forming galaxies with different AGN fractions, see Sect. 3 for details)
considered as an input. The choice of using SPRITZ is justified by the fact
that it is more difficult to define a clean sample of type 2 AGN. Indeed, even
if type 2 AGN display strong narrow high-ionisation emission lines (with ${\rm
FWHM}<2000$ km s-1) and weak continua (Zakamska et al. 2003) in the rest-frame
optical spectra, the literature works often refer to type 2 AGN as
“candidates”, as their selection strongly depends on the diagnostic (and thus
the available emission lines) considered (e.g., Hao et al. 2005; Reyes et al.
2008; Alexandroff et al. 2013; Mignoli et al. 2013).

We thus want to test whether our type 2 composite derived from a semi-
empirical library is consistent with what is currently available in the
literature. Figure 22 shows the comparison of the semi-empirical template
spectra of type 2 AGN utilised in this work (shown with the black solid line)
with the empirical one published by Yuan et al. (2016). The blue line
represents the composite spectrum of all type 2 quasars in the baryon
oscillation spectroscopy survey (BOSS) analysed by Yuan et al. (2016). The
excess flux at $\lambda>6500$ Å is due to the host galaxy emission, which also
seems to be dominant at bluer wavelengths. Therefore, the difference between
the composite by Yuan et al. (2016) and ours is mainly due to the different
host galaxy. The red line is derived by normalising the composite spectrum by
Yuan et al. (2016) to a continuum obtained by spline-interpolating between
relatively line-free regions. This stack preserves the equivalent widths of
the emission lines, although the intrinsic shape of the continuum is lost (see
their Fig. 15 and their Sect. 3.7). Overall, there is good consistency between
the two composites regarding the H $\alpha$+[N ii] complex and the spectral
region around the H $\beta$ and [O iii]. There are instead some differences in
the strength of the emission lines in the UV at $\lambda<4000$ Å, such as [O
ii], [Ne iii] and H $\delta$, with a few spectral features missing in our
template (e.g., [Ne v]).

To this comparison we also included the composite published by Mignoli et al.
(2013, green line in Fig. 22) obtained from a sample of 94 narrow-line AGN in
the redshift range $0.65<z<1.20$. Their sample is selected from the 20k-Bright
zCOSMOS galaxy sample by the detection of the high-ionisation [Ne
v]$\lambda$3426 line. This composite shows strong, high-ionisation narrow
lines, such as [Ne v], that is not present in our stack, and a very faint Mg
ii emission line consistent with the emission seen in our composite. We stress
that these differences do not affect any result in our analysis, as our
comparison is mainly focused on the H $\alpha$ spectral region rather than the
UV and the overall shape of the continuum. In the future, we will implement a
more data-driven composite for the type 2 AGN to improve the incident template
library for type 2 AGN in the UV.

Figure 22: Comparison of the semi-empirical template incident spectra of type
2 AGN utilised in this work (shown with the black solid line) with the
empirical one published by Yuan et al. (2016). The blue line represents the
composite spectrum of all BOSS type 2 quasars in the sample analysed by Yuan
et al. (2016), whilst the red line is derived by normalising the composite
spectrum to a continuum obtained by spline-interpolating between relatively
line-free regions. The green line represents the spectral stack of 94 narrow-
line AGN published by Mignoli et al. (2013). For sake of comparison, the
literature composites are normalised at the 5100 Å flux of our stack. Main
emission lines are marked. The bottom panels are two zoom-ins around the H
$\beta$ (_left_) and the H $\alpha$ (_right_) the spectral regions. Stellar
absorption features are also included and visible in our type 2 AGN spectrum
between H $\gamma$ and H $\beta$ (see also Fig. A1 by Vidal-García et al. 2017
and Figs. 9 e 10 by Bruzual & Charlot 2003), which will be blended to the
continuum in any moderate (low) resolution spectrum.

##  Appendix F Analysis of the flux loss

Here we present some additional test we performed to investigate whether a
fraction of the missing flux could be lost at the slitless spectra extraction
level (see Sect. 5.1.3 in Euclid Collaboration: Gabarra et al. 2023, for
details). We first compared the distribution of the ratios $r$ of the median
observed flux values of the incident spectra $F_{\rm inc}$ and the simulated
spectra $F_{\rm sim}$ (i.e., $r=\langle F_{\rm sim}\rangle/\langle F_{\rm
inc}\rangle$, where the angle brackets $\langle\rangle$ represent medians);
see Fig. 23. The median of this distribution is about 12% for type 1 AGN and
2% for type 2 AGN, with a percentage error on the distributions of spectrum
flux median values (defined as $1.253\,\sigma/\sqrt{N}$, where $\sigma$ and
$N$ represent the standard dispersion of the distribution and the number of
objects, respectively) at the 2.5% level for type 1 and 1% for type 2. As an
example, we present a comparison between the incident (black) and simulated
(red) spectra for a representative type 1 (left) and a type 2 (right) AGN at
the top row of Fig. 23. The inset plot is a zoom-in of a continuum (emission
line free) region. The overall flux of the simulated spectra are
systematically lower for both the type 1 and 2 AGN. For completeness, we also
carried out the same exercise of comparing median luminosity density values
for the best-fit models of the incident and simulated spectra to their
respective data [i.e. ratio = median(data) / median(model)], finding that the
QSFIT best-fit model is close to the data, on average, with the biggest
discrepancy for type 1 simulated spectra at $\simeq$2% level. This analysis
implies that flux loss could partly explain the remaining fraction of flux
missing in the comparison between simulated and incident flux values, although
most of the difference on the H $\alpha$$+$[N ii] emission between the
simulated and incident data is due to the assumptions made in the spectral
modelling of the emission lines. Nonetheless, the OU-SIR team is working on
the spectral extraction pipeline and flux losses will be alleviated by the
optimal extraction routine that is implemented in the upcoming version of the
OU-SIR pipeline.

Figure 23: Top row: Example of a comparison between the incident (black) and
simulated (red) spectra for a type 1 (_left_) and a type 2 (_right_) AGN
around the H $\alpha$ region. The inset plot is a zoom-in of a continuum
(emission line free) region. Bottom row: distribution of the ratios of the
median flux value (i.e., the median of the observed flux of the total
spectrum) of the incident and simulated data for both type 1 (black) and type
2 (blue) AGN. The median values for the two distributions are also reported.
The percentage error on the distributions of the spectrum flux median values
are computed as $1.253\,\sigma/\sqrt{N}$, where $\sigma$ and $N$ represent the
standard dispersion of the distribution and the number of objects,
respectively.


# Combing a double helix

Thomas Bolton Plumb-Reyes  School of Engineering and Applied Sciences, Harvard
University, Cambridge, MA 02138  Nicholas Charles  School of Engineering and
Applied Sciences, Harvard University, Cambridge, MA 02138  L. Mahadevan
School of Engineering and Applied Sciences, Departments of Physics and
Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138
<EMAIL_ADDRESS>
###### Abstract

Combing hair involves brushing away the topological tangles in a collective
curl. Using a combination of experiment and computation, we study this problem
that naturally links topology, geometry and mechanics. Observations show that
the dominant interactions in hair are those of a two-body nature,
corresponding to a braided homochiral double helix. Using this minimal model,
we study the detangling of an elastic double helix via a single stiff tine
that moves along it, leaving two untangled filaments in its wake. Our results
quantify how the forces of detangling are correlated with the magnitude and
spatial extent of the link density, a topological quantity, that propagates
ahead of the tine. This in turn provides a measure of the maximum
characteristic length of a single combing stroke, and thus the trade-offs
between comfort, efficiency and speed of combing in the many-body problem on a
head of hair.

Long-haired people are familiar with a well-known strategy for combing their
hair: start combing away the tangles close to the free hair ends, and work
steadily upward towards the scalp. This allows for the untangling of a
collective curl to proceed more efficiently from the free end, minimizing pain
but at the expense of time. But how does a comb work its way through a curl?
This quotidian problem which lies at the intersection of mechanics, geometry
and topology has many cousins—the carding of textiles and felts, and the
spontaneous tangling and detangling of polymers in a flow, of flux lines in
superconductors and of magnetic fields in solar coronae. In the context of
hair, there has been a recent resurgence of interest in characterizing the
effective properties of fiber assemblies and packings Hearle ; Pan ;
Panaitescu ; Kabla ; Leaf ; Goldstein2018 ; Durand2009 ; Sacks2017 , inspired
by simple curiosity, technological applications to fields such as robotics,
and even the need to model hair animation Ward2007 ; Bao2017 . Much of these
studies neglect individual hair-hair interactions, and little is known about
the dynamics of detangling in complex packings of fibers and hair.

The complexity of the combing problem is potentially associated with the many-
body nature of interacting filaments (hairs) and their potential for long-
range interactions. As an example, observations of a curl of horse hairs (Fig.
1a) show that they are mostly straight, enhanced by gravity that helps
straighten the hairs; only near the free end can the hairs adopt their natural
curvature (see SI for detailed analysis). To quantify the nature of these
interactions, we digitally color each strand to track their interactions with
their neighbors in a curl (Fig. 1b), segment the curl into sections, and then
count and characterize the internal interactions of each strand using the
interaction density in Fig. 1c. We define an interaction to be where strands
cross each other, which is an upper bound on the number of interactions (since
it does not account for the sign of the interaction, i.e. if a hair moves over
or under another). In Fig. 1d, we plot the number of N-body interactions, and
see that pair-wise interactions form a plurality of the tangle types in our
analyzed curls.

Figure 1: Tangles in hair. (a) Curl of horse hair. (b) Colored curl of 12
human hairs, clamped at one end. (c) Schematics of N-body interactions
($N=2$-$5$). (d) Histogram of N-body interactions for a sample similar to (b)
with $\rho=$ number of interactions per unit length (in meters). We segment
the curl into 20 sections, count interaction types defined in (c) within each
section and average.

Given the dominance of two-body interactions in a hair curl, we first consider
a minimal model of the comb-curl system: two homochiral entwined helices
clamped at the top end and hanging freely at the bottom. The filaments are
made of nylon (heated to force them to conform to 3-D printed helically
grooved cylinders, and then cooled). This assembly is then pierced at the
midpoint of the double helix centerline by a single stiff rod (the comb) which
moves downward to detangle the curl. We use an Instron 5566 material testing
machine to measure the force extension curves of the rod-helix combing system.
In Fig. 2a we show the response of the helix that can be associated with
either kinking or winding. Kinking arises when there is some sliding and shear
between the free-end sides of the two helices, leading to a characteristic
bent state (Fig 2a,b). Over- and under-winding describes the stretching out of
the helices behind the tine and a concomittant compression of the free end. As
the tine moves through and along the double helix, the force-extension profile
of the process evolves. We examine this in terms of the helix radius $R$,
filament radius $r$, tine radius $t$, and helix pitch $P$ (Fig. 2b), expressed
in terms of the dimensionless ratios $\pi_{1}=P/r$, $\pi_{2}=R/r$ and
$\pi_{3}=t/r$, deferring variation of $\pi_{3}$ to the supporting information
since $t$ appeared to have little impact on the nature of helix unwinding.

Figure 2: Combing a double helix. (a) As the tine moves down the double helix,
it forms a kink about which the helix whirls as link flows out of the bundle.
(b) Schematic of combing a double helix with pitch $P$, helix radius $R$ (half
the distance between individual filament centerlines), filament radius $r$,
tine radius $t$ and helix height $L$. (c-d) Scaled force
$f=\frac{l_{0}^{2}F}{B}$ applied by tine over scaled distance combed
$x=\frac{d}{D}$, where $F$ is unscaled force, $d$ is unscaled distance combed,
$l_{0}=0.01$m is a characteristic length, $B$ is single filament bending
rigidity and $D$ is total tine displacement required to detangle the helices.
Dashed curves show theoretical force predicted for stretching a single helix;
black curves show a single representative combing. We scan the phase space
varying (c) $\pi_{1}=P/r$ and (d) $\pi_{2}=R/r$. See SI for details and
variation of $\pi_{3}=t/r$. (e) Change in link density $\Delta\lambda$
normalized by initial link density $\lambda_{0}$ for $\pi_{1}=18$,
$\pi_{2}=1$, plotted for several normalized tine locations, showing flow of
link from clamped to free side and eventually out free end by twirling of the
free ends about one another. In all experiments, $r=4.25\cdot 10^{-4}$ m and
helix height $h=0.15$ m.  Figure 3: Numerical results. (a) Simulated curl of
12 hairs modeling those in Fig. 1b. (b) Hair interactions as in Fig. 1d for
simulations similar to (a), where $\rho=$ number of interactions per unit
length (in meters). $L_{0}=0.2$ m, $r=7.5\cdot 10^{-5}$ m and $E=1$ GPa. See
SI for 3-dimensional curl results. (c-d) Scaled force $f=\frac{l_{0}^{2}F}{B}$
applied by tine over scaled distance combed $x=\frac{d}{D}$, where parameters
are defined as in Fig. 2. Dashed curves show theoretical force predicted for
stretching single helix an equivalent distance; black curves show single
representative combing. We scan the phase space varying (c) $\pi_{1}=P/r$ and
(d) $\pi_{2}=R/r$. See SI for details and variation of $\pi_{3}=t/r$. (e)
Change in link density $\Delta\lambda$ normalized by initial link density
$\lambda_{0}$ for helix with $\pi_{1}=18$, $\pi_{2}=1$, plotted for several
tine locations, qualitatively reproducing experimental trends from Fig. 2e.
Filament parameters are same as in Fig. 2, except $E$ values are scaled down
by 0.078. See SI for simulation settings.

In Fig. 2d,e we show the scaled force on the tine $f=Fl_{0}^{2}/B$ as a
function of the scaled tine displacement $x$. Smaller pitch and radius helices
(more tightly wound) lead to larger forces required to detangle. There is
generally an initial rise in the force extension curve before a leveling off,
corresponding to overwinding of the helix in front of the tine. After the
initial rise, the force peaks when the tine jams, falling once the tine breaks
through, although there is no rise in the force or tine jamming for loose
helices, i.e. $P/r>c\approx$$20$. We overlay the force required to
equivalently stretch a single helix for comparison (initial angle $\alpha$ and
helix radius $R$ extended to final radius angle $\alpha_{1}$ and helix radius
$R_{1}$ Love ; see SI for details). In contrast with the force-extension curve
for a single helix wherein the force required to stretch diverges as the helix
is straightened out, combing through a double helix shows that the force
flattens out before either helix is significantly stretched. Instead, the
moving tine untangles the two filaments, eventually decoupling the helix
strands as the force becomes vanishingly small.

The unlinking of the homochiral helices during this process can be quantified
in terms of the Calugareanu-Fuller-White (CFW) theorem Fuller1971 ;
Calugareanu1959 ; White1969  which states that $Lk=Tw+Wr$, where Link ($Lk$)
quantifies the oriented crossing number of the two filaments averaged over all
projection directions (or Gauss Linking integral) and effectively counts the
number of full turns one filament makes around the other Ricca2011 , Twist
($Tw$) is the integrated rotation of one filament around the two-filament-
centerline, and Writhe ($Wr$) is the (negative) integral of the geometric
torsion of the centerline. To quantify the topology of the double helix, we
treat the two interwoven filaments as two edges of a ribbon, and follow the
local link density $\lambda(s)$, defined as link per unit length along the
double helix centerline as a function of centerline arclength $s$, and the
twist density $\tau(s)$. For a relatively straight double helix, $Wr\approx
0$, so that $\lambda(s)\approx\tau(s)$ Kamien1997 . In Fig. 2e we show the
evolution of the link density $\lambda$ calculated from images of the helix
taken as the tine moved. We see that a relatively uniform $\lambda(s)$ changes
to a step-like $\lambda(s)$ as the tine induces a flow of link toward the free
end of the curl. This leads to overwinding in front of the tine and
underwinding behind while link flows through the helical braid from the
clamped to the free end. For tine displacements $x\gtrsim 0.2$, the scaled
jump in the link density $\Delta\lambda/\lambda_{0}>0$ across the tine
($\lambda_{0}$ being the initial link density) increases faster than the rate
it which is expelled from the free end. This accumulation eventually reaches a
plateau as the tine gets closer to the free end, and then induces a flux of
link at the free end that unlinks the filaments, the end goal of combing.

We now turn to correlate the experimentally observed spatio-temporal evolution
of link density associated with the motion of the tine to the evolution of the
force on the tine. For a double helix that is tightly wound initially, as the
tine propagates, $\lambda(s)$ ahead of the tine increases, and the scaled
pitch $\pi_{1}=P/r$ decreases locally. This causes the double helix to stiffen
and eventually the tine jams when the local link density is larger than a
threshold that depends on the diameter, the nominal helix pitch and radius of
the filaments as well as the coefficient of friction. When one of the
filaments slides relative to another, the relative shear often leads to the
kinked configuration seen in Fig. 2a,b accompanied by a peak in applied force.
If the filament does not break, eventually the tine breaks through and the
force decreases, before the same cycle repeats again. During this process, the
kinked double helix twirls about the axis of tine motion (see SI video-S1). If
the double helix is loose enough initially, link propagates more easily from
the tine location to the free end, $\lambda(s)$ never crosses the threshold
needed for kinking, and the force required to comb never peaks (as seen for
$\pi_{1}=48,\pi_{2}=1$). These are the only ways by which link flows out of
the free end, via untwisting of the free ends of the individual filaments, and
via the twirling rotation of the kinked portion of the double helix.

To quantify our experiments on the non-linear topological mechanics of
interacting filaments, we use a numerical approach that models each hair using
the Kirchhoff-Cosserat theory Cosserat1909 ; OReilly2017  and solve the
equations using the discretization, contact-force and numerical integration
scheme described in Gazzola2016 . In the limit of very thin filaments such as
hair, the model naturally reduces to Kirchhoff-Love theory for inextensible,
unshearable filaments Love . We define $s\in[0,L_{0}]$ as the material
coordinate (also the arc length) of the rod of rest length $L_{0}$,
$\boldsymbol{{x}}(s)$ as the position vector of the center-line, and a triad
of orthonormal directors
$\boldsymbol{{d}}_{1}(s),\boldsymbol{{d}}_{2}(s),\boldsymbol{{d}}_{3}(s)=\partial_{s}\boldsymbol{{x}}$
that defines the cross-section orientation. Then any body-convected vector
$\boldsymbol{v}$ with lab-frame coordinates $\boldsymbol{\bar{v}}$ may be
written as $\boldsymbol{v}=\boldsymbol{Q}\boldsymbol{\bar{v}}$, where
$\boldsymbol{Q}(s)\in SO(3)$ is a rotation matrix, and the bending and twist
strain vector is given by $\boldsymbol{\kappa}={\rm
vec}(\partial_{s}\boldsymbol{Q}^{T}\boldsymbol{Q})$ ($\boldsymbol{\kappa}_{0}$
is rest-state curvature). If $\boldsymbol{N}$(s) is the internal force
resultant, $\boldsymbol{f}_{g}$ as gravitational force line density, the
equilibrium equations for the boundary value problem are OReilly2017 ;
Gazzola2016

$\displaystyle 0=$
$\displaystyle\partial_{s}\boldsymbol{{N}}+\boldsymbol{{f}}_{g}$ (1)
$\displaystyle 0=$
$\displaystyle\partial_{s}\left(\boldsymbol{B}\left(\boldsymbol{\kappa}-\boldsymbol{\kappa}_{0}\right)\right)+\partial_{s}\boldsymbol{{x}}\times\boldsymbol{n}$
(2)

where $\boldsymbol{B}$ is the matrix of bending and twisting stiffnesses,
subject to the boundary conditions of the filaments being clamped at one end
and free at the other.

To simulate the initial state of a curl of hairs such as that depicted in Fig.
1b, we start with a collection of clamped filaments hanging in a gravitational
field $|\boldsymbol{{f}}_{g}|=\rho Ag$ ($g\approx 9.8$ N/kg, $\rho$ is the
filament mass density and $A$ is the filament cross-sectional area), resulting
in a curl such as that shown in Fig 3a. We then introduce a intrinsic
curvature at each node, randomly drawn from a Gaussian distribution with mean
and variance matching the distribution of curvatures (see SI for details),
shown in Fig. 1b. Finally, we let the hairs relax elastically to their new
rest configurations, determined by a competition between nonzero intrinsic
curvature and gravitational straightening (see SI). After the hairs relax, we
count interactions using the same method as used in experiments, leading to
the histogram of interactions per unit length in Fig. 3b. We find that
pairwise interactions dominate in agreement with experimental results (see SI
where we show that these interactions do not change qualitatively in a 3d
array).

To follow the combing of an elastic double helix as in Fig. 2, we start with a
random distribution of initial internal strains and anneal the pair of
filaments into a double helix, clamped at one end. We then insert a rigid rod
as a tine between the two helices close to the clamped end and move it quasi-
statically towards the free end. In Fig. 3c-d we show the numerically computed
force applied by the tine during combing, and see that the results are similar
to our experiments shown in Fig. 2c-d 111The high-frequency oscillations seen
in the mean force curves are due to the oscillation in contact force between
the tine and double helix filaments as the tine passes by each discretization
node in the double helix filaments. . Comparing the evolution of the helix
topology computed using the algorithms described in solenoids , Fig. 3e shows
that this also matches the experimental pattern of overwinding and
underwinding shown in Fig. 2e; the oscillations in $\lambda(s)$ on the clamped
side of the tine come from slight periodic shearing of the clamped side
filament segments with respect to each other, a phenomenon that occurs on too
fine a resolution to be seen in experiments. The small discrepancies between
the quantitative values observed in the numerical and experimental results may
come from our scaling of the simulated filaments’ bending stiffness 222Note
that the numerical calculations presented here neglect contact friction.
However, we tested a subset of these simulations with contact friction and
obtained force-displacement curves with the same shape as the ones shown here.
Hence, for simplicity, we neglect contact friction in all simulations shown
here..

Figure 4: (a-b) Scaled force $f$ over scaled distance combed $x$ for (a)
experimental and (b) simulated double helices with varying number of links.
Scaling of $x$ is different for each curve since the total raw tine
displacement needed to comb each curl is different. Below a certain link
threshold, combing finishes without kink formation. Black curves show a single
representative combing for $q=25$. Filament parameters are the same as in Fig.
2. (c) Cost $C$ to comb a 10m double helix over number of downstream pitches
$q$ detangled per combing, using the strategies shown in (a-b). Note that $q$
is intrinsic to combing strategy. $C$ combines effective pain and time
required to comb; $\gamma=0$ considers only time and corresponds to straight
hair; $\gamma=1$ considers only pain and corresponds to curly hair. Simulation
and experiment how optimal combing strategy shifts depending on hair
curvature.

Complementing our analysis of the topological mechanics of combing a double
helix, we now consider how a curl’s resistance to combing varies with the
style of combing, e.g. short strokes versus long strokes. Noting that a single
helix pitch corresponds to one helix filament revolving once around the other
helix filament, we fix $R/r=1$ and $P/r=24$ and vary $q$, the number of
pitches that initially separate the tine from the free end of the double
helix. In experiment (Fig. 4a) and simulation (Fig. 4b) we find that
significant tine jamming—as indicated by the plateaus in the force-
displacement curves—occurs only when we start combing more than 4-5 helix
pitches away from the free end 333Quantitative differences in the shape of
experimental and numerical force-displacement curves in Fig.4 likely stem from
our model’s neglect of contact friction..The differences between the two sets
of curves is likely due to not accounting for friction in the computations,
and explains the larger number of plateaus seen in the large-$q$ experiments
relative to the more smooth force evolution seen in computations.
Nevertheless, we see that starting to comb nearer to the free end allows link
to be expelled more easily, leading to complete untangling before the
differential link density across the tine surpasses the threshold for kink
formation and jamming, consistent with experience.

To quantify this intuitive result, we define a cost $C(q)$ for combing a
double helix of length $L$ using each strategy, balancing the effects of
$f_{\text{max}}$, the maximum dimensionless force during combing, and the
amount of detanglement quantized in terms of the number of strokes $q_{0}/q$
need to complete the combing process, where $q_{0}=25$ a fixed number of
pitches characteristic to the $q$-values used in these combing strategies.
Writing $C(q)=\gamma f_{\text{max}}+(1-\gamma)\frac{q_{0}}{q}$ ( see SI for
details), we can vary the relative cost of the two effects by varying
$\gamma$. Varying $\gamma$ allows us to optimize a combing strategy based on
the initial nature of the double helix; for very curly hair, choosing
$\gamma\sim O(1)$ is more sensible, while for straight hair, choosing
$\gamma\sim 0$ is better. Using the results for the maximum force from Fig.
2-3, we calculate the cost $C(q)$ for the case of curly and straight hair. In
Fig. 4c, we see that for straight hair ($\gamma\sim 0$), the cost decreases as
$q$ increases, i.e. the number of strokes decreases, while for curly hair
($\gamma\sim 1$), the cost is lowest for small $q$ as it is biased to minimize
the cost associated with the maximum force.

Our study has shown that the many-body problem of combing hair can be
simplified to the problem of combing a double helix, and in this context, we
have provided a measure of link density during combing as a function of helix
geometry and combing procedure. We have connected topology, geometry and
mechanics by quantifying the relation between flow of link through the tine
and out the free end, to the time-varying force felt on the tine. Our results
also suggest that the two-body problem also has the ability to capture the
correct optimal strategy of combing a tangle by balancing the cost of many
short strokes relative to longer, potentially more painful ones. Going
forward, a natural problem is to account for the strong frictional anisotropy
of the hairs as well as their response to combing as a function of humidity
and temperature.

Acknowledgements. For partial financial support, we thank the National Science
Foundation grants NSF DMR 20-11754, NSF DMREF 19-22321, and NSF EFRI 18-30901.

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# PAD: a graphical and numerical enhancement of structural coding to
facilitate thematic analysis of a literature corpus

Etienne-Victor Depasquale<EMAIL_ADDRESS>Humaira Abdul Salam
<EMAIL_ADDRESS>Franco Davoli<EMAIL_ADDRESS>
###### Abstract

We suggest an enhancement to structural coding through the use of (a) causally
bound codes, (b) basic constructs of graph theory and (c) statistics. As is
the norm with structural coding, the codes are collected into categories. The
categories are represented by nodes (graph theory). The causality is
illustrated through links (graph theory) between the nodes and the entire set
of linked nodes is collected into a single directed acyclic graph. The number
of occurrences of the nodes and the links provide the input required to
analyze relative frequency of occurrence, as well as opening a scope for
further statistical analysis. While our raw data was a corpus of literature
from a specific discipline, this enhancement is accessible to any qualitative
analysis that recognizes causality in its structural codes.

###### keywords:

Structural coding, thematic analysis, graph theory, nodes, links.,

††journal: MethodsX

[label1]organization=Department of Communications and Computer Engineering,
University of Malta,city=Msida, postcode=MSD2080, country=Malta

[label2]organization=High-Energy Physics Research (FH) - IT, Deutsches
Elektronen-Synchrotron (DESY),city=Hamburg, postcode=22607, country=Germany

[label3]organization=Department of Electrical, Electronic and
Telecommunications Engineering and Naval Architecture (DITEN), University of
Genoa,city=Genoa, postcode=16145, country=Italy

##  1 Motivation

###  1.1 Primary challenge

The undertaking of a survey is subject to the risk of degenerating into a
ramble around the corpus of literature in scope. The use of data structures,
such as tables and lists, gives all stakeholders a sense of value, but these
summaries pose a different problem: each such data structure represents a
single perspective on the corpus. While these may very well be highly
informative and of great value, they fall short in the overarching objective
of the survey: a holistic abstraction of the literature that succinctly
presents the state of knowledge on the field of study.

This problem may be tackled through the use of analytical methods that are
recognized as leading to such an abstraction. We have used thematic analysis
in surveying, and justify our selection of this form of analysis on:

  1. 1.

the coherence amongst widely-cited texts [1, 2, 3, 4, 5] in recommending
thematic analysis as an introduction to the methods of qualitative analysis,
and

  2. 2.

its application in [6].

###  1.2 Derivative challenge

"The excellence of the research [(qualitative analysis)] rests in large part
on the excellence of the coding" [7, p. 27]. Therefore, choice of coding
method requires careful consideration of its fitness for the purpose of
arriving at the desired abstraction. This is the challenge we faced, and this
paper is our response to it. Our enhancement of structural coding converts
text into:

  * •

numeric data, which is then processed using metrics (section 7) that normalize
dataset sizes and thereby lead to frequency analysis;

  * •

graphical objects (section 6), which are then organized into graphical
overviews.

.

These facilitate both objectivity in, and critique of, the insights obtained
from the thematic analysis.

##  2 Contents

  1. 1.

In section 3, the concept of the "code" is revisited(section 3.1). We declare
our use of structural coding as the fundamental coding technique and specify
the series of questions applied to the raw data(section 3.2). This is followed
by an explanation of the reasoning that guided the extraction of the codes
(section 3.3).

  2. 2.

In section 4, we introduce the node and link concepts from graph theory
(section 4.1) and proceed to relate the causally-bound codes to the node and
link. An algorithm for categorization is presented(section 4.2). We then
extend the causal linkage inherent to the mined codes to the resulting
categories.

  3. 3.

Section 5 develops the application of nodes in this method by defining
attributes that support further processing of the nodes.

  4. 4.

In section 6, the graphical means for communicating a survey’s results are
described.

  5. 5.

Statistics produced by the method, as well as intuition on their significance,
are described in section 7.

  6. 6.

An explanation of how the method facilitates thematic analysis is presented in
section 8.

  7. 7.

Observations on the application of the method are presented through case notes
in section 9, where we draw an outline of how this method was applied to a
recent work.

  8. 8.

We conclude by identifying benefits and limitations (section 10.1) of the
method and summarize the prescriptions of the method in section 10.2.

##  3 Coding: the core technique

###  3.1 What are codes?

Codes are terse, dense, textual representations of a verbose articulation of a
concept embedded in the raw data. The prescriptions of sound qualitative
analysis for systematic review require a choice of coding method. The process
of coding collates the diversity of the surveyed set of papers through the
formation of smaller codes. We refer to each paper included within the scope
of the survey as a research unit (RU), i.e. a publication (excluding surveys)
in conference proceedings and journals. Codes must have the following
characteristics.

  1. 1.

They must be semantically rigorous i.e. the meaning a code represents must be
clear and use (application) of the code must be unconfusable.

  2. 2.

They must be universally applicable across RUs, i.e. they must provide a
uniform means of dissecting publications. Use of more than one coding system
(i.e. two or more non-universal coding systems) may create a split in the
coded data with incomparable parts across the split.

###  3.2 How are codes formed?

We satisfy these two requirements through the elemental coding method of
structural coding. Structural coding poses a series of questions relevant to
the inquiry in hand and is well suited to any problem which can be described
using a standardized set of questions. We deconstruct the problem of
literature review into a standardized question-and-answer protocol that can be
directly converted into a structural coding approach. The questions are the
following:

  1. 1.

What is the problem which the researcher(s) saw as an opportunity for study?

  2. 2.

What approach(es) did the researchers take in an attempt to solve the problem?
Or: how did they go about it … what did they do?

  3. 3.

What development(s) and/or contributions derive from the researcher(s) work?

The corresponding structural codes are:

  1. 1.

Problems (or challenges)

  2. 2.

Approaches

  3. 3.

Developments (or contributions)

We refer to this protocol as the PAD review protocol. Each RU (paper) is mined
for:

  1. 1.

“the problem which the researcher(s) saw as an opportunity for study”
(P-codes),

  2. 2.

the “approach(es) … [taken by] the researchers … in an attempt to solve the
problem” (A-codes) and

  3. 3.

the “developments(s) and/or contributions deriv[ing] from the researcher(s)
work” (D-codes).

(quoted text is taken from the questions enumerated above).

###  3.3 Applying the structural code: identifying problems, approaches and
developments

We now express a generalized understanding of the reasoning we follow to
identify the problems, approaches and developments. In terms of thematic
analysis, this is the role which an expert would play to transform the raw
data into codes.This applies a first abstraction to the RUs:

  * •

from detail specific to the RU

  * •

to abstractive problem-, approach- and development-codes

that meaningfully represent the individual RUs.

####  3.3.1 Problem codes

Research is rooted in “the problem which the researcher(s) saw as an
opportunity for study”. Key techniques in identification of the problem
include:

  * •

a focus on the abstract and the introduction,

  * •

making a high-level summary, e.g. by answering questions such as: what is this
paper trying to solve/concerned with, and

  * •

recognition of revealing phrases like "in this paper" or "in this work".

When the scope of a survey is relatively narrow, the problems in the set may
not be fully independent of one another. They may diverge from one another
only as aspects (we could also say that they are derivatives) of a core
challenge. In such a case, each RU would be rooted in this core challenge, but
the derivative problems (the P-codes) addressed differ from one RU to another.

####  3.3.2 Approach codes

Approach is used in the sense defined by the Oxford English Dictionary as
“figurative. A way of considering or handling something, esp. a problem.” This
meaning is broad yet it is purposely cited here to convey some of the depth of
the difficulty we faced in assessing whether an observation candidate as an
approach should be included (or not). Key techniques in identification of the
approach include:

  * •

a focus on the method section

  * •

choices that focus technique

Codes of this type are the hardest to extract from RUs, as they are heavily
dependent on a good grasp of the research space. This difficulty is severe
enough to count as a limitation of our method (see section 10.1). We emphasize
that the “way of considering or handling something” is never (we dare say)
obtained through a single approach, but, rather, through an entire delta (fan-
out) of component approaches that are combined to bring efforts to a yield:
the development.

####  3.3.3 Development codes

Developments include what are commonly referred to as contributions, but we
can afford to broaden our scope for inclusion of less significant products of
research. Therefore, we go beyond simply parsing RUs in search of the familiar
“contributions of our paper” or “in this paper” phrases, and glean those
useful bits that, taken alone, do not qualify as the scope of a paper.

##  4 Categorization: clustering the codes

The act of coding is carried out as an intermediate step on the way towards
categorization. We first introduce our use of the concept of the node. We then
proceed to a detailed treatment of how, through iterations, a set of problem-,
approach- and development-codes can be categorized.

###  4.1 Nodes - our use of a graph theory concept to complement structural
coding

We use the concept of node from graph theory. The product of mining an RU is
one or more P- , A- and D-codes. Each code is encapsulated within a node.
Since problem, approach and development are linked to one another in a causal
chain, this causal relationship can be represented through linked nodes. Each
such causal chain is represented by three nodes; thus we call it a triad. Our
application of the node concept is beneficial for the following reasons:

  1. 1.

An RU’s collection of triads (of nodes) is a synthetic representation of the
RU, that facilitates a good apprehension thereof.

  2. 2.

These characterizing triads facilitate the task of locating this RU within the
greater landscape of research.

###  4.2 The iterative process of categorization

For every RU in the corpus, we iterate through a number of steps to categorize
a code.

  1. 1)

Given an RU, a set of problem, approach and development codes is collected and
added to a pool of ungrouped codes.

  2. 2)

For every code, associate the code, say $\alpha$, with a numerical, integral
co-attribute, say $a$, and add both the text and number as attributes of a
node. The code is, therefore, an aspect, or attribute of the node.

  3. 3)

Seek a pre-extant code (of the same type, i.e. P/A/D) nearest in meaning, say
code $\beta$, with numerical, integral co-attribute $b$.

  4. 4)

If there is no neighbor (in meaning) sufficiently close to group $\alpha$
with,

    1. a)

then this iteration of categorization terminates for code $\alpha$ and it is
kept within the pool of ungrouped codes.

    2. b)

else:

      1. i)

If $\beta$ is not already grouped within a category,

        1. (1)

then:

          1. (a)

create a category, with proximating meaning $\Gamma$ common to both $\alpha$
and $\beta$, and with numerical, integral co-attribute $C$ (upper case, to
distinguish this as a category);

          2. (b)

replace numerical co-attribute $b$ by $C.1$ while $a$ is replaced by $C.2$,
and

          3. (c)

add both the text $\Gamma$ and number $C$ as attributes of a category-node.

        2. (2)

else: (i.e. if $\beta$ is already grouped within a category)

          1. (a)

revise the scope (therefore, the text) of the proximating meaning to take
$\alpha$ into account;

          2. (b)

if the proximating meaning (i.e. $\Gamma$) cannot be reasonably revised to
include code $\alpha$,

            1. (i)

either spawn a new category, containing codes $\alpha$ and $\beta$;

            2. (ii)

or keep $\alpha$ as an orphan (i.e. without a grouping category), within a
pool of ungrouped codes.

Note that:

  * •

we refer to the proximating meaning as the category code;

  * •

if $\beta$ had already been grouped within a category, its numerical co-
attribute would already be of the float type (i.e. include a decimal
separator), as it would have been previously transformed from numerical
integer $b$ to some float type that includes the digit(s) of its grouping
category.

The category code describes the salient meaning common to all codes grouped
under it. It is the abstraction that sacrifices some detail for the sake of
facilitating a holistic view of the research space. The resulting
categorization is conducive to (a) assimilation by a viewer, as well as (b)
further rationalization.

##  5 Node attributes

The process of categorization results in a set of problem-category-, approach-
category- and development-category-nodes. Each type of category-node groups
nodes of the same type (P/A/D). This section describes the attributes of the
individual nodes and those of the category-nodes.

###  5.1 Node attributes

  1. 1.

a short textual description – the code – of the observed
problem/approach/development and

  2. 2.

a unique alphanumeric identifier (a label) of the form P/A/Dx.yz. In this
labelling scheme,

    1. (a)

P, A and D represent the structural codes,

    2. (b)

x is a single digit that represents the observed problem/approach/development
(P/A/D) category,

    3. (c)

y is a single digit that represents an observed sub-category of the P/A/D
category, if any such sub-category is observed, and

    4. (d)

z is a single digit that represents the individual, observed
problem/approach/development.

We chose the notation x.yz instead of x.y.z to type the variable represented
by this identifier as a numeric float type.

###  5.2 Category-node attributes

  1. 1.

a short textual description – the category code – of the observed P/A/D
category;

  2. 2.

a unique alphanumeric identifier (a label) of the form P/A/D$x$. In this
labelling scheme,

    1. (a)

P, A and D represent the structural codes,

    2. (b)

$x$ is an integer that represents the observed problem/approach/development
category.

  3. 3.

a set of member nodes.

Where major sub-clusters need to be identified, we extend the labelling to
P/A/Dx.y ,where y is a second integer representing the sub-cluster. In these
cases, as we show in the next sub-sub-section, the node identifier takes the
form P/A/Dx.yz.

##  6 Graphical maps of the research space

The workings of the method culminate in the production of several graphic
devices, which we describe below.

###  6.1 The causality DAG

The P-, A- and D- category-nodes are linked according to the triads mined from
the RUs, to produce a graphic device which may be tersely and aptly referred
to as a directed acyclic graph (DAG) of causality \- the causality DAG. This
graphic device is an important part of the product of our method. It shows a
bird’s-eye view of the dynamics of research in our chosen scope.

In particular, the DAG is an encoding of surveyed research units, that strives
to relieve a profile not only of current knowledge (the developments) but also
of what has been found a fruitful pursuit (the approaches) thereof. This is
obtained through the relationships that are illustrated in the DAG between the
problems addressed, the approaches to solutions, and the knowledge obtained in
pursuit of solutions.

The DAG may show links between category-nodes, rather than individual nodes,
to minimize clutter and improve readability. Such a reduction is strongly
dependent on truly representative categorization (see section 4), through
groupings meaningful to the survey’s scope. In this form of the DAG, the
inter-category links are aggregators: they include all links between any two
nodes within their respective categories. Line thickness is an excellent way
to represent the size of the set of aggregated links.

###  6.2 Triads graphic

We also highlight triads. Triads are represented by lines passing through a
single combination of a single problem, a single component of approach and a
single development to which the approach (component) led (not necessarily on
its own; indeed, rarely so). The triads graphic identifies the dynamics of
research at a glance, indicating the most highly used triads by line
thickness.

As with the DAG (and for the same reason), the triads graphic may show causal
bindings between category-nodes, not individual nodes. Each triad is a serial
connection of one link from a P-category to an A-category, and one link from
the latter A-category to a D-category. Each triad is illustrated with a bend
where links meet on A-category nodes.

###  6.3 P-A dyads graphics: challenges (P) and associated approaches (A)

The causality DAG’s section showing P-A category links is partitioned into a
set of P-A dyad-graphics. Each such graphic illustrates the set of all
approach components that have been used to tackle a particular problem
category. The P-A dyads graphics complement the causality DAG and the triads
graphic by a focus on how individual challenges have been tackled in published
research. Each P-A dyads graphic provides, from the perspective of a specific
challenge, the same information on frequency (of applied approaches) as the
(global) causality DAG.

###  6.4 Taxonomies of problems, approaches and developments

Within each of the three divisions (i.e. problems, approaches and
developments) of categories, frequency of occurrence of categories is expected
to communicate meaningful information about the state of the art. Furthermore,
within each category-division, it may be possible to find relationships
between categories that convey additional meaning and encourage structural
formations that gather the categories into taxonomies. For example: if two
categories of approaches are proximal in meaning, a super-category might be
formed that abstracts the differences in meaning between the two and
represents them both from the perspective of the common, salient meaning.

##  7 Statistics

The frequency of occurrence of aspects of data collected is examined here. We
combine the structural codes in various ways in order to obtain useful
statistics for a quantitative grasp of the field. In the following sub-
sections, we suggest and describe how these statistics can be collected from
the data.

###  7.1 Frequency of occurrence of a challenge category in RUs

We start by proposing the following simple relationship.

$F_{P_{k}}=\frac{\sum_{j=1}^{N_{RU}}P_{k}^{(j)}}{{N_{RU}}}$ (1)

where:

  1. 1.

$P_{k}^{(j)}$ is a binary variable that represents the presence (or lack
thereof) of a problem in category $P_{k}$,

  2. 2.

within a single RU $RU_{j}$ over the corpus of $N_{RU}$ unique RUs.

This simply indicates the number of times in which a (derivative) challenge-
category appears within the RUs in the corpus, as a fraction of the total
number of RUs. The numerator of $F_{P_{k}}$ is incremented once by $RU_{j}$
for a given $P_{k}$ if a problem in this category is tackled in $RU_{j}$.

###  7.2 Research interest: frequency of occurrence of a challenge category
among all occurrences of challenge categories

Here, we define research interest, denoted by $R_{P_{k}}$, in a given
challenge (problem) category $P_{k}$, as its frequency of occurrence within
the set of all the challenges tackled in all research units. It is computed as
the total number of times in which problems in category $P_{k}$ have been
tackled in RUs, as a fraction of the sum of the total number of times in which
(problems in) all observed challenge categories have been tackled. We suggest
this as a metric of the attention, or research interest, which this challenge
is receiving. The numerator of $R_{P_{k}}$ is incremented once for $RU_{j}$
for a given $P_{k}$ if a problem in this category is tackled in $RU_{j}$.

$R_{P_{k}}=\frac{\sum_{j=1}^{N_{RU}}P_{k}^{(j)}}{\sum_{i=1}^{N_{P}}\sum_{j=1}^{N_{RU}}P_{i}^{(j)}}$
(2)

where:

  1. 1.

$P_{k}^{(j)}$ is a binary variable that represents the presence (or lack
thereof) of a problem in category $P_{k}$,

  2. 2.

within a single RU $RU_{j}$ over the corpus of $N_{RU}$ unique RUs,

  3. 3.

with a total of $N_{P}$ unique, identified challenge/problem categories.

Note that:

$\frac{F_{P_{k}}}{R_{P_{k}}}=\frac{\sum_{i=1}^{N_{P}}\sum_{j=1}^{N_{RU}}P_{i}^{(j)}}{N_{RU}}$
(3)

This ratio is a constant; therefore, both $F_{P_{k}}$ and $R_{P_{k}}$ have
identical distributions, but the statistics differ. Specifically, the ratio
$\frac{F_{P_{k}}}{R_{P_{k}}}$ is the average number of challenges tackled per
RU.

###  7.3 Metric of observed approach diversity: Weighted challenges

We measure the diversity of approaches through which a challenge is tackled.
The set of all unique problem-approach (PA) pairs (dyads) in the triads is
collected first. Therefore, a particular PA dyad is counted once, regardless
of the number of occurrences of that dyad. Then, for each problem, we add up
the total number of dyads within which that problem is found. We suggest a
normalized diversity metric, $W_{P_{k}}$, as follows:

$W_{P_{k}}=\frac{\sum_{j=1}^{N_{PA}}P_{k}^{(j)}}{\sum_{i=1}^{N_{P}}\sum_{j=1}^{N_{PA}}P_{i}^{(j)}}$
(4)

where:

  1. 1.

$P_{k}^{(j)}$ is a binary variable that represents the presence (or absence)
of a problem category $P_{k}$,

  2. 2.

within a single P – A dyad $PA_{j}$, over the set of all $N_{PA}$ unique P – A
dyads within the corpus,

  3. 3.

with a total of $N_{P}$ unique, identified challenge/problem categories.

###  7.4 Frequency of occurrence of an approach category among all occurrences
of approach categories

Unlike problems in challenge categories, approaches within an approach
category do not, most commonly, mutually exclude one another. Therefore, the
count of occurrences of an approach category may be incremented more than once
per RU and thus a metric like $F_{P_{k}}$ lacks a good normalization basis. As
it is useful to learn how widely exploited an approach is, a different
normalization basis must be selected. We therefore use a metric somewhat
similar to $R_{P_{k}}$, i.e. frequency of occurrence of an approach category
in all RUs, among the set of all occurrences of approach categories in all
RUs. We obtain the metric $R_{A_{k}}$, as follows:

$R_{A_{k}}=\frac{\sum_{j=1}^{N_{RU}}\sum_{m=1}^{\lvert
A_{k}\rvert}A_{k_{m}}^{(j)}}{\sum_{i=1}^{N_{A}}\sum_{j=1}^{N_{RU}}\sum_{m=1}^{\lvert
A_{i}\rvert}A_{i_{m}}^{(j)}}$ (5)

where:

  1. 1.

$A_{k_{m}}^{(j)}$ is a binary variable that represents the presence (or lack
thereof) of approach $A_{k_{m}}$,

  2. 2.

where $A_{k_{m}}$ is a member of approach category $A_{k}$, of cardinality
$\lvert A_{k}\rvert$,

  3. 3.

within a single RU $RU_{j}$,

  4. 4.

over the corpus of $N_{RU}$ unique RUs,

  5. 5.

with a total of $N_{A}$ unique, identified approach categories.

Note that occurrence of category $A_{k}$ within an RU is counted as many times
as its members $A_{k_{m}}$ appear in the RU.

###  7.5 Metric of utility of an approach: Weighted approaches

We also analyze approaches in terms of their utility, i.e. how useful they are
in the overall motion between problems and developments, and denote this
metric as $U_{A_{k}}$. The numerator is incremented each time a particular
approach is a component of a triad within an RU. Therefore, a single RU may
increment the metric several times. The utility metric of a specific approach
$A_{k}$ is the normalized metric:

$U_{A_{k}}=\frac{\sum_{j=1}^{N_{RU}}\sum_{l=1}^{N_{triads_{RU_{j}}}}A_{k}^{(l)}}{\sum_{i=1}^{N_{A}}\sum_{j=1}^{N_{RU}}\sum_{l=1}^{N_{triads_{RU_{j}}}}A_{i}^{(l)}}$
(6)

where:

  1. 1.

$A_{k}^{(l)}$ is a binary variable that represents the presence (or lack
thereof) of approach category $A_{k}$,

  2. 2.

within any of the $N_{triads_{RU_{i}}}$ triads in a single research unit
$RU_{j}$,

  3. 3.

over the corpus of $N_{RU}$ unique RUs,

  4. 4.

with a total of $N_{A}$ unique, identified approach categories.

We emphasize that a single research unit may be described by several such
triads that include approach $A_{k}$.

###  7.6 Frequency of occurrence of categories of development

Development statistics are distributed thinly unless developments are
categorized. However, when grouped into meaningful clusters (categories),
conclusions can be drawn about the frequency with which developments take
place in sub-spaces of this research space. We obtain $R_{D_{k}}$, the
normalized frequency of occurrence of categories, as follows:

$R_{D_{k}}=\frac{\sum_{j=1}^{N_{RU}}\sum_{m=1}^{\lvert
D_{k}\rvert}D_{k_{m}}^{(j)}}{\sum_{i=1}^{N_{D}}\sum_{j=1}^{N_{RU}}\sum_{m=1}^{\lvert
D_{i}\rvert}D_{i_{m}}^{(j)}}$ (7)

where:

  1. 1.

$D_{k_{m}}^{(j)}$ is a binary variable that represents the presence (or lack
thereof) of development $D_{k_{m}}$,

  2. 2.

where $D_{k_{m}}$ is a member of development category $D_{k}$, of cardinality
$\lvert D_{k}\rvert$,

  3. 3.

within a single RU $RU_{j}$,

  4. 4.

over the corpus of $N_{RU}$ unique RUs,

  5. 5.

with a total of $N_{D}$ unique, identified development categories.

Thereby, a prospective researcher is guided through grounded insight into
works covering this space.

##  8 Deriving themes

The concept of a theme is consistently described as a pattern that emerges
from the raw data; see, e.g. [2, p. 82] and [1, p. 4]. The PAD enhancement of
structural coding facilitates thematic analysis by accentuating themes through
the abstractive function of causally-bound categories of codes. Both
quantitative and qualitative aspects of analysis are possible.

###  8.1 Quantitative analysis

The results of application of the PAD method are powerfully conducive to a
quantitative aspect of thematic analysis:

  1. 1.

the frequency of individual categories (i.e. whether problem-, approach- or
development-categories) is itself meaningful;

  2. 2.

it is possible to attempt an interpretation of the frequency of occurrence of
pairs (dyads) of problem – approach categories, and

  3. 3.

the frequency of a triad within the overall set identified, is highly
representative of the state of the art. We find it more summative to observe
triads of category-nodes than triads of individual nodes, as the latter
disperse frequency of occurrence too thinly. Categories act as bins that
aggregate frequency usefully.

###  8.2 Qualitative analysis

Qualitative analysis inheres in the very processes of coding and
categorization, and it is further accentuated by the causal linkages between
the categories of the structural code. The observed categories, dyads, triads
and their relative frequencies are fertile grounds for grounded reflection
about the state of research. Some examples are given in the context of the
case notes (section 9.4).

This culminates our thematic analysis.

##  9 Case notes

We have used the PAD method during a survey of research into power modeling
and measurement in virtualized environments. The surveyed body of papers was
gathered from the ACM, IEEE and other sources (the “corpus”). The corpus is
our raw qualitative data.

###  9.1 Paper selection criteria

Every surveyor will decide on the relevance (in/out of scope) of a paper by
considering certain criteria. A generally valid criterion is to exclude other
surveys from the corpus, as a survey is not itself comparable with the works
within its scope. This does not exclude surveys from consideration, since a
prospective surveyor would be well advised to learn about what ground other
surveyors have covered and results obtained from their coverage. However, the
surveys do not themselves constitute raw qualitative data: they contain
results obtained from the processing of raw data.

In our case, the criterion that proved most effective in sorting RUs into
relevant or irrelevant was the challenge undertaken (or what we may now call:
the P-node). It seems fair to extrapolate this observation to surveys in
general, or at the very least, to a major category of surveys. Many surveys of
the state of the art in some field of a discipline are oriented towards the
progress achieved in tackling the field’s challenges. Hence, the
generalization we suggest would hold for this category of surveys. An
illustration of how the field of study emanates categories of challenges (the
P-category-nodes) is included among the files on our study’s GitHub
site111https://github.com/ijqm/pad/blob/main/TheCoreChallengeanditsDerivatives.pdf

###  9.2 Identifying codes and categorizing them

Problem codes are frequently identified using the abstract and certain key
phrases like "in this paper", or "this paper", or "in this work". As the
number of RUs perused increases, nuances start emerging and codes that at
first appeared to be somewhat distinct are recognized as factually
indistinguishable, even before actual categorization begins. These codes are
assigned a number and integrated with their numbers in nodes (node = code +
number).

As with problem codes, the abstract of an RU is a good source of development
codes. At least some development codes are usually self-evident (i.e.
semantic) here, since researchers are keen to point out their primary
contribution(s) (developments) and hold on to readers’ notoriously volatile
attention. However, thorough harvesting was only obtained with an organic
growth in familiarity with the field, as contributions that were less
conspicuous or weighty started to emerge as more RUs were perused. These
secondary contributions were distributed throughout the length of papers.
Therefore, the pace of harvesting development codes was slower than its
problem-codes counterpart, as their transparency, number and distribution were
less favourable.

Most problematic were the approach codes. These require a difficult movement
from the general ("a way of considering or handling something, esp. a
problem") to the domain-specific. A useful generalization is that these codes
describe the empirical setup and RU sections titled "method" or "methodology"
are good sources of codes. Indeed, this would be the "way of … handling
something." However, there is the rather latent aspect of researchers’ "way of
considering … a problem". For example, we observed that models developed are
at least in part the result of an approach towards modelling, and
justification of the selected form of power model is strongly dependent upon
the inputs and parameters of operation.

Among the files on our study’s GitHub site:

  * •

we cross-reference a comprehensive list of harvested codes(labels) and node
numbers222https://github.com/ijqm/pad/blob/main/NodeAttributes.xlsx, and

  * •

we present all node triads333https://github.com/ijqm/pad/blob/main/Triads.xlsx
harvested from the corpus.

Categorization of codes proceeded as described in section 4, with P-, A-, and
D-nodes handled separately. While the final categorization of each code type
required several iterations over the full set of codes, it was a comparatively
lightly taxing endeavour. The process of coding directly led our coding work
through the considerations that drive categorization: inspection, comparison,
contrast, etc., all while noting down codes.

Figure 1: Frequency of occurrence of approaches to tackling P1, shown in line
thickness and as percentage Figure 2: Frequency of occurrence of approaches to
tackling P2, shown in line thickness and as percentage

###  9.3 Graphical maps

The causality DAG derived from our study is included among the files on our
study’s GitHub
site444https://github.com/ijqm/pad/blob/main/CausalityDAGWithLinksBetweenCNodesOnly.pdf.
The concern with problems in category P7 (resource use and measurement) stands
out in the DAG. The primary category of approaches taken to tackle this
problem is A10 (through instrumentation of computing resources).

The triads graphic derived from the study is also
included555https://github.com/ijqm/pad/blob/main/TriadsWithLinksBetweenCNodesOnly.pdf
on our study’s GitHub site . In this graphic, it can be seen that the
development that has emerged most frequently out of the P7-A10 dyad, is that
of models that regress use of computing resources onto a linear relationship
with power consumption.

Finally, we include herein two examples of dyad graphics from our study. We
have chosen direct inclusion of these graphics as they are the least dense of
all the devices and thus well suited to serving as a handy example. Fig. 1
shows that the challenge of obtaining broad indications of the effect of
architecture on power consumption, is tackled through a wide variety of
approaches. Fig. 2 shows that cross-comparison of power consumption of
virtualization genres and technologies (the challenge) has been primarily
tackled using workloads that target specific resources (34.8%), rather than
workloads that represent real use (4.4%).

Several other graphic devices were compiled using this method’s products. It
has proven to be well suited to the compilation of taxonomies of challenges,
approaches and developments in the surveyed field.

###  9.4 The emergence of themes

Our knowledge of the surveyed domain developed as we progressed in parsing RUs
and discussing the coding and categorization thereof. These processes -
coding, categorization, linking and discussion - led to a thorough qualitative
analysis, which was itself amenable to classification. Thus, we were able to
give a bird’s-eye view (see Fig. 3) of:

  * •

the state-of-the-art, including trends in research;

  * •

research gaps;

  * •

pitfalls and fallacies, and

  * •

domains which demand research into models and measurement of power
consumption.

Figure 3: An overview of the qualitative aspect of thematic analysis of the
research space

##  10 Conclusion

###  10.1 Benefits and Limitations

Thematic analysis with structural coding provides at least four distinct
benefits.

  1. 1.

It facilitates the discovery of patterns in research, through the causality
DAG. The causality DAG is a map that illustrates a structural encoding of
primary research sources. This is a robust base for the development of a set
of themes that represents the ongoing effort of discovery within the field of
study. Patterns may be found in the horizontal dimensions of the DAG. A
pattern in the horizontal dimension can consist of a PAD triad that recurs
several times in the data. It can also consist of a group of triads that share
both the P-category-node and the A-category-node. Such a group of triads
comprises a variety of developments that arise out of the same approach to the
same problem. A looser but nonetheless interesting grouping comprises those
triads that share the P-category-node. This would be useful to a researcher
seeking to learn how others who have addressed the same problem.

  2. 2.

It facilitates the evaluation of the novelty of research proposals. Whether
the proposal has reached the stage of problem identification or selection of
approach, the DAG is a useful tool in the assessment of the likelihood of
developing successful research out of the proposal. This follows because
inspection of the map leads to an indication of the density of research in and
near the space under consideration for research.

  3. 3.

Unlike simpler, ad hoc surveying, the processes used by this method are open
to scrutiny through identifiable and tangible proceedings. Whilst still a
subjective (see limitations) method, its workings are more amenable to a
reader’s analysis than other, less open techniques.

  4. 4.

Structural coding leads the surveyor to process his/her corpus systematically.
The generalizable, parsing technique reduces the verbosity of text to a
regular set of codes that aptly and succinctly describe the unit of research.

Two limitations have emerged.

  1. 1.

Clustering coarsens resolution. The causality DAG links category-code-nodes.
This results in an apparent linkage between any node within one category-node
and any node within the category-node at the other end of the link. However,
this is not necessarily reflected in the RUs we have mined and is a result of
the loss of resolution that accompanies clustering. The effects of this
generalization can be mitigated by ensuring that the clustering notion has
narrowly defined meaning. Such narrowness in meaning reduces the scope for
interpretative error.

  2. 2.

The need for multiple iterations. Identification of codes is limited by the
reviewer’s breadth of vision of the field under review. Since the work may be
carried out with the very purpose of gaining a broad view, it may seem that a
Catch 22 is embedded in the method. This is not expected to be as frustrating
as it may appear. It is expected that a reviewer has some background within
the field. The same ability used to search the field may be exploited during
an initial coding iteration. As the review proceeds, the reviewer’s breadth of
vision expands and the set of codes is grown through refinement of existing
codes and addition of new ones. The limitation may be experienced in any of
the P-, A- and D- categories. In particular, during this survey, perception of
approaches improved as more RUs were parsed. Indeed, this progression is
recognized as part of the labor of research: it is partly undertaken during an
initial familiarization [2, p. 87] and partly as a cyclical re-evaluation of
RUs in the light shed by discovery of new codes. [5, p. 8].

Figure 4: A systematic approach to critical review of a corpus of literature

###  10.2 Method Summary

Using thematic analysis with structural coding, a literature review can
process a diverse corpus of RUs to produce a succinct, graphical, numerical
and analytical representation of the research space. Proceedings may be
divided into the following separate tasks. Tasks 1 - 5 are illustrated in Fig.
4.

  1. 1.

Papers are mined (parsed) for triads of problems, approaches and developments,
using the review protocol described in section 3.2.

  2. 2.

The codes are clustered around their structural codes, while keeping the links
between the codes that organize them into triads.

  3. 3.

The clusters are divided into a number of tightly-packed clusters. These
clusters are identifiable as categories.

  4. 4.

A category-node is obtained from of each such category, as described in
section 4.

  5. 5.

The category-nodes are linked sequentially (horizontally) according to the
mined triads to form a causal chain that proceeds from problem to development.
The complete set of triads forms the research space’s directed acyclic graph
of causality (causality DAG).

  6. 6.

Quantitative analysis is possible through the suggested metrics, which provide
statistical information on the field. These are complemented by the causality
DAG, triads graphic, P-A dyads graphics and taxonomies, which combine to give
a multi-faceted profile of the field.

  7. 7.

Qualitative analysis is facilitated through the means for grounded reflection
provided by the causality DAG and the quantitative analysis.

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