film
but stop when the film thickness becomes small relative to the skyrmion size
as shown in the $H=$30\text{\,}\mathrm{mT}$$ subimage of Fig. 2(a). As we
increase the field to $60\text{\,}\mathrm{mT}$ we see an increase in skyrmion
density, which we attribute to the effect of decreasing skyrmion size with
increasing external field [36]. The reduction of skyrmion size also allows for
them to extend further into thinner regions of the film. At fields of
$H=$110\text{\,}\mathrm{mT}$$ and $120\text{\,}\mathrm{mT}$ (which correspond
to $H_{z}$ =19 mT), IP skyrmions begin to disappear altogether, saturating
into either a ferromagnetic or an out-of-plane cone state, as is suggested by
the magnetic phase diagram in Figure 1(c).

The proposed mechanism for the in-plane skyrmion nucleation we observe is
shown in Fig. 2(b). Here, the skyrmions that run vertically through the bulk
reservoir are interrupted, where they transition from the bulk and are drawn
into the wedge. Based on this model, we propose that the reservoir acts as a
topological charge bath for the attached thin-film wedge, providing a means to
overcome the large energy barrier associated with skyrmion nucleation. Having
established the nucleation and migration of in-plane skyrmions with increasing
field magnitude but fixed orientation, we next investigate the behavior of the
skyrmion flop transition as a function of film thickness and external field
angle.

Figure 3:  (a) LTEM image sequences of a CoZnMn TEM wedge sample for an out of
plane field of $B=$120\text{\,}\mathrm{mT}$$ and varying film angle, $\alpha$.
At high angles, the perpendicular component of the field suppresses the OOP
phase and the IP is present in the thicker regions of the film as indicated by
the lack of OOP on the left side of the film and the lack of IP on the right
side of the film for all values of $\alpha$. Close to $\alpha=4\degree$, the
IP undergoes a flop-transition into an OOP phase that is only stable in the
thinner regions of the wedge. The colormap in the $\alpha=1.8\degree$ image
shows the phase reconstruction for OOP skyrmions as detailed in the
supplemental information. The color indicates the direction of the IP
magnetization and confirms the Bloch character of the skyrmions. (b)
Simulations performed via mumax3 showing the energy dependence as a function
of $\alpha$ for different thicknesses of the wedge. For intermediate
thicknesses ($t=$150\text{\,}\mathrm{nm}$$), the simulations reveal a
crossover angle where the IP phase will flop to the OOP variety, which agrees
with our experimental observation that the skyrmions flop in the intermediate
film thickness region. (c) Equilibrium simulations of IP and OOP skyrmions
with a tilted field in a geometrically confined thin film.

###  III.2 Skyrmion flop transition

The principle of skyrmion axis reorientation was predicted by Vlasov et al. in
Ref. [28], where the authors showed that skyrmions can change orientation via
an intermediate state with two mutually orthogonal skyrmions. Figure 3(a)
shows a series of LTEM images that demonstrates the flop transition as a
function of the tilt angle and film thickness. The thickness of the TEM foil
was measured from cross-section orientation images acquired by SEM and is
labeled by the vertical dotted lines at the top of Figure 3(a). The leftmost
part of the image is the thickest region of the foil, which connects to the
bulk reservoir.

An applied field of $120\text{\,}\mathrm{mT}$ is aligned along the film normal
and swept between angles of $\alpha=15\degree$ \- $0\degree$. A video
demonstrating the magnetic texture evolution as the field is continuously
tilted is provided in the online supplementary material. For high tilt angle,
$\alpha=12.2\degree$, IP skyrmions nucleate at the thicker side of the film,
close to the bulk reservoir (Fig. 3(a)). In these thicker regions of the film,
all skyrmions observed are in-plane. The IP skyrmions exhibit capped ends,
similar to those observed by Refs [37, 38]. For $\alpha=7.9\degree$, the IP
skyrmions move into thinner regions of the film. As the field angle approaches
that of the film normal at $\alpha=3.9\degree$, we approach the critical
angle, $\alpha^{*}$. We find that as $\alpha$ decreases, the IP skyrmions
shift toward thinner regions of the film, where the transition between the two
types is observed as a mixture of IP and OOP skyrmions in the image. For the
small angle of $\alpha=1.8\degree<\alpha^{*}$, the image shows only small,
pearl necklace-type OOP skyrmions.

The magnetization map inset in the $\alpha=1.8\degree$ image shows a vortex of
magnetic moments, confirming the chirality and Bloch-like character of the OOP
skyrmions. The magnetization maps are generated from the LTEM images via a
procedure detailed in the SI.

Micromagnetics calculations shown in Figure 3(b) plot the expected impact of
sample thickness on the skyrmion orientation by calculating the energy density
of the two orientations of skyrmions as a function of applied field angle for
three different film thicknesses—$190$, $150$ and $100\text{\,}\mathrm{nm}$.
The results of the numerics agree with the experimental data in Fig. 3(a): the
calculations predict IP to be the stable phase for large film thicknesses
while for thin thicknesses OOP becomes the favorable phase due to confinement
effects. Notably, for intermediate thicknesses, the micromagnetics predicts a
critical angle, $\alpha^{*}$, where the energies of the two orientations are
the same, and the skyrmion flop transition can therefore occur. Examples of
the equilibrium magnetization distributions used to generate the phase diagram
and curves are depicted in 3c, for both OOP and IP skyrmions, respectively.

Interestingly, all OOP phases observed in these experiments were arranged
together into a 1D chain, suggesting the presence of an attractive skyrmion-
skyrmion interaction. One possible explanation for this morphology could be
stabilization via the ‘zip-locking’ mechanism proposed by Vlasov _et al._
[28], where an IP guides the OOP skyrmions along an axis.

Figure 4:  A schematic for a skyrmion writing component utilizing the
mechanisms demonstrated in this work. The blue material represents the
Co8Zn8Mn4 material, the gold represents a current carrying stripline, the
orange arrows represent an applied field generated by the striplines and the
red represents a skyrmion core. (a) The first step is to inject in-plane
skyrmions through the mechanism described in Fig. 2(a). Gold wires may be used
to trim the field strength and angle. (b) The injected IP skyrmions may be
transformed into OOP skyrmions via the flop transition shown in Fig. 3(a). The
inset shows how the field is tilted with respect to the wedge to induce the
IP-OOP flop. (c) After the OOP skyrmions are formed, they can be manipulated
along the track through an applied current.

Wang et al. [27] demonstrate a skyrmion nucleation method similar to the one
illustrated here with a few key differences. First, they are studying a
centrosymmetric material without DMI, and second, they show a transition
between magnetic helix to skyrmion-in-helix states.

###  III.3 Device governed by skyrmion flop transition

We have shown two properties of the Co8Zn8Mn4 alloy shaped as a thin wedge
sample in a tilted field. First, we have observed the generation of IP
skyrmions near the reservoir-film interface (Figure 2). Second, we have
demonstrated that depending on the host film thickness, the IP skyrmions can
undergo a flop transition into the OOP orientation induced by sweeping the
field angle (Figure 3). Based on these two principles, we propose a new
concept for a skyrmion generation writing component in a racetrack memory
device shown in Figure 4. The operating principle of the device relies on the
skyrmion injection method from a bulk reservoir, followed by a field-tilt-
induced flop transition from IP to OOP in a thin region of the attached wedge-
shaped wire. To control the field magnitude and the field angle, two gold
striplines close to the region of interest are proposed.

The first step in generating skyrmions would be to apply a tilted field far
from the critical angle for the flop transition. Using 30 mT as in Figure 2(a)
will be sufficient to produce a semi-localized IP phase in the wedged portion
of the device. Subsequent trimming of the field may change the field angle or
magnitude, flopping the IP into OOP, which is the desired skyrmion
configuration for high densities of nanobits, schematically shown in Figure
3(b).

Finally, with the stable OOP skyrmions generated along the device, a current
may be applied along the wedge illustrated by the green arrow in Figure 4(c).
Due to skyrmion motion under an applied current [39, 40, 12], the nucleated
skyrmions can be transported along the track, for example to a reader head.
This reservoir-wedge configuration then acts as a ‘skyrmion-writer’ element of
a larger skyrmion memory device.

##  IV Conclusions

We demonstrate the existence and regions of stability for in-plane skyrmions,
IP, in thin-film Co8Zn8Mn4 as well as show the field-angle induced skyrmion
flop transition from an IP to an OOP spin-texture. In-plane skyrmions, while
less studied than their out-of-plane counterparts, provide a path toward
controlled nucleation of skyrmions in a geometrically confined reservoir-wedge
configuration and a room-temperature device. We show that the flop transition
is repeatable across two different devices and that control over field angle
and wire geometry provides a method of switching between the types. Lastly, we
propose a component of a larger race-track device utilizing these principles
that could pave the way for current-controlled writing of skyrmion bits in
drastically reduced scales and low-power data storage components.

##  V Acknowledgements

This work was supported by the Materials Research Science and Engineering
Center (MRSEC) program of the National Science Foundation (NSF) through
DMR-1720256 (IRG-1), and employed the shared facilities of the MRSEC at UC
Santa Barbara, a member of the Materials Research Facilities Network. Work at
the Molecular Foundry was supported by the Office of Science, Office of Basic
Energy Sciences, of the U.S. Department of Energy under Contract No. DE-
AC02-05CH11231. SAM acknowledges the support of the Natural Sciences and
Engineering Research Council of Canada (NSERC), [AID 516704-2018] and the UCSB
Quantum Foundry.

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# Streamlined approach to mitigation of cascading failure in complex networks

Karan Singh  School of Physics, Indian Institute of Science Education and
Research Thiruvananthapuram, Thiruvananthapuram, 695551, Kerala, India.  V.K.
Chandrasekar  Department of Physics, Center for Nonlinear Science and
Engineering, School of Electrical and Electronics Engineering, SASTRA Deemed
University, Thanjavur, 613401, Tamil Nadu, India.  D.V. Senthilkumar
<EMAIL_ADDRESS>School of Physics, Indian Institute of Science Education
and Research Thiruvananthapuram, Thiruvananthapuram, 695551, Kerala, India.

###### Abstract

Cascading failures represent a fundamental threat to the integrity of complex
systems, often precipitating a comprehensive collapse across diverse
infrastructures and financial networks. This research articulates a robust and
pragmatic approach designed to attenuate the risk of such failures within
complex networks, emphasizing the pivotal role of local network topology. The
core of our strategy is an innovative algorithm that systematically identifies
a subset of critical nodes within the network, a subset whose relative size is
substantial in the context of the network’s entirety. Enhancing this
algorithm, we employ a graph coloring heuristic to precisely isolate nodes of
paramount importance, thereby minimizing the subset size while maximizing
strategic value. Securing these nodes significantly bolsters network
resilience against cascading failures. The method proposed to identify
critical nodes and experimental results show that the proposed technique
outperforms other typical techniques in identifying critical nodes. We
substantiate the superiority of our approach through comparative analyses with
existing mitigation strategies and evaluate its performance across various
network configurations and failure scenarios. Empirical validation is provided
via the application of our method to real-world networks, confirming its
potential as a strategic tool in enhancing network robustness.

Mitigation, Cascading Failure, Complex Network, Graph Theory.

††preprint: APS/123-QED

## Introduction

Our everyday routines rely extensively on the operation of numerous natural
and artificial networks, including neural and genetic regulatory networks, as
well as communication systems, social networks, transportation networks, and
electric power grids [1, 2, 3]. However, these networks are susceptible to
cascading failures, where the failure of a single node can lead to a domino
effect, causing widespread breakdowns and disruptions in the entire network
[4]. Understanding the resilience of these networks to random failures and
targeted attacks is crucial for avoiding system failures with serious
consequences [5]. Cascading phenomena can have either beneficial or adverse
effects, depending on their practical usefulness. Why do certain works, such
as books, movies, and albums, gain widespread popularity despite limited
marketing efforts [6], while similar endeavors fail to attract attention? Why
do fluctuations occur in the stock market without any apparent significant
news driving them [7]? How do grassroots social movements originate without
centralized coordination or widespread public communication [8]? These
occurrences exemplify what economists term as information cascades [9], where
individuals in a population exhibit herd-like behavior, basing their decisions
on the actions of others rather than on their own information about the issue.
Studies have also looked at how networks of adopters grow, particularly
through social contagion [10, 11] processes like adopting opinions, behaviors,
emotions [12, 13, 14, 15] or innovations, which can happen quite rapidly under
the influence of social pressure. Social influence is significant in this
scenario, with individuals being influenced by the viewpoints of their peers
[12, 13]. This impact is represented in threshold models [16, 4, 17], that
propose an individual becomes an adopter when the proportion of their already
adopting neighbors reaches a critical level specific to their sensitivity.
Such phenomena underscore the complexities of human behavior and decision-
making processes, particularly in contexts where social influence plays a
significant role. One of the examples of the collapse of the top Online Social
Networks(OSNs) is the Hungarian social networking site. The Hungarian social
networking site iWiW [18] was operational from 2004 to 2013. In late 2010,
there was a noticeable increase in the number of users departing from iWiW,
resulting in significant loss and ultimately leading to the collapse of the
entire network. The main reason cited [19] for this downfall was that
individuals left when most of their friends had already departed.
Understanding the cascade effect is crucial for achieving various applications
such as marketing, advertising, and spreading information. Identifying
critical nodes where resources can be strategically allocated to effectively
disseminate information across the network is key. However, these processes
can also have negative implications in areas like finance, healthcare, and
infrastructure by causing undesired or detrimental outcomes for the community.
Therefore, it is important to mitigate the cascading failure in complex
networks to minimize the potential negative consequences and ensure the
stability and resilience of these systems.

Figure 1: Illustration of the fractional cascade process. Node size
corresponds to its degree, with only labelled nodes relevant to this
description. (a) The gradient (gray color) indicates the impacted node to be
removed. (b) After removal, its edges are disconnected and eliminated. (c)
Nodes are classified as high degree (1), low degree (6 and 7), and medium
degree (3, 4, and 5). Low-degree nodes with few neighbors have minimal impact
on the cascade. High-degree nodes remain unaffected by a single neighbor’s
failure, while medium-degree nodes are susceptible to fractional failure and
possess significant neighbors that will be impacted by their failure, hence
intensifying the cascade. This subset of nodes exacerbates the cascading
failure process. (d) to (h) depict the cascade progression, ultimately leaving
only the high-degree node unaffected and surviving the failure.

In this work, we introduce a refined strategy for mitigating cascading
failures, leveraging local environmental information of nodes, specifically
focusing on the node’s nearest neighbors [20]. Our novel methodology involves
the identification and enhancement of fragile nodes, the safeguarding of which
ensures near-complete network survivability. To achieve this, we utilize graph
coloring to ascertain the critical average degree, employing it as a threshold
for refining the set of fragile nodes. Our investigation demonstrates that the
proposed strategy is optimized in terms of both the fraction of fragile nodes
and the probability of survival.

The effectiveness of our approach is first evaluated in theoretical models,
namely Erdős-Rényi (ER) networks [21] and Scale-free networks [22], which
emulate various real-world network structures. Subsequently, we assess its
applicability in practical scenarios by demonstrating its efficacy in social
networks, collaboration networks, and the US power-grid network. Our method
not only adeptly mitigates cascading failures in simple theoretical models but
also exhibits robustness in intricate real-life network contexts. We
illustrate the versatility of our approach across a wide spectrum of network
topologies, emphasizing its adaptability to more complex settings. Our
findings underscore the efficacy of the proposed method in averting network
collapse across diverse types of networks. Understanding the dynamics of
cascading failure in complex networks is crucial for developing effective
mitigation strategies. As we continue to rely on and integrate complex
networks into daily life, understanding and safeguarding against cascading
failures becomes increasingly imperative.

Figure 2: Visualization of the mitigation scheme. (a) The graph is colored
(ignoring the unlabelled nodes as they are not relevant). We are applying
color to the periphery of the nodes instead of fully coloring them. (b)
Dotted-circumferenced nodes (2, 3, 4, 5, 6, and 7) are identified as fragile
nodes under the fractional failure mechanism. (c) Among the fragile nodes, the
green gradient nodes (2, 3 and 5) with degrees exceeding the average lowest
degree (critical degree) of color groups are selected for protection,
designated as immune nodes. When node 2 is impacted (refer to Fig. 1), its
inclusion in the protected set prevents its removal. Consequently, its fragile
neighbors (3 and 5) remain unaffected, and so on, ensuring the system’s
complete survival.

## The Models of Cascading Failure

Many different failure models have been extensively studied and applied for
specific purposes in simulating the propagation of failures. The k-core [23,
24]method and the fractional threshold are two common approaches heavily used
to represent real-life scenarios. In the k-core method, the threshold is
determined by the absolute number of active neighbors, making it particularly
applicable in epidemiological settings where an individual’s actual number of
contacts is crucial. On the other hand, when the fraction of active neighbors
holds more significance than their absolute number, it becomes known as the
fractional threshold [25, 4, 26, 27] model. This model finds application in
various fields such as opinion formation, social dynamics, infrastructures,
economics, and finance. In the context of opinion formation, an individual
adopts a certain behavior if a fraction $m$ of its friends $k$ adopt it. In
this case, the threshold is $m/k$ and not $m$. This results in individuals
with many friends needing more of their neighbors to become adopters to change
their state. This process is illustrated in Fig. 1.

Watts [4] presented a simple model on global cascades, implementing the
fractional threshold model for infinitesimally small impact. Later, Gleeson
and Cahalane [28] modified it for finite seed size and studied its effects on
the cascades.

We used the Gleeson et al. [28] framework for analytic calculations of
cascading failure processes. Each node, having degree $k$ with the degree
distribution $p_{k}$ such that $\sum p_{k}=1$, is an agent that belongs to the
undirected network and can be in a binary state, called active or inactive.
All agents are assigned a frozen random threshold $r$ chosen from the
distribution with $F(r)$ denoting the probability that an agent has a
threshold less than $r$. The cascade is initiated by seeding the network by
activating a randomly chosen fraction of nodes $\rho_{0}$ out of total $N$
nodes. Nodes update their state, and the average final fraction $\rho$ of
active nodes is given by [28]

$\rho=\rho_{0}+(1-\rho_{0})\sum_{k=1}^{\infty}p_{k}\sum_{m=0}{k}\begin{pmatrix}k\\\
m\\\ \end{pmatrix}q_{\infty}^{m}(1-q_{\infty})^{k-m}F\left(\frac{m}{k}\right)$
(1)

where $q_{\infty}$ is the steady state or fixed point of the recursion
relation

$q_{n+1}=\rho_{0}+(1-\rho_{0})G(q_{n})\hskip 28.45274ptn=0,1,2,\dots$ (2)

and the generating function $G$ is defined as

$G(q_{n})=\sum_{k=1}^{\infty}\frac{k}{z}p_{k}\sum_{m=0}^{k-1}\begin{pmatrix}k-1\\\
m\\\ \end{pmatrix}q_{n}^{m}(1-q_{n})^{k-1-m}F\left(\frac{m}{k}\right)$ (3)

Here $z$ is the network’s average degree $z=\sum kp_{k}$.

Figure 3: Critical Mass: In every network exists a critical mass
$\Omega_{critical}$ where numerous vulnerable nodes are interconnected. These
nodes only require one of their friends to activate them so they can also
switch on. Just beyond the critical mass component lies the triggering
component $\Omega_{trigger}$, where certain nodes are connected to critical
nodes. When these connecting nodes (depicted as red nodes in Fig. 3) become
active, they trigger someone within the critical component and initiate a
network collapse. $\Omega_{final}$ is the fraction of failed nodes after the
cascade is settled, and $\Omega_{network}$ is the entire network.

## Methods

### Mitigation strategy

As discussed in a recent study [20], our focus will be on identifying critical
nodes that play a key role in propagating cascading effects instead of
measuring a node’s importance based on the damage it causes when removed. As
shown [20], the node’s importance in terms of the damage it causes doesn’t
help from the mitigation perspective beacause at a critical threshold, even an
exceedingly small impact [20] (as minor as removing a single node) could
initiate (see Fig. 3) a cascade effect leading to network collapse.

We adhere to the concept [20] of a fragile node within a failure mechanism,
$M$, wherein the failure of the node occurs upon the removal of a minimal
significant number, $K$, of its links. The fractional failure under the
removal of a single edge takes the form $(k-1)/k$, and the node will be
considered to fail if $(k-1)/k<\theta$. We will now present the algorithm to
select the potent nodes based on the basic percolation argument: to facilitate
the cascade, the node must be fragile under the fragility condition (mentioned
above) and possess at least some fragile neighbours.

Algorithm: Outputs nodes to be protected
---
1. Scan each node 2. Check if it is fragile as per the above condition 3. If yes, check if it has at least two fragile neighbours 4. Color the graph and calculate the average degree of each color subgroup 5. Protect only those fragile nodes (fulfilling 2nd and 3rd condition) with degrees greater than the lowest average degree color group.

The first two steps are basics and are to determine the fragile nodes, with
the rationale being that, as per the fragility defined above, i.e., a node
fails under the removal of the single edge, but if the node remains intact, it
is unlikely to be affected at the initial cascade stage as it is not fragile
enough as per the definition. Additionally, the cascade doesn’t aggravate if
the node’s fragile neighbours are fewer than two in number. The last two steps
are to reduce/refine the number of fragile nodes. It’s found (see the Results
section) that there is no need to protect all the fragile nodes, but
protecting a fraction(which we achieve after the application of our last step)
would ensure the full survivability of the network. We refer to this
reduced/refined set as immune nodes (the set of nodes to be protected). We use
the graph coloring [29] technique to reduce the number of fragile nodes (more
discussion in the Result section). Upon examining the algorithm, it is
observed that nodes with high degrees are not suitable for inclusion in the
immune set because they remain resilient even under very high thresholds. On
the other hand, low-degree nodes become fragile relatively quickly, but they
do not effectively transmit the shock and, therefore, cannot be considered for
the immune set. The ideal candidates for immune nodes lie between high and low
degrees – these medium-degree nodes are sufficiently weak to be fragile and
well-connected enough to propagate the shock significantly. Selecting this
group of nodes offers an effective mitigation strategy due to their limited
number.

Figure 4: Critical degree: Coloring the graph transforms it into $k$-partite
graph. Its vertices are partitioned into $k$ different independent sets (we
refer to them as color groups), where $k$ is the chromatic number, for
example, $k=2$ is the bipartite graph. $<k>_{c}$ is the lowest average degree
among the color groups. Here, we have taken a $k=3$ example for the
demonstration purpose .

## Results

### Application on standard networks

The primary objective of the algorithm is to pinpoint the critical nodes in
terms of failure propagation. The selection of nodes relies on understanding
their local surroundings, making it a highly effective mitigation strategy
with minimal and readily accessible information. Using this group of
vulnerable nodes, we apply graph coloring [29] techniques to minimize this
number and create a more cost-effective strategy. We refer to this subset as
immune nodes (as the nodes to be protected; see green nodes in Fig. 2).
Coloring the graph and dividing it into subgroups (colour groups), allows us
to determine the connectivity of a fragile node with the other colour groups
based on the average degree of that particular colour group. For example,
suppose the average degree of a certain colour group is high. In that case,
each node is highly connected with other colour groups but not at all
connected among the same colour group. If this node is determined to be
fragile, upon removal of this node, it will affect the other nodes in
different color groups, thereby endangering their fractional threshold.
Therefore, by choosing a threshold above which the node designated for
protection possesses the lowest average degree among the color groups, we
effectively and automatically safeguard the fragile nodes characterized by
high interconnectivity between different color groups.
We first apply our algorithm to the standard network models, Erdos-Renyi and
Scale-free networks, to demonstrate its effectiveness. Figures 5 (a) and (c)
show the survivability of the system against the fractional threshold. The
value on the y-axis represents the probability of the system survival, and
value on the x-axis denotes the fractional threshold, that means the fraction
of functional neighbors required for a node in question for its survival. The
blue line represents full protection (1.0), which means protecting all the
nodes (immune nodes) selected by our algorithm. The dashed pink line,
corresponding to the full protection case, indicates the fraction of protected
nodes compared to the entire network. The orange (0.7) and green (0.4) lines
indicate partial protection - that is, protecting the immune nodes 70% or 40%
of the times respectively. Finally, the red line shows no protection; it
depicts how the system evolves without any form of safeguarding in place. The
system always survives until the fractional threshold of 0.86, regardless of
the protection. Therefore, for the ER case, 0.86 is considered the critical
threshold. As for scale-free networks, it’s close to 0.835. Near the critical
threshold, it just requires a small fraction of nodes to be protected for the
complete survival of the network, as shown by the mitigation strategy (blue
line) and basic cascading process (yellow line) in Figures 5 (a) and (c).

Figures 5 (b) and (d), for ER and Scalefree networks respectively, show the
system’s survivability for all protection probabilities plotted for four
different fractional thresholds depicted by four colours. The blue lines
correspond to a fractional threshold that is lower than their respective
critical threshold, ensuring the system always survives. As the threshold
increases, the probability of survival decreases for the orange, green, and
red lines. Increasing the protection probability enhances the survivability of
the system. To guarantee that the system always survives, it is necessary to
protect the immune nodes with a protection probability of 1.0 (as depicted in
the Fig. 5 (b) and (d)).

We have demonstrated the efficacy of our algorithm on ER and Scalefree
networks. It identifies the fragile nodes and substantially reduces their
number, obtaining what we call immune nodes. These immune nodes are designated
for the protection and ensuring the full survival of the system. To show that
the reduce/refine step of our algorithm is optimised in terms of both survival
probability and the protected node size, we scale the critical degree
$<k>_{c}$ (See Fig. 4) by lowering and raising its value to test whether this
automatic selection of the critical degree threshold is optimised or not. In
Figure 6, the critical degree threshold for (a) ER and (b) Scale-free networks
is scaled. The green solid line in Fig.6(a) represents the survival
probability for a fully protected network with the original critical threshold
degree, while the corresponding green dashed line denotes the protected set.
The blue and yellow lines indicate thresholds scaled to 95% and 97% of the
original threshold degree, respectively, allowing more nodes (as shown by
corresponding dashed lines) to be included in the protected set, resulting in
improved survival probability. However, raising the original threshold to 102%
or above (as seen by solid red or magenta lines) results in poor survivability
due to the reduction of nodes in the protected set (shown by corresponding
dashed lines).

Figure 5: Mitigation of cascading failure on standard ER and Scalefree
networks. (a) The plot shows the surviving probability vs Fractional threshold
for ER network ($N=20000$ and $<k>=8$) with four protection probabilities -
Blue, orange, green and red representing probabilities 1.0, 0.7, 0.4, and 0.0,
respectively. The blue curve represents a fully protected network, while the
red line indicates no protection. The orange and green curves represent
partial protection. Additionally, the pink dashed line denotes the size of
immune nodes (only for fully protected case) as a fraction of the entire
network (right y-axis). (b) Surviving probability is plotted against all
protection probabilities for an ER network using four different fractional
thresholds. (c,d) Similar configurations are plotted for Scalefree network.
Figure 6: Scaling the critical degree threshold for (a) ER and (b) scalefree
networks for full protection case. The solid green lines in both figures have
the original critical threshold (i.e., the lowest average degree of the color
groups). The dashed lines represent the respective protected set as the
fraction of the entire network size (right axis). The solid yellow and blue
lines represent the lower critical threshold, allowing more nodes (see dashed
yellow and blue lines) to be protected and resulting in a higher survival
probability. The solid red and magenta lines represent an increase in the
critical threshold, allowing fewer nodes (see dashed red and magenta lines) to
be protected and thus reducing the survival probability.

Similar scaling was tested for scale-free networks, revealing that ER is more
sensitive than Scale-free networks in terms of scaling factor. So, this
demonstrates that our refining/reducing step in the algorithm effectively
establishes the threshold degree to which fragile nodes with a degree greater
than that should be protected to enhance the system’s surviving probability.

Figure 7: Comparison with the standard mitigation techniques. (a,b) ER (c,d)
Scalefree

### Comparison with standard mitigation techniques.

Fig. 7 illustrates the effectiveness of our algorithm in comparison to
standard intuitive and centrality-based mitigation strategies. Initially, we
compare our proposed technique with intuitive algorithms. In Fig. 7 (a), we
compare our proposed protection for ER networks with simple intuitive
strategies using the same set size of protected nodes for a fair comparison.

We start by randomly selecting and protecting nodes to verify the
effectiveness of the proposed strategy compared to random selection,the point
here is to verify if the proposed strategy is actually effective or is it just
another random selection. As shown clearly, it’s much more than the random
selection as it outperforms it. The next obvious candidate is to protect the
high-degree nodes; the reason behind this is that since it is highly connected
with the other nodes, keeping them safe would protect their neigbors. That,
too clearly is not the reasonable set to protect, as clearly shown in the
figure. Protecting the lowest degree nodes, the rationale behind it is that
saving the most vulnerable nodes in the network ensures the better survival of
the network.

Fig. 7 (b) compares our proposed protection for the ER networks with the
centrality-based strategies. A better insightful class for selecting nodes
would be different centrality measures: closeness centrality, Katz centrality,
and betweenness centrality. Closeness centrality measures how close a node is
to all other nodes in a network, indicating nodes that can quickly interact
with others. Katz centrality quantifies the influence of a node in a network
by considering the number of immediate neighbors as well as the neighbors’
influence recursively throughout the network, and Betweenness centrality
identifies nodes that act as bridges between different parts of a network,
measuring the fraction of shortest paths that pass through a node, indicating
its importance in facilitating communication or information flow. Once again,
our proposed strategy stands out, providing enhanced survival chances compared
to these centrality measures techniques. We obtained the same results for the
Scalefree network as shown in the Fig.7 (c,d).

### Application to the Real-World networks

#### 1\. LastFM Asia Social Network

We proceed to apply our mitigation framework to assess its efficacy in a real-
world scenario. First, we examine its application within the LastFM Asia
Social Network—a collection of LastFM users sourced from the public API in
March 2020. In this network, nodes represent LastFM users hailing from Asian
countries, while edges denote mutual follower relationships between them. The
number of user recorded is $N=7626$, and the average degree $<k>=7.29$. Fig. 8
(a,b) shows the result of the application of our mitigation strategy on the
real-world data of the social network site. (a) System’s survivability is
plotted against the fractional threshold for four different protection
probabilities, and in (b), the same is plotted for all protection
probabilities.

Figure 8: Application to the Real-World Networks. (a-b) LastFM Social Network,
where the nodes are the users and mutual follower relationships are the links.
(c-d) US Power Grid network where nodes within the network correspond to
essential components such as generators, transformers, or substations, while
edges symbolize the interconnected power supply lines.(e-f) Condensed matter
physics collaboration network, authors are represented as nodes. If authors
have collaborated, there exists a link between them. When more than two
authors (denoted as $k$) are involved, a connected subgraph is created among
these $k$ authors (a) The plot shows the surviving probability vs Fractional
threshold for the LastFM Social Network ($N=7626$ and $<k>=7.29$) with four
protection probabilities - Blue, orange, green and red representing
probabilities 1.0, 0.7, 0.4, and 0.0, respectively. The blue curve represents
a fully protected network, while the red line indicates no protection. The
orange and green curves represent partial protection. Additionally, the pink
dashed line denotes the size of immune nodes (only for fully protected case)
as a fraction of the entire network (right y-axis). (b) Surviving probability
is plotted against all protection probabilities for the LastFM Social Network
using four different fractional thresholds. (c,d) The same analysis is carried
out for the US Grid network ($N=4941$ and $<k>=2.67$).

The fractional threshold model proves pertinent in delineating the dynamics of
collapse within Online Social Networks (OSNs). Users tend to disengage from
these platforms when a substantial portion of their social connections has
already withdrawn from active participation. The waning interest of an
individual in a social networking site may stem from its failure to furnish
compelling features, content, or interactive activities. Consequently, users
may curtail or cease their habitual sharing of daily experiences on the
platform, or altogether abandon its usage. This decline in activity can
precipitate a ripple effect, causing the user’s network connections to lose
interest due to the absence of their friend’s contributions or the inability
to engage in real-time interactions. If a user observes that a significant
proportion, denoted by $m$ out of $k$, of their network connections have
exited the platform, it may prompt their own departure. This phenomenon can
propagate through the network, culminating in a cascade failure. Notably, the
Hungarian social networking site iWiW [18] fell victim to such a cascade
failure. Operating from 2004 to 2013, iWiW witnessed a surge in user
departures towards the end of 2010, resulting in substantial attrition and
eventual network collapse. Research [19] attributes this decline to
individuals departing once their social circles had largely vacated the
platform. In response, major OSNs such as Facebook and Instagram have
strategically integrated engaging features like platform games, activity
sharing, story updates, and enhanced chat functionalities to mitigate similar
attrition trends and bolster user engagement.

#### 2\. US Power Grid Network

The power grid network is mostly based on the redistribution of load [30].
Here, we model it very basics by fractional threshold model. That is, the grid
fails if more than $m$ out of its $k$ connecting grids fails. We are utilizing
the US grid network with nodes $N=4941$ and average connectivity $<k>=2.67$.
Fig. 8 (c,d) shows the result of the application of our mitigation strategy on
the real-world data of the US Grid network. (a) System’s survivability is
plotted against the fractional threshold for four different protection
probabilities, and in (b), the same is plotted for all protection
probabilities. In this context, nodes within the network correspond to
essential components such as generators, transformers, or substations, while
edges symbolize the interconnected power supply lines. In the event of a
transformer or substation malfunction, the repercussions extend to the
connected transformers, as they assume the burden of redistributed loads.
Employing a fractional threshold to gauge the vulnerability of a transformer
based on the proportion of its neighboring components could be a pragmatic
approach. By leveraging such a strategy, the algorithm can proactively
identify susceptible transformers for preemptive upgrades, mitigating the risk
of future load fluctuations or overload failures.

#### 3\. Collaboration Network

Collaboration networks are primarily Scalefree in nature. The fractional
threshold model could be highly useful for analyzing the failure of an
institution, industry, or researcher. Here, we utilize the condensed matter
physics collaboration data with nodes $N=23133$ and average degree $<k>=8.08$
that is derived from the e-print arXiv and encompasses scientific
collaborations among authors who have submitted papers to the Condensed Matter
category. In this network, if author $i$ has co-authored a paper with author
$j$, an undirected edge is included between $i$ and $j$. Moreover, when a
paper is co-authored by $k$ authors, this results in a fully connected
subgraph comprising $k$ nodes.

The rationale is that an individual researcher needs at least some of their
collaborators to be present and participate in the publication. If this does
not happen, it could result in a decrease in the rate or quality of
publications, or publication may even stop altogether. Collaboration failure
can occur due to various reasons such as severed ties, illness or death of a
collaborator, non-compliance with mutual agreements, issues related to credit
sharing, and more. As depicted in Fig.8 (e), the red line shows that randomly
removing a few researchers (two in this case, in order to start the cascade)
from the network affects their immediate neighbors and leads to a cascade of
failures until the network collapses. On the other hand, the blue line
demonstrates that protecting a fraction of nodes (pink dotted line), such as
collaborators, ensures the survival of the network. The orange and green lines
represent probabilistic protection of critical nodes by 70% and 40%,
respectively, significantly increasing the probability of survival for
partially protected networks. And in Fig. 8 (b), the system’s surviving
probability is plotted for all protection probabilities.

## Discussion and Conclusion

In this work, we introduce an algorithm designed to efficiently mitigate
cascading failures in complex networks. This mitigation is achieved by
safeguarding critical nodes that are most effective at propagating and
intensifying failure. Differently from the existing work on the mitigation of
cascading failures in complex networks [20], we have exploited the graph
coloring concept to identify the critical degree threshold, designating nodes
with a degree greater than it for protection. This approach optimally refines
the fragile nodes without compromising the system’s survival probability. Our
findings demonstrate that effective node selection can be accomplished with
only minimal understanding of the local neighborhoods of the nodes and the
underlying failure mechanisms. This strategy significantly enhances the
likelihood of network resilience, even without precise knowledge of the
initial source of impact. We have evaluated our method across various network
configurations and failure scenarios, yielding robust mitigation strategies.
Furthermore, we have successfully implemented this approach in real-world
network settings, including social networks, the US grid network, and
collaboration networks. In each case, our algorithm has effectively mitigated
cascading failures. However, it leaves some open questions for further
research.

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Supplementary Information
Streamlined approach to mitigation of cascading failure in complex networks

##  I Analytical calculation of cascading failure processes

We apply the Gleeson et al. [28] framework for analytic calculations of
cascading failure processes. Each node, having degree $k$ with the degree
distribution $p_{k}$ such that $\sum p_{k}=1$, is an agent that belongs to the
undirected network and can be in a binary state, called active or inactive.
All agents are assigned a frozen random threshold $r$ (uniform threshold for
our case) chosen from the distribution with $F(r)$ denoting the probability
that an agent has a threshold less than $r$. The cascade is initiated by
seeding the network by activating a randomly chosen fraction of nodes
$\rho_{0}$ out of total $N$ nodes. Nodes update their state, and the average
final fraction $\rho$ of active nodes is given by [28]

$\rho=\rho_{0}+(1-\rho_{0})\sum_{k=1}^{\infty}p_{k}\sum_{m=0}{k}\begin{pmatrix}k\\\
m\\\ \end{pmatrix}q_{\infty}^{m}(1-q_{\infty})^{k-m}F\left(\frac{m}{k}\right)$
(S1)

where $q_{\infty}$ is the steady state or fixed point of the recursion
relation

$q_{n+1}=\rho_{0}+(1-\rho_{0})G(q_{n})\hskip 28.45274ptn=0,1,2,\dots$ (S2)

and the generating function $G$ is defined as

$G(q_{n})=\sum_{k=1}^{\infty}\frac{k}{z}p_{k}\sum_{m=0}^{k-1}\begin{pmatrix}k-1\\\
m\\\ \end{pmatrix}q_{n}^{m}(1-q_{n})^{k-1-m}F\left(\frac{m}{k}\right)$ (S3)

Here $z$ is the network’s average degree calculated as $z=\sum kp_{k}$. The
improved cascade condition developed by Gleeson et al. [28] for a finite
initial impact,

$(C_{1}-1)^{2}-4C_{0}C_{2}+2\rho_{0}(C_{1}-C_{1}^{2}-2C_{2}+4C_{0}C_{2})<0$
(S4)

where $C_{l}$’s are the coefficients of power series
$G(q)=\sum_{l=0}^{\infty}C_{l}q^{l}$.

Fig. S1: Demonstration of the fractional cascade process on a ER network with
$N=20000$ nodes and average degree $<k>=8$.

Fig. S1 illustrates the process of cascading failure and simulation results
are in good fit with the analytical result (Eq. S1) of an ER network. A
collapse of the network occurs at the critical threshold around 0.86. In Fig.
S2, the critical threshold for various initial average degrees is plotted,
showing good alignment with the analytical result (S4). As the average degree
of the network increases, so does its critical threshold; consequently, it
will collapse at a higher threshold level.

Fig. S2: Analytical and simulation results of the critical fractional
threshold compared to the initial average degree of ER network.  Fig. S3:
Frequency distribution of node types are observed at four different fractional
thresholds for the ER network consisting of $N=20000$ nodes and average degree
$<k>=8$. The sky blue bars represent the unprotected or non-fragile nodes,
while the yellow bars indicate fragile nodes. A red dashed line denotes the
critical degree above which the fragile nodes should be protected. (a) At a
fractional threshold of 0.85, no protection is required as all fragile nodes
are below the critical threshold (dashed-red line). (b-d) From thresholds 0.86
to 0.89, an increasing number of fragile nodes lie above the critical
threshold and therefore need to be protected (green bars).  Fig. S4: Similar
analysis for Scalefree network consisting of $N=20000$ nodes and average
degree $<k>=6$.(a) At a fractional threshold of 0.82, no protection is
required as all fragile nodes are below the critical threshold (dashed-red
line). (b-d) From thresholds 0.835 to 0.89, an increasing number of fragile
nodes lie above the critical threshold and therefore need to be protected
(green bars).

##  II Critical threshold degree and degree distribution of different types of
nodes

The critical threshold degree is the cutoff degree above which fragile nodes
are designated for protection for effective mitigation. This threshold is
determined by coloring the graph [29], calculating the average degree of each
color group, and selecting the lowest average degree as the critical threshold
degree. Fragile nodes with a degree greater than this critical value are
assigned for protection. Figures S3 and S4 illustrate the distribution of
degrees among fragile, protected, and unprotected nodes in ER and Scalefree
networks at four different fractional thresholds respectively. Fragile nodes
(depicted in yellow) are those that fail under a specific failure mechanism
defined in step 3 of our algorithm; meanwhile, protected nodes (displayed in
green) refer to fragile nodes with a degree greater than the critical
threshold (represented by a red-dashed line), making them eligible for
protection. Finally, unprotected nodes are the rest of the nodes that are not
fragile(displayed in sky blue).