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Restructured from flat .tex/.pdf files into per-paper subdirectories. Added Antje Worring as co-author on all papers. Synced PDFs to pdfs/ for download site.
29 lines
1.0 KiB
TeX
29 lines
1.0 KiB
TeX
\section{Decentralization Metrics}
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\label{sec:decentralization}
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\subsection{Nakamoto Coefficient}
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\begin{definition}[Nakamoto Coefficient]
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The Nakamoto coefficient $\mathsf{NC}$ is the minimum number of validators whose combined stake exceeds $W/3$:
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\begin{equation}
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\mathsf{NC} = \min \left\{ k : \sum_{i=1}^{k} s_{\pi(i)}^{\text{eff}} \geq W/3 \right\}
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\end{equation}
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where $\pi$ sorts validators by descending effective stake.
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\end{definition}
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\begin{proposition}[NC Lower Bound Under Stake Cap]
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With delegation multiplier $\omega$, $\mathsf{NC} \geq \lceil W / (3 \omega s^{\min}_{\text{val}}) \rceil$.
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\end{proposition}
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\subsection{Gini Coefficient of Stake Distribution}
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\begin{definition}[Stake Gini]
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\begin{equation}
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G = \frac{\sum_{i=1}^{n} \sum_{j=1}^{n} |s_i - s_j|}{2n \sum_{i=1}^{n} s_i}
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\end{equation}
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$G = 0$ indicates perfect equality; $G = 1$ indicates maximum concentration.
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\end{definition}
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Target: $G \leq 0.60$ for a healthy validator set. The delegation cap ensures $G$ cannot exceed $G_{\max} = 1 - 1/\omega \approx 0.93$.
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