Author: Lucas Luhur Last updated: 2026-06-26

1. Background: what is $k$?

The consensus model draws each node’s raw stake $w_i$ from a Pareto distribution and normalises it to a relative stake $\alpha_i = w_i / \sum_j w_j$ (see Consensus Model §2). The Pareto has two parameters:

• shape $k$
• scale $x_m$

But the scale cancels under normalisation, so $k$ is the only real tuneable parameter.

Why the scale cancels: The scale $x_m$ enters the Pareto purely as a multiplicative factor: if $w_i \sim \mathrm{Pareto}(k, x_m)$ then
$w_i = x_m \cdot u_i$ where $u_i \sim \mathrm{Pareto}(k, 1)$ is the same draw with unit scale. Substituting into the normalisation, the common factor $x_m$ pulls out of both the numerator and every term of the denominator and divides away:

$$ \alpha_i = \frac{w_i}{\sum_j w_j} = \frac{x_m\,u_i}{\sum_j x_m\,u_j} = \frac{x_m\,u_i}{x_m \sum_j u_j} = \frac{u_i}{\sum_j u_j} $$

The relative stakes $\alpha_i$ depend only on the unit-scale draws $u_i$, i.e. on $k$ alone — so we fix $x_m = 1$ as a convention and tune only $k$.

What $k$ controls is stake inequality, i.e. how unevenly stake is split across nodes:

• small $k$ $\Rightarrow$ a heavy tail — a few large nodes hold most of the stake
• large $k$ $\Rightarrow$ the distribution concentrates onto its lower bound, so every node tends toward an equal share $\alpha_i \approx 1/N$.

So choosing $k$ is choosing how unequal the simulated validator set is — the single number that fixes the shape of the entire stake distribution.

Why do we need an “optimum” $k$?: The earlier baseline $k = 2$ was a placeholder. Because $k$ sets the whole shape of the stake distribution, an arbitrary value makes the whole simulation arbitrary. Calibrating $k$ to a real chain therefore adds realism to the simulator: the stake distribution it runs on is no longer a guess but a match to an actual deployed proof-of-stake network (Cardano).

2. The shape parameter $k$ and the Gini coefficient

We need a bridge from “a real chain’s inequality” to “a value of $k$”. That bridge is the Gini coefficient $G \in (0,1)$ — one interpretable scalar ($0$ = perfectly equal, $\to 1$ = one node owns everything). For the classical Pareto(I) distribution the Gini has a closed form (Arnold, Pareto Distributions, 2nd ed. 2015, p. 150):

$$ G = \frac{1}{2k - 1}, \qquad\Longleftrightarrow\qquad k = \frac{1 + G}{2G} $$

Inequality depends on $k$ through this.

The same page also gives the Lorenz curve $L(u)$: rank nodes poorest to richest, then $L(u)$ is the cumulative share of total stake held by the bottom fraction $u$. Perfect equality is the diagonal $L(u)=u$; the curve’s sag below it is the inequality (the Gini is twice the area between them). For the Pareto (Arnold, Pareto Distributions, 2nd ed. 2015, p. 150):