# Uncertainty in Grain Size Estimations of Volatiles on Trans-Neptunian
Objects (TNOs) and Kuiper Belt Objects (KBOs)

A. Emran AR Center for Space and Planetary Sciences, University of Arkansas,
Fayetteville, AR 72701, USA  V. F. Chevrier AR Center for Space and Planetary
Sciences, University of Arkansas, Fayetteville, AR 72701, USA

###### Abstract

We analyze the uncertainty in grain size estimation of pure methane (CH4) and
nitrogen saturated with methane (N2:CH4) ices, the most abundant volatile
materials on trans-Neptunian objects (TNOs) and Kuiper belt objects (KBOs). We
compare the single scattering albedo, which determines the grain size
estimation of outer solar system regolith (Hansen, 2009), of these ices using
the Mie scattering model and two other Hapke approximations (Hapke, 1993) in
radiative transfer scattering models (RTM) at near-infrared (NIR) wavelengths
(1 – 5 µm). The equivalent slab (Hapke Slab) approximation model predicts
results much closer to Mie scattering over the NIR wavelengths at a wide range
of grain sizes. In contrast, even though the internal scattering model (ISM)
predicts an approximate particle diameter close to the Mie model for particles
with a 10 µm radii, it exhibits higher discrepancies in the predicted
estimation for larger grain sizes (e.g., 100 and 1000 µm radii). Owing to the
Rayleigh effect on single-scattering properties, neither Hapke approximate
models could predict an accurate grain size estimation for the small particles
(radii $\mathrm{\leq}$ 5 µm). We recommend that future studies should favor
the equivalent slab approximation when employing RTMs for estimating grain
sizes of the vast number of TNOs and KBOs in the outer solar system.

Classical Kuiper belt objects (250) — Trans-Neptunian objects (1705) —
Plutoids (1268) — Radiative transfer (1335) — Surface ices (2117)

##  1 Introduction

An abundance of methane (CH4) and nitrogen (N2) ices, among others, have been
detected on trans-Neptunian objects (TNOs) and Kuiper belt objects (KBOs) such
as Triton (Cruikshank et al., 1984, 1993) and Pluto (Owen et al., 1993). Eris,
a dwarf planet in the Kuiper belt, also exhibits a prevalence of N2 and CH4
ices on its surface (Dumas et al., 2007). Of these ices, solid CH4 ice has
several strong absorption bands in near-infrared (NIR) wavelengths (Cruikshank
et al., 2019). The physical and chemical properties of these ices on TNOs and
KBOs have been determined from NIR observations using the radiative transfer
models (RTM) of Hapke (1981, 1993). The RTMs enable estimation of abundances
and grain size of the constituent mixtures based on albedo or reflectance of
single scattered light by an average surface grain (Mustard & Glotch, 2019).
For instance, Protopapa et al. (2017) estimated the global scale spatial
abundance and grain size distribution of ices (both volatile and non-volatile)
on Pluto from Ralph/LEISA infrared spectrometer (Reuter et al., 2008) onboard
New Horizons.

Varied results have been reported in grain size estimation of water ice at
outer solar system icy bodies using different scattering models (Hansen,
2009). Thus, it is reasonable to anticipate that the estimation of grain sizes
of CH4 and N2 ices (pure and mixture) on TNOs and KBOs surfaces will have
inconsistencies owing to different models used. Hansen (2009) argued that the
differences in grain size estimation primarily arise from models used for
single scattering albedo calculation rather than bidirectional reflectance
models. Following the same approach, we use scattering models to NIR optical
constants of CH4 and N2 ices (pure and mixture) at temperatures relevant to
TNOs and KBOs surface conditions. Single scattering albedo can exactly be
calculated using Mie theory (Mie, 1908) for simple geometric grains or can be
estimated using Hapke approximation models (Hapke, 1981, 1993) from optical
constants of materials. Accordingly, we analyze the single scattering albedo
to estimate the inconsistencies in grain sizes using Mie theory and two
approximation models from Hapke (1993) that have been widely used in the
existing literature.

The constituent volatiles ices of TNOs and KBOs surfaces exhibit different
thermodynamic phase transitions at surface conditions of the planetary bodies.
For instance, the crystalline $\alpha$ (cubic) - $\beta$ (hexagonal) solid-
phase transition of N2 occurs at 35.6K (Scott, 1976), while CH4I - CH4II
solid-phase transition occurs at $\mathrm{\sim}$21K (Prokhvatilov &
Yantsevich, 1983). On TNOs and KBOs surfaces, the CH4 and N2 ices form solid
solutions and do not coexist as pure substances (Cruikshank et al., 2019). CH4
and N2 ices are completely miscible in one another and show two different
solid solutions such as N2 diluted in CH4 (CH4:N2) and CH4 diluted in N2 (N2:
CH4) (Trafton, 2015). In this study, we use the NIR optical constant of pure
CH4-I ice measured at 39K, as a proxy of methane saturated with nitrogen
(CH4:N2), and nitrogen saturated with methane (N2:CH4) measured at two
different temperatures of 35K and 38K.

The rationale of using pure CH4 ice as the proxy CH4:N2 is that at a
temperature of 40K, the marginal saturation limit of N2 in CH4 ice is
$\mathrm{\sim}$0.035 (Cruikshank et al., 2019). In the CH4:N2 binary system,
the wavelength shift of the CH4 band is very small on the order of
$\mathrm{\sim}$2 x 10 ${}^{-4\ }$µm (Protopapa et al., 2015, 2017). Using
optical constants of pure CH4 ice as the proxy of the CH4:N2 system is
somewhat valid below 40K (Protopapa et al., 2017) since the saturation limit
of N2 in CH4 ice is proportional to temperature changes (Prokhvatilov &
Yantsevich, 1983). At the 35K temperature, the crystalline $\alpha$ (cubic)
phase of N2 is saturated with the CH4-I, while at the 38K, the crystalline
$\beta$ (hexagonal) phase of N2 is saturated with the CH4-I${}_{\ }$ (Tegler
et al., 2010).

##  2 Methods

###  2.1  Single scattering albedo

The single scattering albedo, w, refers to the ratio of the amount of photon
scattered to the combined amount of light scattered from and absorbed by a
particle:

$w=\frac{Q_{sca}}{Q_{sca}\ +\ Q_{abs}},$ (1)

where $Q_{sca}$ is the scattering efficiency and $\ Q_{abs}$ is the absorption
efficiency. The sum of $Q_{sca}$ and $Q_{abs}$ is termed the extinction
efficiency, $Q_{ext}.$ A highly absorbing material likely exhibits a w = 0
whereas a transparent material is more likely to show a w = 1 (Shepard, 2017).

In most space science applications, the w is assumed to be the average
properties, such as optical characteristics, grain size, and to some extent
shape and internal structure, of particles that make up planetary regolith
(Hapke, 1981). The w is a function of optical constants/ indices of refraction
(real, n and imaginary, k) of particles or regolith medium (Mishra et al.,
2021). A particle with a larger size and moderate to larger k tends to absorb
more incident light, and therefore, exhibits a lower w (Shepard & Helfenstein,
2007). A variety of approximate models has been presented by Hapke (1993) to
calculate the w from particle refraction indices (Hansen, 2009). Of these
models, the equivalent slab model (Hapke Slab) and internal
scattering/scatterer model (ISM) are widely used in different planetary bodies
(Li & Li, 2011). Subsequent studies (Hapke, 2001, 2005, 2012) also presented
versions of ISM for surface scattering function that was originally derived
from Hapke (1981). We use a version of the approximate ISM and the Hapke Slab
models to calculate w from the optical constants of the ices found on TNOs and
KBOs.

###  2.2 Optical constants

The optical constant (OC) of pure CH4 and N2:CH4 were chosen based on the
thermodynamics equilibrium of solid methane and nitrogen ices at different
temperatures relevant to TNOs and KBOs surface conditions. As the available
data permits, we use the optical constants of $\alpha$-N2:CH4 at 35K (1- 3.97
µm), $\beta$-N2:CH4 at 38K (1 - 5 µm), and CH4 as the proxy of CH4:N2 at 39K
(1 - 5 µm). The optical constants of pure CH4 were collected from Grundy et
al. (2002) and the N2:CH4 system from Quirico & Schmitt (1992). For detail
about the optical constants used in this study see Table 1. Note that the
$\alpha$-N2:CH4 and $\beta$-N2:CH4 systems are the solid solutions with a
concentration of $\mathrm{<}$2% CH4 and the absorption coefficient is
normalized to a concentration of 1 for the diluted CH4 in solutions (for
detail refer to Quirico & Schmitt (1997)).

Table 1: Optical constant used in this study.

Materials | Temperature (K) | References${}^{\ a}$ | Wavelength range (µm) | Notes
---|---|---|---|---
CH4 | 39 | Grundy et al. (2002) | 1 - 5 | As CH4:N2${}_{\ }$b
N2:CH4 | 35 |  Quirico et al. (1999); Quirico & Schmitt (1997) | 1- 3.97 | N2 in $\alpha$ phase c
N2:CH4 | 38 |  Quirico et al. (1999); Quirico & Schmitt (1997) | 1 - 5 | N2 in $\beta$ phase d

b Filename: optcte-Vis+NIR+MIR-CH4cr-I-39K
c Filename: optcte-NIR-CH4-lowC-alpha-N2-35K
d Filename: optcte-NIR-CH4-lowC-beta-N2-38K-cor

Note. — a Optical constants are available at https://www.sshade.eu/

###  2.3 Mie calculation

The w for particles with simple geometrics (i.e., spherical shape) can exactly
be calculated from Maxwell’s equations using Mie theory (Mie, 1908) if
refractive indices and particle size parameters are known (Moosmüller &
Sorensen, 2018). However, scattering models of highly asymmetric phase
functions demand the treatment of diffraction effects from Mie w (Hansen,
2009). Thus, we use the $\delta$-Eddington corrected Mie single scattering
albedow’as compared to the estimations of w from Hapke approximations. The
$\delta$-Eddington approximated Mie, w’ can be calculated as (Wiscombe &
Warren, 1980):

$w^{\prime}=\frac{\left(1-{\xi}^{2}\right)\ w}{1-{\xi}^{2\ }w}$ (2)

where $\xi$ is the asymmetry factor, which refers to the ratio of the forward-
scattered light to the back-scattered light, calculated by Mie theory. We
employ Mie w calculation following the method described in Wiscombe (1979)
using miepython routine, a Python module licensed under the terms of the
Massachusetts Institute of Technology (MIT) license. The calculated Mie w
result was then adjusted to $\delta$-Eddington corrected Mie w’ following Eq.
2.

###  2.4 Hapke Slab and ISM calculations

The Hapke Slab approximation model is applied if the imaginary part of the
optical constant k$<$ $<$1 so that w can be approximated as (Hansen, 2009):

$w=\frac{1}{\alpha DR\left(n\right)+1}$ (3)

where D is the “particle diameter” and $\alpha$ is the absorption coefficient,
given by:

$\alpha=\frac{4\pi k}{\lambda}$ (4)

where $\lambda$ is the wavelength. R(n) is the reflection function of the real
part (n) of the OC. The R(n) can be derived as (Hansen, 2009):

$R\left(n\right)=\frac{1-S_{e}}{1-S_{i}}$ (5)

where S${}_{e\ }$and Si are the average Fresnel reflection coefficient for
externally and internally incident light, respectively (Hapke, 1993). For the
case of the slab model (k$<$ $<$1), Se can be approximated as (Hapke, 2001):

$S_{e}=\frac{{\left(n-1\right)}^{2}}{{\left(n+1\right)}^{2}}+0.05$ (6)

and the approximation for Si can be written as (Lucey, 1998):

$S_{i}=1.014-\frac{4}{n{\left(n+1\right)}^{2}}\ $ (7)

We use the ISM (Hapke, 1981) as the second approximation method where the w
can be reproduced as (Hapke, 2001):

$w=S_{e}+\left(1-S_{e}\right)\frac{\left(1-S_{i}\right)\textrm{$\Theta$}}{1-S_{i}\textrm{$\Theta$}}\
$ (8)

$\Theta$ is the internal-transmission function of the particle is given by:

$\textrm{$\Theta$}=\frac{r+\mathrm{exp}\mathrm{}(-\sqrt{}(\alpha\left(\alpha+s\right))\langle
D\rangle}{1-r\
\mathrm{exp}\mathrm{}(-\sqrt{}\left(\alpha\left(\alpha+s\right)\right)\langle
D\rangle}$ (9)

where the near-surface internal scattering coefficient, s = 0, and the
internal hemispherical (diffused) reflectance, r can be given as:

$r=\frac{1-\ \sqrt{}\alpha\left(\alpha+s\right)}{1+\
\sqrt{}\alpha\left(\alpha+s\right)}$ (10)

The average distance travel by transmitted ray i.e., mean free path of photon
$\langle$D$\rangle$ as a function of n and particle diameter, D for perfectly
spherical particle can be written as (Hapke, 2005):

$\langle D\rangle\
=\frac{2}{3}\left(n^{2}-\frac{1}{n}{\left(n^{2}-1\right)}^{\frac{3}{2}}\right)D$
(11)

In the ISM model, Si is derived from Eq. 7 while Seis a function of both n and
k, as given by:

$S_{e}=\frac{{\left(n-1\right)}^{2}+k^{2}}{{\left(n+1\right)}^{2}+k^{2}}+0.05$
(12)

##  3 Results

###  3.1 Calculated single scattering albedo

To evaluate how the Mie and Hapke models produce different w at a specific
particle diameter, we compare w of a 10 µm radius particle for pure CH4 at
39K, $\alpha$-N2:CH4 at 35K, and $\beta$-N2:CH4 at 38K (Fig. 1). The
calculated w exhibits small spikes over the wavelengths, and therefore we
smoothen our calculated w curves by applying the Savitzky-Golay filters
(Savitzky & Golay, 1964), an algorithm typically used for signal processing,
with a 3rd order of the polynomial fit.

Figure 1: Plots for single scattering albedo of pure CH4 at 39K (a),
$\alpha$-N2:CH4 at 35K (b), and $\beta$-N2:CH4 at 38K (c) for a grain 10 µm
radius. The dashed solid line presents $\delta$-Eddington corrected Mie,
dashed yellow represents ISM, and the dotted green represents Hapke Slab
results. The plots were smoothened using the Savitzky-Golay filter (Savitzky &
Golay, 1964).

The result of w from Hapke Slab and ISM calculated for pure CH${}_{4\ }$ice
shows a varying degree of closeness to the result of Mie calculation at
different NIR wavelengths (Fig 1a). More specifically, at shorter wavelengths
up to 2 µm, both ISM and slab model exactly follow and mimic the results of
Mie calculation. However, at wavelengths of 2.13 and 2.37 µm, the slab model
results show slightly higher w than the Mie calculation. This indicates that
the Hapke Slab model predicts a slightly smaller pure CH4 grain size compared
to the Mie model at these wavelengths (Hansen, 2009). The ISM w is closer to
the Mie result at longer wavelengths at 3.3 and 3.5 µm – indicating a similar
grain-size prediction by ISM to the Mie at these wavelengths. In contrast, the
Hapke Slab predicts a smaller and larger pure CH4 ice grain compared to the
Mie results at 3.3 and 3.5 µm, respectively. At 3.82 µm, both approximation
models predicate a slightly higher grain size than the Mie results.

The N2:CH4 system results (Fig. 1b and 1c) show that the Hapke Slab model,
overall, produces w values much closer to the Mie model than ISM at both 35
and 38K temperatures, except at $\mathrm{\sim}$ 3.3 µm wavelength for 38K
where the ISM’s w gets closer to Mie. In the $\alpha$-N2:CH4 ice (Fig. 1b),
the w calculated from the Hapke Slab model is much closer to the Mie model
while ISM predicted a much higher w than Mie calculations over the entire NIR
wavelengths. This indicates a much smaller $\alpha$-N2:CH4 ice grain-size
prediction by ISM. Likewise, in the case of the $\beta$ -N2:CH4 system (Fig.
1c), the slab model result follows much closer to the Mie result over the
wavelengths, except for the 3.3 µm where ISM gets closer to Mie than slab
result. This implies that the Hapke Slab model produces a slightly smaller
grain size around 2.3, 3.3, and 3.5µm at 35K and 2.3 and 3.5 µm at 38K than
the Mie model. Around the wavelength of 3.3µm at 38K, the ISM predicts a
slightly smaller grain size than the Mie compared to Hapke Slab to Mie.
Overall, for the N2:CH4 systems, ISM predicts a much smaller grain size than
the Mie compared to the slab model to Mie over the NIR wavelengths for the
particles with a 10 µm radii.

###  3.2 Discrepancies in grain size estimation

We compare the relative grain size predicted by the approximated Hapke models
to the Mie model over the NIR wavelengths. To end that, we first estimate the
$\delta$-Eddington corrected Mie w’ at grain radii of 1, 10, 100, and 1000 µm.
Then we estimate the grain sizes corresponding to these Mie w’ values by
applying the inverse ISM and Hapke Slab models. The w is a non-linear function
of diameter (D) in both Hapke approximation models (Eq. 3 and 8). We solve the
non-linear equations of Hapke Slab and ISM for D using Powell’s hybrid
(dogleg) method (Powell, 1970; Chen & Stadtherr, 1981). Lastly, we estimate
the relative discrepancies in grain size determination by normalizing the
estimated grain sizes from the Hapke Slab and ISM models to the Mie grain size
for pure CH4, $\alpha$-N2:CH4, and $\beta$-N2:CH4 (Fig. 2).

Figure 2: Grain sizes determined from the Hapke Slab (left column) and ISM
(right column) using spectra calculated using the Mie model at different
particle radii of 1, 10, 100, 1000 µm for pure CH4 ice at 39K (top row),
$\alpha$-N2:CH4 at 35K (middle row), and $\beta$-N2:CH4 at 38K (bottom row).
The resulting grain sizes are normalized to the input grain sizes. The plots
were smoothened using the Savitzky-Golay filter (Savitzky & Golay, 1964).

For pure CH4 ice, the Hapke Slab model (Fig. 2a) better predicts the grain
sizes than the ISM model (Fig. 2b). Overall, the predicted grain sizes by the
Hapke Slab model are within $\mathrm{\sim}$ 20% of the grain size of Mie,
whereas the discrepancies in the predicted grain sizes by the ISM model are
much higher. However, around the 3.3 µm region, neither model did predict
grain sizes very well to Mie’s results. Even though the slab model can predict
twice as much as the Mie results at some wavelengths, for instance, at around
2.3$\mathrm{\sim}$2.4 µm region for larger grain sizes (e.g., 1000 µm radii),
the ISM predicated results still show much more discrepancies than slab model
results at these wavelengths. The continuous rise of the 1 µm curve is due to
largely the Rayleigh effect (Hansen, 2009) from the Mie model that the Hapke
Slab method does not model. Likewise, the Rayleigh effect on single-scattering
properties is not modeled by the ISM. Both Hake Slab and ISM provide a better
prediction at 10 µm grain size, within $\mathrm{\sim}$30% of the grain size of
Mie. However, larger grain-sized were not modeled very well by the ISM and
predicted the grain sizes that are many times the grain size of Mie (Fig. 2b).

The N2:CH4 system results (Fig. 2c, d for 35K and Fig. 2e, f for 38K) show
that, overall, the Hapke Slab model has better a prediction, and thus lower
discrepancies in grain size estimation to the Mie model than the ISM model.
Owing to the Rayleigh effect on single-scattering properties modeled by Mie
theory (Hansen, 2009) but not by either of the approximation models for
smaller grain sizes (e.g., 1 µm), there is a trend of continuously increasing
discrepancies in grain sizes estimation by both Hapke Slab and ISM. Similar to
the case of CH4 ice, both Hapke Slab and ISM models fit best at 10 µm, where
it is within $\mathrm{\sim}$30% of the Mie result, for the $\alpha$-N2:CH4 and
$\beta$-N2:CH4 ices. For larger grain radii (e.g., 100 and 1000 µm) the Hapke
Slab exhibits comparatively lower discrepancies than ISM results and predicted
grain sizes within the twice of the Mie over most of the NIR wavelengths,
except at 3.3µm for both temperatures. In contrast, though ISM models a good
fit at 10 µm, the larger particles (e.g., 100 and 1000 µm) exhibit higher
degrees of discrepancies compared to the Hapke Slab results to Mie. This
characteristic result from the N2:CH4 system is also consistent with the
result for water ice grain sizes of Enceladus (Hansen, 2009).

###  3.3 Effect of the absorption coefficient

The distribution of absorption coefficients ($\alpha$) of pure CH4 and CH4
saturated with N2 ices over the NIR wavelength region is given in Fig. 3.
Absorption coefficients of pure CH4 show a peak around the 3.3 µm wavelength
(Fig. 3a). Though absorption coefficients are quite similar for N2:CH4 systems
both 35 and 38K, at the latter temperature (Fig. 3c), absorption coefficients
at around 3.3 µm wavelength are much higher than the former temperature (Fig.
3b). Pure CH4 and $\beta$-N2:CH4 ice show a similar higher absorption
coefficient at 3.3 µm wavelength. The anomalies in grain size prediction of
larger particles by the Hapke Slab method for pure CH4 and $\beta$-N2:CH4 at
3.3 µm are, perhaps, due to the larger k value at this NIR wavelength region.

Figure 3: Absorption coefficient of pure CH4 ice at 39K (a), CH4 saturated
with $\alpha$-N2 ice at 35K (b), and CH4 saturated with $\beta$\- N2 ice at
38K (c). Pure CH4 has an absorption coefficient peak at 3.3 µm. The absorption
coefficient is much higher over the wavelength around 3.3 µm at 38K compared
to 35K for CH4saturated with N2 ices.

The absorption coefficient peak at 3.3 µm at 38K and 39K are consistent with
the fact of lower single scattering albedo or higher absorption. This is also
evident in the relative grain-size curve for 10 µm, the best grain size
prediction by the Hapke approximated models. The grain-size curves at 38K
(Fig. 2a, b) and 39K (Fig. 2e, f) follow a continuum over the NIR wavelengths
except for a “dome” in the 3.3 µm region – indicating a predicted larger grain
size corresponding to higher absorption. We compare $\alpha$ from our result
to that of water (H2O) ices in outer solar system bodies as given in Fig. 4 of
Hansen (2009). Water ice has a higher absorption coefficient in most NIR
wavelength regions compared to both pure CH4 and N2:CH4 systems. This implies
that pure CH4 and N2:CH4 ice grains have higher reflectance and lower
absorption compared to water ice grains over the NIR wavelengths. One possible
interpretation of this comparison is that grain size estimation of H2O ice
using the Hapke approximation models may predict relatively larger grains
compared to pure CH4 and N2:CH4 ices grains at the outer solar systems bodies.
However, this interpretation is based on the distribution of absorption
coefficient over NIR wavelengths, while other factors are involved, different
interpretations are also plausible.

###  3.4 The interplay between albedo, absorption coefficient, and grain size

Figure 4: Plot of the single scattering albedo against absorption coefficient
for pure CH4 ice particle at 39K (left column), $\alpha$ -N2:CH4 particle at
35K (middle column), and $\beta$ -N2:CH4 particle at 38K (right column). The
subplots show the distribution of w at grain radii at 1 µm (upper row), 10 µm
(2nd row), 100 µm (3rd row), and 1000 µm (bottom row). In all subplots, the
scattering albedo from the Mie calculation (blue), ISM (yellow), and Hapke
Slab (green). The Mie calculation follows two different paths due to the
Rayleigh scattering effect at lower grain-size particles.

We analyze characteristic distribution w as a function of $\alpha$ for Mie,
ISM, and Hapke Slab models. The w verses $\alpha$ for pure CH4 (left column),
$\alpha$-N2:CH4 (middle column), and $\beta$-N2:CH4 (right column), ices at
different grain radii (in rows) are given in Fig. 4. In 1 µm gain size graphs
(upper row of Fig. 4), the Mie calculation follows two different paths, and
their separation is largely due to the Rayleigh scattering effect at smaller
grain-sized particles (Hansen, 2009). At all temperatures, each approximation
model follows a relatively linear path while one of the Mie paths shows a
slightly exponential fall of albedo for pure CH4 and $\beta$-N2:CH4 ices
plots. In the N2:CH4 systems, the Hapke Slab and ISM models follow a
relatively similar path. However, one of the routes of Mie plots follows
closely to the Hapke Slab and ISM plots for pure CH4. Most of the weakly
absorbing points with higher single-scattering albedos are located below the
absorption coefficient of $\mathrm{\sim}$ 0.07 µm-1 in pure CH4 and
$\beta$-N2:CH4 ices, whereas below 0.03 µm-1 in $\alpha$-N2:CH4 ice.

There is an exponential drop in the single scattering plots for 10 µm graphs
at all temperatures and thermodynamics ice phases of methane and nitrogen (2nd
row of Fig. 4). The Rayleigh scattering effect (i.e., two separate routes of
Mie plots) is also evident here, but in the weakly separated routes. The plots
of Hapke Slab and ISM approximate models follow along (similar trend) one of
the Mie paths for all thermodynamic pure and mixture ice phases –indicating a
close fit of these approximates models to Mie result at this grain size. There
are few points beyond the absorption coefficient of 0.07 µm-1, that we
consider roughly the exponential breakpoint, for pure CH4 and $\beta$-N2:CH4,
whereas this breakpoint is $\mathrm{\sim}$0.03 µm-1 for $\alpha$-N2:CH4.

For the 100 µm plots (3rd row of Fig. 4), there is a steep decline of the
plots for all phases. Neither of the approximation models consistently follow
the Mie route; at some absorption coefficient values, ISM gets closer to Mie
while at other points Hapke Slab model closely follows the Mie path. The
breakpoint of the steeply declined plots is roughly around 0.01 µm-1 and most
of the absorption points reside below this threshold where single scattering
albedo value varies in a wide range (e.g., w = 0.1 – 1.0). In the 1000 µm
plots (lower row of Fig. 4), the single scattering albedo plots more steeply
decline at all thermodynamics ice phases of methane and nitrogen. However, the
breakpoint of the albedo plots is $\mathrm{<}$ 0.01 µm-1, meaning the single
scattering albedo steeply falls at a lower absorption coefficient for larger
grain sizes. Similar to the 100 µm plots, in 1000 µm plots neither of the
approximation models consistently follow the Mie route.

##  4 Discussion and conclusion

Amidst inconsistent results in the grain size estimation of water ice on outer
solar system bodies (Hansen, 2009), we analyze the relative differences of
grain size estimation for pure CH4 and N2:CH4 ices relevant to TNOs and KBOs.
We calculate the single scattering albedo using Mie and two other Hapke
approximations models for these ices at NIR wavelengths. Compared to ISM, the
Hapke Slab approximation model predicts much closer results to Mie scattering
results. In pure CH4 ice, the overall estimated grain size differences between
the Hapke Slab and ISM are about 10% for the particles with different grain
radii, except in the case of larger particles and longer wavelengths where the
imaginary part of the refractive index is much larger. Similarly, in N2:CH4
systems, the average differences between the estimated grain sizes from Hapke
Slab and ISM are around $\mathrm{\sim}$20% for wide ranges of grain size.

Both Hapke Slab and ISM were found to be appropriate models for a grain size
radius of 10 µm. For smaller grain-sized particles (radii of $\mathrm{\leq}$5
µm), neither approximate model predicts an accurate grain size due to the
Rayleigh effect. For larger grains at longer wavelengths, particularly at
wavelengths with higher absorption coefficient values, the ISM predicted grain
sizes exhibit larger anomalies compared to the Mie result. Overall, the
results estimated prediction at different grain sizes indicate that the Hapke
Slab model is the more well-predicted model to the Mie result over the NIR
optical constants of pure CH4 and N2:CH4 systems while ISM’s predictions show
higher discrepancies. Existing literature indicates that the discrepancy in
grain-size determination for larger particles at longer wavelength using the
ISM can be mitigated by fine-tuning the value of free parameters. For
instance, Roush et al. (2007) used an increased value of s in their Eq. 2 and
3 from 10-17 to 1.25 cm-1 to fit the modeled spectrum of gypsum power to
measured spectrum (Hansen, 2009).

The particle diameter in the Hapke Slab model is associated with a scaling
factor that varies from 3/4 to 4/3 (Hansen, 2009). In ISM, similar uncertainty
in selecting the mean free path of photon $\langle$D$\rangle$ by using
different scale factors to effective grain size D. For instance, for spherical
particles, $\langle$D$\rangle$ can be approximated to $\mathrm{\cong}$ 0.9D
(Hapke, 2012), 2D/3 (Melamed, 1963), etc., while for irregular particles
$\langle$D$\rangle$ = 0.2D (Shkuratov & Grynko, 2005). This study uses a scale
factor of 1 in the Hapke Slab model. We use the mean free path of photon
$\langle$D$\rangle$ calculated from effective grain size (D) and the real part
(n) of the refractive index using Eq. 12. The Hapke Slab and IMS can,
therefore, be improved by fine-tuning the scale factors to the approximate
models. The slab model accounts only n while the ISM considers both the n and
k parts of the refractive indices in the calculation of the average Fresnel
reflection coefficients. If the internal scattering coefficient, s is set to
0, the internal hemispherical (diffused) reflectance, r equals 0. However, the
relationship between the internal scattering coefficient (s) and effective
particle diameter (D) has also been expressed as s = 1/D  (Sharkey et al.,
2019). This relationship also defines that the number of scattering events
within a single grain is set to 1. Consequently, this relationship indicates
that the value of s cannot be 0.

The application of the scattering (and absorption) properties of Mie spheres
has been shown to be satisfactory for varied non-spherical particle shapes
(Grenfell & Warren, 1999; Neshyba et al., 2003; Grenfell et al., 2005). The
Mie formulation accurately predicts the scattering properties of equivalent
spheres of particles, even it can produce satisfactory scattering results for
irregular particles (Neshyba et al., 2003). Thus, the size of spherical
particles from the Mie model can somewhat be analogous to non-spherical
particles (Hansen, 2009). Moreover, the Mie scattering formulation properly
accounts for the Rayleigh effects of scattering properties for smaller
particle sizes that are ignored by the Hapke approximation models. Therefore,
based on our results, we recommend using the Mie calculation for radiative
transfer modeling to unknown spectra of TNOs and KBOs. Our results show that
the Hapke Slab approximation model, overall, well predicts the grain size to
the Mie model over the NIR wavelengths. Thus, if the Hapke approximation
models are to choose, we suggest using the equivalent slab model over the
internal scattering model in estimating the pure CH4 and N2:CH4 ice grain
sizes on trans-Neptunian objects and Kuiper belt objects. Our study provides a
guideline for the future application of RTM in estimating the ice grain sizes
at TNOs and KBOs.

The authors would like to thank anonymous reviewers for their useful comments.

##  Appendix A Notation

List of notation and symbols used in this paper

$\langle$D$\rangle$ mean free path of a photon

D particle diameter

$\xi$ asymmetry parameters of Mie theory

k imaginary part of the refractive index

n real part of the refractive index

r internal diffused reflectance

R(n) reflection function

s internal scattering coefficient

Se Fresnel reflection coefficient for externally incident light,

Si Fresnel reflection coefficient for internally incident light,

w single scattering albedo

w’ $\delta$-Eddington Mie single scattering albedo

$\alpha$ absorption coefficient

$\lambda$ wavelength

$\Theta$ internal transmission coefficient

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identity $e_{H}$ of $H$ for all but finitely many $y\in G$. The group
operation is given by

$(\varphi_{1},x_{1})(\varphi_{2},x_{2})=(\varphi_{1}(x_{2}\varphi_{2}),x_{1}x_{2}),$

where $x\varphi(y)=\varphi(x^{-1}y)$ and $(\varphi\psi)(y)=\varphi(y)\psi(y)$.

It is well-known that if $G$ and $H$ are finitely generated, then $H\wr G$ is
finitely generated. In fact, given a finite generating set $S$ of $G$ and a
finite generating set $T$ of $H$, the canonical generating set of $H\wr G$ is
defined as follows: For every $t\in T$ let $\varphi_{t}\in H^{(G)}$ be the
function with $\varphi_{t}(e_{G})=t$ and $\varphi_{t}(x)=e_{H}$ for all $x\neq
e_{G}$. Moreover, let $e_{H^{(G)}}\colon G\to H$ be the function which is
equal to identity everywhere. Then

$\Bigl{\\{}(e_{H^{(G)}},s)\colon s\in
S\Bigr{\\}}\sqcup\Bigl{\\{}(\varphi_{t},e_{G})\colon t\in T\Bigr{\\}}$

is a finite generating set for $H\wr G$, see e.g. [44].

###  7.2. Wreath products over free groups.

In the following we restrict our attention to one particular example, the
wreath product $\Gamma=\mathbb{Z}_{2}\wr\mathbb{F}_{2}$, the classical
lamplighter group over the free group on two generators. This is only for
simplicity of the exposition and all arguments extend in straightforward
manner to the general setting of Theorem 7.1.

Let ${\mathcal{G}}$ denote the Cayley graph of $\Gamma$ with respect to the
canonical generating set. We can picture this group as the $4$-regular tree
with a lamp attached to each vertex. An element $(\varphi,x)\in\Gamma$ is then
interpreted as lamps being switched on at the support of $\varphi$, denoted by
${\rm supp}(\varphi)$, and a marker standing at position $x$. If $a,b$ denote
the canonical generators for $\mathbb{F}_{2}$, then the generators

$\Bigl{\\{}(\mathbf{0},a),(\mathbf{0},a^{-1}),(\mathbf{0},b),(\mathbf{0},b^{-1})\Bigr{\\}},$

where $\mathbf{0}(x)=0$ for all $x$, correspond to moving the marker in the
direction of the appearing generator while leaving the configuration of lamps
untouched. The remaining generator
$({\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{e_{\mathbb{F}_{2}}},e_{\mathbb{F}_{2}})$
corresponds to switching the state of the lamp at the current position of the
marker. For this Cayley graph it is well-known how to describe the graph
distance: Given two elements $(\varphi,x)$ and $(\psi,y)$ of $\Gamma$, a
shortest path in ${\mathcal{G}}$ is such that the marker starts at position
$x$, visits the position of all lamps which need to be switched and switches
these configurations accordingly, and finally moves to position $y$. Note that
the positions of the lamps which need to be switched are given by ${\rm
supp}(\varphi\psi)$ and that $|{\rm supp}(\varphi\psi)|$ many switches have to
be made. Therefore

$d((\varphi,x),(\psi,y))=|{\rm supp}(\varphi\psi)|+\ell(x,{\rm
supp}(\varphi\psi),y),$

where $\ell(x,{\rm supp}(\varphi\psi),y)$ denotes the length of a shortest
path in the $4$-regular tree which starts at $x$, visits ${\rm
supp}(\varphi\psi)$ completely and ends at $y$. In other words, the distance
is essentially determined by the solution to the above travelling salesman
problem in the Cayley graph of $\mathbb{F}_{2}$. When $\mathbb{F}_{2}$ is
replaced with a different finitely generated group and the tree is replaced
with the corresponding Cayley graph, this description remains correct.
However, the particularly nice connection between walks in the base graph and
walls in the lamplighter graph we describe next makes use of the tree-
structure.

The following description of a proper, left-invariant structure of space with
walls on $\Gamma$ is due to de Cornulier, Stalder and Valette [15]. We adapt
their notation to our more restrictive setting in order to readily obtain
quantitative estimates of the associated wall distance. Let $T_{4}$ denote the
canonical Cayley graph of $\mathbb{F}_{2}$, i.e. the $4$-regular tree, whose
root is given by the vertex corresponding to $e_{\mathbb{F}_{2}}$. Note that
every edge $e\in E(T_{4})$ determines a wall by splitting $T_{4}$ into the two
connected components of $T_{4}\setminus\\{e\\}$. For our purposes it is
convenient to consider the set $E^{\to}(T_{4})$ of oriented edges of $T_{4}$.
For every oriented edge $e$, let $A_{e}$ denote the connected component of
$T_{4}\setminus\\{e\\}$ which it points away from. Given a finitely supported
function $\psi\colon A_{e}^{c}\to\mathbb{Z}_{2}$, define

$E(e,\psi)\coloneqq\Bigl{\\{}(x,\varphi)\in\Gamma\colon x\in
A_{e},\varphi_{|A_{e}^{c}}=\psi\Bigr{\\}}.$

A family of walls on $\Gamma$ is then defined by

${\mathcal{W}}\coloneqq\Bigl{\\{}\\{E(e,\psi),E(e,\psi)^{c}\\}\colon e\in
E^{\to}(T_{4}),\psi\colon A_{e}^{c}\to\mathbb{Z}_{2}\,\,\mbox{finitely
supported}\Bigr{\\}}.$

It was proven in [15, Theorem 1] that $(\Gamma,{\mathcal{W}})$ is a space with
walls on which $\Gamma$ acts properly via left translations. We are interested
in a quantitative version, which is given by the following lemma and was also
observed in a more general context in [16, Proposition 7.2].

######  Lemma 7.2.

Let $\Gamma\coloneqq\mathbb{Z}_{2}\wr\mathbb{F}_{2}$. Consider the proper
action of $\Gamma$ on the space with walls $(\Gamma,{\mathcal{W}})$ with

${\mathcal{W}}\coloneqq\Bigl{\\{}E(e,\psi)\colon e\in
E^{\to}(T_{4}),\psi\colon A_{e}^{c}\to\mathbb{Z}_{2}\,\,\mbox{finitely
supported}\Bigr{\\}}.$

Set $o\coloneqq(\mathbf{0},e_{\mathbb{F}_{2}})$ and consider the canonical
Cayley graph of $\Gamma$. Then, for every $(\varphi,x)\in\Gamma$,

$w(o,(\varphi,x))\geq\frac{1}{4}|(\varphi,x)|-\frac{1}{4}.$

###### Proof.

Let $(\varphi,x)\in\Gamma$ and recall from above that

$|(\varphi,x)|=d(o,(\varphi,x))=|{\rm
supp}(\varphi)|+\ell(e_{\mathbb{F}_{2}},{\rm supp}(\varphi),x).$

To shorten notation, set $\ell=\ell(e_{\mathbb{F}_{2}},{\rm
supp}(\varphi),x)$. Let $\rho=\\{e_{1},\ldots,e_{\ell}\\}$ be a shortest path
of edges from $e_{\mathbb{F}_{2}}$ to $x$ which visits ${\rm supp}(\varphi)$
completely. Note that every position in ${\rm supp}(\varphi)$ except the
origin has to be visited by $\rho$, thus

$|{\rm supp}(\varphi)|\leq 1+\ell.$

In particular

$|(\varphi,x)|\leq 2\ell+1.$ (7.1)

We claim that

$\ell\leq 2w(o,(\varphi,x)).\vskip 6.0pt plus 2.0pt minus 2.0pt$ (7.2)

To see this, we describe a subset $\rho^{\prime}$ of $\rho$, consisting of at
least half of the original edges, such that for each in $\rho^{\prime}$ there
is a unique wall separating $o$ and $(\varphi,x)$: Let $e_{i}$ be an edge in
$\rho$ which appears for the first time. It is clear from the assumptions on
$\rho$ that each edge can appear at most twice and we treat the resulting two
cases separately.

  1. (i)

Assume that $e_{i}$ appears a second time, say at $e_{j}$ with $i\leq
j\leq\ell$. Since the walk takes place on a tree, it follows that every edge
$e_{k}$ with $i\leq k\leq j$ appears exactly twice. Treating each $e_{k}$ as
an oriented edge pointing in the direction of the next appearing edge in
$\rho$ or towards $x$ in case $i=\ell-1$, we see that this case describes the
following situation: The path $\rho$ starts at endpoint $y$ of $e_{i}$,
proceeds in the direction of $e_{i}$ and explores a part of the corresponding
branch of the tree, before returning to $y$ and leaving in the direction of
one of the remaining branches originating from $y$. In particular the edge
$e_{i}$ points away from the final position $x$ of the marker. The same holds
for every $e_{k}$ with $i\leq k\leq j$ which appears for the first time. Now
fix any such edge $e_{k}$. Clearly the only reason for its appearance can be
that there is an element $z\in{\rm supp}(\varphi)$ which lies in the direction
it is pointing. Recalling that $A_{k}\coloneqq A_{e_{k}}$ denotes the
connected component of $T_{4}\setminus\\{e_{k}\\}$ which $e_{k}$ points away
from, we therefore have that $x\in A_{k}$ and that $z\in{\rm
supp}(\varphi_{|A_{k}^{c}})$. It follows that the set

$E(e_{k},\varphi_{|A_{k}^{c}})=\Bigl{\\{}(v,\psi)\in\Gamma\colon v\in
A_{k},\psi_{|A_{k}^{c}}=\varphi_{|A_{k}^{c}}\Bigr{\\}}$

contains $(x,\varphi)$, but not $o=(\mathbf{0},e_{\mathbb{F}_{2}})$ because
$\varphi_{|A_{k}^{c}}(z)=1\neq\mathbf{0}(z)$. We have thus exhibited for half
of the edges $e_{i},\ldots,e_{j}$ a unique wall separating $o$ and
$(\varphi,x)$.

  2. (ii)

Assume that $e_{i}$ appears exactly once. We again treat $e_{i}$ as an
oriented edge pointing towards the next appearing edge in $\rho$ or towards
$x$ in case $i=\ell-1$. Since the path started at the identity and since
$e_{i}$ appears for the first time, $e_{i}$ points away from
$e_{\mathbb{F}_{2}}$, i.e. $e_{\mathbb{F}_{2}}\in A_{i}\coloneqq A_{e_{i}}$.
Since $e_{i}$ is not traversed twice, the final position $x$ of the marker
lies in $A_{e_{i}}^{c}$. In particular

$E(e_{i},\mathbf{0}_{|A_{i}^{c}})=\Bigl{\\{}(v,\psi)\in\Gamma\colon v\in
A_{i},\psi_{|A_{i}^{c}}=\mathbf{0}_{|A_{k}^{c}}\Bigr{\\}}$

contains $o=(\mathbf{0},e_{\mathbb{F}_{2}})$, but not $(x,\varphi)$ because
$x\notin A_{i}$. We have thus exhibited for this edge a unique wall separating
$o$ and $(\varphi,x)$.

Applying the above procedure iteratively proves Inequality (7.2). Combined
with Inequality (7.1), this concludes the proof of Lemma 7.2. ∎

Proof of Theorem 7.1. Combining Lemma 7.2 with Theorem 4.1 yields the desired
result. ∎

##  8\. Isometry groups of Hyperbolic space: Proof of Theorem 1.8

In this section we discuss a relevant class of examples in which the
percolations constructed using Lemma 3.6 can be understood in a precise
geometric sense. These are finitely generated groups with a proper isometric
action on real hyperbolic space. In dimension two, we use the additional
geometric information to describe the number of infinite clusters completely,
i.e. we prove Theorem 1.8 which, as a special case, yields that canonical
Cayley graphs of co-compact Fuchsian groups admit invariant bond percolations
with marginals arbitrarily close to $1$ but with exponential decay of the two-
point function and infinitely many infinite clusters a.s.

###  8.1. Hyperbolic space as a space with measured walls.

The following observations are due to [11, Section 3] building on work of
Robertson [51], see also [10, Example 3.7] for the notation we adopt here. Let
$\mathbb{H}^{n}$ be real $n$-dimensional hyperbolic space and let ${\rm
Isom}(\mathbb{H}^{n})$ be its isometry group. Let
${\mathcal{W}}_{\mathbb{H}^{n}}$ denote the set of all closed or open
geometric half-spaces in $\mathbb{H}^{n}$ with boundary an isometric copy of
$\mathbb{H}^{n-1}$. Each half-space defines a wall by partitioning the space
into the half-space itself and its complement. The group ${\rm
Isom}(\mathbb{H}^{n})$ is unimodular, acts transitively on
${\mathcal{W}}_{\mathbb{H}^{n}}$ and the stabilizer of a given wall is
unimodular. Thus ${\mathcal{W}}_{\mathbb{H}^{n}}$ carries a non-zero ${\rm
Isom}(\mathbb{H}^{n})$-invariant Borel measure $\mu_{\mathbb{H}^{n}}$. Note
that the set ${\mathcal{W}}_{\mathbb{H}^{n}}(z,w)$ of walls separating two
points $z,w\in\mathbb{H}^{n}$ is relatively compact, which implies
$\mu_{\mathbb{H}^{n}}({\mathcal{W}}_{\mathbb{H}^{n}}(z,w))<\infty$. Hence the
$4$-tuple
$(\mathbb{H}^{n},{\mathcal{W}}_{\mathbb{H}^{n}},{\mathcal{B}}({\mathcal{W}}_{\mathbb{H}^{n}}),\mu_{\mathbb{H}^{n}})$
defines a space with measured walls. By Crofton’s formula [51, Proposition
2.1], there exists a constant $\lambda>0$ such that, for every
$z,w\in\mathbb{H}^{n}$,

$\mu_{\mathbb{H}^{n}}({\mathcal{W}}_{\mathbb{H}^{n}}(z,w))=\lambda d(z,w),$
(8.1)

where $d(z,w)$ denotes the hyperbolic distance.

###### Remark 8

It is much more difficult to define a structure of space with measured walls
on $n$-dimensional complex hyperbolic space $\mathbb{H}^{n}_{\mathbb{C}}$ such
that it is invariant under the action of the corresponding isometry group; the
difficulties are described e.g. in [51]. The fact that this is possible is a
result of Chatterji, Drutu and Haglund [10, Corollary 6.28]: In fact,
$\mathbb{H}^{n}_{\mathbb{C}}$ may be equipped with an invariant structure of
space with measured walls such that the corresponding wall distance is the
square-root of the hyperbolic distance and such that all walls are convex with
respect to the wall distance.

Example 8.1  From the above considerations, we obtain many examples of groups
which satisfy property (ii) in Theorem 4.1, namely that the Haagerup property
is witnessed by two-point functions of invariant percolations with exponential
decay. More precisely, this is true for every finitely generated group
$\Gamma$ which has an isometric action on $\mathbb{H}^{n}$ such that for some
$z_{0}\in\mathbb{H}^{n}$ we have $d(z_{0},gz_{0})\geq c|g|$ for all
$g\in\Gamma$. Particular examples are provided by discrete subgroups of ${\rm
Isom}(\mathbb{H}^{n})$ which act co-compactly (by the Milnor-Schwarz Lemma,
see e.g. [53, Theorem 1.18]).

###  8.2. Hyperplane percolation.

We now focus on the hyperbolic plane $\mathbb{H}^{2}$. Suppose that
${\mathcal{G}}$ is a graph embedded in $\mathbb{H}^{2}$ such that the
automorphisms of ${\mathcal{G}}$ extend to isometries of $\mathbb{H}^{2}$. Let
$\eta$ be a Poisson process on ${\mathcal{W}}_{\mathbb{H}^{2}}$ with intensity
measure $\gamma\mu_{\mathbb{H}^{2}}$ for some $\gamma>0$. Define a bond
percolation on ${\mathcal{G}}$ as in Lemma 3.6, i.e. for every half-space
contained in $\eta$, delete all edges which intersect its boundary. We refer
to this bond percolation as the hyperplane percolation on ${\mathcal{G}}$
induced by $\eta$. Note that ${\rm Isom}(\mathbb{H}^{2})$-invariance of $\eta$
implies ${\rm Aut}({\mathcal{G}})$-invariance of the corresponding hyperplane
percolation.

######  Theorem 8.2.

Let ${\mathcal{G}}$ be a transitive, nonamenable, planar graph with one end.
Then for every $\alpha<1$, there exists an ${\rm
Aut}({\mathcal{G}})$-invariant bond percolation $\mathbb{P}$ on
${\mathcal{G}}$ with $\mathbb{E}[\deg_{\omega}(o)]>\alpha\deg(o)$, such that
$\tau(o,\cdot)$ has exponential decay and such that it has infinitely many
infinite clusters a.s.

###### Remark 9

Theorem 8.2 applies to planar Cayley graphs of nonamenable one-ended groups.
In particular it applies to the examples considered by Lalley [38, 39], namely
planar Cayley graphs of co-compact Fuchsian groups. For these groups, it was
shown in [9, 38] that Bernoulli bond percolation exhibits a double phase
transition. In this sense, these groups behave quite differently from
infinitely ended ones, which all have uniqueness threshold $p_{u}=1$. With
respect to general invariant bond percolations, this difference between one-
ended and infinitely-ended nonamenable planar Cayley graphs disappears in the
sense that there exist invariant percolations with marginals arbitrarily close
to $1$ but with infinitely many infinite clusters a.s.

Example 8.3  We specialize Example 8.1 to the setting of the hyperbolic plane:
Consider a co-compact Fuchsian group $\Gamma$. Recall that there exists a
compact fundamental polygon $T\subset\mathbb{H}_{2}$ for $\Gamma$, i.e. a
compact, connected subset, bounded by finitely many geodesics and such that
the translates $\\{gT\\}_{g\in\Gamma}$ tesselate $\mathbb{H}_{2}$. In
particular there are finitely many group elements for which $gT$ and $T$
intersect. These are called side-pairing transformations and constitute a
generating set for $\Gamma$. Let ${\mathcal{G}}$ be the corresponding Cayley
graph, which can be identified with the dual graph of the tiling $\\{gT\\}$.
Since the tiling is locally finite, we have that for every radius $r>0$, there
exists a constant $0<D(r)<\infty$ such that no hyberbolic ball of radius $r$
intersects more than $D(r)$ tiles. Now pick $z_{0}\in T\setminus\partial T$
and set $\rho\coloneqq\inf_{g,h\in\Gamma}d(gz_{0},hz_{0})$. It follows by
elementary arguments that

$\frac{\rho}{2D(\rho)}|g|\leq d(z_{0},gz_{0})\leq{\rm
diam}(T)|g|\quad\mbox{for all} \,\,g\in\Gamma\setminus\\{o\\},$

see e.g. [39, Proposition 1.10].

###### Proof of Theorem 8.2.

By [9, Proposition 2.1], the assumptions guarantee that ${\rm
Aut}({\mathcal{G}})$ is discrete, hence unimodular, and that ${\mathcal{G}}$
can be embedded in the hyperbolic plane $\mathbb{H}^{2}$ in such a way that
the action of ${\rm Aut}({\mathcal{G}})$ extends to an isometric action on
$\mathbb{H}^{2}$. Moreover, the embedding may be taken such that the dual
graph is quasi-transitive, hence the embedding is a quasi-isometry. Identify
${\mathcal{G}}$ with its embedding. Let $p\in(0,1)$, let $\eta_{p}$ be a
Poisson process on ${\mathcal{W}}_{\mathbb{H}^{2}}$ with intensity measure
$(1-p)\mu_{\mathbb{H}^{2}}$ and let $\xi_{p}$ denote the induced hyperplane
process on ${\mathcal{G}}$. Let $\mathbb{P}_{p}$ denote the law of $\xi_{p}$,
which defines an ${\rm Aut}({\mathcal{G}})$-invariant bond percolation. Since
the embedding is a quasi-isometry, it follows from Crofton’s formula, see
Equation (8.1), that $\tau_{\xi_{p}}(o,\cdot)$ has exponential decay and that

$\lim_{p\to 1}\mathbb{E}[\deg_{\xi_{p}}(o)]=\deg(o).$

Hence it suffices to show that $\xi_{p}$ has infinitely many infinite clusters
for $p$ sufficiently close to $1$. By [9, Theorem 3.1], the number of infinite
clusters of $\xi_{p}$ is a.s. constant and equal to $0,1$ or $\infty$. Note
that since ${\rm Aut}({\mathcal{G}})$ is transitive and unimodular,
nonamenability of ${\mathcal{G}}$ implies nonamenability of ${\rm
Aut}({\mathcal{G}})$ by [7, Theorem 5.3] . The generalization [7, Theorem 5.1]
of Theorem Theorem A to general automorphism groups of graphs implies that any
invariant percolation on ${\mathcal{G}}$ with marginals sufficiently close to
$1$ has infinite clusters with positive probability. Therefore it suffices to
rule out the possibility of a unique infinite cluster. Note that this seems
obvious because we already know that the two-point function of $\xi_{p}$
vanishes at infinity. However, we cannot directly apply the standard Harris-
FKG-Inequality because the percolation $\xi_{p}$ may be highly dependent.

We present two alternative proofs, one specific to the setting of hyperbolic
geometry and one based on the Harris-FKG-Inequality for the Poisson process,
which therefore applies quite generally and yields some additional information
about the infinite clusters of the percolations in Theorem 3.5, see Remark 10
below.

Completing the proof using the hyperbolic setting. This proof uses only the
following result.

######  Theorem 8.4 ([42, Theorem 8.48 & 8.49]).

Let ${\mathcal{G}}$ be a quasi-transitive graph embedded in hyperbolic space
$\mathbb{H}^{n}$ in such a way that the embedding is a quasi-isometry. Let
$\omega$ be the configuration of an ${\rm Aut}({\mathcal{G}})$-invariant
percolation on ${\mathcal{G}}$. If $\omega$ has a unique infinite cluster
a.s., then a.s. the set of points $z$ in the ideal boundary
$\partial\mathbb{H}^{n}$ for which there is a path in $\omega$ with limit $z$
is dense in $\partial\mathbb{H}^{n}$.

This is a generalization of results of Benjamini and Schramm [9, Theorem 4.1 &
Lemma 4.3], stated in [42]. Note that it applies not just in dimension two.
However, in general there might be a lack of ergodicity in dimensions three
and higher. Looking back at our model, we observe that $\eta_{p}$ is a.s. non-
zero, hence the set of points $z$ in the ideal boundary for which there is a
path in $\xi_{p}$ with limit $z$ is a.s. not dense in
$\partial\mathbb{H}^{2}$. Theorem 8.4 then implies that $\xi_{p}$ does not
a.s. have a unique infinite cluster.

An alternative proof using the Harris-FKG-Inequality for the Poisson process.
By [40, Corollary 3.7], we may assume that our Poisson process is proper.
Moreover, the intensity measure $\mu_{\mathbb{H}^{2}}$ is $\sigma$-finite. To
see this, take for example a countable dense set $D$ in $\mathbb{H}^{2}$ and
consider the countable covering of ${\mathcal{W}}_{\mathbb{H}^{2}}$ by the
compact, thus of finite $\mu_{\mathbb{H}^{2}}$-measure, sets
$\\{\overline{{\mathcal{W}}(z,w)}\colon z\neq w\in D\\}$. For $g\in\Gamma$,
let $A_{g}$ denote the event that the cluster of vertex $g$ in the hyperplane
process $\xi_{p}$ is the unique infinite cluster. Since adding hyperplanes to
the process $\eta_{p}$ can only result in removing more edges, the function
${\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{A_{g}}$
is decreasing on all of ${\mathcal{W}}_{\mathbb{H}^{2}}$. By Theorem 2.5, for
every $g,h\in\Gamma$,

$\tau_{\xi_{p}}(g,h)\geq\mathbb{P}_{p}[A_{g}\cap A_{h}]=
\mathbb{E}[{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{A_{g}}(\eta_{p}){\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{A_{h}}(\eta_{p})]\geq\mathbb{P}_{p}[A_{g}]\,\mathbb{P}_{p}[A_{h}]=\mathbb{P}_{p}[A_{o}]^{2}.$

Now suppose there is a unique infinite cluster $\mathbb{P}_{p}$-as., then
$\mathbb{P}_{p}[A_{o}]$ is positive and thus $\tau_{\xi_{p}}$ is bounded away
from zero. This contradicts the fact that $\tau_{\xi_{p}}$ vanishes at
infinity. ∎

###### Remark 10

The Poisson processes in the proof of Theorem 3.5 also have $\sigma$-finite
intensity measure. By a slight modification of the above argument, we obtain
that it is impossible that these percolations have a unique infinite cluster
with positive probability. It follows that the percolations appearing in the
statement of Theorem 3.5 may always be assumed to have no unique infinite
cluster a.s. However, in the general context there is neither ergodicity nor
planar hyperbolic geometry, hence the number of infinite clusters could be a
non-trivial random variable with values in
$\\{0,2,3,\ldots\\}\cup\\{\infty\\}$.

Acknowledgement: The research of both authors is funded by the Deutsche
Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC
2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure.

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# Coalitions in International Litigation: A Network Perspective

R. Mastrandrea  IMT Alti Studi Lucca, Piazza S. Francesco 19, 55100 Lucca,
Italy<EMAIL_ADDRESS>G. Antuofermo  Centro Studi Giuridici
Francesco Carrara, Lucca, Italy  M. Ovadek  Centre for Empirical
Jurisprudence, University of Leuven, Tiensestraat 45, 3000 Leuven, Belgium
T.Y.-C. Yeung  Centre for Empirical Jurisprudence, University of Leuven,
Tiensestraat 45, 3000 Leuven, Belgium  A. Dyevre  Centre for Empirical
Jurisprudence, University of Leuven, Tiensestraat 45, 3000 Leuven, Belgium  G.
Caldarelli  DSMN Ca’Foscari, University of Venice, Via Torino 155, 30171
Venezia Mestre, Italy  ECLT Ca’Foscari, University of Venice, Dorsoduro 3246,
30123 Venice, Italy

###### Abstract

We apply network science principles to analyze the coalitions formed by
European Union (EU) nations and institutions during litigation proceedings at
the European Court of Justice. By constructing Friends and Foes networks, we
explore their characteristics and dynamics through the application of cluster
detection, motif analysis, and duplex analysis. Our findings demonstrate that
the Friends and Foes networks exhibit disassortative behavior, highlighting
the inclination of nodes to connect with dissimilar nodes. Furthermore, there
is a correlation among centrality measures, indicating that member states and
institutions with a larger number of connections play a prominent role in
bridging the network. An examination of the modularity of the networks reveals
that coalitions tend to align along regional and institutional lines, rather
than national government divisions. Additionally, an analysis of triadic
binary motifs uncovers a greater level of reciprocity within the Foes network
compared to the Friends network.

## Introduction

The digital revolution[1] has ushered in an era of unprecedented access to
vast amounts of data, revolutionizing social-scientific research and opening
up new avenues for quantitative approaches.[2] Network science, a powerful
tool for representing and visualizing complex systems [3], has been
increasingly applied across various domains, from the study of scientific
advancements[4] to finance[5] and social systems[6].

Although the application of network science to social behaviour and human
institutions predates the digital revolution as illustrated by Jonathan L.
Moreno’s famous sociograms[7] and pioneering studies on self-organised
segregation phenomena[8] and social cooperation [9], law has long been
perceived as a field beyond the reach of quantitative modelling. However, with
digitalisation making considerable progress, the legal field has also
witnessed growing interest in network science methods to model case citation
dynamics[10, 11, 12, 13, 14, 15], the evolution and structure of legislation
[16, 17] as well as professional networks of judges [18] and law professors
[19]. While untangling the complexity of normative structures[20, 17], this
literature has delivered new insights on the operations of legal
institutions[21] and the possibility to predict case citations [15].

In this context, we present a network analysis of coalitions in litigation
before the Court of Justice of the European Union (CJEU). The CJEU adjudicates
disputes over the legality of EU acts, the interpretation of EU treaties and
legislation and state compliance with EU policies. Due to the impact and
implications of its rulings across the bloc, the CJEU is regarded as one of
the world’s most powerful judicial bodies. National governments and EU
institutions may appear before the Court as complainant or as defendant. They
may also intervene in proceedings to express support or opposition to the EU
member state or the EU institution party to the case.

We use network analysis to study the coalition patterns emerging from this
data. We construct networks to model the web of directed "friendly" and
"unfriendly" connections between intervening states and parties. We examine
centrality, modularity and triadic motifs both in the "Friends" and "Foes"
networks over time. Additionally, we conduct a multiplex analysis and merge
the two networks to gain further insights.

We highlight the following findings. Firstly, Friends and Foes networks (see
below) display a disassortative behaviour, implying a tendency for nodes to
connect with dissimilar nodes rather than similar ones. Secondly, strong
correlations among centrality measures suggest that member states and
institutions with a higher number of connections simultaneously play a
prominent role in bridging the nodes. Thirdly, the modularity of networks
points to alignments along regional lines and divisions between EU
institutions and member states consistently with results from social science
research on European integration. Finally, we find a greater degree of
reciprocity within the Foes network compared to the Friends network,
suggesting a higher level of mutual opposition and conflict among nodes in the
Foes network.

##  1 Materials and Methods

###  1.1 Data

Our data set consists of 625 cases filed with the CJEU between 1977 and 2018.
We use web-scraping methods to identify and extract cases with at least one
third party intervention from the EUR-Lex website (www.eur-lex.eu). Cases in
our data set are either initiated by a member state against an EU institution
(annulment action) or initiated by the European Commission against a member
state (infringement action). We only consider cases involving coalitions, i.e.
cases which feature at least one intervention or in which two states or more
act as plaintiff/defendant. While governments are also allowed to present
observations in cases passed on to the CJEU by national courts (so-called
?preliminary rulings? in EU law jargon), these cases only address matters of
interpretation and do not determine the final outcome of the legal dispute at
hand. The observations presented by intervening governments in preliminary
rulings are ambiguous and do not clearly indicate which side they are meant to
support. For these reasons, they are excluded from our analysis.

The number of third party interventions varies between 1 and 20 per case. In
this data, 29 cases do not feature interventions but are also included as they
feature two or more countries/institutions as main parties on the same side of
the dispute. While governments have enjoyed the right to intervene in CJEU
proceedings since the inception of European integration in the 1950s, the
first intervention occured in 1977. To explore the evolution of coalition
dynamics over time, we divided the data in eight periods: 1977-1981,
1982-1986, 1987-1991, 1992-1996, 1997-2001, 2002-2006, 2007-2011 and
2012-2018.

###  1.2 Network Structure

We constructed two directed, weighted networks. The first is a same-side
network, which we refer to as Friends network; the second an opposite-side
network, which we refer to as Foes network. In our two networks, a Node
represents a country or an EU institution involved in a case either as
plaintiff/defendant or as intervening third party. An Edge is drawn between
two nodes if they are on the same (Friends network) or opposite (Foes network)
side of the case, whereby the intervening party is the source while the
plaintiff/defendant and their co-interveners are the target. We make two
exceptions to this rule. First, we omit to draw an edge between the Commission
and the main party to the case in infringement proceedings initiated by the
European Commission. Similarly, we do not draw any edge between the case
initiator and the Council of the European Union in annulment proceedings.
Because infringement actions are brought by the Commission and all annulment
cases are directed against legislation approved by the Council, these edges
would merely reflect the proportion of infringement and annulment cases in our
data. For the purpose of investigating coalition dynamics, these edges are,
therefore, uninformative.

Edges are weighted according to the number of cases involving the two nodes as
friend/foe in the corresponding period. We then define two adjacency matrices
of size $N$: our $fr$ and $fo$ matrices. They represent the weighted adjacency
matrices associated to, respectively, the Friends and Foes network and we
indicate the element of such matrices with $fr_{ij}$ and $fo_{ij}$. i.e.

$\left[W^{fr}\equiv(w^{fr}_{ij})_{1\leq i,j\leq N}\quad\quad
W^{fo}\equiv(w^{fo}_{ij})_{1\leq i,j\leq N}\right]$ (1)

###  1.3 Node Importance and Network Organisation

Node importance is assessed using various centrality measures. Some of those
centrality measures are for example designed to capture the ability of a node
to spur or stop epidemic processes within a network, others to measure the
bridging value of a connection and therefore we lack a unique all-purpose
centrality measure. For this reason, the importance of a given centrality
metric is often a matter of the context. In our study, we focus on four
measures of node centrality that highlight different aspects of our system:
(i) degree centrality, which represents a simple (binary), local measure
considering only the first order connections of a node; (ii) strength
centrality. which as a natural extension of degree centrality, includes the
effect of edge-values on node importance; (iii) betweenness centrality, which
is defined as a higher-order measure of node importance taking into account
the shortest paths connecting any two nodes in the network, thus focusing on
its possible key role in diffusion processes; and (iv) page-rank centrality
[22], which functions as a global measure recursively taking into account the
centrality of all the node’s neighbours – in other words, it assigns a measure
of importance to a node according to the centrality of its neighbours.

We examine triangular closure and compare motif occurrence to a null model. To
detect community structures, we apply the Louvain algorithm. Our multiplex
analysis models the two networks as layers of a duplex, with the analysis
focusing on the overlapping node degree/strength (sum of node degree/strength
over the two layers), the multiplex participation coefficient with respect to
a chosen local quantity and the z-score of the overlapping degree/strength to
identify - if any - peculiar nodes[23].

Finally, we merged the two networks into a single network by computing the
frequency of relationships for each pair of nodes:
$w^{fr}_{ij}/(w^{fr}_{ij}+w^{fo}_{ij})$. We consider this network
representation complementary to the multiplex perspective – offering a more
intuitive visualization of the relationships occurring among
countries/institutions and allowing the study of additional topological
properties, although at the price of losing some features of the two layers.

##  2 Results

We first examine basic topological local properties (size, volume, degree,
strength) before turning to centrality measures and community structures.

###  2.1 Basic Network Properties

Our entire data set features 36 countries and institutions (see Table in
Appendix). Yet the network size never exceeds 31 nodes for any of the eight
periods, as shown in Figure 1. Successive increases in network size reflect
the impact of enlargement (19 countries joined the EU between 1981 and 2013),
treaty revisions (which created new institutions, such as the European Central
Bank) and more frequent litigation, which has provided governments and EU
institutions with more opportunities to intervene in judicial proceedings.
Whereas Friends and Foes networks increase at the same pace (Fig.1, top), the
density of the Friends network appears higher in the last 3 periods after
starting from similar values (Fig.1, middle). The total number of cases rises
in tandem with the number of agents involved, but with greater speed for the
Friends network (Fig.1, bottom).

Figure 1: Network properties. Size (top); density (middle); total
weight/volume (bottom).

Both Friends and Foes networks exhibit binary disassortativity when modelled
as binary, although not when modelled as weighted. Figure 2 shows the scatter
plot for the binary model for the last period in our data set, 2012-2018.
Disassortativity suggests a tendency for countries and institutions involved
in a high number of lawsuits to be connected with countries and institutions
less active in the litigation process. This property is a manifestation of
structural disparities in the involvement of institutions and member states in
EU-level legal disputes. Some institutions (e.g. ECB) have authority over
narrow policy domains, limiting the range of cases in which they may be
involved, while member states differ considerably in economic size, political
influence and familiarity with EU law litigation [24, 25, 26, 27].

Figure 2: Disassortativity Total node degree versus the Average Neighbor Node
Degree for Friends and Foes Network in the period 2012-2018.

###  2.2 Node Centrality

Figure 3 and 4 report node rankings according to our four centrality measures
over the eight periods for both Friends and Foes. Here the analysis is
restricted to the countries and institutions appearing in all periods in both
networks to ensure meaningful temporal comparisons. France dominates the
Friends rankings in the first three periods, whereas later periods are
dominated by the UK, the Czech Republic, Finland and the European Commission.
For the Foes network, centrality rankings are dominated by the UK and the
European Commission. The fact that the UK and the Commission score high on
out-degree centrality as well as on the other three measures indicate that
they are both initiators and targets of hostile interventions.

Figure 5 report node rankings for the period 2012-2018. Ranking for the
Commission differ little across centrality in both the Friends and Foes
networks. In the Friends network Germany ranks highest on out-degree
centrality, indicating frequent friendly interventions. The UK’s in-degree
score in the Friends network and out-degree score in the Foes network reveal
active intervention both against and in support of other EU actors.

(a) In degree centrality (b) Out degree centrality

(c) Betweenness centrality (d) Page rank centrality

Figure 3: Centrality ranking for Friends network Only ten
countries/institutions are included in the figures as the most involved in
both Foes and Friends networks in the whole period (BE = Belgium; COM =
European Commission; DE = Germany; FR = France; GR = Greece; IE = Ireland; IT
= Italy; LU = Luxembourg; NL = Netherlands; UK = United Kingdom.)

(a) In degree centrality (b) Out degree centrality

(c) Betweenness centrality (d) Page rank centrality

Figure 4: Centrality ranking for Foes network Only ten countries/institutions
are included in the figures as the most involved in both Foes and Friends
networks in the whole period (BE = Belgium; COM = European Commission; DE =
Germany; FR = France; GR = Greece; IE = Ireland; IT = Italy; LU = Luxembourg;
NL = Netherlands; UK = United Kingdom.)

(a) Friends network (b) Foes network

Figure 5: Centrality ranking comparison Ranking for the four centrality
measures in the last period under study for (a) Friend network and (b) Foes
network.Only ten countries/institutions are included in the figures as the
most involved in both Foes and Friends networks in the whole period (BE =
Belgium; COM = European Commission; DE = Germany; FR = France; GR = Greece; IE
= Ireland; IT = Italy; LU = Luxembourg; NL = Netherlands; UK = United
Kingdom.)

The scatter plots in figure 6 display correlation values among the four
centrality measures for the last period. We observe high correlations between
all centrality measures (for the sake of simplicity, we report total degree
rather than in and out-degree). High correlation values are not a necessary
property of every network. The central node of a star network, for instance,
will possess both high degree and high betweennes centrality, whereas a node
connecting the central node of two star networks will exhibit low degree but
high betweenness centrality. Correlation values thus indicate the extent to
which nodes tend to play the same role in the network. More specifically, in
our networks, the degree-betweenness correlation highlights the fact that
countries/institutions with a great number of neighbours also a fundamental
role in bridging the network.

Figure 6: Correlation centrality measures. Friends (left); Foes (right).

###  2.3 Motifs

To detect patterns of coalition formation, we performed an analysis of
recurrent triadic binary motifs (Fig. 7, top), comparing their occurrences in
our networks with null models sharing the same degree distribution (see
Methods for details). We compare the three periods 2002-2006, 2007-2011 and
2012-2018, which are sufficiently comparable in terms of network size so as
not to affect the z-score.

Figure 7 shows the z-scores for each motif in the three periods. As regards
the Friends network, motif 1 and 4 appear to be overestimated by the null
model, open triangles with either two exiting or two entering links are less
frequent than expected for the period 2002-2006. Motif 5 and 10, by contrast,
are more frequent in the Friends network than in the null models for two
periods 2002-2006 and 2012-2018. The frequency of motif 10 in these two
periods suggests limited reciprocation in friendly support. Interestingly,
reciprocation seems more pronounced in the Foes network. Motif 8 and 10, both
triangles with reciprocated links, are abundant, especially in the period
2007-2011 (motif 8) and 2002-2006. These patterns suggest that member states
and institutions tend to reciprocate hostile interventions.

Figure 7: Binary motifs. Friends network (top) and Foes network (bottom) for
the last three periods analyzed.

###  2.4 Communities

Fig.8 displays community structures, denoted by node colour, in the Friends
and Foes networks for the last period under study (2012-2018). Communities in
the Friends network reveal a clear separation between member states and pro-
integration institutions countries as well as East-West and North-South
divides, which social scientists have documented in legislative settings [27].
The European Commission and the European Parliament, both in the light-blue
community, tend to advocate federalist policies, whereas member states, along
with the Council (which serves to represent the interests of national
governments) typically advocate greater deference to domestic decision making
[24]. The purple community is mostly comprised of southern member states
(France, Italy, Spain, Greece, Portugal) along with the Council. The orange
community is a cluster of predominantly northern European member states
(Sweden, Finland, Netherlands, UK). Eastern member states (including Poland,
Hungary, Slovakia and Romania) form a separate (green) community.

The position of the European Commission in the Foes network reflects its role
as "guardian of the treaties". The Commission frequently intervenes to defend
EU legislation against legal challenges brought by national governments. The
European Parliament and the Council, who often defend opposite policy
positions, find themselves in the same community (orange nodes). So do the UK,
France and Spain – a pattern largely driven by reciprocal hostility between
the UK, on the one hand, and France and Spain, on the other.

Figure 8: Network communities for 2012-2018. Friends (left); Foes (right).

###  2.5 Multiplex Perspective

This section reports the results of our duplex analysis, which helps better
understand the role of nodes as supporters/rivals. We focus on node degree,
computing the overlapping degree between the two layers[28] and the
participation coefficient with respect to them[23]. The overlapping degree
simply sums the node degrees over the two layers, while the participation
coefficient quantifies the distribution of nodes presence in the two networks:
it ranges in [0,1] and is equal to 0 if all edges of a node belong to just one
layer, while it is exactly 1 if the edges are equally distributed over the
two.

Figure 9: Role of nodes from a multiplex perspective. Top: box plot of
participation coefficient distribution for the 8 periods under study and the
in-degree (left)/out-degree (right); Bottom left:the participation coefficient
for the in-degree (top) and out-degree (bottom) versus the related z-score. We
reported the results only for the last three periods under study. Bottom
right: two examples from the out-degree case for the last period: the ego
network of country SK and the institution COM (nodes red circled in the
related plots).)

A first inspection of node scores according to in/out overlapping degree
reveals high variability in node behaviour, as shown in fig9. The same figure
also reports the participation coefficient (with respect to in/out degree)
versus the related z-scores, highlighting the role played by nodes in the two
layers. Also displayed are the ego network of the Slovakia (SK) and the
European Commission (COM). They were chosen to contrast the role of node in
the duplex. Slovakia can be considered as a "focused" node, as its outgoing
edges mainly belong to the Friends network (6, only one link in the other
layer), whereas the European Commission exhibits a proper multiplex behaviour,
behaving as hub in both layers with 16 outgoing links in the Friends and 25 in
the Foes network.

###  2.6 Merged Network

(a) 2002-2006 (b) 2007-2011 (c) 2012-2018

Figure 10: Networks of EU institutions and countries in litigation before the
European Court of Justice. Node size is directly proportional to in-degree;
node colors is directly proportional to out-degree (from light to dark blue);
edge color goes from blue to red proportionally to the number of cases
involving the two nodes as supporters with respect to the total cases
interesting them.

In this section we define an alternative network merging information from both
Friends and Foes connections among EU countries and institutions.

A link between any two nodes is weighted according to number of cases
involving them as Friends divided by the total amount of cases between them:

$\centering W^{mrg}\equiv(w^{mrg}_{ij})_{1\leq i,j\leq
N},\quad\text{with}\quad
w^{mrg}_{ij}=w^{fr}_{ij}/(w^{fr}_{ij}+w^{fo}_{ij})\@add@centering$ (2)

The edge-direction indicates the source and the target of intervention. In
other words, a link from A to B with weight 0.8 indicates that in 80% of cases
A supported B (regardless of whether A initiated the case or intervened
later), while a weight equal to 0.1 means that only in 10% of cases A
supported B.

Figure 10 illustrates the merged networks for the last three periods under
study. Node size is proportional to in-degree centrality, while lighter shades
of blue indicate lower and darker shades of blue higher out-degree centrality.
Edge colour captures weights as defined above, with darker shades of red
denoting more thoroughly supportive behaviour. The blue and green shades of
the edges linking institutions and member states clearly show the divide
opposing EU institutions and national governments in EU litigation. EU
institutions typically promote pro-integration laws which national governments
seek to contain [24].

In figure 11 we show for a select group of countries and institutions, how the
frequency of supporting behaviour are distributed over all the
country/institution present in the dataset. On the top we report out-going
links, therefore the frequency associated to the country/institution indicated
by the color as starter of the supportive action, while on the bottom we
consider in-going link, therefore consider the node as target of the action
started by the country/institution indicated by the column.

(a) Out-going links (b) In-going links

Figure 11: Frequency of supportive behaviour in 2012-2018. Out-going (top) and
in-going edge value of the novel network MRG: for each of the ten
country/institution selected before we show the frequency of supporting
behaviour towards a specific country/institution computed as
$fr_{ij}/(fr_{ij}+fo_{ij})$

Figure 11 (a) shows interesting information about the tendency to be or not
supportive in the last period under study. For example, COM appears generally
not supportive as source of an intervention as most of exiting link values are
small, except for connections with CON, ECHA, EDPS and EP, all European
Institutions. The UK and Luxembourg direct supportive behaviour only towards
some countries and institutions, respectively Czech Republic, Denmark,
Finland,DK,FI,IE,LU,NL and SE for UK; EP, HU, NL,SE and UK for LU. Figure 11
(b) reports information about receiving supportive interventions in the last
period of our sample.. Most of the ten countries/institutions reported here
receive supportive behaviour from the others, with some exceptions for DE
(small incoming edge links from COM, EE, ES, FI, LT), UK (small incoming edge
links from COM and EP). These figures could help in attempting a
classification for country/institution as mainly supportive or not, be
supported or not and grouping agents accordingly.

##  3 Conclusions

Our network analysis of third interventions before the CJEU provides insights
into international litigation dynamics. The disassortative behaviour displayed
in both networks indicates a tendency for nodes to form connections with
dissimilar nodes rather than similar ones. The strong correlations among
centrality measures suggest that certain member states and institutions hold a
prominent role in litigation as source and target of interventions and in
bridging the networks’ communities. The modularity analysis revealed
alignments along regional lines and divisions between EU institutions and
member states, consistent with previous social science research on European
integration. Lastly, the higher degree of reciprocity observed within the Foes
network compared to the Friends network suggests a greater level of mutual
opposition and conflict among nodes in the Foes network.

While our analysis remains exploratory, we hope to have showed that
international litigation provides as a suitable context for network analysis,
allowing researchers to navigate the complexity of the underlying coalition
patterns.

## Acknowledgements

RM acknowledges support from the Italian "Programma di Attività Integrata"
(PAI) project "PROsociality COgnition and Peer Effects" (PRO.CO.P.E.), funded
by IMT School for Advanced Studies Lucca and the European Union ? Horizon 2020
Program under the scheme ?INFRAIA-01-2018-2019 ? Integrating Activities for
Advanced Communities?, Grant Agreement n.871042, ?SoBigData++: European
Integrated Infrastructure for Social Mining and Big Data Analytics?
(http://www.sobigdata.eu).

## Author contributions

RM, AD and GC conceived the study; RM analyzed the data, produced the figures
and drafted the manuscript; RM, AD, GC interpreted the results. All authors
critically revised the manuscript.

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