# Integral of scalar curvature on manifolds with a pole

Guoyi Xu  Department of Mathematical Sciences
Tsinghua University, Beijing
P. R. China, 100084<EMAIL_ADDRESS>
###### Abstract.

On any complete three dimensional Riemannian manifold with a pole and non-
negative Ricci curvature, we show that the asymptotic scaling invariant
integral of scalar curvature, is equal to a term determined by the asymptotic
volume ratio of this Riemannian manifold.
Mathematics Subject Classification: 53C20, 53C21.

The author was partially supported by Beijing Natural Science Foundation
Z190003, NSFC 11771230 and NSFC 12141103.

##  1\. Introduction

Since the well-known Gauss-Bonnet Theorem, there are a lot of studies around
the integral of scalar curvature. Especially, Cohn-Vossen [CV] proved:

######  Theorem 1.1.

If $(M^{2},g)$ is a finitely connected, complete, oriented Riemannian
manifold, and assume $\int_{M^{2}}R$ exists as an extended real number, then

$\displaystyle\int_{M^{2}}R\leq 4\pi\cdot\chi(M^{2}),$

where $R$ is the scalar curvature and $\chi(M^{2})$ is the Euler
characteristic of $(M^{2},g)$ respectively.

Motivated to get a generalization of the Cohn-Vossen’s inequality, Yau [Yau,
Problem $9$] posed the following question:

######  Question 1.2.

Given a $n$-dimensional complete manifold $(M^{n},g)$ with $Rc\geq 0$, let
$B_{p}(r)$ be the geodesic ball around $p\in M^{n}$ and $\sigma_{k}$ be the
$k$-th elementary symmetric function of the Ricci tensor, is it true that
$\displaystyle\lim\limits_{r\rightarrow\infty}r^{-n+2k}\int_{B_{p}(r)}\sigma_{k}<\infty$?

Related to the above question, Shi and Yau [ShiYau] gave a scaling invariant
upper bound estimate for the average integral of the scalar curvature (note
that the scalar curvature is $\sigma_{1}$ in Question 1.2), on Kähler
manifolds with bounded, pinched, nonnegative holomorphic bisectional
curvature. Also see [Petrunin] for discussion with respect to manifolds with
lower bound of sectional curvature.

Later Yang [YangBo] constructed examples, which answered Question 1.2 for
$k>1$ negatively. But the case $k=1$ is left open, where $\sigma_{1}$ is the
scalar curvature $R$. We reformulate the rest case of Question 1.2 as follows.

######  Question 1.3 (Yau).

For any complete Riemannian manifold $(M^{n},g)$ with $Rc\geq 0$, any $p\in
M^{n}$, is it true that

$\displaystyle\lim\limits_{r\rightarrow\infty}r^{2-n}\int_{B_{p}(r)}R<\infty?$

Let $\omega_{n}$ be the volume of the unit ball in $\mathbb{R}^{n}$, we recall
the definition of the asymptotic volume ratio of the manifold $M^{n}$ with
$Rc\geq 0$:

(1.1)
$\displaystyle\mathrm{V}_{M^{n}}\vcentcolon=\lim\limits_{r\rightarrow\infty}\frac{\mathrm{Vol}(B_{p}(r))}{\omega_{n}r^{n}}\in[0,1].$

Using the monotonicity formulas of Colding and Minicozzi [CM], we [Xu]
obtained: for a complete non-compact, non-parabolic Riemannian manifold
$M^{3}$ with $Rc\geq 0$, there is

$\displaystyle\varlimsup_{r\rightarrow\infty}\frac{\int_{b\leq r}R\cdot|\nabla
b|}{r}\leq 8\pi\big{[}1-\mathrm{V}_{M^{3}}\big{]},$

where $b=G^{-1}$ and $G$ is the positive Green’s function of $(M^{3},g)$.

If there is $p\in(M^{n},g)$ such that $\exp_{p}:T_{p}M^{n}\rightarrow M^{n}$
is a diffeomorphism, we call $(M^{n},g)$ is a Riemannian manifold with a pole
(also see [GW]). Among other things, for $3$-dim Riemannian manifolds with a
pole and $Rc\geq 0$, Zhu [Zhu-1], [Zhu] proved

$\displaystyle\varlimsup_{r\rightarrow\infty}r^{-1}\int_{B_{p}(r)}R\leq
20\pi.$

In this note, we prove the following theorem, which answers Question 1.3 for
$3$-dim manifolds with a pole and $Rc\geq 0$.

######  Theorem 1.4.

If $(M^{3},g)$ is a Riemannian manifold with a pole and $Rc\geq 0$, then

$\displaystyle\lim_{r\rightarrow\infty}\frac{\int_{B_{q}(r)}R}{r}=8\pi(1-\mathrm{V}_{M^{3}}),\quad\quad\quad\forall
q\in M^{3}.$

Recall that $(M^{n},g)$ is called Ricci-pinched if there is $\epsilon>0$ such
that

$\displaystyle Rc\geq\epsilon\cdot R\cdot g\geq 0.$

Hamilton has proposed the following conjecture:

######  Conjecture 1.5.

If $(M^{3},g)$ is a complete, non-compact Riemannian manifold and Ricci
pinched, then $(M^{3},g)$ is flat.

The conjecture has been verified with additional assumption by [CZ] and
[Lott]. For recent progress on Conjecture 1.5, see [DSS] and [LT] for
discussion using Ricci flow (also see [HK] for argument by the inverse mean
curvature flow).

In this note, we prove the following result as a byproduct of the proof of
Theorem 1.4, which is a special case of Conjecture 1.5.

######  Theorem 1.6.

If $(M^{3},g)$ is a Riemannian manifold with a pole and Ricci pinched, then
$(M^{3},g)$ is flat.

We conclude this section with the sketch of the proof of Theorem 1.4 and the
structure of the paper. By the Gauss-Codazzi equation, we firstly decompose
the scalar curvature into three parts.

The first part is the intrinsic scalar curvature of geodesic sphere, whose
integral can be calculated by Gauss-Bonnet Theorem; because geodesic sphere is
diffeomorphic to $\mathbb{S}^{2}$ in our case.

The second part and third part involve the Ricci curvature along the radial
direction and the principal curvatures of the geodesic spheres. By Calabi’s
classical computation, we reduce these two parts as the Ricci curvature along
the radial direction and the radial derivative of geodesic spheres’ area.

One difficulty we need to overcome is estimating the average integral of the
Ricci curvature along the radial direction. We firstly observe the derivative
of volume elements of geodesic sphere along the radial direction is non-
negative on manifolds with a pole. This observation and Gromov-Bishop’s Volume
Comparison Theorem, reduce the estimate of radial Ricci curvature to the
estimate of the radial derivative of geodesic spheres’ area.

Then we expand (or shrink) the geodesic ball suitably, such that the radial
derivative of suitable geodesic sphere’ area can be represented as the
difference of two geodesic spheres’ area, by the help of the Mean Value
Theorem. This trick is partially motivated by the argument in [Petrunin].
Again combining the Gromov-Bishop’s Volume Comparison Theorem, we get the
estimate of the radial derivative of geodesic spheres’ area. Furthermore, the
average integral of the Ricci curvature along the radial direction vanishes.

The structure of the paper is as follows. The estimate of the radial
derivative of suitable geodesic sphere’ area is discussed in Section 2. Then
we present the estimate of Ricci curvature along the radial direction in
Section 3. Finally, the proof of Theorem 1.4 and Theorem 1.6 is provided in
Section 4.

##  2\. The volume of geodesic spheres

Without loss of generality, we assume $p$ is one pole of $(M^{3},g)$ in the
rest argument. And we always use $\rho(x)=d(p,x)$ unless otherwise mentioned.

For any $q\in M^{n}$, we can write $q=(r,\theta)$ in terms of polar normal
coordinates at $p$, where $r=d(p,q)$. And it is well known that we can write
the volume element of $(M^{n},g)$ as

(2.1) $\displaystyle d\mu_{g}=J(t,\theta)dtd\theta,$

where $d\theta$ is the area element of the unit $(n-1)$-sphere. The area
element of geodesic sphere $\partial B_{p}(t)$ is given by
$J(t,\theta)d\theta$.

Now we recall the Bishop-Gromov Volume Comparison Theorem for manifolds with a
pole (see [LiBook] etc.) as follows:

######  Theorem 2.1 (Bishop-Gromov).

If $(M^{n},g)$ is a Riemannian manifold with a pole and $Rc\geq 0$, assume
$s\geq t>0$, then

$\displaystyle\frac{J(s,\theta)}{J(t,\theta)}\leq\frac{s^{n-1}}{t^{n-1}},\quad\quad\frac{J^{\prime}}{J}(t,\theta)\leq\frac{n-1}{t},\quad\quad
J(t,\theta)\leq t^{n-1},$
$\displaystyle\frac{V(B_{p}(s))}{V(B_{p}(t))}\leq\frac{s^{n}}{t^{n}},\quad\quad\quad
V(B_{p}(t))\leq\omega_{n}t^{n}.$

∎

Define $\mathscr{A}(t)=V(\partial B_{p}(t))$, where $t>0$. From (1.1) and
Theorem 2.1, we get

(2.2) $\displaystyle\lim_{r\rightarrow\infty}\frac{\mathscr{A}(r)}{4\pi
r^{2}}=\lim_{r\rightarrow\infty}\frac{\int_{\mathbb{S}^{2}}J(r,\theta)d\theta}{4\pi
r^{2}}=\mathrm{V}_{M^{3}}.$

Fixing $\epsilon\in(0,1)$, for any $s>0$, there are
$b_{s}\in((1-\epsilon)s,s),c_{s}\in(s,(1+\epsilon)s)$ such that

$\displaystyle\mathscr{A}^{\prime}(b_{s})=\frac{\mathscr{A}(s)-\mathscr{A}((1-\epsilon)s)}{\epsilon
s},\quad\quad\mathscr{A}^{\prime}(c_{s})=\frac{\mathscr{A}((1+\epsilon)s)-\mathscr{A}(s)}{\epsilon
s}.$

For any $s>a$, we define $A_{a,s}=B_{p}(s)-B_{p}(a)$ in the rest argument.

######  Lemma 2.2.

If $(M^{3},g)$ is a Riemannian manifold with a pole $p$ and $Rc\geq 0$, then
for any $a>0$, we have

$\displaystyle\varlimsup_{s\rightarrow\infty}\int_{A_{a,c_{s}}}\frac{-J^{\prime\prime}}{s\cdot
J}=4\pi\mathrm{V}_{M^{3}}(-2-\epsilon),\quad\quad\varliminf_{s\rightarrow\infty}\int_{A_{a,b_{s}}}\frac{-J^{\prime\prime}}{s\cdot
J}=4\pi\mathrm{V}_{M^{3}}\cdot(-2+\epsilon).$

Proof: Now using (2.2), we get

$\displaystyle\varlimsup_{s\rightarrow\infty}\frac{1}{s}\int_{A_{a,c_{s}}}(-\frac{J^{\prime\prime}}{J})=\varlimsup_{s\rightarrow\infty}\frac{\mathscr{A}^{\prime}(a)-\mathscr{A}^{\prime}(c_{s})}{s}=\varlimsup_{s\rightarrow\infty}\frac{\mathscr{A}(s)-\mathscr{A}((1+\epsilon)s)}{\epsilon
s^{2}}$ $\displaystyle=4\pi\mathrm{V}_{M^{3}}(-2-\epsilon).$
$\displaystyle\varliminf_{s\rightarrow\infty}\int_{A_{a,b_{s}}}\frac{-J^{\prime\prime}}{s\cdot
J}=\varliminf_{s\rightarrow\infty}\frac{\mathscr{A}^{\prime}(a)-\mathscr{A}^{\prime}(b_{s})}{s}=\varliminf_{s\rightarrow\infty}\frac{\mathscr{A}((1-\epsilon)s)-\mathscr{A}(s)}{\epsilon
s^{2}}.$ $\displaystyle=4\pi\mathrm{V}_{M^{3}}\cdot(\epsilon-2).$

∎

##  3\. The Ricci curvature along the radial direction

######  Lemma 3.1.

If $(M^{3},g)$ is a Riemannian manifold with a pole and $Rc\geq 0$,

$\displaystyle
0\leq\frac{J^{\prime}}{J}(t,\theta)\leq\frac{2}{t},\quad\quad\quad\quad\forall\
t\in\mathbb{R}^{+},\theta\in\mathbb{S}^{n-1}.$

Proof: From [Calabi], we have

$\displaystyle\frac{J^{\prime}}{J}=\lambda_{1}+\lambda_{2},\quad\quad\quad(\frac{J^{\prime}}{J})^{\prime}=-(\lambda_{1}^{2}+\lambda_{2}^{2})-Rc(\nabla\rho,\nabla\rho),$

where $\lambda_{i}$ are principal curvature of $\rho^{-1}(t)$ where $i=1,2$.
Therefore we get

(3.1) $\displaystyle
Rc(\nabla\rho)=2\lambda_{1}\lambda_{2}-\frac{J^{\prime\prime}}{J}.$

By (3.1), we have

(3.2) $\displaystyle
Rc(\nabla\rho)\leq\frac{1}{2}(\lambda_{1}+\lambda_{2})^{2}-\frac{J^{\prime\prime}}{J}=\frac{1}{2}(\frac{J^{\prime}}{J})^{2}-\frac{J^{\prime\prime}}{J}.$

From (3.2), we obtain

(3.3)
$\displaystyle(-\frac{1}{2}\frac{J^{\prime}}{J})^{\prime}\geq\frac{1}{2}Rc(\nabla\rho)+(\frac{1}{2}\frac{J^{\prime}}{J})^{2}\geq(\frac{1}{2}\frac{J^{\prime}}{J})^{2}.$

Note $J(t)>0$ and $J^{\prime}(t)$ is well-defined for any
$t\in\mathbb{R}^{+}$, because $M^{3}$ is a Riemannian manifold with a pole.
Hence from (3.3) we know that

$\displaystyle J^{\prime}(t)\geq 0,\quad\quad\quad\quad\forall
t\in\mathbb{R}^{+}.$

Otherwise $\frac{-J^{\prime}}{J}$ will be positive infinity for some finite
$t_{0}>0$ because of (3.3).

Combining Theorem 2.1, we get the conclusion. ∎

######  Lemma 3.2.

If $(M^{3},g)$ is a Riemannian manifold with a pole $p$ and $Rc\geq 0$, then

$\displaystyle\lim_{r\rightarrow\infty}\frac{\int_{B_{p}(r)}Rc(\nabla\rho)}{r}=0.$

Proof: Step (1). By (2.2), for $\epsilon>0$, we can find $a>0$ such that
$\displaystyle\frac{\mathscr{A}(a)}{a^{2}}\leq
4\pi\mathrm{V}_{M^{3}}(1+\epsilon)$. By Theorem 2.1, we know that
$\frac{J(t,\theta)}{t^{2}}\leq\frac{J(a,\theta)}{a^{2}}$ for any $t\geq a$.
Therefore, integration by parts yields

$\displaystyle\varlimsup_{s\rightarrow\infty}\frac{1}{s}\int_{A_{a,c_{s}}}\frac{J^{\prime}(t,\theta)}{t\cdot
J(t,\theta)}=\varlimsup_{s\rightarrow\infty}\frac{1}{s}\int_{\mathbb{S}^{2}}d\theta\int_{a}^{c_{s}}\frac{J^{\prime}(t,\theta)}{t}dt$
$\displaystyle=\varlimsup_{s\rightarrow\infty}\frac{1}{s}\cdot\Big{\\{}\int_{\mathbb{S}^{2}}\frac{J(c_{s},\theta)}{c_{s}}d\theta+\int_{\mathbb{S}^{2}}d\theta\int_{a}^{c_{s}}\frac{J}{t^{2}}dt\Big{\\}}$
$\displaystyle=\varlimsup_{s\rightarrow\infty}\frac{\mathscr{A}(c_{s})}{s\cdot
c_{s}}+\frac{c_{s}-a}{s}\frac{\mathscr{A}(a)}{a^{2}}$
$\displaystyle\leq(1+\epsilon)\cdot\varlimsup_{s\rightarrow\infty}\frac{\mathscr{A}(c_{s})}{c_{s}^{2}}+(1+\epsilon)\cdot
4\pi\mathrm{V}_{M^{3}}(1+\epsilon)$ (3.4) $\displaystyle\leq
8\pi\mathrm{V}_{M^{3}}(1+\epsilon)^{2}.$

Step (2). Now from (3.2) and Lemma 3.1, we get

(3.5) $\displaystyle
Rc(\nabla\rho)(t,\theta)\leq\frac{J^{\prime}(t,\theta)}{t\cdot
J(t,\theta)}-\frac{J^{\prime\prime}}{J}(t,\theta).$

Using (3.5), (3.4) and Lemma 2.2, we get

$\displaystyle\varlimsup_{s\rightarrow\infty}\frac{1}{s}\int_{B_{p}(s)}Rc(\nabla\rho)=\varlimsup_{s\rightarrow\infty}\frac{\int_{A_{a,s}}Rc(\nabla\rho)}{s}\leq\varlimsup_{s\rightarrow\infty}\frac{\int_{A_{a,c_{s}}}Rc(\nabla\rho)}{s}$
$\displaystyle\leq\varlimsup_{s\rightarrow\infty}\frac{\int_{A_{a,c_{s}}}\frac{J^{\prime}(t,\theta)}{t\cdot
J(t,\theta)}}{s}+\varlimsup_{s\rightarrow\infty}\int_{A_{a,c_{s}}}\frac{-J^{\prime\prime}}{s\cdot
J}\leq 4\pi\mathrm{V}_{M^{3}}(3\epsilon+2\epsilon^{2}).$

Let $\epsilon\rightarrow 0$ above, we have

$\displaystyle\varlimsup_{s\rightarrow\infty}\frac{1}{s}\int_{B_{p}(s)}Rc(\nabla\rho)\leq
0.$

On the other hand we know that $Rc(\nabla\rho)\geq 0$, therefore the
conclusion follows. ∎

##  4\. The integral of the scalar curvature

######  Theorem 4.1.

If $(M^{3},g)$ is a Riemannian manifold with a pole and $Rc\geq 0$, then

$\displaystyle\lim_{r\rightarrow\infty}\frac{\int_{B_{q}(r)}R}{r}=8\pi(1-\mathrm{V}_{M^{3}}),\quad\quad\quad\forall
q\in M^{3}.$

Proof: Step (1). Recall that $\lambda_{i}$ are principal curvature of
$\rho^{-1}(t)$ where $i=1,2$. From the Gauss-Codazzi equation we have

(4.1) $\displaystyle R(M^{3})(t,\theta)=R(\partial
B_{p}(t))(t,\theta)+2Rc(M^{3})\Big{(}\nabla\rho,\nabla\rho\Big{)}-2\lambda_{1}\lambda_{2},$

where $R(\partial B_{p}(t))(t,\theta)$ is the intrinsic scalar curvature of
$\partial B_{p}(t)$ at $\exp_{p}(t\theta)$ and $\exp_{p}$ is the exponential
map of $M^{3}$.

Note $\rho^{-1}(t)$ is diffeomorphic to $\mathbb{S}^{2}$. By the co-area
formula and the Gauss-Bonnet Theorem for the compact surface $\rho^{-1}(t)$,
we get

(4.2) $\displaystyle\int_{\rho^{-1}[a,s]}R(\partial B_{p}(t))dx$
$\displaystyle=\int_{a}^{s}dt\int_{\rho^{-1}(t)}R(\partial
B_{p}(t))=8\pi(s-a).$

From (4.1) and (3.1), we get

$\displaystyle\int_{\rho^{-1}[a,s]}R(M^{3})$
$\displaystyle=\int_{\rho^{-1}[a,s]}\Big{\\{}R(\partial
B_{p}(t))+2Rc(M^{3})\Big{(}\nabla\rho,\nabla\rho\Big{)}-2\lambda_{1}\lambda_{2}\Big{\\}}$
(4.3)
$\displaystyle=8\pi(s-a)+\int_{\rho^{-1}[a,s]}\Big{\\{}Rc(\nabla\rho)-\frac{J^{\prime\prime}}{J}\Big{\\}}.$

Step (2). From (4.3) and $Rc\geq 0$, using Lemma 2.2 and
$\frac{b_{s}}{s}\in[1-\epsilon,1]$, we have

$\displaystyle\varliminf_{s\rightarrow\infty}\frac{1}{s}\int_{A_{a,s}}R\geq\varliminf_{s\rightarrow\infty}\frac{1}{s}\int_{A_{a,b_{s}}}R\geq
8\pi(1-\epsilon)+\varliminf_{s\rightarrow\infty}\int_{A_{a,b_{s}}}\frac{-J^{\prime\prime}}{s\cdot
J}$ $\displaystyle=8\pi(1-\epsilon)+4\pi\mathrm{V}_{M^{3}}\cdot(\epsilon-2).$

Let $\epsilon\rightarrow 0$ in the above, we get

(4.4)
$\displaystyle\varliminf_{s\rightarrow\infty}\frac{1}{s}\int_{B_{p}(s)}R=\varliminf_{s\rightarrow\infty}\frac{1}{s}\int_{A_{a,s}}R\geq
8\pi(1-\mathrm{V}_{M^{3}}).$

Step (3). Using (4.3), Lemma 3.2 and Lemma 2.2, also note
$\frac{c_{s}}{s}\in[1,1+\epsilon]$, we gave

$\displaystyle\varlimsup_{s\rightarrow\infty}\frac{1}{s}\int_{B_{p}(s)}R=\varlimsup_{s\rightarrow\infty}\frac{1}{s}\int_{A_{a,s}}R\leq\varlimsup_{s\rightarrow\infty}\frac{1}{s}\int_{A_{a,c_{s}}}R$
$\displaystyle\leq
8\pi(1+\epsilon)+\varlimsup_{s\rightarrow\infty}\frac{c_{s}}{s}\cdot\frac{1}{c_{s}}\int_{B_{p}(c_{s})}Rc(\nabla\rho)+\varlimsup_{s\rightarrow\infty}\frac{1}{s}\int_{A_{a,c_{s}}}(-\frac{J^{\prime\prime}}{J})$
$\displaystyle\leq 8\pi(1+\epsilon)+4\pi\mathrm{V}_{M^{3}}(-2-\epsilon).$

Let $\epsilon\rightarrow 0$ in the above, we obtain

(4.5)
$\displaystyle\varlimsup_{s\rightarrow\infty}\frac{1}{s}\int_{B_{p}(s)}R\leq
8\pi(1-\mathrm{V}_{M^{3}}).$

From (4.4) and (4.5), we get

$\displaystyle\lim_{r\rightarrow\infty}\frac{\int_{B_{p}(r)}R}{r}=8\pi(1-\mathrm{V}_{M^{3}}).$

Because $\displaystyle\lim_{r\rightarrow\infty}\frac{\int_{B_{p}(r)}R}{r}$
exists and is finite, it is easy to see that

$\displaystyle\lim_{r\rightarrow\infty}\frac{\int_{B_{q}(r)}R}{r}=\lim_{r\rightarrow\infty}\frac{\int_{B_{p}(r)}R}{r}=8\pi(1-\mathrm{V}_{M^{3}}),\quad\quad\quad\forall
q\in M^{3}.$

∎

######  Corollary 4.2.

If $(M^{3},g)$ is a Riemannian manifold with a pole $p$ and $Rc\geq 0$, then

$\displaystyle\lim_{s\rightarrow\infty}\frac{\mathscr{A}^{\prime}(s)}{s}=8\pi\cdot\mathrm{V}_{M^{3}}.$

Proof: From (4.3), using Lemma 3.2 and Theorem 1.4, we have

$\displaystyle\lim_{s\rightarrow\infty}\frac{\mathscr{A}^{\prime}(s)}{s}$
$\displaystyle=\lim_{s\rightarrow\infty}\int_{\rho^{-1}[a,s]}\frac{J^{\prime\prime}}{s\cdot
J}=8\pi+\lim_{s\rightarrow\infty}s^{-1}\Big{\\{}\int_{\rho^{-1}[a,s]}Rc(\nabla\rho)-R(M^{3})\Big{\\}}$
$\displaystyle=8\pi+\lim_{s\rightarrow\infty}s^{-1}\Big{\\{}\int_{B_{p}(s)}Rc(\nabla\rho)-R(M^{3})\Big{\\}}=8\pi\cdot\mathrm{V}_{M^{3}}.$

∎

######  Theorem 4.3.

If $(M^{3},g)$ is a Riemannian manifold with a pole and Ricci pinched, then
$(M^{3},g)$ is flat.

Proof: From Ricci pinched assumption, there is $\epsilon>0$ such that
$Rc\geq\epsilon\cdot R\cdot g$. Now by Lemma 3.2 and Theorem 1.4, we get

$\displaystyle
0=\lim_{r\rightarrow\infty}\frac{\int_{B_{p}(r)}Rc(\nabla\rho)}{r}\geq\epsilon\cdot\lim_{r\rightarrow\infty}\frac{\int_{B_{p}(r)}R}{r}=\epsilon\cdot
8\pi(1-\mathrm{V}_{M^{3}}).$

Therefore, we obtain that $\mathrm{V}_{M^{3}}=1$, which implies that $M^{3}$
is isometric to $\mathbb{R}^{3}$ by the rigidity part of Bishop-Gromov’s
Volume Comparison Theorem. ∎

## References


Thesis Introduction

Contribution: Improving the Utility and Privacy of LDP protocols

Contribution: Differentially Private Machine Learning Predictions

Conclusion & Perspectives


# Measuring the Concentration of Control in Contemporary Ethereum

Simon Brown  ConsenSys Software Inc.
<EMAIL_ADDRESS>
###### Abstract

Ethereum is undergoing significant changes to its architecture as it evolves.
These changes include its switch to PoS consensus and the introduction of
significant infrastructural changes that do not require a change to the core
protocol, but that fundamentally affect the way users interact with the
network. These changes represent an evolution toward a more modular
architecture, in which there exists new exogenous vectors for centralization.
This paper builds on previous studies of decentralization of Ethereum to
reflect these recent significant changes, and Ethereum’s new modular paradigm.

###### Index Terms:

blockchain, Ethereum, decentralization, cryptocurrency, cryptoeconomics

##  I Introduction

The contribution of this paper is to propose a model for measuring
decentralization that accommodates structural changes in wider network
topology over time. As web3 and cryptocurrencies are a relatively nascent
socio-technological innovation, they are in a phase of initial rapid
innovation, in which the architecture and topology of the networks are
evolving significantly. This is a well understood phenomenon in technological
innovation, which was documented as early as the 1960s [1], in which the
innovation and adoption of new technologies develop in an “S-Curve” shape,
involving compressed stages of very rapid innovation followed by a period
where innovation plateaus for a time. Ethereum is an example of a technology
that is in the rapid innovation phase, in which there are significant changes
to the topology of the overall network, both intrinsic and extrinsic to the
core protocol. Whereas previous research [2, 3, 4] focused on measuring
decentralization at the various layers of a vertical stack within a monolithic
system, our model views Ethereum as a socio-technological ecosystem in which
significant components of the network develop outside the core protocol.

The paper is organized as follows: in section II we deliver an overview of how
Ethereum is evolving and the challenges faced when attempting to measure its
level of decentralization. In section III we outline the various dimensions
that we propose to measure with our model, and our data sources. In section IV
we describe our methodology, including the various indices that are applied to
our data. In section V we deliver a breakdown of results, and we close with
our conclusions in section VI.

##  II Background

It can be argued that protocols that are built on top of the base layer of
Ethereum do not pose a direct threat to the base layer itself, even when they
are highly centralized, and should therefore not be a factor in quantifying
the network’s level of decentralization. Once the base layer is sufficiently
decentralized, any number of protocol designs can be implemented on top of it,
and ideally the base layer should not be aware of them, or be adversely
affected by them. However, as Ethereum evolves, users increasingly interact
with the network through abstracted layers of infrastructure that overlay the
core protocol, and as such it can be conversely argued that such protocols
could potentially affect the security and/or performance of the overall
network under certain conditions, and should therefore be considered within a
holistic model of measurement of the networks’ level of decentralization. Our
criteria for inclusion within our model is that the component being measured
does not just serve a single use case or application, but is a protocol
through which users interact with a substantial number of other dapps and
protocols.

Any infrastructure that assumes a significant role in Ethereum can pose a
centralization risk to the overall network based on two critical factors:

  1. a.

the size of the infrastructure compared to the base layer, as measured in
either the amount of base layer transactions that flow through the
infrastructure and/or the Total Value Locked (TVL) compared to the base layer.

  2. b.

the potential effect on the base layer should the infrastructure be
compromised or develop misaligned incentives, whether this effect is a level
of effective degraded performance of the network, or an increased level of
censorship.

Ethereum is not a static ecosystem, and other innovations will likely assume a
prominent role within the ecosystem in the future, e.g. EigenLayer [5], DVT
[6]. As such, any model that we develop should account for the changing
topology of the ecosystem and allow us to incorporate new infrastructure into
the model at a future date, while still being able to track the changes of
effective decentralization over time.

##  III Selection of Data Points

###  III-A Overview of Data Model

We use as a base for our model the measurement of decentralization in
blockchain first described by Balaji Srinivasan as the Minimum Nakamoto
coefficient [7]. This model considers a blockchain network as being composed
of a number of subsystems, which are important in terms of maintaining
decentralization within an ecosystem, allowing it to remain resistant to
capture by any one party or group. Srinivasan describes any blockchain as
being only as decentralized as the least decentralized subsystem, and his
original model loosely defines a number of discrete subsystems to measure.

We have adapted Srinivasan’s model to the contemporary PoS Ethereum topology
and introduced several other dimensions that represent exogenous vectors for
potential centralization. Our model thus extends Srinivasan’s original model
from 6 dimensions to 12. These dimensions of measurement are listed below, and
are followed by a detailed explanation of the rationale for each.

  * •

Based on original Nakamoto Coefficient subsystems:

    * –

Consensus nodes by client

    * –

Consensus nodes by country

    * –

Execution nodes by client

    * –

Execution nodes by country

    * –

Distribution of native asset by amount

    * –

Amount staked by pool / staking service provider

  * •

Metrics pertaining to PBS:

    * –

Blocks proposed by builder

    * –

Blocks proposed by relay

  * •

Metrics pertaining to Account Abstraction:

    * –

Number of user operations per bundler

    * –

Number of wallets per deployer

  * •

Miscellaneous Metrics:

    * –

Effective inflation rate adjusted for burn

    * –

Percentage of total supply staked

    * –

Layer 2 rollups by relative TVL

    * –

Stablecoins by relative TVL

###  III-B Metrics based on the Nakamoto Coefficient Subsystems

We have adapted the Nakamoto Coefficient model through a number of
modifications to the original model. These changes include removing Mining
Decentralization and Developer Decentralization.

The “Mining Decentralization” metric is no longer relevant in PoS Ethereum and
as such has been replaced by the “Amount staked by pool” metric, which
measures the relevant share of the staked ETH by staking service provider.

The “Developer Decentralization” metric is no longer an applicable metric for
PoS Ethereum. The rationale for this change is the fact that nodes on the
network run a number of different client implementations, each with its own
distinct development team. In this context, and considering a priori that
developers are unique to each team, it is sufficient to measure the level of
client diversity among nodes on the network rather than the relative
contributions of individual developers.

The “Exchanges by Supply” metric is not employed in our model as there is not
a strong enough argument as to relevance of this metric to decentralization in
Ethereum.

In terms of client diversity, it is also necessary to update the model to
reflect the fact that Ethereum is now technically two merged blockchains that
operate in unison, the Beacon Chain which handles consensus, and the execution
layer, which is the P2P layer that gossips transactions and handles execution.
For this reason, the original “Client Decentralization”, and “Node
Decentralization” metrics have been replaced by “Consensus / Execution nodes
by client / country” metrics.

The only metric that has been retained in its original form is “Distribution
of native asset by amount”, which measures wealth inequality in terms of
ownership of ETH.

###  III-C Metrics pertaining to Proposer Builder Separation

Our model introduces two new metrics that pertain to Proposer Builder
Separation (PBS), which are “Blocks proposed by builder” and “Blocks proposed
by relay” respectively.

PBS is a network topology that has not been implemented at the protocol level,
but has been implemented via the mev-boost middleware developed by Flashbots
[8], and which came online at the time of Ethereum’s switch to PoS.

Fundamentally, PBS allows for the separation of concerns between block
building and block proposing [9], whereas currently the protocol assigns the
responsibility of both to the validator. Ethereum’s PoS protocol requires
validators to broadcast a valid block of transactions to the network when they
are selected as a proposer. As per the specification, validators will build a
block locally by requesting their local execution client to collate pending
transactions from the public P2P network. However, validators can install mev-
boost and can request blocks from third party specialist block builders via
public relays, instead of building one themselves [10].

This has several benefits, from lowering the resource requirements for running
a validator node, to reducing centralizing economics of MEV in staking pools
[11]. However, it also introduces a number of other actors into Ethereum’s
infrastructure topology, i.e. block builders and relays, creating new vectors
for potential centralization.

Figure 1: High level mev-boost architecture

Our model applies a weighting to the PBS metrics when considering the
measurement in the context of Ethereum’s overall level of decentralization.
This is because any byzantine behavior of the mev-boost middleware will result
in validator nodes falling back to local block production, thus preventing any
safety or liveness fault within the core protocol [12].

However, it has been well documented [13] that a number of prominent mev-boost
builders and relays actively censor transactions according to specific
criteria. This effectively results in those transactions experiencing a
potentially significant delay in being included in a block, (about 68% longer
than regular transactions according to Yang et al. [14]).

This can effectively create a two-tier network with transactions associated
with certain addresses becoming “less privileged” than other transactions.
Ironically the more transactions that are censored in this way, the harder it
is to censor them, as block builders will need to bid higher than the combined
value of those transactions in order to have their blocks proposed, resulting
in an effective per-block fee for censoring transactions [15]. However, the
higher the level of centralization within the block builder / relay
infrastructure, the greater the risk for censorship, creating increased
barriers to participation in the network for affected users.

###  III-D Metrics pertaining to Account Abstraction

Our model introduces two metrics that pertain specifically to account
abstraction, including “Number of user operations per bundler” and “Number of
wallets per deployer”. Account Abstraction has been a goal of Ethereum since
its inception, and there have been a number of previous proposals that were
not implemented [16, 17, 18], which all involved some change to the core
protocol. The breakthrough came with “ERC-4337: Account Abstraction Using Alt
Mempool” [19], which does not require a protocol change, but which introduces
new roles within the ecosystem topology: bundlers and paymasters.

ERC-4337 specifies a specific transaction type called a user operation, or
“userop”. User operations are submitted to bundlers, who batch them into a
single transaction to a global entrypoint contract, which iterates over the
userops in the batch, passing them to their respective smart contract wallets
along with the userop’s calldata for the contract wallet to execute (e.g. send
ETH or call a function on some specific smart contract).

Figure 2: ERC-4337 Account Abstraction Architecture

Our model weights these metrics lower than other metrics within the model, as
the risk posed from centralization within this class of actors is lower than
other parts of the infrastructure we have included. Although bundlers can
choose to censor specific transactions, the censored sender can simply decide
to send their transaction directly to the entry-point contract, or to their
smart contract wallet directly, if its design allows. This represents a
relatively weak form of censorship, but it does require some user
sophistication in order to bypass. Over-centralization in this part of
Ethereum’s infrastructure can potentially lead to censorship risks, resulting
in an effective two-tier network, with some addresses being less privileged
than others in terms of their access to the network. We posit that this is a
reasonable basis for including this metric in our model, albeit with an
adjusted weighting.

###  III-E Layer 2 Rollups by Relative TVL

Our model introduces a metric to measure L2 Rollups by TVL relative to the TVL
of the base layer. As Ethereum progresses through its “rollup-centric roadmap”
[20], the TVL of L2 rollups as proportionate to the overall network becomes
more significant within the composition of the ecosystem.

Our model applies a weighting to this metric, taking into account the extent
of the risk that centralization within these protocols pose, i.e. they may not
cause a safety or liveness fault in the underlying protocol, but may
nevertheless potentially cause loss of funds or significant delays should they
be compromised.

Consider as an example an L2 rollup with a centralized sequencer that
experiences a significant liveness fault, in which users with funds on that
network are no longer able to transact as they would under normal conditions.
In this scenario, it may be possible that the users can force a withdrawal
through the L2’s base layer smart contract bridge. However, if this rollup
contains a substantial number of user accounts, it may result in significant
congestion on the base layer [21]. This would likely cause an increase in base
fee and a prolonged delay in transaction inclusion. Furthermore, in the case
of tokens that are minted natively on an L2, it may not be possible to
withdraw them to the base layer at all.

There is also a theoretical risk to Ethereum’s underlying social consensus
from having a single dominant rollup, which is described by Buterin [22], as
forming a broad assumption within the ecosystem that ”if there is a bug that
causes funds to be stolen, the losses will be so large that the community will
have no choice but to fork to recover the users’ funds”.

###  III-F Miscellaneous Metrics

As part of our model we measure the “effective inflation rate adjusted for
burn”. This is an important metric for any PoS base layer protocol, including
Ethereum. A high rate of issuance of the network’s native asset via validator
rewards, has the effect of diluting the circulating supply, effectively
decreasing the asset’s value. This incentivizes the network’s users to stake
the native asset in order to counteract the dilutionary effects of issuance,
which forms a self-reinforcing cycle, leading to eventual hyperinflation even
if that process takes a number of years. Polynya describes a number of
examples of this phenomenon that have been observed in practice [23].

Our model thus incorporates a simple metric for measuring the inflation rate
and adjusting it for the amount of ETH that is burned through the EIP-1559
mechanism, in which the base fee, which is adjusted for every block by the
protocol itself, and that each transaction must pay at a minimum in order to
be included in a block, is subsequently burned when a block is proposed. This
has the effect of creating a negative issuance rate once transaction volume
surpasses critical threshold, which will likely decrease the total supply over
time [24].

We have also incorporated a closely related metric which is the “percentage of
total supply staked”. This is directly related to the effective inflation rate
metric with respect to maintaining an economic equilibrium between the
issuance rate and circulating supply, allowing the asset to hold its value
over time [25]. It is worth pointing out that Ethereum’s economics are
designed to maintain this equilibrium by reducing issuance as more validators
come online [26], which theoretically reduces the incentive to stake once the
percentage of staked assets reaches a certain threshold. However, there is
always the possibility that innovations such as EigenLayer may disrupt this
equilibrium over time.

Another meaningful metric that we have introduced into our model is the
measurement of “Stablecoins by relative TVL on Ethereum”. There exists both
algorithmic stablecoins, (which are backed by a number of other assets and
which rely on networks of decentralized oracles, and which are at least
notionally decentralized by nature), and there also exists stablecoins which
are backed largely by fiat deposits, and which are issued by a centralized
authority. The latter type of stablecoin is relevant to our model, insofar as
while these stablecoins are not part of the infrastructure of the network,
they are a significant part of the ecosystem with which users interact with
other dapps. They also pose a centralization risk in terms of censorship as
there have been a number of cases where stablecoins have been frozen from
specific addresses [27].

##  IV Methodology

The measurement of inequality in distributions is a well understood area of
statistics that has found many applications in the fields of economics and
social sciences. As our model aims to measure the level of decentralization
across a number of different dimensions with different qualities, it is
necessary to incorporate a number of different statistical measurements.

We use as a base measurement, the Gini index and the Herfindahl-Hirschman
index. The Gini index is arguably the most widely used index with regards to
measuring wealth inequality, and is well suited for measuring the distribution
of a network’s native asset, (i.e. ETH), while the Herfindahl-Hirschman index
(HHI) is more often used for measuring the level of competition in specific
industrial sectors, making it more suitable for measuring the degree of
decentralization within the block builder market. The base measurement is
applied to each measurement dimension, but our results will reference one or
the other based on the respective qualities of each.

The Shannon index is used to complement and cross reference any indicative
result from our base measurement. This is useful because the Shannon Index is
based on a different approach to the Gini Index and the HHI. The Atkinson
Index is used to fine-tune the results from our base measurement by updating
the index parameters to make it more sensitive to changes at different
extremes of the distribution. This can be useful if we want to qualify our
results with more detail on any part of a distribution at certain times.

Tail ratios are used to inform any decisions about the parameterization of the
Atkinson Index. These include the P90:P10, and P50:P10 ratios, as well as the
Palma ratio. These ratios describe inequality within a distribution by
comparing percentiles, for example, the P90 represents the level of resources
allocated to $>$ 90% of the population, while the P10 represents the level of
resources allocated to $>$ 10% of the population. Our model adjusts the
parameterization of the Atkinson index with respect the to the values of these
tail ratios.

The Jensen-Shannon Divergence is used to track changes to the distribution
over time, by comparing distributions of respective measurement dimensions
between specific intervals, such as 30, 60 and 90 day intervals.

Our model also incorporates a master index in order to track the changes in
each measurement dimension over discrete intervals. This master index is an
aggregate of other relevant indices, that is derived from calculating the
normalized weighted geometric mean of the respective index’s value across all
other relevant metrics.

The indices that we have employed in our model are as listed below and are
described in detail in the following sections. Each index has characteristics
and trade-offs associated with the underlying approach or model that they are
based on.

Underlying Approach | Index
---|---
Deviations model | Gini index
Combinatorics model | Herfindahl-Hirschman index
Entropy model | Shannon index
Social welfare model | Atkinson index
Tail ratios | Palma ratio, Pareto ratio
Divergence measures | Jensen-Shannon Divergence
Weighted geometric mean | Master Index

###  IV-A Gini Index

The Gini Index was developed by Corado Gini [28] as a mechanism for measuring
inequality of income / wealth in a population, and is arguably the most
commonly used measurement of inequality across a number of fields. It is
employed as the basis of the original Nakamoto Coefficient model, and has been
used in several previous studies of decentralization in Bitcoin and/or
Ethereum [29, 30, 31, 32, 33, 34, 2, 4, 35, 36], which allows for some
comparison with the results of previous studies.

The Gini Index is derived from the Lorenz curve [37], which allows us to plot
the individual shares of the distribution in relation to the overall total
distribution. This becomes particularly useful for visualizing inequality
within a distribution at a high level, and can also be very useful for
comparing inequality between two distributions easily.

Figure 3: Visualization of Lorenz Curve

There are a number of different methods of calculating the Gini Index
(Tutberidze et al. describe four [38]), though most methods are based on the
calculation of the Lorenz Curve of a population, whereby the Gini index is
calculated as the area between the line of equality and the Lorenz curve,
divided by the total area under the line of equality, i.e.: $G=\frac{A}{A+B}$

While the area under a curve is commonly calculated using the Newton-Leibnitz
formula [39] , where $L(x)$ is the Lorenz curve. It can also be approximated
as the sum of the areas of a series of trapeziums, correlating in width to the
unit interval being measured.

$G=1-2\int_{0}^{1}L\left(x\right)dx$

While the calculation of the Gini Index using the Lorenz Curve is useful for
visualizing the distribution and level of inequality on a chart, it is also
possible to calculate directly using an approach that is based on the
empirical mean difference of the values in the dataset [40], which is the
approach that we have implemented:

$G=\frac{\sum_{i=1}^{n}\sum_{j=1}^{n}\left|x_{i}-x_{j}\right|}{2n^{2}\mu}$

The Gini index gives us a value of between 0 and 1, where 0 indicates perfect
equality of distribution of resources within the population, and 1 is total
inequality, i.e. one single entity controls 100% of the resources.

###  IV-B Herfindahl-Hirschman Index

The Herfindahl-Hirschman Index is commonly used as a measurement of
competition within a certain industry sector. It has found applications in
regulation, particularly with antitrust authorities [41]. It is calculated as
the sum of the square of the percentage market share of each entity in a
sector.

As the HHI is based on percentage shares of the market, it becomes close to
zero for a market that has been commoditized, having a large number of
participants with a relatively equal share. Conversely, the HHI approaches
10,000 for a highly concentrated market, with 10,000, (or $1\times 100^{2}$),
being a single entity monopoly.

Our model adapts the standard HHI by re-scaling it to make it comparable with
other indices employed in the model, by dividing the HHI by $10^{4}$ so that
it falls in the interval $0<=\text{HHI}<=1$. As such, the re-scaled HHI,
denoted by $\theta$, is expressed via the following formula, where $n$ is the
total number of participants in the market, $P$ is the total number of units
produced in the entire market, and $p_{i}$ is the number of units consumed
from $i^{th}$ participant.

$\theta=\sum_{i=1}^{n}\frac{(p_{i}/P\cdot 100)^{2}}{10^{4}}$

As an example, in measuring concentration in the block building market, $P$
would be the total number of blocks produced in an interval, $p_{i}$ would be
the blocks proposed to the network that are built by the $i^{th}$ block
builder.

The US DoJ generally classifies markets within three discrete categories [41]:

Unconcentrated | $\text{HHI}<1,500$
---|---
Moderately Concentrated | $1,500<=\text{HHI}<=2,500$
Highly Concentrated | $\text{HHI}>2,500$

Because the Herfindahl-Hirschman Index is commonly used for identifying and
measuring the presence of monopolies in industry, it is more suited to the
data points in our model that measure the infrastructure that is provided as a
service or public good by a relatively small number of actors, as opposed to
measuring the distribution of ownership or control of some asset. This makes
it particularly useful for measuring middleware such as block builders or
relays, or ERC-4337 bundlers.

###  IV-C Shannon Index

The Shannon Index [42] is part of a family of measurements that are derived
from information theory and which are based on the concept of entropy. Other
measurements in this category include the Generalized Entropy measurement, and
the Theil indices. As the Shannon Index is based on a very different approach
from either the Gini index or HHI, it is a useful measurement to cross-
reference results against.

The Shannon index is a measure of the amount of entropy in a dataset. It was
intended to be used to measure the amount of information content in a signal,
where information content can be considered as a measurement of the
unexpectedness of a particular value occurring at a specific point in the
signal, and entropy is the average level of unexpectedness within a signal.

Shannon gives examples of strings of characters, where the more characters and
randomness there is, the lower the probability of predicting the next
character in the string, and the more unexpected it is when that value occurs
as predicted. The less often the next character can be predicted correctly,
the higher the entropy.

A basic example of this concept is a coin toss, where one party chooses heads,
and where there is a 50% chance of the expected value (i.e. heads) occurring
as predicted. As this has a relatively high probability, $p(heads)=0.5$, coin
tosses have low entropy. If we roll a die, the entropy increases as the
probability of an expected value decreases, e.g. $p(6)=0.16$.

Entropy as applied within the field of economics was systematized by Theil
[43], and has found applications in measuring inequality within a distribution
of resources, and later within several studies of decentralization in
cryptocurrencies [35, 2, 32].

The Shannon index is commonly expressed using the following formula, where
$N_{i}$ is the frequency of a each value in the dataset divided by $N$, the
total number of distinct values in the dataset, i.e. $N_{i}$ is the number of
entities within the population that have a $i$ amount of resources, in
proportion to $N$ total number entities in the population.

$H^{\prime}=-\sum_{i=1}^{N}\left(\frac{N_{i}}{N}\right)\log_{e}\left(\frac{N_{i}}{N}\right)$

The Shannon index was designed for use with categorical data, as opposed to
continuous data, for which the GE index is better suited [44]. However, for
the purposes of measuring distribution of native asset (i.e. ETH), we have
applied the Shannon index to ranges of amounts of ETH. In this context,
$p_{i}$ is the proportion of the asset owned by the $i^{th}$ percentile of the
population.

The Shannon index has a range between 0 and the logarithm of the number of
categories in the dataset, i.e. $0<=H^{\prime}<=log(n)$. The more centralized
a system becomes, the closer the Shannon index will be to zero [32].

###  IV-D Atkinson Index

The Gini Index is useful as a base for calculating inequality, but it has
limitations in describing the qualities of inequality. As the Gini Index is
based on the ratio of total areas under the curve, it does not account for
variance or skewness, and it also means that two different distributions can
potentially have the same Gini index, which could potentially affect the
tracking of changes over time.

For this reason, our model employs the Atkinson index [45] as a further
measure of decentralization, to allow us to cross-reference our Gini index
against a measurement that can be fine-tuned to our requirements, and which
can be used to capture any nuance in the distribution of different
measurements.

The Atkinson index is based on the social-welfare approach, which infers the
amount of resources that would need to be redistributed to achieve a certain
level of equality. The Atkinson index is calculated using the following
formula:

$\displaystyle
A(\varepsilon)=1-\left(\frac{1}{N}\sum_{i=1}^{N}\left(\frac{y_{i}}{\mu}\right)^{1-\varepsilon}\right)^{\frac{1}{1-\varepsilon}},\quad\quad\varepsilon\neq
1$ $\displaystyle
A\left(\varepsilon\right)=1-\frac{\prod_{i=1}^{N}\left(y_{i}^{\frac{1}{N}}\right)}{\mu},\quad\quad\varepsilon=1$

where the parameters include:

$\epsilon$ | inequality aversion parameter where $\epsilon>0$
---|---
$n_{i}$ | number of people in the $i^{th}$ income group
N | total number of people
$y_{i}$ | average income of the $i^{th}$ income group
$\mu$ | average income of the total population

The Atkinson index is very closely linked to the generalized entropy index,
and other related entropy based indices, such as the Theil index (i.e. GE(1)).
The Atkinson index results in a value between 0 and 1, and takes as a
parameter $\epsilon$, which allows us to fine-tune the formula by increasing
the value of $\epsilon$ in order to make the index more sensitive to changes
at lower end of the distribution. The value of $\epsilon$ is referred to as
the inequality aversion, and the index yields a higher value when $\epsilon$
is given a value closer to 1.

###  IV-E Tail Ratios

According to Atkinson [45], the Gini index is affected by changes closer to
the median of the distribution more than it is affected by changes at tail
ends of the distribution. In order to account for this characteristic of the
Gini index, we employ a series of tail ratios to highlight any changes in the
shape of the distribution that affect the lower and upper percentiles, and as
a means to inform decisions about the inequality aversion parameter in our
Atkinson Index.

Our model incorporates the Palma Ratio[46], which is the ratio of the share of
resources allocated to the top 10% of the distribution to the lower 40% of the
distribution. Palma concludes that changes in the level of inequality tend to
happen more at either end of the distribution, with the middle being affected
less.

We complement the Palma ratio with other interdecile ratios that can be used
to qualify the character of the inequality in the distribution, including the
P90:P10, P50:P10 ratios. Without loss of generality, Px represents the level
of resources allocated to greater than x% of the population, and is used to
compare the gap between allocation of resources to two discrete percentiles
within the distribution.

These inter-decile ratios are calculated using the linear interpolation method
[47] described below, where $P_{x}$ denotes the desired percentile, and $N$ is
the population size, and $v$ represents the value at position $i$ in an
ascending ordered dataset, where $i$ is calculated using the following
formula:

$i=\frac{P_{x}\left(N+1\right)}{100}$

This gives us the rank position of the desired percentile, assuming the
dataset is sorted in ascending order. We then interpolate between the value at
this rank position in the ordered dataset, and the value at the subsequent
position, in order to attain each percentile in our desired inter-decile
ratio:

$\displaystyle P=\left(v_{\left\lfloor i+1\right\rfloor}-v_{\left\lfloor
i\right\rfloor}\right)(i\ \text{mod}\ 1)+v_{\left\lfloor i\right\rfloor}\ ,$
$\displaystyle\ \ i\ \text{mod}\ 1\neq 0$ $\displaystyle P=v_{i}\ ,$
$\displaystyle\ \ i\ \text{mod}\ 1=0$

By themselves these ratios are not useful measurements as the range within the
results are not bounded, as with the Gini index or HHI. They can be useful to
reference when deciding which end of a distribution warrants further scrutiny.

###  IV-F Jensen-Shannon Divergence

As we are interested in measuring the changes to levels of decentralization
over time, we employ a method described by Lin [48] as the Jensen-Shannon
Divergence, which is a method to measure of the level of similarity or
divergence between two probability distributions. The JSD is useful because it
has a defined upper bound, which means it can be scaled for comparison with
other index values. In order to scale the JSD, we normalize it by dividing the
result by the upper bound of the possible range of results, which in our
model, is $log_{2}$.

We calculate the JSD between two intervals by comparing the distribution of
each one across each measurement dimension respectively. As such, we define
$p$ and $q$ as two probability distributions, taken from two discrete
intervals of a single metric, represented as vectors of sizes $n$ and $m$
respectively. The normalized JSD is then calculated as follows:

$D_{JS}(p||q)=\frac{1}{\log_{2}}\left(\frac{1}{2}D_{KL}(p||m)+\frac{1}{2}D_{KL}(q||m)\right)$

where $m=\frac{1}{2}(p+q)$ is the average of $p$ and $q$, and $D_{KL}(p||m)$
and $D_{KL}(q||m)$ are the Kullback-Leibler divergences of $p$ and $q$ from
$m$ respectively, defined as:

$D_{KL}(p||q)=\sum_{i=1}^{n}p(i)\log\frac{p(i)}{q(i)}$

It is important to note that both vectors are the same size, therefore if
$n<m$, we pad $p$ with zeros until it matches the size of $q$, or if $m<n$, we
pad $q$ with zeros until it matches the size of $p$. As our data is
categorical in nature, we also align values each distribution such that $p$
and $q$ are ordered by category.

The normalized JSD is calculated across each measurement dimension for both
the 1 day, 30, 60 and 90 day intervals in order to track and quantify any
changes in decentralization in specific subsystems over time.

###  IV-G Deriving a Master Index

As we are measuring Ethereum’s level of decentralization over time, we need to
account for its modular topology, in which significant components of its
infrastructure will change. For example, we might measure the concentration in
the mev-boost relay market, but later decide to remove this metric as
innovations like enshrined PBS make them redundant [49].

Our model calculates an aggregate set of indices across a number of dimensions
of measurement, that can be used as further indicator of the relative level of
decentralization in Ethereum as it changes over time. Each relevant index,
(i.e. Gini, HHI, Atkinson), contributes to an aggregate index that is derived
from calculating the normalized weighted geometric mean of the respective
index’s value across all relevant metrics, i.e.:

$\gamma=\frac{{\left(\prod_{i=1}^{n}(\beta_{i}\times\omega_{i})\cdot
100\right)^{\frac{1}{n}}-\text{min}\left(\beta\right)}}{{(\text{max}\left(\beta\right)-\text{min}\left(\beta\right))\cdot
10^{-2}}}$

where $\beta=\\{\text{metric}_{1},\text{metric}_{2},...,\text{metric}_{n}\\}$
i.e. the set of relevant metrics, $\omega$ is the respective weighting for
each metric, and $n=\left|\beta\right|$, the number of metrics being measured.

This master index can be used to see changes on a daily basis over a period of
time and be used in conjunction with the normalized JSD, which measures
changes at specific points of 30, 60, and 90 days.

The relevant metrics that are included in each aggregate index, along with
their respective weightings, are listed below. The weightings are assigned
based on the qualitative properties of the infrastructural component being
measured. The actual weightings used in the formula are the percentage of each
weighting relative to the sum of all weightings.

Metric | Weight
---|---
Consensus nodes by client | 1
Consensus nodes by country | 1
Execution nodes by client | 1
Execution nodes by country | 1
Distribution of ETH by amount | 1
Amount staked by Pool | 1
Blocks proposed by builder | 0.7
Blocks proposed by relay | 0.7
Number of userops per bundler | 0.2
Number of wallets per deployer | 0.2
Layer 2 rollups by relative TVL | 0.5
Stablecoins by relative TVL | 0.3

This results in a series of aggregate indices for each interval within the
time-range our sample data is taken from. The aggregate indices relate to the
Gini, Atkinson, Normalized HHI and Normalized Shannon indices. This should
allow us to track the changes in decentralization over time, while allowing us
to examine an individual index at specific intervals in order to further
understand any changes that occur.

##  V Results

In this section we discuss the results of the application of our model to a
sample that was recorded at 24 hour intervals across a period of 90 days,
between the 23rd May 2023 to the 23rd of August 2023. We discuss how the
results demonstrate our hypothesis that decentralization is a dynamic quality
that changes over time, rather than a static quality that is maintained in
continuous equilibrium.

###  V-A Discussion of Results

As can be seen from the table below, different indices give divergent results
when applied to the same data. This highlights the need to leverage more than
one approach to measuring decentralization, and also highlights the fact that
different indices are more suitable for applying to certain dimensions of
measurements over others.

Figure 4: 90 day averages across all metrics

The values in the table in figure 4 are color coded for legibility, with
colors closer to red indicating higher levels of concentration, and colors
closer to green indicating higher levels of decentralization. We observe that
the index values for several metrics diverge significantly, particularly
between the Gini index and HHI. In most cases this is expected, since the HHI
is more focused on measuring concentration at the upper end of a distribution.
Consider, without loss of generality, a distribution with a large number
entities that have a low share of resources. In this distribution the Gini
index will increase significantly, while the HHI will remain largely
unaffected. If the same distribution contains a small number of entities that
control nearly all the resources but that each have an equal share, the HHI
will be relatively low, but will increase as the share between these
controlling entities becomes less equal.

We observe this phenomenon clearly in several metrics in the results table,
for example in the Amount Staked by Pool metric, where 2 entities control 60%
of the market[50] (Lido controls 31% while solo stakers control 29%), where
the distribution of these two entities is almost equal, and the distribution
between entities controlling the other 40% of the market is also relatively
equal, which results in a re-scaled HHI of 0.2, indicating only moderate
concentration, while the Gini index gives a value of 0.91, which more
accurately reflects the level of concentration in this area.

While the Gini index and the HHI are each better suited to specific dimensions
of measurement, there can often be nuances that they fail to capture by
themselves. An illustrative given by Buterin [51] is where there are two
hypothetical societies with profound levels of inequality, one in which half
the population equally shares all the resources while the other half has none,
and the other in which one person has half of all the resources, everyone else
equally shares the remaining half. In Buterin’s example, both distributions
would result in the same Gini index value, despite having radically different
characteristics.

These limitations form the rationale behind in the inclusion of the Shannon
index within our model. The Shannon index is sensitive to both size of the
distribution and the diversity of different values in the distribution, and is
therefore able to highlight differences between distributions that the Gini
index does not capture.

This is meaningful in considering the level of concentration in areas where
the distribution is smaller, for example with ”Stablecoins by Tvl”. In the
result for this metrics, the Gini index is very high, indicating high
concentration, but the Shannon Index shows a relatively low concentration, and
this is a reflection of the fact there are fewer stablecoins, and hence less
chance for a decentralized market to emerge.

As we can see there is broad agreement between the Atkinson and Shannon
indices, and this is not surprising considering they are both based on the
entropy model. However, the Atkinson index is not used by our model except for
certain specific dimensions of measurements, because of its non-standard
inequality aversion parameter, which makes it unsuitable for comparison with
other studies. It can be useful however, in fine tuning certain measurements
when needed.

The observations of the recorded results have been organized into the
following sections which broadly correspond to the categories of metrics as
described in section I.

###  V-B Results based on original Nakamoto Coefficient subsystem selection

####  V-B1 Network Nodes

When examining the data recorded across the dimensions of measurement that are
based on original Nakamoto Coefficient subsystem selection, we first applied
the Gini index to the metrics pertaining to network nodes. The results of this
analysis is displayed in figure 5, which plots the Gini index of these metrics
across our sample time period.

We observe a relatively high Gini index value for both Execution Nodes by
Country and Consensus Nodes by Country, which both maintain an average Gini
index of 0.85 to 0.79. When examining the data more closely, this was found to
be caused by an out-sized number of nodes that are situated in the USA, having
34% of Consensus nodes and 44% of Execution nodes. Both numbers are
significant in terms of having crossed a key consensus threshold of 33%, which
is enough to cause disruption to liveness under adverse conditions.

We observe that measuring Consensus Nodes by Client yields an average Gini
index of 0.57, which is a reasonably safe level of client diversity, while
Execution Nodes by Client shows a surprisingly dynamic Gini index, which
changes between 0.66 to 0.81. Reaching 0.8 is unhealthy for the network, as a
bug in a single client version can have consequences for the network, as
occurred in May 2023 when a bug in a consensus client caused a lack of
finality for a period of time [52].

Figure 5: Gini Indices for Network Nodes

We also refer to the HHI as measurement of client diversity, which may be a
more more accurate measurement than the Gini index, due to the fact that the
population size of the distribution. In both cases the HHI suggest a highly
concentrated distribution, and therefore a lack of healthy diversity, with a
slight over-concentration in consensus nodes (mainly driven by prism and
Lighthouse [53]), and a pronounce over-concentration in execution nodes
(driven by Geth).

####  V-B2 Amount Staked by Staking Pool

In examining the Amount Staked by Staking Pool, we observe that the level of
concentration stays relatively static with a Gini index of 90% to 91% during
the sample period. This suggests a very high level of centralization amount
staking pools, however upon scrutinizing the the underlying data, we observe
that 30% of the total ETH staked on the network is controlled by a single
entity, which is Lido. While this seems concerning at first, it is to be noted
that Lido itself is a DAO, and as such is a decentralized entity. Lido
commissions 29 separate independent node operators to manage their validators,
with a relatively even distribution of stake between each node operator, as
documented on the VaNoM portal[54].

Rather than accounting for Lido’s decentralized quality by breaking down the
composition of its node operators within the original dataset, we leverage the
Atkinson index, by modifying the inequality aversion parameter to account for
its level of decentralization. To do this we multiply the standard inequality
parameter of $0.5$ by a weighting derived from Lido’s market share, denoted by
$\omega$, i.e.:

$\alpha^{\prime}=\alpha\cdot\left(1-\omega\right)=0.5\cdot\left(1-0.3\right)=0.35$

By using an inequality aversion parameter of $\alpha^{\prime}=0.35$ we obtain
an Atkinson index of $0.6$ to $0.61$ which remains relatively static through
the sample period, and which represents a much lower level of centralization
than suggested by applying the Gini index. The value of the Atkinson index
indicates a moderate level of centralization, which is likely caused by a
number of entities with larger of market shares, such as Coinbase, which has a
share of 10%.

####  V-B3 Distribution of Native asset by Amount

We observe that there is an indicative level of centralization with regards to
the distribution of the native asset, ETH. The most widely used index for
measuring the distribution of wealth within an economic system is the Gini
index, which gives an average value of 0.76 for the distribution of ETH.

While this represents a relatively high level of centralization with regards
to control of ETH, we must make some consideration to the qualities of the
entities at the top end of the distribution, that control large amounts of the
asset, which could include the smart contracts of DeFi dapps, bridges etc..
One previous study by Glassnode [55] reported that 22.8% of all ETH is stored
in smart contracts. This suggests that actual level of centralization in terms
of ETH ownership is considerably less than is suggested through the
application of the Gini index. While a portion of the smart contracts that
hold ETH are wallets of private individuals, rather than dapps, the relative
proportions of each is a question for further study.

###  V-C Metrics pertaining to PBS

In analyzing the data with regards to the metrics pertaining to Proposer-
Builder-Separation, we examined both the metrics for Blocks proposed by
Builder and the Blocks proposed by Relay. For these metrics we applied the
Herfindahl-Hirschman index as a more suitable index, due to the fact that the
builder and relay space is naturally more concentrated, and comprised of a
number of known entities, making it similar in some ways to more traditional
economic sectors (as opposed to 500,000 pseudonymous validators for instance).
The results of our analysis is visualized in figure 6.

As can be observed from the visualization, the market concentration ranges
from a re-scaled HHI of 0.16 (or 1,600), to 0.24 (or 2,400) for relays,
whereas the builders display a HHI that ranges from 0.15 (1,500) to 0.26
(2,600).

To refer to the description of the HHI in section II, we recall that the US
DoJ regards any sector with a HHI of less than 1,500 as being unconcentrated,
whereas anything above 2,500 as being highly concentrated.

Using these broad categories as a reference, we observe that the block builder
and relay markets range from almost unconcentrated to highly concentrated
within the sample period. We also observe that the level of concentration
changes with significant relevance for the above categorization, even within a
90 day period.

Figure 6: HHI of Block Builders and Relays

When examining the Jensen-Shannon Divergence between the first and last 24
hour intervals in the dataset for both relevant metrics, we see a relatively
high value compared to all other metrics, which indicates that there are
significant changes in the number of blocks proposed by individual builders
and relays, as well as overall changes to level of concentration within the
market. This is a useful index to cross-reference as it is capable of
capturing nuance that other indices sometimes fail to capture, for example in
cases where two or more entities can frequently reverse their respective
market shares, with each claiming an outsized portion of the market for a
period of time. In this scenario, the overall level of inequality and
centralization remains static but the relative share of individual
participants can change radically. This is a quality that the JSD index can
help to capture.

Metric | JSD
---|---
Amount Staked by Pool | 0.0150451
Execution Nodes by Country | 0.0073133
Execution Nodes by ClientBase | 0.0118433
Consensus Nodes by Country | 0.0058399
Consensus Nodes by Client | 0.0077753
Blocks by Relays | 0.1324419
Blocks by Builder | 0.2201886
Stablecoins by Tvl | 0.0103413
Rollups by Tvl | 0.1058467
Native Assets by Address | 0.0005811

It is worth noting that over a longer time-frame, our model should be able to
account for changes to the ecosystem from newer innovations and architectural
changes. These include enshrined PBS or the introduction of distributed block
building, and there are a number of such proposals under development
currently.

###  V-D Miscellaneous Metrics

####  V-D1 Rollups by TVL

We observe a pronounce level of centralization when we examine Rollups by TVL,
with a 90 day average Gini index value of 0.87, and a re-scaled HHI of 0.45.
Both of these values make sense when we observe the underlying data, in which
a single rollup has 54.3% of TVL across all rollups, which is Arbitrum One
(down from 64.5% at the start of the sample period). This is closely followed
by Optimism that has a TVL of 25.9%. There are 12 rollups with a TVL share of
less than 1%, and 7 with a TVL of more than 1%, the latter group exhibiting
significant variance, from which we would expect a high HHI value.

Interestingly, the one other metric that also show a high JSD value for the
sample period is Rollups By TVL. This could be explained by the launch of the
Base network on August 9th which significantly altered the relevant market
share of prominent rollups.

We observe that the level of centralization in the rollup market at the time
the data was collected, is at an unhealthy level for the Ethereum ecosystem.
The TVL across all rollups is 8.1B USD, which is approximately 0.3% of the
26.562B USD TVL of Ethereum [56], and of which one single rollup accounts for
0.2% of Ethereum’s TVL.

It is worth noting again that these figures will more than likely change over
time as a number of zk-EVM rollups will enter the market, which will dilute
the marketplace and will likely affect market concentration significantly,
underpinning the need for ongoing measurement of decentralization over time.

####  V-D2 Stablecoins by TVL

We observe the highest average value for the Gini index in the “Stablecoins by
TVL” metric, which also has a markedly high HHI value. This suggests a very
high level of concentration within the stablecoin market, which is obvious
when we examine the underlying data, that shows that just two stablecoins
comprise over 80% of the market, i.e. USDC and USDT. As both stablecoins are
issued by corporations, this represents a significant area of centralization
within Ethereum. Stablecoins are used by many people to interact with a wide
array of dapps, and both USDC and USDT are vulnerable to censorship, and have
previously frozen token holders funds on request from authorities [57] [58].

According to DefiLlama [56], the combined market cap for stablecoins on
Ethereum is 9.2B USD, which represents a third of the TVL on Ethereum. This
could arguably be the weakest point of Ethereum’s overall decentralization.

####  V-D3 Effective inflation rate adjusted for burn

As discussed in section III, the Effective inflation rate adjusted for burn is
an important metric to measure, as a high inflation rate can undermine the
security of the network by diluting supply, and the devaluing the asset,
potentially leading to misaligned incentives for the validators.

Since the introduction of EIP-1559, which introduces the concept of the ”base
fee”, which is set deterministically via protocol consensus and which is
burned in every block, Ethereum’s effective issuance rate becomes negative
when the network usage / gas price crosses a certain threshold [59].

According to data from the Ultra Sound Money resource [60], the effective
level of inflation of ETH since the merge has been -0.94%, and ETH is
projected to continue to be a deflationary asset in the long term, which is
positive for the network’s overall security and level of decentralization.

####  V-D4 Percentage of total supply staked

The percentage of total supply staked is related to the effective inflation
rate for the same reason as previously outlined, in that a high issuance has a
dilutionary effect on the circulating supply, which incentives asset holders
to stake in order to counteract the dilution, resulting in a feedback cycle
that can undermine the network’s security. Ethereum’s economics are designed
to reduce issuance as more validators come online [26], which theoretically
reduces the incentive to stake once the percentage of staked assets reaches a
certain threshold. Currently the amount of ETH staked is 24,932,109 ETH
according to beaconcha.in [61], out of 120,218,472 ETH total supply according
to Etherscan [62], resulting in a ratio of 20%, which is well within the
bounds of what is a safe level of total supply staked.

###  V-E Metrics pertaining to Account Abstraction

The metrics that pertain to Account Abstraction, which are User Operations by
Bundler and Wallets by Deployer are both striking insofar as they have
especially low averages for the Gini index but very high values for the HHI
values. This indicates that there is an issue with the dataset that warrants
further scrutiny. Upon investigation we observe that the 90 day Gini indices
for User Operations by Bundler has a range of 0 to 1, with a median of 0.6 and
a standard deviation of 0.36. This highly unusual range and variance can be
attributed to the nascent nature of the account abstraction space, and
indicates that the infrastructure has yet to see significant maturity or
adoption.

Upon inspecting the underlying data, we observe that the maximum number of
user operations processed in a single day during the sample period is only
301. At this nascent stage, there is significant market concentration, with
one specific bundler having 76% market share out of 13 total active bundlers.

It is worth noting however that this data looks at Ethereum mainnet itself,
and does not consider any of the L2 rollups, on which there may be a higher
level of adoption of account abstraction infrastructure, and which is an area
for future study.

The Wallets by Deployer metric shows very similar characteristics to the User
Operations by Bundler and follows much the same pattern of distribution
throughout the sample period, with one specific deployer having an 87% market
share.

The various metrics outlined in this paper will be useful in tracking the
changes in decentralization over time as the account abstraction space matures
and continues to see more adoption. The influence of erc-4337 bundlers on the
effective level of decentralization of the network as a whole is somewhat
limited, so even a moderately high level of centralization is acceptable.

###  V-F Master Indices

For every index that we track, we derive a master index that is a aggregate
value derived from the value of each dimension of measurement for that index.
This means that we can derive a single value for every day in the sample
period, rather than a 12 separate values for each day, (i.e. one for each
dimension of measurement).

Figure 7: Master Indices of Gini Index and Herfindahl-Hirschman Index

We are specifically interested in the master index for both the Gini Index and
Herfindahl-Hirschman Index for each day in the sample period. We have sub-
sampled 18 data points from the 90 day period and have charted them in figure
7.

The master index establishes a measurement that can be referenced to itself as
it changes over time, in order to derive a crude high-level indicator of the
overall effective level of measurement of the network, and in theory we should
be maintain a single relative measurement even as the dimensions of measure
change over time.

As can be seen from the sample above, we can see that the overall effective
level of decentralization changes over time, and this is a reflection of the
changes within the various dimensions of measurement, specifically the Gini
index and HHI index in this case. Overall we can see a trend in a reduction in
the value of the Gini across all dimensions of measurement from 21% to 14%,
conversely the trend in the HHI for across dimensions of measurement increases
slightly from 7.5% to 10%. Note that the anomalous reduction in the level of
centralization around the July 8th to July 13th period are due to gaps in the
source data. Also, we have excluded the values for metrics related to account
abstraction in order to get a clearer representation of the overall trends in
the data.

It is important to note that the values of the master are not like other
indices, insofar as it is not the case that values closer to 0 indicate more
decentralization and values closer to 1 indicate more centralization. These
values are relevant in comparison the same value at other points in time.

##  VI Conclusion and Discussion

Our results clearly demonstrate that the overall Ethereum ecosystem displays
elements of concentration of control that are arguably less decentralized than
the community would aspire to. The results suggest the need for diligence in
efforts to develop and maintain a healthy level of decentralization. This can
seen in areas such as client diversity (in terms of network nodes), market
share of staking pools, market share of rollups, and stablecoins. While
stablecoins have a much lower TVL compared to Ethereum as a whole, this belies
the important function stablecoins serve in the ecosystem, and thus presents a
red flag for centralization concerns.

As the ecosystem evolves, there will be changes in the important components of
the ecosystem’s overall infrastructure beyond the core Ethereum protocol, and
there will also be changes to the extent of their impact on the overall effect
level of decentralization. As Barnabé Monnot states: ”There are things that
the protocol does not see, but cares about” [63]. As such, we should expect
our model to adapt to these changes over time.

Not only will the components that we measure change over time, but also our
model will evolve as we learn more about the ecosystem and how to measure it.
While we have demonstrated that attempting to measure decentralization using a
single index yields inconclusive and incomplete results, using a number of
different indices together can paint a more complete picture, giving us a more
holistic view of the system.

While we have attempted to use a series of metrics that can be used to cross
reference each other to paint a larger picture, there are still some areas
that can be explored further. One potential area of exploration is to study
the methodologies used in other areas that face a similar challenge. One
example of research that can be explored is the OECD’s methodologies to
measure market competition [64], in which they explore the use of numerous
models and indices under two broad categories of structural measures and
performance measures.

Another potential direction for research is to develop a way to measure
potential collusion between entities within a population, rather than assuming
that each entity in a population is totally independent, as our current model
assumes. Vitalik Buterin discussed this approach [65] and described it as
being done by assigning pairwise coordination coefficients to all distinct
pairs of entities within a population. This would mean that any two pairwise
entities within a population would have coefficient between 0 and 1, where 1
implies they are so tightly coordinated and aligned, they should be measured
as a single actor, and a 0 implies they are completely independent. The
properties that attribute the level of coordination and alignment, and
resultant coordination coefficient, could be as simple as being the same
country, or being on the same network etc.

Obviously an approach such as this would require much more data about the
individual entities in the population, and would present a challenge to
gathering the data and ensuring data quality, which is an important
consideration when deciding to develop a model based on that data.
Nevertheless, this represents an interesting direction for future research.

One important factor in the measurement of decentralization, is that while our
models become more complex, the ability for them to be widely understood
remains an objective that should not be lost. If the model itself becomes
complex enough to preclude analysis from a diverse audience, then it loses its
ability to act as a reference point for communication and discussion, which
remains its primary objective.

##  VII Acknowledgements

Special thanks to the people that contributed valuable feedback and
suggestions for future improvements to the model and areas for further study,
including Tina Zhen, Vitalik Buterin, Stephane Gosselin and Griffin Howlett.

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##  VIII Appendix: List of Data Sources

The following table lists the various sources of data for each data point that
is referenced in our model. The data from these sources were recorded
programmatically and compiled into a database using software that was
specifically designed for that purpose.

Based on original Nakamoto Coefficient subsystem selection:
---
Consensus nodes by client | https://migalabs.es/api/v1/client-distribution
Consensus nodes by country | https://migalabs.es/api/v1/geo-distribution
Execution nodes by client | https://www.ethernodes.org/
Execution nodes by country | https://www.ethernodes.org/countries
Distribution of native asset by amount | https://data.messari.io/api/v1/assets/ethereum/metrics
Amount staked by Pool / Staking Service Provider | https://api.dune.com/api/v1/query/2394100/results
Metrics pertaining to Proposer Builder Separation:
Blocks proposed by builder | https://mevboost.pics/data.html
Blocks proposed by relayer | https://mevboost.pics/data.html
Metrics pertaining to Account Abstraction:
Number of user operations per bundler | https://dune.com/queries/2193933/3599135
Number of wallets per deployer | https://dune.com/queries/2434102/3999582
Miscellaneous Metrics:
Rollups by relative TVL / size / volume | https://l2beat.com/scaling/tvl
Stablecoins | https://stablecoins.llama.fi/stablecoins