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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.
45 lines
1.9 KiB
TeX
45 lines
1.9 KiB
TeX
\section{The Field Protocol}
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\label{sec:protocol}
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\subsection{Core Protocol}
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\begin{algorithm}[H]
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\caption{Field Protocol --- Multi-Valued Consensus}
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\label{alg:field-core}
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\begin{algorithmic}[1]
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\Require Validator $i$, $d$-valued conflict with initial preference $\mathbf{p}_i$
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\Require Parameters: $k$ (sample size), $\alpha$ (alignment threshold), $\beta$ (acceptance threshold), $\eta$ (update rate)
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\State $\text{count}[c] \gets 0$ for all $c \in \{1, \ldots, d\}$
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\While{$\max_c \text{count}[c] < \beta$}
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\State $S \gets \text{RandomSample}(V \setminus \{i\}, k)$
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\State Query each $j \in S$ for $\mathbf{p}_j$
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\State $\bar{\mathbf{p}} \gets \frac{1}{k} \sum_{j \in S} \mathbf{p}_j$ \Comment{sample mean preference}
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\State \Comment{Update preference vector toward sample mean}
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\State $\mathbf{p}_i \gets (1 - \eta) \mathbf{p}_i + \eta \bar{\mathbf{p}}$ \Comment{field alignment}
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\State $\mathbf{p}_i \gets \mathbf{p}_i / \|\mathbf{p}_i\|_1$ \Comment{renormalize to simplex}
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\State $c^* \gets \arg\max_c \bar{p}^{(c)}$ \Comment{dominant color in sample}
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\If{$\bar{p}^{(c^*)} \geq \alpha$} \Comment{supermajority for $c^*$}
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\State $\text{count}[c^*] \gets \text{count}[c^*] + 1$
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\EndIf
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\If{$\max_c \text{count}[c] \geq \beta$}
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\State \textbf{finalize} value $\arg\max_c \text{count}[c]$
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\EndIf
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\EndWhile
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\end{algorithmic}
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\end{algorithm}
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\subsection{Simplified Binary Mode}
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When $d = 2$, the preference vector reduces to a scalar $p_i = p_i^{(1)}$ (with
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$p_i^{(0)} = 1 - p_i$), and the protocol reduces to the Wave dynamics with continuous
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preferences. The update becomes:
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\[
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p_i \gets (1 - \eta) p_i + \eta \bar{p}
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\]
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which converges to $p_i \in \{0, 1\}$ under the same conditions as Wave.
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%% -----------------------------------------------------------------------
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%% 4. CONVERGENCE VIA FIELD POTENTIAL
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%% -----------------------------------------------------------------------
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