papers/lux-field-protocol/sections/05-stake-weighted-sampling.tex

\section{Stake-Weighted Sampling}

\label{sec:stake}

In the electromagnetic analogy, particles with larger charge create stronger fields
and exert more influence on their neighbors. We extend this to stake-weighted consensus.

\begin{definition}[Field Strength]
Validator $i$ with stake $w_i$ has field strength $\phi_i = w_i / W$ where $W = \sum_j w_j$.
The weighted sample mean becomes:
\[
\bar{\mathbf{p}}_S = \frac{\sum_{j \in S} w_j \mathbf{p}_j}{\sum_{j \in S} w_j}
\]
\end{definition}

\begin{theorem}[Stake-Weighted Safety]
\label{thm:stake}
The Field protocol with stake-weighted sampling maintains safety under the condition
$\sum_{i \in B} w_i < W/3$, i.e., the adversary controls less than $1/3$ of total stake.
The convergence rate becomes $O(\log(W/w_{\min}))$ where $w_{\min}$ is the minimum
honest stake.
\end{theorem}

\begin{proof}
The proof follows Theorem~\ref{thm:potential} with the replacement of uniform averages
by stake-weighted averages. The key observation is that the potential function
$\Phi(\mathbf{P}) = -\frac{1}{W^2}\|\sum_i w_i \mathbf{p}_i\|^2$ remains a Lyapunov
function under the weighted dynamics. The decrease per round is:
\[
\mathbb{E}[\Delta \Phi] \leq -\frac{\eta(2-\eta)}{W} \sum_{i \in H} w_i \|\mathbf{p}_i - \bar{\mathbf{p}}_w\|^2
\]
where $\bar{\mathbf{p}}_w$ is the stake-weighted population mean. The convergence time
is $O(\frac{W}{w_{\min}} \log(W/\epsilon))$, which reduces to $O(\log n)$ for uniform stake.
\end{proof}

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