# On the Kinematic Morphology around Halos

Xiaolin Luo 1,3, Xiangling Xu 4, Xin Wang 1,2
1School of Physics and Astronomy, Sun Yat-Sen University, No.2 Daxue Rd,
519082, Zhuhai, China
2 CSST Science Center for the Guangdong-Hong Kong-Macau Greater Bay Area,
SYSU, China
3 Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai, 200240, China
4 Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

In this paper, we report an interesting kinematic phenomenon around the halos’
edge related to the ‘splashback’ radius. After the shell-crossing, cosmic flow
exhibits various rotational morphologies via stream-mixing. Vorticity is
generated in a particular way that coincides with the large-scale structure.
Notably, one specific flow morphology, which is spiraling inward and
compressing in the third direction, concentrates around halos. A detailed
examination that reveals a sharp change in the logarithmic derivative of its
volume fraction, coincides with the location of the splashback radius $R_{\rm
sp}$ defined as the outermost caustic structure. Such a feature encodes
valuable phase space information and provides a new perspective on
understanding the dynamical evolution of halos. As a volume-weighted quantity,
the profile of flow morphology is purely kinematic. And unlike other related
studies, the rotational flow morphologies capture the anisotropic phase
structure in the multi-stream region.

###### keywords:

cosmology: large-scale structure of universe, cosmology: dark matter

††pubyear: 2022††pagerange: On the Kinematic Morphology around Halos–On the
Kinematic Morphology around Halos

##  1 Introduction

The evolution of dark matter halos plays a critical role in structure
formation. Over the past several decades, extensive efforts have been made to
understand the details of such a process, from the simple spherical collapse
model (Gunn & Gott, 1972) to the phase-space Vlasov simulation (Colombi,
2021). The density profile of a halo is well described by Navarro–Frenk–White
profile (Navarro et al., 1996) or Einasto profile (Einasto, 1965). Recent
numeric studies have revealed a sharp drop in the slope of the density profile
around halos’ edge. This feature, whose location is known as the splashback
radius, has attracted much interest recently (Diemer & Kravtsov, 2014). The
splashback radius provides a natural definition of the halo’s boundary,
reducing the ambiguity of identifying clusters (Ryu & Lee, 2021). It has been
proposed to provide constraints in cosmic expansion history (Adhikari et al.,
2018), modified gravity theories (Adhikari et al., 2018; Contigiani et al.,
2019), and neutrino mass (Ryu & Lee, 2022).

Theoretically, it has been shown that the splashback radius matches the
outermost caustic predicted by the secondary infall model (Bertschinger,
1985), corresponding to the apocenter of orbital particles. Its location is
universal and only mildly depends on the accretion rate and redshift. It was
soon discovered that such a feature reveals itself in various physical
quantities. In the phase space, the splashback radius corresponds to the
position where the slope of the phase space sheet becomes vertical (Adhikari
et al., 2014; Sugiura et al., 2020). Okumura et al. (2018) first detected a
similar sharp steepening in the momentum correlation function associated with
the splashback radius. Furthermore, in the velocity space, a clear boundary
was found between radial and tangential velocity distribution Aung et al.
(2021).

All these results suggest a deep, complex physical process hidden in this
splashback feature. Particularly in this paper, we are interested in the rich
information encoded in the kinematic structure. At the large scale, the cosmic
web structure could be identified with the so-called ’V-web’ (Hoffman et al.,
2012), i.e., using eigenvectors of the symmetric velocity shear tensor, in
comparison with the dynamically classified ’T-web’ (Forero-Romero et al.,
2009) using the tidal tensor. At the non-linear scale, as the three-
dimensional cold dark matter sheet winds up, the multi-streaming renders the
initial potential flow to generate vorticity (Peebles, 1980; Bernardeau et
al., 2002; Hahn et al., 2015a). Such a rotational degree of freedom also
encompasses rich information about structure formation. For example, after
defining various rotational flow morphologies with the asymmetric velocity
gradient tensor, Wang et al. (2014) demonstrated that individual rotational
morphology also traces different component of the cosmic web. Consequently,
these categories of flow types provide a unique tool for examining complicated
phase space dynamics. In particular, one specific rotational morphology
(spiraling inward and collapsing in the third direction) is mainly distributed
around dark matter halos (Wang et al., 2014). Given such an association, it
would be interesting to investigate the detailed distribution of this flow
morphology and its connection with density distribution.

Figure 1: Illustration of four types of rotational flow morphology (upper
panels) and their spatial distribution in a cosmological simulation (lower
panels). With the velocity gradient tensor (Eq. 1), the fluid trajectory is
classified from two parameters shown in Table 1. The underlying density map is
randomly selected from a N-body simulation with the box size of $100~{}{\rm
Mpc/h}$ at redshift $z=0$. Here both density and velocity fields are
constructed with the DTFE algorithm on a $512^{3}$ grid. The flow
classification is calculated with a smoothed velocity field with Gaussian
smoothing scale of two pixels. The grey points in each panel represents
corresponding morphology type, and we have slightly enlarged its size for
better visual identification. As one can see, type SFS (inward spiral -
stretching) mainly traces filament, SFC (inward spiral - collapsing)
concentrates around nodes and UFS (outward spiral - stretching) are likely to
locate in lower density regions.

The paper is organized as follows. Section 2 briefly reviews the rotational
morphologies defined from the velocity gradient tensor and their spatial
distribution at the large scale. Section 3 presents our main result, showing
the radial profile of one specific morphology, closely tracing the splashback
signature. And finally, we conclude in section 4. Throughout this paper, we
utilize two sets of N-body simulations: one ‘large-scale’ single realization
simulation with the box size of $L=100~{}{\rm Mpc/h}$ and particle mass of
$m_{\rm part}=6.1\times 10^{8}\rm M_{\sun}/h$ for demonstrating large-scale
cosmic-web structures, and another set of 10 realizations with the box size
$L=50~{}{\rm Mpc/h}$ and $m_{\rm part}=7.6\times 10^{7}\rm M_{\sun}/h$ for all
measurements of halo statistics. Here, dark matter halos are selected with
Rockstar code (Behroozi et al., 2013). Since the kinematic morphologies depend
on spatial derivatives of the velocity field, traditional field estimator like
the Cloud-In-Cell is susceptible to numerical errors. On the other hand,
methods like the Delaunay tessellation (DTFE) (Bernardeau & van de Weygaert,
1996; Schaap & van de Weygaert, 2000) continuously interpolate the particle-
based sample on a regular grid and are more suitable for our application. In
this paper, we construct the density and velocity field using a publicly
available code111https://www.astro.rug.nl/$\sim$voronoi/DTFE/dtfe.html.

##  2 Classification of Cosmic Flow Morphology

Our calculation in this paper is based on the classification method of
kinematic morphologies first introduced by Wang et al. (2014) in a
cosmological setting. In this section, we will briefly review such
classification, more details can be found in Chong et al. (1990) and Wang et
al. (2014). Denoting the peculiar velocity as $\mathbf{u}(\mathbf{x},\tau)$,
one defines the velocity gradient tensor as

$\displaystyle A_{ij}(\mathbf{x},\tau)=\frac{\partial u_{i}}{\partial
x_{j}}(\mathbf{x},\tau)$ (1)

where $\mathbf{x}$ is Eulerian position and $\tau$ the comoving time. Instead
of this coordinate dependent form $A_{ij}$, it is more convenient to work with
some rotational invariants. One choice is eigenvalues of the velocity gradient
tensor $A_{ij}$. For example, in the V-web classification, the symmetrized
tensor $A^{\rm s}_{ij}=(A_{ij}+A_{ji})/2$

$\displaystyle\mathbf{A}^{\rm s}=\mathbf{R}^{-1}~{}{\rm
diag}(\lambda_{1},\lambda_{2},\lambda_{3})~{}\mathbf{R}$ (2)

has three real eigenvalues $\lambda_{i}\in\mathbb{R}$, where $\mathbf{R}$ is
the rotation matrix. Depending on the sign of
$\lambda_{i}^{\prime}=\lambda_{i}-\lambda_{\rm th}$222As in Wang et al.
(2014), here we set the threshold value $\lambda_{\rm th}=0$., one could
classify each position as nodes (all three negative $\lambda$), filaments (two
negative and one positive), walls (one negative and two positive) or voids
(all three positive $\lambda$).

Equivalently, an alternative is the coefficients of the characteristic
equation of $A_{ij}$: $\det[\mathbf{A}-\lambda I]=0$, i.e.

$\displaystyle\lambda^{3}+s_{1}\lambda^{2}+s_{2}\lambda+s_{3}=0.$ (3)

Similar to eigenvalues, the coefficients $s_{i}=\\{s_{1},s_{2},s_{3}\\}$ are
also rotational invariant. They are defined as (Chong et al., 1990; Wang et
al., 2014)

$\displaystyle s_{1}=-{\rm tr[\mathbf{A}]},\quad
s_{2}=\frac{1}{2}\left(s_{1}^{2}-\rm tr[\mathbf{A}^{2}]\right),\quad
s_{3}=-\det[\mathbf{A}]$ (4)

From the characteristic equation (3), there is a one-to-one correspondence
between $\lambda_{i}$ and $s_{i}$. However, the advantage of using
coefficients $s_{i}$ is that, after the shell-crossing, the generation of
vorticity introduces assymetric $A_{ij}$ with complex eigenvalues $\lambda$,
whereas $s_{i}$ are always real, enabling a continuous description of the
process.

In the canonical form, an asymmetric tensor $A_{ij}$ can be expressed as
(Chong et al., 1990; Wang et al., 2014)

$\displaystyle\mathbf{A}=\mathbf{R}^{-1}\left(\begin{array}[]{ccc}a&-b&\\\
b&a&\\\ &&c\\\ \end{array}\right)\mathbf{R},$ (8)

here $a,b,c$ are all real numbers, and the complex eigenvalues
$\lambda_{1,2}=a\pm ib$, $\lambda_{3}=c$. The local trajectory is then
governed by the linear equation

$\displaystyle\frac{d}{d\tau}\mathbf{x}=\mathbf{A}\mathbf{x}.$ (9)

Particularly in this case (Eq. 8), the eigenvectors corresponding to the
complex eigenvalues $\lambda_{1,2}$ spans a plane, in which the spiral curve

$\displaystyle r=r_{0}e^{m\alpha}$ (10)

resides. Here $(r,\alpha)$ are polar coordinates, the factor $m=a/b$ denotes
the rate of spiral, and $r_{0}$ is a constant depending on the initial
condition. For $m$ approaching a larger value, the trajectory spiral rate will
decrease, and vice versa. Along the direction outside the spiral plane, the
trajectory is then controlled by the value of the eigenvalue $\lambda_{3}=c$.

Consequently, depending on the value of $m$ and $c$, and neglecting degenerate
cases (Chong et al., 1990), there are four major categories of rotational
flow. Specifically, negative $m$ represents the trajectory spiraling inwards
to the origin of the plane, which named as the ‘stable focal flow’ (SF).
Conversely, positive $m$ corresponds to the trajectory spiraling outwards,
i.e., the ‘unstable focal flow’ (UF). Meanwhile, if $c>0$, mass elements flow
away from the plane, i.e. ‘stretching’; and $c<0$ corresponds to fluid element
flow towards the plane, corresponding to the ‘compressing’. In summary, we
have total of four categories of rotational flow shown in Table (1).

Table 1: Classification of rotational morphologies with two dimensional parameter space: $\\{m=a/b,c=\lambda_{3}\\}$, where $a,b$ and $c$ are canonical form (Eq. 8) of asymmetric velocity gradient tensor $A_{ij}$.  |  $m<0$ (inward spiral) |  $m>0$ (outward)
---|---|---
$c>0$ (Stretching) | Stable Focal | Unstable Focal
| Stretching (SFS) | Stretching (UFS)
$c<0$ (Compressing) | Stable Focal | Unstable Focal
| Compressing (SFC) | Compressing (UFC)

Please see Chong et al. (1990) and Wang et al. (2014) for more details of the
flow classification.

To illustrates all these flow morphologies, we plot the trajectories in the
upper panels of Figure 1. Similar to potential flows, these rotational
morphologies also trace the cosmic web. In the lower panels, we show these
four categories on top of a slice of matter distribution. Grey points
highlight each morphology type at corresponding pixels, and we have slightly
enlarged their sizes for better visual identification. As one can see, the SFS
type tends to track filamentary structures, and SFC concentrates on the nodes
of the web. On the other hand, UFS seems to populate in void regions, and UFC
scatters around in both high and low-density regions. In the following, we
will concentrate on the detailed distribution of SFC around clustering nodes.

Figure 2:  The distribution of density (upper panels) and flow morphology
(middle and bottom panels) of two halos, with halo masses $M=1.8\times
10^{14}~{}\rm M_{\sun}/h$ (left column) and $M=9\times 10^{13}~{}\rm
M_{\sun}/h$ (right column). Here the depth of the projection is one pixel of
the $512^{3}$ grid, i.e. $50{\rm Mpc/h}/512\approx 0.098~{}{\rm Mpc/h}$. Both
left and right columns have the same pixel size but different physical scales.
For better comparison, a standard $2~{}{\rm Mpc/h}$ length scale is displayed
in the corner of the top panels. The flow morphology is categorized from
Gaussian smoothed velocity field, with a smoothing length of $4$ pixels
(middle panels) and $2$ pixels (middle panels), corresponding to $0.39~{}{\rm
Mpc/h}$ and $0.195~{}{\rm Mpc/h}$ respectively. Similar to Figure 1, both
density and velocity fields are constructed with DTFE on a $512^{3}$ grid. Red
circles indicate their splashback radii $R_{\rm sp}$, obtained by rescaling
the viral radii with factors estimated from the the peak position of the
stacked density slope of the corresponding mass bin. As shown, with more
smoothing, the flow types inside halos are dominated by nodes and SFC types.
On the other hand, morphology distribution from the fine-grained velocity
field would be much noisier; therefore, we do not present it here.

##  3 Rotational Flows around Halos

###  3.1 Rotational Morphologies around Halos

This section focuses on flow morphologies around halos and all statistical
measurements are made with our $10$ realizations of $50~{}{\rm Mpc/h}$
simulations. Figure 2. exhibits the detailed density (top panels) and flow
type (middle and bottom panels) distribution of two (almost-)randomly selected
halos (each column represents one halo). From the matter density distribution,
one could see that these are massive halos ($M=1.8\times 10^{14}~{}\rm
M_{\sun}/h$ and $M=9\times 10^{13}~{}\rm M_{\sun}/h$ respectively) as they
both connect with multiple filaments. In each panel, a red circle was drawn to
indicate the splashback radius $R_{\rm sp}$, obtained by rescaling the viral
radius with a factor estimated from the peak position of the stacked density
slope of the corresponding mass bin. Both left and right columns have the same
pixel size but different physical scales. For better visual clarity, the
morphology is constructed from a Gaussian smoothed velocity field with the
filter length of $4$ times (middle) and twice (bottom) of the pixel size,
corresponding to $0.39~{}{\rm Mpc/h}$ and $0.195~{}{\rm Mpc/h}$ respectively.

As shown, morphology distribution with a larger smoothing length appears more
organized. Within the halo, both SFC and node type dominate over other
morphologies. Following along the filaments connected to central halos, one
starts to see more SFS flows. This spatial correspondence between rotational
flow morphologies and the cosmic web structures is consistent with physical
intuition. For example, the type of vorticity generated along the filaments
involves inward winding and stretching, i.e., the SFS type. Moreover, within
halos, the radial contraction and the energy transfer from infalling to
orbiting lead to inward spiraling and compression, corresponding to the SFC
type. With the definition of flow classification, Wang et al. (2014) suggested
that the relative region of each flow category in the parameter space $s_{i}$
could partially explain such a phenomenon. From the flow classification in the
invariants space $s_{i}$ (see Wang et al., 2014, Figure 1), one could see that
the region corresponding to filaments is adjacent to the region of SFS type,
and similarly the region categorized as the nodes in the invariants $s_{i}$
space resides just inside the SFC regime.

Table 2: The mass range and halos’ number of each mass bins for calculating density and morphology profiles.  name | halo mass range | number of halos
---|---|---
Mass bin - 1 | $3\times 10^{12}\rm M_{\sun}/h<m<5\times 10^{12}\rm M_{\sun}/h$ | $378$
Mass bin - 2 | $5\times 10^{12}\rm M_{\sun}/h<m<10^{13}\rm M_{\sun}/h$ | $321$
Mass bin - 3 | $m>10^{13}\rm M_{\sun}/h$ | $291$

However, this spatial correspondence is not as strong as one might expect. As
one can see, flow morphology with less smoothing looks noisier within halos;
large regions of SFC and nodes now fragment into pieces of various types. The
distribution without any smooth filtering would appear very random, and it is
hard to identify any physical properties visually; therefore not plotted here.
Instead, we will study the stacked statistics of unsmoothed morphology in the
next subsection. From the lower panels, we can see that many flow types,
including nodes, SFS, and SFC, all seem to be presented inside halos,
reflecting particles’ complicated movement within the virialized structure.

Figure 3:  The radial profile of volume fraction for the SFC type (upper) and
its logarithmic derivative $\gamma_{\rm sfc}$ (lower, Eq. 13). For comparison,
the density profile and its slop are shown as dash-dotted line. Different
colors present three groups of halos we stacked. The details of the mass bin
are shown in Table (2); each bin has rough $\sim 300$ halos. The color band in
the top panel indicates the $68\%$ tolerance interval around the mean. In the
lower panel, dotted vertical lines correspond to the splashback radius $R_{\rm
sp}$ identified from the density profile. All curves are smoothed with the
fourth-order Savitzky-Golay algorithm. As can be seen, for more massive halos,
the density splashback radius location coincides with the $\gamma_{\rm sfc}$
peak very well.

For a deeper insight, we could write down the averaged velocity gradient
tensor $A_{ij}(\mathbf{x})$ at an Eulerian location $\mathbf{x}$ as the
combination of each stream’s density, velocity, and their spatial derivatives
(Pichon & Bernardeau, 1999; Wang et al., 2014; Hahn et al., 2015b)

$\displaystyle\frac{1}{\rho(\mathbf{x})}\sum_{\rm
stream~{}s}\left[\left(\rho^{s}A^{s}_{ik}J^{-1}_{kj}\right)(\mathbf{q}^{s})+\left(u^{s}_{i}\partial_{j}\rho^{s}\right)(\mathbf{q}^{s})\right]-\left(\frac{u_{i}}{\rho}\partial_{j}\rho\right)(\mathbf{x}).$
(11)

This is obtained by taking the derivative $\partial/\partial x_{j}$ of the
density-weighted average of velocity
$u_{i}(\mathbf{x})=\left(\sum_{s}\rho^{s}u^{s}_{i}\right)/\rho(\mathbf{x})$ .
Here the summation is over individual streams denoted by superscript $s$,
$\rho(\mathbf{x})=\sum_{s}\rho^{s}$, $\partial_{i}=\partial/\partial x_{i}$,
$A^{s}_{ij}=\partial u^{s}_{i}(\mathbf{q}^{s})/\partial q^{s}_{j}$ is the
velocity gradient of the stream $s$, and $J^{-1}_{ij}$ is the inverse of the
Jacobian matrix from the Lagrangian $\mathbf{q}$ to the Eulerian coordinate
$\mathbf{x}$. Eq. (11) shows that the rotational morphology is determined by
how streams are mixed, as well as the properties of individual streams, such
as $\rho^{s},\mathbf{u}^{s}$ and $A^{s}_{ij}$. Consequently, we can infer that
the morphology should generally follow the cosmic-web structure as
$\rho^{s},\mathbf{u}^{s}$ and $A^{s}_{ij}$ do correlate with the cosmic-web.
Meanwhile, the morphology should also behave somewhat randomly since these
components are statistically distributed. Therefore, Wang et al. (2014)
constructed a simple statistical toy model of multi-streaming, in which
irrotational flows sampled with Zel’dovich approximation were randomly mixed
together. The result (see Wang et al., 2014, Table 1) reveals a complicated
physical picture: only about a quarter of the node and a fifth of filament
types turn into SFC and SFS, respectively. The rest rotational types were not
dominant but will still be generated 333A substantial fraction of mixed flows
were still potential..

Given such complexity, it is even more intriguing to see the structure
revealed by the coarse-grained flow morphology, i.e., the gradual emergence of
SFC around halos, which could be interpreted as a simplified first-order
conclusion. Stick to this overly simplified picture; there then exists a
‘boundary region’ describing the transition to the SFC type when infalling
dark matter is compressed at the outskirt of halos. Naturally, this raises the
question of its relation with the splashback radius $R_{\rm sp}$, which we
will examine in the next subsection.

###  3.2 Morphology Profile of SFC Type

To understand the SFC transition towards the center of halos in more detail,
we define the spherical averaged volume fraction density of SFC as

$\displaystyle f_{\rm sfc}(r)=\frac{V_{\rm sfc}}{V_{\rm shell}}(r)$ (12)

where $V_{\rm shell}=4\pi r^{2}dr$ is the volume of each spherical shell and
$V_{\rm sfc}$ is the Eulerian volume of the fluid that flows as the SFC type.
Unlike the flow morphology displayed in Figure (2), here, we do not perform
any smoothing in the velocity field before classifying flow morphologies.

In contrast to the particle-based density profile $\rho(r)$, our morphology
classification is defined on a mesh and therefore is susceptible to the
numerical resolution of the grid. To alleviate the resolution effect, we
interpolate the velocity field on a finer $1024^{3}$ grid, as the particle
density is much higher around halos. Moreover, we adopt a Monte Carlo sampling
procedure to measure the radial profile of kinematic morphologies with the
following steps:

  1. 1.

Construct the velocity field $u_{i}(\mathbf{x})$ from the N-body particle data
with the Delaunay Tessellation (DTFE) (Bernardeau & van de Weygaert, 1996;
Schaap & van de Weygaert, 2000).

  2. 2.

Calculate the velocity gradient tensor $A_{ij}$ and the rotational invariants
$s_{1},s_{2}$ and $s_{3}$. Compared with other velocity estimation methods,
e.g., the cloud-in-cell, DTFE will greatly reduce the numerical noise for the
velocity gradient tensor. At each mesh point, we then classify and assign a
kinematic morphology type based on the criteria introduced in Wang et al.
(2014).

  3. 3.

For each halo, we then measure the spherically averaged volume fraction of the
SFC type in each radial shell $[r,r+dr]$ (Eq. 12). Since the classification is
defined on a finite grid, we calculate the profile $f_{\rm sfc}$ by generating
random samples in each $dr$ shell (uniformly distributed in the shell), and
then counting the fraction of which falls in the specific morphology mesh
(e.g., the SFC), assuming all sample points within each grid cell carries the
same morphology type.

  4. 4.

A averaged profile $f_{\rm sfc}$ is then obtained by taking the median of
roughly $\sim 300$ halos with radial distance re-scaled to the viral radius,
i.e., $r/R_{\rm vir}$. The exact number of halos in each mass bin is listed in
Table (2). Finally, all curves are smoothed with the fourth-order Savitzky-
Golay algorithm.

In the top panel of Figure 3., we show the radial profile of volume fraction
$f_{\rm sfc}(r)$. For comparison, the density profile is also displayed as the
dash-dotted line. Different color lines indicate results from three halo mass
bins we have used (Table 2). As shown in the upper panel, $f_{\rm sfc}$ is a
decreasing function with respect to the radius $r$. At the center, roughly
one-third of the region is dominated by SFC flow, and this number decreases to
about $10\%$ towards the edge of the halo, indicating a more chaotic flow. Our
measurement also shows a halo mass dependence where dark matter seems to flow
less likely as SFC type within more massive halos. However, this could at
least partially be caused by the worse resolution of low massive halos, since
as shown in Figure. 2 larger smoothing length does appear to increase the SFC
fraction in the halo. In the plot, we also display the $68\%$ tolerance
interval among stacked halos, much larger than that of the density profile.
This is understandable since $f_{\rm sfc}$ is volume-weighted and $V_{\rm
shell}\to 0$ as $r$ approaches to zero.

Figure 4:  The radial profile of spiral rate $m$ (upper) and logarithmic
derivative (lower). The mass bins and calculation method are the same as
Figure. 3. Here, only the $|m|$ of SFC type are included, all other morphology
types are set $m=0$. As can be seen for the largest mass bin (black curve),
the bump in the log derivative coincides with the splashback radius $R_{\rm
sp}$ very well, and there exists a little mis-alignment for smaller halos.

Following the $f_{\rm sfc}$ curve along to the larger radius, one sees that
the decrease slowed down and then formed a small bump between one and two
times of $R_{\rm vir}$. To examine the detailed structure, analogous to the
splashback radius, we define the logarithmic derivative of $f_{\rm sfc}$ with
respect to $\ln r$

$\displaystyle\gamma_{\rm sfc}(r)=\frac{d\ln f_{\rm sfc}}{d\ln r}(r).$ (13)

The result is shown in the lower panel of Figure. 3, where one could see that
$\gamma_{\rm sfc}$ vary from $\sim-0.8$ to zero and display a clear bump
around between one and twice of the virial radius $R_{\rm vir}$. For
comparison, we also plot the matter density profiles and their logarithmic
derivatives in the dash-dotted lines. We follow the standard practice from
e.g. (Diemer & Kravtsov, 2014). After measuring the spherically averaged
density profiles $\rho(r)$ and their slopes $\gamma_{\rho}=d\ln\rho/d\ln r$,
we then take the median and smooth the curves with the fourth-order Savitzky-
Golay algorithm.

Remarkably, the peak locations in $\gamma_{\rm sfc}$ coincide almost perfectly
with the splashback radius $R_{\rm sp}$, i.e., troughs of the density slope
$\gamma_{\rho}$, which are displayed as dotted vertical lines. Among all three
mass bins, only the lowest mass bin, i.e., bin-1, appears a little misaligned
with $R_{\rm sp}$, which is likely caused by the relatively lower resolution
of the velocity field for smaller halos. Therefore above results suggest that,
besides being the apocenter of orbital particles, the splashback radius also
corresponds to the transition to the SFC type. From Eq. (11) and the
discussions thereafter, different types of stream-mixing would generate
different rotational morphologies. Therefore outside $R_{\rm sp}$, dark
matters gravitate towards halos, so morphology is largely affected by their
previous environments, such as SFS or filament. Once they enter the periphery
of halos and encounter outbound particles, SFC types start to be generated.

Along with the generation of the SFC morphology type, one could further
characterize the process with the spiraling rate $m$ of the trajectory. From
the morphology definition, the ‘stable’ flow has a negative $m$ value, we plot
the absolute value $|m|$ in Figure 4. Here we only focus on the SFC type,
which means that all other flow types have been set to $m=0$. From Eq. (10),
the larger the value of $|m|$, the slower the trajectory spirals until
approaching to non-rotational nodes in the limit of $|m|\to\infty$. Therefore,
we can see from the upper panel of Figure 4 that the average $|m|$ gradually
increases towards the edge of the halos, corresponding to a slowing down of
the spiral rate. Around $R_{\rm sp}$, $|m|$ suddenly accelerates and then
drops, producing a small hillock. Following the same procedure of defining the
splashback radius, we also examine the derivative of logarithmic $|m|$ with
respect to $\ln r$

$\displaystyle\gamma_{m}=\frac{d\ln|m|}{d\ln r}.$ (14)

As shown in the lower panel of Figure (4), for the most massive halos, i.e.,
mass bin-3, the peak in the slop $\gamma_{m}$ coincides with the splashback
radius $R_{\rm sp}$ pretty well. As for lower mass bins, there exists a slight
misalignment in their relative locations. Again, since the splashback radius
is tied in with the caustic structure at the edge, this sudden change in the
spiraling rate indicates a similar phenomenon in the phase space as well.

##  4 Conclusion and Discussion

With the velocity gradient tensor, the cosmic flow can be classified into
various potential or rotational flow morphologies. Previous studies have shown
that both categories of flows are deeply connected with the cosmic-web
structure. Specifically, the symmetric part of the tensor leads to the V-web
categorization, whereas the asymmetric tensor shows an even more interesting
connection between the multi-streaming and cosmic structures. Most notably,
the SFS type (spiral-inward and stretching) mainly follows the filaments, and
the SFC type (spiral-inward and compressing) concentrates around halos. It is
then interesting to follow the cosmic fluid and examine how its flow
morphology changes.

This paper focuses on the dark matter halos and studies the flow morphology
distribution around this region. A visual inspection of a couple of selected
halos confirms the generation SFC type within and around halos. This is
particularly evident in the coarse-grained field. However, with little or no
smoothing, the flow morphologies distribution appears much noisier, reflecting
complicated flows in this highly non-linear region.

To obtain a more quantitative description, we measure the spherically averaged
volume fraction of the SFC type $f_{\rm sfc}$ and its logarithmic derivative
$\gamma_{\rm sfc}$. Interestingly, the derivative curve displays a clear bump
at the location that coincides with the splashback radius defined by the
density profile. Furthermore, a similar feature is also identified in the
radial profile of spiral rate $m$. These results suggest that the rotational
flow morphologies capture the anisotropic phase structure in the multi-stream
region.

Unlike the density profile, our morphology classification is grid-based and
therefore is susceptive to the resolution of the velocity field and its
spatial gradients. We believe the slight mismatch between the splashback
radius and the feature location in $\gamma_{\rm sfc}$ is due to such
resolution effects. We defer the detailed examination for low mass halos in
the future.

## Acknowledgements

X.W. would like to thanks the support from the science research grants from
the China Manned Space Project with NO.CMS-CSST-2021-B01.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

## References

  * Adhikari et al. (2014) Adhikari S., Dalal N., Chamberlain R. T., 2014, J. Cosmology Astropart. Phys., 2014, 019
  * Adhikari et al. (2018) Adhikari S., Sakstein J., Jain B., Dalal N., Li B., 2018, Journal of Cosmology and Astroparticle Physics, 2018, 033
  * Aung et al. (2021) Aung H., Nagai D., Rozo E., García R., 2021, MNRAS, 502, 1041
  * Behroozi et al. (2013) Behroozi P. S., Wechsler R. H., Wu H.-Y., 2013, ApJ, 762, 109
  * Bernardeau & van de Weygaert (1996) Bernardeau F., van de Weygaert R., 1996, MNRAS, 279, 693
  * Bernardeau et al. (2002) Bernardeau F., Colombi S., Gaztañaga E., Scoccimarro R., 2002, Phys. Rep., 367, 1
  * Bertschinger (1985) Bertschinger E., 1985, ApJS, 58, 39
  * Chong et al. (1990) Chong M. S., Perry A. E., Cantwell B. J., 1990, Physics of Fluids A, 2, 765
  * Colombi (2021) Colombi S., 2021, A&A, 647, A66
  * Contigiani et al. (2019) Contigiani O., Vardanyan V., Silvestri A., 2019, Phys. Rev. D, 99, 064030
  * Diemer & Kravtsov (2014) Diemer B., Kravtsov A. V., 2014, ApJ, 789, 1
  * Einasto (1965) Einasto J., 1965, Trudy Inst. Astrofiz. Alma-Ata, 5, 87
  * Forero-Romero et al. (2009) Forero-Romero J. E., Hoffman Y., Gottlöber S., Klypin A., Yepes G., 2009, MNRAS, 396, 1815
  * Gunn & Gott (1972) Gunn J. E., Gott J. Richard I., 1972, ApJ, 176, 1
  * Hahn et al. (2015a) Hahn O., Angulo R. E., Abel T., 2015a, MNRAS, 454, 3920
  * Hahn et al. (2015b) Hahn O., Angulo R. E., Abel T., 2015b, MNRAS, 454, 3920
  * Hoffman et al. (2012) Hoffman Y., Metuki O., Yepes G., Gottlöber S., Forero-Romero J. E., Libeskind N. I., Knebe A., 2012, MNRAS, 425, 2049
  * Navarro et al. (1996) Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462, 563
  * Okumura et al. (2018) Okumura T., Nishimichi T., Umetsu K., Osato K., 2018, Phys. Rev. D, 98, 023523
  * Peebles (1980) Peebles P. J. E., 1980, The large-scale structure of the universe. Princeton University Press
  * Pichon & Bernardeau (1999) Pichon C., Bernardeau F., 1999, A&A, 343, 663
  * Ryu & Lee (2021) Ryu S., Lee J., 2021, The Astrophysical Journal, 917, 98
  * Ryu & Lee (2022) Ryu S., Lee J., 2022, The Astrophysical Journal, 933, 189
  * Schaap & van de Weygaert (2000) Schaap W. E., van de Weygaert R., 2000, A&A, 363, L29
  * Sugiura et al. (2020) Sugiura H., Nishimichi T., Rasera Y., Taruya A., 2020, MNRAS, 493, 2765
  * Wang et al. (2014) Wang X., Szalay A., Aragón-Calvo M. A., Neyrinck M. C., Eyink G. L., 2014, ApJ, 793, 58


# Analysis of Polkadot: Architecture, Internals, and Contradictions

Hanaa Abbas  College of Science and Engineering
Hamad Bin Khalifa University
Doha, Qatar
<EMAIL_ADDRESS>Maurantonio Caprolu  College of Science and Engineering
Hamad Bin Khalifa University
Doha, Qatar
<EMAIL_ADDRESS>Roberto Di Pietro  College of Science and Engineering
Hamad Bin Khalifa University
Doha, Qatar
<EMAIL_ADDRESS>
###### Abstract

Polkadot is a network protocol launched in 2020 with the ambition of unlocking
the full potential of blockchain technologies. Its novel multi-chain protocol
allows arbitrary data to be transferred across heterogeneous blockchains,
enabling the implementation of a wide range of novel use cases. The Polkadot
architecture is based on the principles of sharding, which promises to solve
scalability and interoperability shortcomings that encumber many existing
blockchain-based systems. Lured by these impressive features, investors
immediately appreciated the Polkadot project, which is now firmly ranked among
the top 10 cryptocurrencies by capitalization (around 20 Billions USD).
However, Polkadot has not received the same level of attention from academia
that other proposals in the crypto domain have received so far, like Bitcoin,
Ethereum, and Algorand, to cite a few. Polkadot architecture is described and
discussed only in the grey literature, and very little is known about its
internals.
In this paper, we provide the first systematic study on the Polkadot
environment, detailing its protocols, governance, and economic model. Then, we
identify several limitations—supported by an empirical analysis of its
ledger—that could severely affect the scalability and overall security of the
network. Finally, based on our analysis, we provide future directions to
inspire researchers to investigate further the Polkadot ecosystem and its
pitfalls in terms of performance, security, and network aspects.

###### Index Terms:

blockchain, multi-chain, cryptocurrency, Proof of Stake, Polkadot,
decentralization, scalability, DeFi, governance

##  I Introduction

Interest in cryptocurrencies has been steadily rising ever since the
introduction of Bitcoin and blockchain—the foundation of all existing
crypotocurrencies—in 2009. As of April 2022, the global cryptomarket
capitalization is estimated at more than 1.88 trillion US dollars. Not only
did blockchain enable secure financial transactions, but it also laid the path
for transforming cooperation over the Internet, allowing “trusted cooperation
between untrusted entities” [1]. As a technology of promising prospects,
blockchain and its applications have continued to witness rapid growth in
terms of development and public attraction over the years.

This paper focuses on a relatively new cryptocurrency—Polkadot—whose network
was only launched mid-May 2020, yet had successfully secured a spot amongst
the top 10 cryptocurrencies, owning a market capitalization of around 18
billion USD [2]. Polkadot is developed by the Web3 Foundation and Parity
Technologies, both founded by Dr. Gavin Wood, who is also known for being a
co-founder and former CTO of Ethereum. In 2016, Dr. Gavin Wood shared his
vision of Polkadot and its technical outline through the white paper [3];
which was, in his words, creating “the next version of Ethereum” that would
complement the aspects for which Ethereum 2.0 is less optimal [4].

Polkadot is a fully “sharded” blockchain whose design is based on sharding—a
database splitting technique—that enables multiple chains to process their
transactions in parallel. Each blockchain shard is called a “parachain” which
connects to the Relay Chain. The Relay Chain is effectively the heart of the
blockchain, acting as the main hub of the system. Furthermore, Polkadot serves
as an interoperability platform; i.e., it allows cross-communication between
heterogeneous blockchains including external ones, such as Bitcoin and
Ethereum. While the Relay Chain orchestrates and ensures correct functioning
of the entire network, parachains can be customized as needed, including to
host smart contracts and a myriad of other use cases. Essentially, Polkadot
was developed with the goal to address some shortcomings of existing
blockchain technologies, namely, scalability, interoperability, and achieving
standard security guarantees across differing trust models. In the rest of
this paper, we will test whether these objectives have been achieved.

Due to the novelty of multi-chains, there is a definite need to investigate
their design and processes, particularly within an active ecosystem. As one of
the first publicly released multi-chain projects, Polkadot is a perfect case
study for evaluating the security properties of cross-chain communications
among heterogeneous blockchains. However, despite achieving a good standing
and wide consideration in the market, Polkadot has not received the same level
of attention from academia enjoyed by other proposals, such as Bitcoin,
Ethereum, and Algorand, to cite a few. In fact, although well described in the
grey literature, the Polkadot architecture and its practice is still little
discussed in the scientific community. To fill this gap, we shed light on the
complex ecosystem of Polkadot through a preliminary analysis of its
architecture, followed by the identification of several contradictions that
could limit the strengths envisaged by its creators. Our main goal is to
propose the first systematic study on the Polkadot ecosystem, providing
researchers interested in this field with a solid knowledge base to understand
the Polkadot network and other blockchains based on multi-chain protocols. The
highlighted limitations, calling for contributions to mitigate their effects,
and the proposed research directions complete the work.

Contributions. In this paper, we provide several contributions:

  * •

We first study and analyze the overall architecture of Polkadot, ranging from
the roles of its network participants, consensus schemes, and governance to
its economic and security models. To the best of our knowledge, this is the
first systematic study to investigate Polkadot’s architecture and protocols.

  * •

Based on our analysis, we highlight the main limitations and contradictions of
Polkadot, calling for further investigations and research to propose viable
solutions addressing the uncovered limitations.

  * •

We validate our assertions by providing an empirical analysis of the
validators’ election data that we collected from the Polkadot blockchain,
starting from the introduction of network validators in June 2020 until April
2022.

Paper Organization. The remainder of the paper is organized as follows.
Section II presents an overview of multi-chain design approaches existing in
the literature and other multi-chain projects publicly available with a market
presence. Section III extensively reviews Polkadot from different
perspectives, including its architecture, consensus protocol, governance, and
security. Further, in Section IV, we discuss the limitations we found in the
Polkadot ecosystem, validated by an empirical analysis of the ledger. We then
conclude the study in Section V, providing possible extensions and directions
for future work.

##  II Related Work

Despite its promising prospects, blockchain technology has encountered some
fundamental limitations over the course of its development. Real-world
applications demand for secure inter-communication between heterogeneous
blockchain systems; for instance, for transferring different assets, product
lifecycle management [5], etc.. Several works (e.g., [6, 7]) implemented two-
way pegged sidechains that enable bidirectional movement of assets and cross-
chain communication between Proof of Work (PoW) sidechains, without the need
for a trusted third party intermediary. Back et al. (2014) [6] implemented a
two-way peg between Bitcoin and other sidechains, such that: (a) transfers are
atomic; and, (b) there should be zero risk to the other party. In their
implementation, a transaction is created on the parent chain to lock the
assets, then another transaction on the sidechain generates the new asset,
given valid proof of possession. Later, Kan et al. (2018) [8] further
developed the two-way peg architecture to enable inter-blockchain
communication between heterogeneous systems. Besides achieving blockchain
interoperability [9], other works relied on the multi-chain architecture for
resolving scalability and throughput issues in traditional blockchains, e.g.,
for uses-cases in IoT [10] and auditing applications [11].

To the best of our knowledge, there are a few multi-chain projects that both
improve transactions’ scalability and implement cross-chain communication. The
first project is Cosmos [12], which launched in March 2019 but did not
approach the same level of market capitalization as Polkadot did. In Cosmos,
there are multiple independent chains operating in ‘zones’ that enjoy trust-
free inter-chain communication via the main hub chain. The second is the
anticipated version of Ethereum (formerly known as Ethereum 2.0); however,
Ethereum 2.0 is still to be launched, approximately, later in 2023. As per
Polkadot, to the best of our knowledge, there is no work in the literature
that analyzes the proposed architecture, having in mind to check its true
abiding to the principles of scalability, interoperability and
decentralization. In this respect, our contribution is the first one to show
that the limitations of the proposed architecture introduce some tensions with
respect to the above discussed objectives.

Figure 1: Polkadot Data Structures and Participants

##  III Polkadot Internals

Polkadot is popular for being a multi-chain (sharded) blockchain network,
specifically built to support the integration of heterogeneous blockchains in
a trust-free environment and under common security guarantees. It also enables
secured communication with external sovereign blockchains (e.g., Bitcoin or
Ethereum); thus, laying the foundations for establishing a truly decentralized
web. In addition to interoperability, Polkadot’s multi-chain design allows
transactions to be processed in parallel instead of sequentially; leading to a
higher throughput and scalability. All these properties are provided owing to
the Relay Chain. It unites a network of heterogeneous blockchains (or
arbitrary state machines), operating in parallel, which include parachains,
parathreads, and bridges. The Relay Chain provides shared security to the
entire network through its consensus mechanisms.

Polkadot adopts Proof of Stake (PoS) consensus, namely a variation of it
called the Nominated Proof of Stake (NPoS). The scheme is referred to as
“nominated” because the validators (Relay Chain maintainers: nodes that are
responsible for authoring and verifying new blocks) are backed up by
nominators who lock their DOT — Polkadot’s native currency — as collateral in
exchange for staking rewards. However, if validators misbehave, their
corresponding nominators are also slashed (i.e., DOT deduction from their
accounts). While validators maintain the Relay Chain, a collator is a node
responsible for maintaining its parachain. All previously described structures
and roles are shown in Figure 1.

###  III-A Paraobjects

In this subsection, we review the paraobjects supported by Polkadot, namely:
parachains, parathreads, and bridges.

####  III-A1 Parachains

These are independent chains that operate in parallel, maintained by
collators, and can communicate with each other via the Cross-Consensus Message
Passing (XCMP) format. Parachains can have customized runtime logic, which
includes project-specific rules for governance, economics and incentive
mechanisms. The key constraint on parachains is that their state transition
function (STF), which is stored on the Relay Chain, must be verifiable by
validators. Proofs of new state transitions (aka, Proof of Validity or PoV)
that occur on a parachain are produced by collators and then validated against
the registered STF by validators. Polkadot is designed to support a maximum of
100 parachain slots, where a slot can be acquired through governance (for
common good parachains) or via candle auctions.

Common Good Parachains offer functionality that benefits the network as a
whole. Since they are not allocated via the auctioning process, their slot
lease does not expire and can only be revoked by the governance body. Examples
of public utility chains include: asset transfer (fungible and non-fungible)
chains, smart contract chains, and bridges.

Slot (Candle) Auctions. In parachain slot auctions, parachain accounts lock
their DOT as bids to secure a slot on the network for a specified lease
duration. The lease duration is in three months increments up to a maximum of
2 years. Throughout the auction’s ending period, a per-block snapshot is taken
to capture the bids and winners status. From the snapshots, the ‘ending block’
from which the winners are announced is randomly selected. This is a security
feature to prevent malicious bidders from ‘sniping’ the auction—waiting out
until the auction is about to terminate to place the highest bid.

Parachain projects can bid with their own funds or source them using the
crowdloan functionality, which is asking the community to stake DOT in support
of their chain. The reserved DOT contributes to the security of the system
just as staking does. Since parachain slots are scarce, placing sufficiently
high bids and getting support from the community ensures that honest projects
get connected.

Parachain Consensus. Parachain consensus is the process of ensuring that only
valid candidate blocks get stored on the Relay Chain. Figure 2 lists the
phases of parachain consensus. The first process, collation, is performed on
the parachain by collators, where they produce parachain blocks and send them
to the assigned set of para-validators for the next phase, backing. Backing is
the process where the validators conduct initial validity checks on the
submitted parachain blocks, and if the checks pass, they dispatch “candidate
receipts” to the Relay Chain transactions pool, where further validation and
consensus decide if the candidate would be included in the Relay Chain block.
Availability scheme ensures that the necessary data is available to all
validators so that they are able to validate the parachain block if needed. In
the final stage, validators indicate their approval or disputes about the
block by voting, and the super-majority vote ($2/3f+1$) wins.

Figure 2: Stages of Parachain Consensus

The four-staged process is explained hereafter:
Collation. Collators aggregate all transactions created on the parachain and
produce a candidate block. Along with the proposed block, the collator
provides PoV block which is a summary of the parachain’s state transition.
Backing. After a para-validator receives the PoV block, it validates it
against the parachain’s registered STF. If the block passes the checks, the
validator gossips it to the remaining para-validators who also attest its
validity. When majority of the para-validators agree on the block, they sign
the PoV’s header and prepare a candidate receipt. The candidate receipt is
gossiped to the other Relay Chain validators. After which, a subset of other
validators also validate the PoV block and submit their attestations of
validity. If the verification fails at any stage, the block is immediately
rejected.
Availability. In addition, the availability of the parachain blocks is checked
by Relay Chain validators. The parachain blob (PoV block and set of outgoing
XCMP messages) need to be available for some time to all validators for the
finality (see Section III-C3) process. The para-validators construct erasure
coding chunks of the parachain blob and gossip a piece to each validator.
Enough validators must report that they posses an erasure code piece to pass
the availability test. At least $1/3+1$ of the code pieces are needed to
construct the full block whenever needed.
Approvals/Disputes. These are collected on the Relay Chain as part of the
block finalization process (GRANDPA consensus). If the block fails any of the
previous checks, the corresponding para-validators/offenders are slashed.

####  III-A2 Parathreads

Unlike parachains, parathreads do not lease a dedicated slot on the Relay
Chain. Instead, they form a pool together that shares a designated number of
slots. In other words, the only difference between parachains and parathreads
is economic. In order for a parathread to submit its block candidate to the
Relay Chain, it must participate in a block-by-block auction competing with
other parathreads. Collators signal to the assigned para-validators their bid
in DOT, and the block author selects the candidates with the highest bids to
include on the Relay Chain. This pay-as-you-go model makes parathreads
suitable for applications that do not require high throughput or run less
frequently. Parathreads are also an alternative for projects that cannot
afford to lease a parachain slot.

####  III-A3 Bridges

A Polkadot bridge is a specialized parachain, also maintained by collators,
that handles the communication between the Relay Chain and an external
sovereign blockchain (e.g., Bitcoin). In this respect, bridges offer two-way
compatibility and interoperability between blockchains; allowing arbitrary
data exchange between two networks.

###  III-B Consensus Roles

In this section, we present the network maintainers, validators and collators,
and nominators—the entities that contribute to the NPoS consensus. Collators
and validators participate in the network by running a full (highly available)
node that stores the network’s data, however nominators rely on a light client
that retrieves only specific validator-related data. The interactions between
the roles of Polkadot participants are depicted in Figure 3, while Table I
shows the limitations set by the protocol on the number of
participants/paraobjects.

####  III-B1 Validators

Polkadot restricts the number of active validators on the network. As the
number of connected parachains increases, the active set will reach the
maximum size of 1,000 validators. This limit is the result of a compromise
between security and performance. The larger the number of validators in the
system, the higher its security; however, the throughput of the system is
reduced since more time is needed to reach consensus between all the nodes.

Validators are backed by self-stake and by nominators who bond a sufficiently
large amount of DOT in their favor. This NPoS model contributes economic
security to the network. Since majority of the actors are assumed to be
honest, and at most a third can be malicious, validators are incentivized to
behave honestly because malicious activities, if detected, are heavily
punished in the form of major slashing of their staked funds. Since slashing
applies to the total stake of a validator, a validator bonding a large self-
stake would communicate higher confidence and trust to potential nominators.

Validators are the maintainers of the Relay Chain. They engage in several
tasks in parallel that include: proposing new relay chain blocks, reaching
consensus about the finalized chain, validating the state transition function
of parachains, and ensuring parachains’ data availability and validity. In
exchange for infrastructure and operational costs involved in running its
node, a validator can set a variable commission rate (0%-100%) — which is a
percentage of the total block rewards that the validator had earned in a
completed era. Era points are gained when the validator issues validity
statements for a parachain block and generates a valid relay chain block. The
commission is initially deducted from the era rewards and the remaining
rewards are paid to the validator and his nominators according to their stake
distribution. Validators are also rewarded for authoring new blocks by
receiving a portion (20%) of the transaction fees and 100% of the (optional)
transaction tips.

####  III-B2 Collators

Collators are responsible for maintaining a full node on their parachain and a
full node on the Relay Chain. Meaning, they can access all necessary
information, including the Relay Chain’s state transitions, needed to author
new blocks. In summary, the collators tasks are to: (a) maintain their
respective parachain and store its data; (b) exchange messages with other
parachains using XCM format; (c) produce parachain block candidates; (d)
generate parachain state transition proof, also known as PoV block, which is a
bundled proof/summary of the parachain’s state transitions; and, (e) in case
of a parathread collator, offer a bid in DOT in the block-by-block auction.

####  III-B3 Nominators

Nominators can nominate up to 16 (trusted) validators daily by bonding a part
of their DOT as collateral (aka, staking). In return, nominators receive part
of the block rewards that their active nominations earned in proportion to the
stake they bonded. However, if the active validator commits a punishable
offense (e.g., being unresponsive or equivocating), the nominator also gets
slashed and loses a portion of his DOT.

Phragmén Election. Validators are elected into the active set based on the
phragmén election which implements proportionally justified representation (a
property in voting theory). Phragmén’s election has two major goals that are
to: (a) select a subset of validators from a larger set based on stake-
weighted votes; and, (b) spread out the stake backing each validator equally
as much as possible. For further technical details about how the election
algorithm works, refer to the research paper in [13] and [14].

Figure 3: Interaction between Polkadot roles

###  III-C Consensus Mechanisms

Polkadot adopts a hybrid consensus mechanism where block production and
finality are two isolated processes. The algorithm responsible for handling
block production is called BABE (Section III-C2), and the finality gadget is
GRANDPA (Section III-C3). We cover these algorithms in more detail later in
this subsection. On the one hand, block production (BABE) is intended to be
probabilistically safe (able to continuously generate new blocks with a
probabilistic assurance that the generated blocks will be finalized after some
time) [15]. On the other hand, GRANDPA provides deterministic and provable
finality (universal agreement on the canonical chain with zero probability of
reversion except by a hard fork) [15]. Provable finality is a desirable
feature specifically for achieving secure interoperability with external
blockchains; it provides guarantees that data stored on Polkadot or its
parachains are final and have almost no chance of being easily reverted. By
combining these two mechanisms, Polkadot allows block production to continue
rapidly, while the slower finality mechanism runs asynchronously without
risking slower processing times or stalling. In what follows, we describe the
NPoS and the hybrid consensus scheme adopted by Polkadot.

TABLE I: Polkadot’s Limitations on the Number of Participants/Paraobjects Entity | Limit
---|---
Validators | 297 (max. 1000)
Nominators | 22,500
Parachains | 14 (max. 100)
Council | 13 (max. 24)

####  III-C1 NPoS (Nominated Proof of Stake)

In NPoS, a large number of nominators back a limited set of validators. NPoS
is comparable to Delegated Proof of Stake (DPoS); however, the key difference
between them is that in NPoS, nominators are subject to slashing if their
validator is malicious, whereas delegators (or witnesses) are not. As the
nominators share slashings as well as economic benefits, they are incentivized
to choose honest validators with a record of good performance. Thus, the NPoS
scheme is expected to contribute to the overall economic security of the
network.

####  III-C2 BABE (Blind Assignment for Blockchain Extension)

BABE is a slot-based block production algorithm. Time is represented as epochs
(4 hours), each consisting of 2400 slots, i.e., each slot is 6 seconds long.
In Polkadot, the target block time is also 6 seconds; meaning that within each
slot, a new block should be produced and appended to the Relay Chain. BABE is
responsible for orchestrating the block production process, which handles the
following mechanisms: (a) selection of block producers; (b) selection of the
canonical chain; and, (c) adjustment of slot time. We describe these three
processes hereafter.

Selection of Block Producers. In each slot, BABE assigns primary author(s) and
a secondary author to produce a Relay Chain block. Primary slot leaders are
selected randomly based on a Verifiable Random Function (VRF). The inputs of
the VRF are: epoch randomness seed, slot number, and author’s private VRF key.
Because of the randomness in the selection process, some slots may end up with
more than one primary producer and, sometimes even, a slot may have zero
primary producers. To handle the case of no primary producers, BABE uses a
fall-back round-robin algorithm to assign secondary slot leaders. In each
slot, a secondary producer is selected and produces a block along with the
primary producer(s), if any. However, if a valid primary block is produced,
the secondary one is ignored.

When multiple validators are primary producers in a given slot, all of them
produce a block and broadcast it in the network, and the block that reaches
most of the network first wins the race. Due to network conditions, forks may
emerge until the finalization algorithm (GRANDPA) selects the canonical chain
and kills other forks.

At the beginning of an epoch, each validator evaluates its VRF for each slot
in the entire epoch’s duration. If the result of the VRF is below a threshold
value defined in the protocol, the validator knows that it is a primary author
for that slot. However, other participants in the network are not aware of
that assignment in advance (hence the name ‘Blind Assignment’). Not knowing
the assignment beforehand adds to the security of the system; unlike round-
robin algorithms, adversaries cannot launch a coordinated attack against the
next author.

Selection of the Canonical Chain. There are two rules for selecting the best
chain in BABE. First, block producers must build on the latest block that has
been finalized by GRANDPA. Second, when there are forks after the finalized
head, BABE builds on the chain with the most primary blocks.

Adjustment of Slot Time. BABE does not rely on a centralized clock authority
for time synchronization. Instead, it aligns slot numbers using a relative
time protocol. When a validator node receives a block, it checks the block’s
timestamp and slot number, compares it to its local time, and forecasts future
slot times. More information about local time synchronization in BABE is
provided in [15] and [16].

####  III-C3 GRANDPA—Finality Gadget

GRANDPA stands for GHOST-based Recursive ANcestor Deriving Prefix Agreement.
It allows a set of nodes to agree on the canonical chain out of many possible
forks. It works under the assumption of a Byzantine Fault Tolerant system: a
partially synchronous network with at most one third Byzantine (dishonest or
unresponsive) nodes. GRANDPA reaches agreements on chains rather than blocks;
in other words, whenever more than two thirds of validators vote on a chain
containing a certain block, all blocks up to that one are finalized at once.
For more details about GRANDPA, refer to [15] and [17].

###  III-D Security Model

The adversarial model of Polkadot assumes that at least two thirds of the
active validators are honest. The model also assumes that parachains are
untrusted clients of the Relay Chain; and hence, there are no specific
security assumptions enforced on the parachains or their collators.
Nonetheless, Polkadot assumes that every parachain has one honest and
reachable collator [15].

Security is shared in Polkadot—all connected parachains automatically benefit
from same level of security provided by the Relay Chain. This is to ensure the
validity of the entire system and that no individual component is corruptible.
Security is independent of the number of parachains; it only depends on the
validators who have a sufficiently large amount of DOT backing them.
Nominators and validators contribute to the security of the network by locking
funds, aka, staking. Nominators share in the rewards, but also the punishment,
of the validators they vote for. Phragmén’s election process picks the winners
by finding the combination that leads to having the most value at stake. In
PoS systems, the biggest indicators of the network’s security are the value at
stake and value at risk. Meaning, the greater the amount of DOT staked by
honest validators and nominators, the higher the amount of DOT an attacker
needs to acquire a validator slot.

The implication of a shared/pooled security is that, in order to revert a
parachain’s block, an attacker would have to revert the entire Polkadot
system, including all other parachains. This is due to the Relay Chain blocks
storing proofs of validity from parachains, meaning that when the chain is
finalized, that parachain block is also finalized. This shared security
provides the necessary guarantees to enable trusted cross-chain communication
between untrusted entities.

###  III-E Economic Model

In this subsection, we describe Polkadot’s economics and incentive layers.

####  III-E1 Transaction Fees

Polkadot utilizes weight-based fees that are charged from the sender’s account
before the transaction is dispatched. Transaction fees are the sum of a weight
fee and a length fee. The block weight represents the time it takes to produce
and validate the block. Hence, each transaction has a base weight that
accounts for its inclusion overhead (e.g., time required to verify sender’s
signature) and a dispatch weight for the execution time. The length fee is the
size of the transaction in bytes multiplied by a per-byte fee. Moreover, the
weight fee is adjustable according to current network conditions. A portion of
the fee (20%) goes to the block producer and the remainder goes to the
treasury. Therefore, producers are likely to prioritize transactions with the
highest fees for maximum rewards. Users can provide an optional tip in order
to increase the transaction’s priority, especially during congested periods.

####  III-E2 Staking

The staking system calculates block rewards for every era based on the total
era points earned by validators. Points do not have a corresponding DOT value
until the end of the era. Validators accumulate points based on their
performance, primarily for signing validity statements and producing valid
blocks. Era points are expected to average out amongst validators, hence
rewards are paid out almost equally (with slight variations) regardless of
validators’ total stake. The slight variations in payouts are due to
probabilistic factors (randomness) in the block production and parachain
validation assignments.

After the commission payment is deducted, the staking rewards are distributed
to the validator and nominators in proportion to their contributed stake.
Thereby, the scheme provides incentives for more nominators to vote for lower-
staked validators. This is because, the lower the validator’s total stake is,
the higher the percentage of the nominator’s stake, and therefore the higher
the rewards the nominator receives. Note that only the top 256 nominators
receive staking rewards for an oversubscribed validator.

####  III-E3 Slashing

If a validator misbehaves, punishment will be enforced on him and his
nominators by losing a percentage of their staked DOT. Slashed DOT are added
to the treasury, rather than being burned, since slashes can be reverted by
the Council by simply returning the DOT from the treasury to the slashed
accounts, given that the slash is protested and found unreasonable (e.g., a
faulty runtime) within 28 days. Generally, slashing depends on the number of
repeated offenses and how many other validators committed the same offense
during the epoch. Meaning, slashing is super-linear wherein as the number of
offenders increases, the slash percentage also increases.

There are several degrees of offense where the percentage of stake deduction
is decided accordingly. A validator is required to be available and to prove
its availability; every session the node will send an ”I’m Online” heartbeat.
If a validator fails to produce any block during the epoch and send the
heartbeat, it is declared unresponsive and might receive slashing. Slashing in
the case of unresponsiveness depends on the number of repeated offenses and
how many other validators were offline during the same epoch.

A more serious offense is equivocation — providing conflicting information to
the network. For example, a validator signs multiple votes in the same GRANDPA
round for different forks that conflict with each other, or a validator
produces two or more blocks on the Relay Chain in the same BABE time slot.
Repeated or concurrent equivocation is unlikely to be accidental, and
therefore is severely slashed. If equivocations are detected from different
validators, it is likely to be a coordinated attack. Offending validators will
be kicked immediately and will lose their nominators.

Some misconduct can be slashed up to 100% of the stake because they pose a
grave security or monetary risk or result in major collusion. An example would
be casting GRANDPA votes for a chain that conflicts with an already finalized
chain. The behavior is considered a definite attack because it is trying to
revert finalized blocks.

####  III-E4 Inflation

Polkadot has no maximum limit for DOT supply; hence, the network follows an
inflation model. Block rewards are maximized when the staking rate (total
amount staked over the total token supply) is at 50% . The remainder of the
rewards due to non-ideal staking rate goes to treasury.

####  III-E5 Accounts

An account’s address in Polkadot is represented by the user’s public key which
is 32-byte (256-bit) long. Polkadot uses SS58 address encoding format which
allows account portability among compatible Substrate-based blockchains.
Different chains have a different version prefix that is used to identify
whether an account belongs to its chain or another. Consequently, a user needs
only to generate a keypair once and use it on all chains, given that only the
address’s format would change from one chain to the other. Although this
practice is discouraged for security reasons, it can be beneficial for
specific use cases, e.g., when setting up a crowdloan account.

Nominators and validators are recommended to set up two separate accounts,
namely the Stash and Controller accounts. The controller account acts on
behalf of the stash account, initiating nominating and validating actions. It
is used to set staking preferences and only needs to contain enough DOT to pay
transaction fees. Whereas, the stash account holds the actual funds bonded for
staking. This keypair separation ensures higher protection for the accounts.

###  III-F Governance

Polkadot’s governance constitutes of three entities: public referenda,
council, and technical committee. Any DOT holder in Polkadot has the power to
submit or endorse a proposal, vote on active referenda, nominate for a seat on
the council, or vote for other council candidates. Council members are elected
candidates whose roles include proposing, voting and vetoing public referenda.
The technical committee is composed of the developer teams who are actively
building Polkadot, e.g., Parity Technologies and Web3 Foundation. They are
given the power to propose an emergency referendum.

####  III-F1 Public Referenda Chamber

Any DOT holder in the system can bond tokens to create or second a public
referendum, where the voters also deposit an amount equal to the original
bond. Every proposal is compiled and its preimage must be submitted to the
blockchain. There can be a maximum of 100 public proposals in the proposal
queue, which is made up of community proposals and council proposals. Every
launch period (28 days), proposals with the highest bond will be moved to the
referendum timetable.

The referendum to be voted upon alternates between the top proposal in either
of the community or council queues. When a referendum is selected, the voting
period (28 days) starts. Votes can be either ‘yes’ or ‘no’, or completely
abstaining from voting. After the voting period, the votes are tallied which
decide whether the referendum passed or not.

####  III-F2 Council

The council is currently composed of 13 members including Gavin Wood, the
founder of Polkadot. In addition, the network plans to accommodate only a
maximum of 24 seats on the council [13]. Council members have a term of 7 days
after which a new Phragmén election takes place, and members are selected
based on the stake backing them. Council members have two main tasks:
proposing referenda and cancelling malicious referenda.

####  III-F3 Treasury

Any DOT holder in the system can bond tokens to create a treasury proposal
suggesting a spend from it. If the treasury proposal is declined by the
council, 1% of the tokens that were bonded to create the proposal are burned.
The treasury can release funds only every 24 days.

##  IV Polkadot Contradictions

For a long time, Polkadot has maintained a steady position in terms of coin
market capitalization, ranking amongst the top 10, meanwhile its network and
architecture are still developing and not fully mature yet. In addition, even
though Polkadot was introduced as a multi-chain interoperability platform,
parachains were only added in November 2021, a year and half after the
network’s launch, and it is not known yet when bridges will become available
to fulfill the interoperability claims. In the following subsections, we recap
from our perspective numerous contradictions in the architecture and design of
Polkadot, our empirical analysis, and discussion of the results.

###  IV-A Polkadot Limitations

Difficulties in becoming a validator. If you want to become a validator on the
Polkadot network, you will have to overcome some hefty hurdles. The network is
built to support a maximum of 1,000 validators. Currently, only 297 validators
are allowed into the active set per era (day); and, this number will slowly
scale up to 1,000 as the number of connected parachains increases. Meaning,
validators must compete against one another to enter the active set on a daily
basis, and only those with the highest stake, i.e., self-stake and nominators
stake combined, win. This competition effectively forces a high minimum stake
requirement on validators, making it extremely challenging for new joiners to
participate and compete. Also, the collective stake is locked in the
validator/nominators accounts, i.e., it is not transferable until an account
decides to un-nominate or un-validate. In this case, the bonded amount remains
locked for a further 28 days until it can be withdrawn. This may create a
fertile ground for only a handful of big corporations who can invest such
large amounts to become a validator, potentially leading to some level of
centralized operation in the future.

Problems with becoming a nominator. Nominators face even greater obstacles.
This is because validators can charge from 0 - 100% commission and can change
the rate at any time without prior notice to nominators. If a validator sets
his commission to 100%, none of the nominators get staking rewards for the
duration of the session. Consequently, validators who charge lesser commission
are more likely to be nominated, possibly causing them to become
oversubscribed. However, only the top 256 nominators, for an oversubscribed
validator, receive staking rewards and the rest lose on their bonded stake.
Moreover, if a validator misbehaves or goes offline, all nominators who staked
him, including those not eligible for rewards, will be slashed. Whist the
block rewards are equally distributed to all active validators, slashing
removes a percentage of the validators’ total stake. For validators who set
aside the minimal self-stake, the nominators are at greater risk; that is,
they could lose more to slashing than what they would earn from the staking
rewards. And, the slash could go up to 100%, depending on the type of offence.

Another major scalability concern is related to the validators phragmén
election. When the number of nominators increased beyond 30,000, a number of
problems occurred on the Polkadot network: the block production time slowed
down and the network suffered high instability. According to the statement
released by Polkadot [18], its runtime crashed on May 24, 2021, with an out of
memory error while trying to build a block containing the large election
mapping results, due to the sizeable nominators set. At that time, validators
were asked to downgrade their nodes’ runtime version while Polkadot worked on
a quick workaround (increasing the Wasm memory bounds). To this day, the
problem persists, including on Kusama (Polkadot’s canary network) where the
nominators set was reduced to around 7,300 participants as the number of
validators reached the maximum 1,000.

To address the aforementioned issue, Polkadot’s governance approved a
temporary solution—unfortunately, at the disadvantage of the nominators, as
explained in the following. First, the nominators set is now limited to 22,500
members. Second, in order to nominate, nominators must bond at least 120 DOT,
which equates to $2,160 at $18/DOT. These restrictions come in addition to the
original limit of allowing an account to nominate a maximum of 16 validators
at a time. Yet, phragmén works in a way that ensures that at most only one of
the validators you voted for will become active, and only if the stake was
high enough; since, validators are currently prioritized according to their
total stake.

Problems with participating in governance. The democracy and decentralization
of the governance system is indeed questionable. The technical committee
consists of Web3 Foundation and Parity Technologies, which are both founded
and run by Gavin Wood, who also happens to be a “prime voter” on the council.
A prime voter is one who decides what the default vote is for any abstaining
voters at the end of the voting period. In effect, Gavin Wood has direct
influence on governance decisions; where as a council member, he holds the
power to control the treasury and veto referenda, and in the technical
committee, he provides the final say about the implementation of any
referendum, even if it passed. To top that, the council is restricted to 13
members only. This brings fears that if only seven members are compromised,
the integrity of the entire governance structure will collapse, such as in the
nefarious Ronin Bridge attack [19], where only five of nine validator keys
were needed to succeed in a large-scale cryptocurrency heist worth around 624
millions USD.

Council members are also elected through phragmén method just like validators;
DOT holders back council candidates with their stake. At the time of writing,
the minimum stake required to be elected as a council member is around 9.5
million111Data taken from Polkadot.js:
https://polkadot.js.org/apps/\\#/council, Accessed on April 2022 DOT, which is
171 million USD at $18/DOT. Moreover, monetary punishment and restrictions
make it unfavorable to participate in Polkadot’s ‘democracy’. For instance,
stake deposited by users is burned in such cases when a proposal is rejected.
In addition, the reserved DOT for creating or endorsing a proposal can never
be released until the proposal is brought to a vote (i.e., tabled), which is
an indeterminate amount of time.

TABLE II: Polkadot’s Parachain Winners in its First Batch of Auctions

| Parachain | Role |  | Locked Stake
---
(million DOT)
| Locked Stake
---
(million USD)
1 | Acala |  | Polkadot-native
---
Stablecoin hub
32 | 576
2 | Moonbeam | Smart contracts | 35 | 630
3 | Astar | Smart contracts | 10.3 | 185
4 | Parallel Finance | Assets transfer | 10.7 | 193
5 | Clover Finance | Smart contracts | 9.7 | 175

Problems with becoming a parachain. Polkadot will support only 100 parachain
slots in total, thus imposing a major limitation on true scalability of the
system. Parachain slots are split between parachains and parathread pools,
where a number of slots are reserved for common good parachains whose lease
never expires, e.g., Statemint and public utility chains. Nonetheless, the
cost to secure a parachain slot in the candle auctions is, simply put, almost
prohibitive. As an example, the winning parachains of Polkadot’s first batch
of auctions222Data taken from Subscan: https://polkadot.subscan.io/auction
(concluded on December 17, 2021) and their stakes are listed in order in Table
II.

The funds are reserved in an account created for the parachain during the
entire lease period, up to 2 years. When the lease expires, the parachain has
to enroll again in the auctions. Since slots are scarce, it is expected that
the rising competition will increase the cost of securing a slot. If the
parachain can no longer secure its slot, it has to simply either downgrade to
a parathread or retire from the network altogether. In addition, any number of
parathreads can exist in the parathread pool, but only a limited number can
execute in each block. As a result, as the number of parathreads increases,
smaller parathread projects will eventually not be able to include their
blocks.

Transparency problems with the validators elections. Polkadot uses a complex
(computationally intensive) algorithm for proportionally justified
representation in its elections. However, since the election is run by off-
chain workers and the results are submitted later to the chain, the
transparency of the results become questionable. Are the election results
being validated by on-chain validators? What proof do the designated off-chain
workers provide to attest the authenticity of their computations? The answers
to these questions were not clear in the online documentations [13] nor in the
overview paper [15].

In addition, at the beginning of each era, a pool of 200 para-validators is
formed from the larger validator set. It is said that the pool members are
selected at random, and then groups of five are again formed randomly. It is
not clear whether said randomness is provable using a VRF like BABE’s block
authoring assignments. By observing Polkadot.js, one of Polkadot’s block
explorers, we believe that these groups are seemingly rotated in a round-robin
fashion, and the group assignments do not change during the epoch (4 hours).
This brings up several security-related concerns. For example, can an
adversary target a known majority in one of the para-validator groups to halt
the operation of a parachain during an epoch?

Figure 4: Validators Minimum Total Stake

Figure 5: Validators with 100% Commission Rate

###  IV-B Empirical Analysis

NPoS scheme is a crucial component of Polkadot that governs economic
incentives and directly impacts overall security of the network. Therefore, we
resorted to quantitative analysis to support the previously highlighted
substantial limitations and contradictions. In what follows, we present our
findings, primarily related to the nomination process, validators selection,
and validator preferences.

First, we had set up a Polkadot full archive node on an Ubuntu 20.04 LTS
virtual environment. We then traced ‘NewSession’ events emitted from June 2020
to April 2022 to obtain the corresponding validators elections data from the
runtime storage. We parsed the data into a MySQL database (v8.0.28) using a
Python (v3.10.2) script that we developed. From the data, we investigated the
claims related to the validators minimum stake requirement and the validators
commission rate.

As shown in Figure 5, the minimum stake required to enter the active set
varies with the number of active validators ($n$). For instance, in the
initial stages where $n<197$, even though the size of the validators set was
small, there was much less competition and traction in becoming a validator;
and therefore, the required stake was negligible. The minimum stake declined
as the number of validators increased. As of April 2022, the minimum amount
required to become a validator is around 1.8 million DOT, and at approximately
$18 per DOT333CoinMarketCap: https://coinmarketcap.com/currencies/polkadot-
new/, Accessed April 2022., that is a total of 32.4 million USD. It is unknown
whether the minimum stake would further decrease as the number of validators
progresses towards 1,000, because there are other factors affecting stake
requirement, e.g., competition.

In terms of validators’ preferences, we find an alarming pattern. That is, the
percentage of active validators that set 100% commission is found to be
increasing over time, where currently it has exceeded 60%. Figure 5 depicts
the fraction of active 100%-commissioned validators over time and proves that
a majority has been involved in this behavior. Typically, the perpetrators put
forward the minimum self-stake and are backed by a few nominator accounts. Our
hypothesis is that the nominator accounts also belong to the validator,
otherwise there is no monetary incentive for backing a 100%-commissioned
validator that has almost no self-stake; risking painful slashing and also
getting no rewards. We also hypothesize that this behavior is done to
discourage other nominators from subscribing to the validator, so that the
validator retains all the rewards for himself.

###  IV-C Discussion

Polkadot sets limits on various parameters (as shown in Table I) to balance
between the performance and the security of the system. These bounds
effectively restrict the actual scalability of the network. Even though the
parameters can be modified later through governance, we notice that
scalability is advertised as a prominent feature of Polkadot and a
distinguishing factor from other blockchains. However, enforcing the mentioned
limits, especially at such an early stage of the network’s lifecycle, becomes
a conceivable problem and makes us question whether the network can expand, as
it continues to grow in size and features.

Polkadot claims that its NPoS economic incentive scheme provides higher
inclusion for network participants than centralized systems [4]. Nevertheless,
the limits induced on the staking system resulted in the presence of “whales”
and market exchange centers (e.g., Binance)—owners of a large amount of the
network’s tokens, which is a typical phenomenon in traditional PoS systems.
Due to the size of their holdings, these whales have the potential to unfairly
influence the network and destabilize the system. Polkadot also claims that
its NPoS system helps preventing the formation of validator pools, like what
is seen in Bitcoin, since a larger number of nominators back a limited number
of validators. However, this is not entirely true since majority of the
validators charge full commission and hypothetically they are the ones staking
DOT in their own favor. Validators who own large stake do not need nominators
to back them up and tend to monopolize the rewards for themselves. In the
future, this may lead to a centralization of power, contradicting with the
concept of decentralization envisioned by Web3 Foundation and described in
Polkadot’s whitepaper [3].

Overall, we find that Polkadot is not so different from other PoS networks in
terms of the “rich gets richer” conundrum. The strict limits enforced on
nominators, the prohibitive high stake required to become a validator, in
addition to the seemingly not so democratic governance, eventually will lead
to heavy centralization of power amongst the top few who own the most DOT. We
believe that the contradictions and limitations discussed above call for
further investigations and solutions.

##  V Conclusion and Future Work

In this paper, we presented the first systematic study on the Polkadot
ecosystem. To the best of our knowledge, we are the first ones to discuss the
complex architecture behind one of the most promising projects in the current
DeFi landscape, its protocols, and economic model. We also provided our
preliminary insights about the analyzed architecture, including various
contradictions regarding how Polkadot is presented to the general public and
the way it is implemented. For instance, according to our results, Polkadot
shows several limitations, at least in its current implementation, that can
lead to heavy centralization of power, excluding regular users from network
maintenance and governance. In fact, the constraints on the number of
validators and nominators, together with the NPoS consensus in use, restrict
the participation in the protocol to a few entities with considerable funds
(around 32 Million USD to become a validator), leaving no room for ordinary
users.

Our results call for a more in-depth analysis of the on-chain data to further
investigate the real potential and functioning of the multi-chain protocols
behind the multi-billion Polkadot project, as well as its systemic weaknesses.

## Acknowledgment

This publication was made possible by an award [GSRA7-2-0527-20101] and
[NPRP-S-11-0109-180242] from Qatar National Research Fund (a member of Qatar
Foundation). The contents herein are solely the responsibility of the author.

## References

  * [1] D. Guo, J. Dong, and K. Wang, “Graph structure and statistical properties of ethereum transaction relationships,” _Information Sciences_ , vol. 492, pp. 58–71, 2019.
  * [2] “Coin market cap,” https://coinmarketcap.com/, accessed: 2022-04-20.
  * [3] G. Wood, “Polkadot: Vision for a heterogeneous multi-chain framework,” _White Paper_ , vol. 21, pp. 2327–4662, 2016.
  * [4] “Polkadot network,” https://polkadot.network/, accessed: 2022-01-30.
  * [5] A. Ismailisufi, T. Popović, N. Gligorić, S. Radonjic, and S. Šandi, “A private blockchain implementation using multichain open source platform,” in _2020 24th International Conference on Information Technology (IT)_ , 2020, pp. 1–4.
  * [6] A. Back, M. Corallo, L. Dashjr, M. Friedenbach, G. Maxwell, A. Miller, A. Poelstra, J. Timón, and P. Wuille, “Enabling blockchain innovations with pegged sidechains,” _URL: http://www. opensciencereview. com/papers/123/enablingblockchain-innovations-with-pegged-sidechains_ , vol. 72, 2014.
  * [7] A. Kiayias and D. Zindros, “Proof-of-work sidechains,” in _Financial Cryptography and Data Security_ , A. Bracciali, J. Clark, F. Pintore, P. B. Rønne, and M. Sala, Eds. Cham: Springer International Publishing, 2020, pp. 21–34.
  * [8] L. Kan, Y. Wei, A. Hafiz Muhammad, W. Siyuan, L. C. Gao, and H. Kai, “A multiple blockchains architecture on inter-blockchain communication,” in _2018 IEEE International Conference on Software Quality, Reliability and Security Companion (QRS-C)_ , 2018, pp. 139–145.
  * [9] R. Belchior, A. Vasconcelos, S. Guerreiro, and M. Correia, “A survey on blockchain interoperability: Past, present, and future trends,” _ACM Computing Surveys (CSUR)_ , vol. 54, no. 8, pp. 1–41, 2021.
  * [10] A. Alkhodair, S. Mohanty, E. Kougianos, and D. Puthal, “Mcpora: A multi-chain proof of rapid authentication for post-blockchain based security in large scale complex cyber-physical systems,” in _2020 IEEE Computer Society Annual Symposium on VLSI (ISVLSI)_ , 2020, pp. 446–451.
  * [11] A. Ahmad, M. Saad, L. Njilla, C. Kamhoua, M. Bassiouni, and A. Mohaisen, “Blocktrail: A scalable multichain solution for blockchain-based audit trails,” in _ICC 2019 - 2019 IEEE International Conference on Communications (ICC)_ , 2019, pp. 1–6.
  * [12] “Cosmos whitepaper,” https://v1.cosmos.network/resources/whitepaper, accessed: 2022-04-24.
  * [13] “Polkadot wiki,” https://wiki.polkadot.network/docs/, accessed: 2022-01-30.
  * [14] A. Cevallos and A. Stewart, “A verifiably secure and proportional committee election rule,” in _Proceedings of the 3rd ACM Conference on Advances in Financial Technologies_ , 2021, pp. 29–42.
  * [15] J. Burdges, A. Cevallos, P. Czaban, R. Habermeier, S. Hosseini, F. Lama, H. K. Alper, X. Luo, F. Shirazi, A. Stewart _et al._ , “Overview of polkadot and its design considerations,” _arXiv preprint arXiv:2005.13456_ , 2020\.
  * [16] H. K. Alper, “Babe,” https://research.web3.foundation/en/latest/polkadot/block-production/Babe.html, accessed: 2022-03-8.
  * [17] A. Stewart and E. Kokoris-Kogia, “Grandpa: a byzantine finality gadget,” _arXiv preprint arXiv:2007.01560_ , 2020.
  * [18] B. Köcher, “A polkadot postmortem - 24.05.2021,” https://polkadot.network/blog/a-polkadot-postmortem-24-05-2021/, accessed: 2022-03-12.
  * [19] “Ronin attack shows cross-chain crypto is a ‘bridge’ too far,” https://www.coindesk.com/layer2/2022/04/05/ronin-attack-shows-cross-chain-crypto-is-a-bridge-too-far/, accessed: 2022-04-26.