path: root/paper/experiments_master.tex

% =============================================================================
% GRAFT — Master experiment notes (all results, grouped by category)
% Standalone .tex; compile with `pdflatex notes/experiments_master.tex` at repo root
% so the \includegraphics paths to ../graft_*.pdf resolve.
% =============================================================================
\documentclass[10pt]{article}
\usepackage[margin=0.9in]{geometry}
\usepackage[table]{xcolor}
\usepackage{tabularx,booktabs,multirow,float,graphicx,hyperref,amsmath,amssymb}
\definecolor{bestg}{HTML}{D6F0DC}
\definecolor{negr}{HTML}{F8D7DA}
\newcommand{\best}[1]{\colorbox{bestg}{$#1$}}
\newcommand{\nega}[1]{\colorbox{negr}{$#1$}}
\graphicspath{{../}{./}}
\hypersetup{colorlinks=true,linkcolor=blue!50!black}

\title{GRAFT — Master Experiment Notes\\\large All experiments grouped by category}
\author{Internal notes (auto-aggregated)}
\date{Last updated: 2026-04-30}

\begin{document}
\maketitle
\tableofcontents

\section*{Reading guide}
Numbers come from \texttt{neurips\_v4\_main.tex} (Tables T1--T12), \texttt{drafts/hero\_table.tex}, \texttt{drafts/hero\_realworld\_L20.tex}, and the \texttt{results/} folder. Figure files are PDFs at the repo root. Categories are in topical order, not story order; each section is self-contained.

\textbf{Default experimental setup (unless noted):} GCN backbone, hidden=64, lr=0.01, 200 epochs, no LR scheduler, no residual / BatchNorm / Dropout, 5\,\%/class semi-supervised split (Planetoid-style), 20 seeds, paired $t$-test BH-corrected, mean$\pm$std on test accuracy. ``Paper setup'' refers to this default. Deviations are stated per table.

\textbf{Main datasets.} Cora, CiteSeer, PubMed (Planetoid), DBLP (CitationFull). Real-world large: CitationFull-CiteSeer (4.2K, deg 2.5, 6-cl), CitationFull-DBLP (17.7K, deg 5.4, 4-cl), CitationFull-PubMed (19.7K biomed, deg 4.5, 3-cl), Coauthor-Physics (34.5K, deg 14.4, 5-cl).

% =============================================================================
\section{Main accuracy (BP vs GRAFT, paper setup)}\label{sec:main}

\subsection{Per-backbone, per-depth (T2)}
Source: \texttt{tab:main}, paper line 217. 4 datasets $\times$ 4 backbones $\times$ \{$L=5,6$\} $\times$ 20 seeds, paired-$t$ BH-corrected. GRAFT improves over BP in \textbf{86 of 96} paired comparisons; all non-GIN settings significant at $q\!=\!0.05$.

\begin{table}[H]
\centering\small
\caption{BP vs GRAFT per (dataset, backbone, depth). GIN excepted because its $(1+\epsilon)I$ identity already provides a residual gradient path.}
\begin{tabularx}{\textwidth}{ll *{4}{>{\centering\arraybackslash}X}}
\toprule
Dataset & Backbone-$L$ & BP & GRAFT & $\Delta$ & $p$ \\
\midrule
\multirow{8}{*}{Cora}
& gcn $L\!=\!5$  & $74.3{\pm 2.5}$  & \best{78.8{\pm 1.0}}  & $+4.5$  & $<\!0.001$ \\
& gcn $L\!=\!6$  & $69.4{\pm 5.7}$  & \best{78.2{\pm 1.1}}  & $+8.7$  & $0.002$ \\
& sage $L\!=\!5$ & $74.4{\pm 2.8}$  & \best{77.9{\pm 0.9}}  & $+3.5$  & $<\!0.001$ \\
& sage $L\!=\!6$ & $69.5{\pm 4.9}$  & \best{78.4{\pm 0.9}}  & $+8.9$  & $<\!0.001$ \\
& appnp $L\!=\!5$ & $74.8{\pm 2.7}$ & \best{79.1{\pm 1.1}}  & $+4.3$  & $<\!0.001$ \\
& appnp $L\!=\!6$ & $66.4{\pm 5.0}$ & \best{77.8{\pm 2.9}}  & $+11.4$ & $<\!0.001$ \\
& gin $L\!=\!5$  & $78.5{\pm 1.3}$  & \best{80.1{\pm 1.0}}  & $+1.6$  & $<\!0.001$ \\
& gin $L\!=\!6$  & $77.8{\pm 1.5}$  & $77.8{\pm 1.5}$       & $+0.0$  & ns \\
\midrule
\multirow{8}{*}{CiteSeer}
& gcn $L\!=\!5$ & $60.6{\pm 3.1}$ & \best{63.7{\pm 1.8}} & $+3.1$ & $0.002$ \\
& gcn $L\!=\!6$ & $55.7{\pm 3.6}$ & \best{63.5{\pm 2.2}} & $+7.7$ & $<\!0.001$ \\
& sage $L\!=\!5$ & $61.2{\pm 3.2}$ & \best{63.9{\pm 1.8}} & $+2.8$ & $0.005$ \\
& sage $L\!=\!6$ & $55.8{\pm 4.8}$ & \best{62.0{\pm 2.1}} & $+6.2$ & $0.007$ \\
& appnp $L\!=\!5$ & $61.3{\pm 2.7}$ & \best{64.6{\pm 1.6}} & $+3.2$ & $<\!0.001$ \\
& appnp $L\!=\!6$ & $53.3{\pm 5.4}$ & \best{64.7{\pm 1.7}} & $+11.4$ & $<\!0.001$ \\
& gin $L\!=\!5$ & \best{66.7{\pm 1.3}} & $65.2{\pm 1.3}$ & $-1.5$ & $<\!0.001$ \\
& gin $L\!=\!6$ & \best{65.1{\pm 1.7}} & $63.1{\pm 2.3}$ & $-2.1$ & $0.004$ \\
\midrule
\multirow{8}{*}{PubMed}
& gcn $L\!=\!5$ & $75.8{\pm 2.1}$ & \best{76.9{\pm 0.7}} & $+1.2$ & $0.032$ \\
& gcn $L\!=\!6$ & $73.2{\pm 2.7}$ & \best{75.8{\pm 1.1}} & $+2.6$ & $<\!0.001$ \\
& sage $L\!=\!5$ & $75.8{\pm 1.8}$ & \best{76.6{\pm 0.4}} & $+0.8$ & ns \\
& sage $L\!=\!6$ & $74.5{\pm 1.8}$ & \best{76.5{\pm 1.0}} & $+2.0$ & $0.001$ \\
& appnp $L\!=\!5$ & $76.9{\pm 1.8}$ & \best{79.1{\pm 0.4}} & $+2.2$ & $<\!0.001$ \\
& appnp $L\!=\!6$ & $73.7{\pm 3.7}$ & \best{78.3{\pm 0.9}} & $+4.6$ & $<\!0.001$ \\
& gin $L\!=\!5$ & $76.6{\pm 0.7}$ & \best{77.7{\pm 0.6}} & $+1.1$ & $<\!0.001$ \\
& gin $L\!=\!6$ & $76.4{\pm 1.3}$ & \best{76.9{\pm 1.0}} & $+0.5$ & ns \\
\midrule
\multirow{8}{*}{DBLP}
& gcn $L\!=\!5$ & $82.1{\pm 0.4}$ & \best{83.1{\pm 0.3}} & $+0.9$ & $<\!0.001$ \\
& gcn $L\!=\!6$ & $81.3{\pm 0.5}$ & \best{82.9{\pm 0.3}} & $+1.5$ & $<\!0.001$ \\
& sage $L\!=\!5$ & $82.4{\pm 0.3}$ & $82.5{\pm 0.4}$ & $+0.2$ & ns \\
& sage $L\!=\!6$ & $81.7{\pm 0.5}$ & \best{82.5{\pm 0.3}} & $+0.8$ & $0.002$ \\
& appnp $L\!=\!5$ & $81.6{\pm 0.4}$ & \best{83.1{\pm 0.4}} & $+1.5$ & $<\!0.001$ \\
& appnp $L\!=\!6$ & $79.6{\pm 1.2}$ & \best{83.2{\pm 0.4}} & $+3.6$ & $<\!0.001$ \\
& gin $L\!=\!5$ & $81.8{\pm 0.4}$ & \best{82.3{\pm 0.4}} & $+0.5$ & $0.001$ \\
& gin $L\!=\!6$ & $81.6{\pm 0.6}$ & \best{82.2{\pm 0.5}} & $+0.6$ & $0.004$ \\
\bottomrule
\end{tabularx}
\end{table}

\subsection{BP vs GRAFT visual summary}
\begin{figure}[H]\centering
\includegraphics[width=0.85\textwidth]{graft_vs_bp_boxscatter.pdf}
\caption{Per-seed scatter+box of GRAFT vs BP across paper-setup configurations (4 datasets, GCN $L=5,6$).}
\end{figure}

% =============================================================================
\section{Backward-method baselines (vs DFA / DFA-GNN / VanillaGrAPE / PEPITA / FF / CaFo)}\label{sec:backwards}

\subsection{Leaderboard (T1, paper)}
Source: \texttt{tab:leaderboard}, paper line 184. GCN $L\!=\!6$, 20 seeds.
\begin{table}[H]\centering\small
\begin{tabularx}{\textwidth}{l *{3}{>{\centering\arraybackslash}X}}
\toprule
Method & Cora & CiteSeer & DBLP \\
\midrule
\multicolumn{4}{l}{\emph{BP $+$ forward-side anti-over-smoothing}}\\
BP (vanilla)            & $68.8{\pm 4.6}$ & $54.0{\pm 4.1}$ & $80.5{\pm 1.0}$ \\
BP $+$ ResGCN           & $77.5{\pm 1.6}$ & $63.0{\pm 2.2}$ & $82.3{\pm 0.4}$ \\
BP $+$ JKNet            & $78.2{\pm 1.0}$ & $64.4{\pm 1.2}$ & $79.9{\pm 0.8}$ \\
BP $+$ PairNorm         & $69.0{\pm 3.2}$ & $55.4{\pm 3.4}$ & $79.0{\pm 0.8}$ \\
BP $+$ DropEdge         & $74.8{\pm 1.8}$ & $64.0{\pm 1.6}$ & $81.6{\pm 0.5}$ \\
\midrule
\multicolumn{4}{l}{\emph{Feedback-alignment baselines (graph-agnostic backward)}}\\
DFA                     & $70.4{\pm 6.8}$ & $60.2{\pm 2.4}$ & --- \\
DFA-GNN                 & $68.1{\pm 5.9}$ & $60.0{\pm 2.2}$ & --- \\
VanillaGrAPE            & $77.5{\pm 1.7}$ & $62.3{\pm 1.5}$ & $82.0{\pm 0.6}$ \\
\midrule
\multicolumn{4}{l}{\emph{GRAFT and combinations}}\\
\textbf{GRAFT}          & $76.7{\pm 1.8}$ & $62.4{\pm 1.9}$ & $82.1{\pm 0.4}$ \\
\textbf{GRAFT $+$ ResGCN}  & $77.8{\pm 1.9}$ & $61.5{\pm 2.2}$ & \best{82.7{\pm 0.6}} \\
\textbf{GRAFT $+$ JKNet}   & \best{78.3{\pm 1.6}} & $61.8{\pm 2.2}$ & $82.4{\pm 0.4}$ \\
\textbf{GRAFT $+$ PairNorm}& $75.8{\pm 1.5}$ & \best{64.3{\pm 2.0}} & $80.7{\pm 0.6}$ \\
\textbf{GRAFT $+$ DropEdge}& $70.8{\pm 3.8}$ & $62.1{\pm 1.8}$ & $80.7{\pm 0.7}$ \\
\bottomrule
\end{tabularx}
\caption{T1: Backward-method leaderboard at $L=6$. (DFA/DFA-GNN DBLP cells filled in T1' below.)}
\end{table}

\subsection{Wide backward-only hero (drafts)}
Source: \texttt{drafts/hero\_table.tex} (4 datasets, 6 backward methods, 20 seeds, $L=6$). PEPITA and FF$+$VN are essentially random-class on these graphs.

\begin{table}[H]\centering\small
\begin{tabularx}{\textwidth}{l *{6}{>{\centering\arraybackslash}X}}
\toprule
Dataset & BP & DFA & DFA-GNN & PEPITA & FF$+$VN & GRAFT \\
\midrule
Cora     & $68.8{\pm 4.6}$ & $70.4{\pm 6.8}$ & $70.1{\pm 6.1}$ & $31.9{\pm 0.0}$ & $25.5{\pm 8.8}$ & \best{76.7{\pm 1.8}} \\
CiteSeer & $54.0{\pm 4.1}$ & $60.2{\pm 2.4}$ & $60.0{\pm 1.8}$ & $18.2{\pm 0.3}$ & $19.0{\pm 2.0}$ & \best{62.4{\pm 1.9}} \\
PubMed   & $73.2{\pm 3.0}$ & $72.4{\pm 2.0}$ & $70.8{\pm 2.0}$ & $41.6{\pm 2.6}$ & $39.7{\pm 5.0}$ & \best{74.4{\pm 1.6}} \\
DBLP     & $80.5{\pm 1.0}$ & $81.5{\pm 1.2}$ & $81.0{\pm 1.1}$ & $47.7{\pm 5.5}$ & $44.7{\pm 0.0}$ & \best{82.1{\pm 0.4}} \\
\bottomrule
\end{tabularx}
\caption{Wide hero (not in paper). DBLP DFA/DFA-GNN cells filled here.}
\end{table}

\subsection{Hidden / deferred baselines (not in hero)}
\begin{itemize}\setlength\itemsep{1pt}
\item \textbf{CaFo$+$CE} (Park et al.\ 2023): Cora 79.5, CiteSeer 66.3, PubMed 76.4, DBLP 81.8 (20 seeds, $L=6$). Beats GRAFT on 3/4 datasets (+2.0 to +3.9). Greedy layer-wise (no gradient chain), different paradigm $\Rightarrow$ hidden from hero per paper-direction. Data: \texttt{results/cafo\_baseline\_20seeds/}.
\item \textbf{ForwardGNN-SF}: deferred (separate conda env + multi-file integration); paper SF on Cora $L=3$ reports $\sim$84.5 (close to BP 86.0).
\end{itemize}

\subsection{Ablation: learned alignment $\times$ topology factor (T3)}
Source: \texttt{tab:ablation}, paper line 273. GCN $L=6$, 20 seeds. Learned alignment dominates accuracy; explicit topology factor is marginal in raw accuracy but causal under intervention (\S\ref{sec:wrong-topo}).
\begin{table}[H]\centering\small
\begin{tabularx}{\textwidth}{l *{4}{>{\centering\arraybackslash}X}}
\toprule
Method & Cora & CiteSeer & PubMed & DBLP \\
\midrule
DFA (random $R$, $P\!=\!I$) & $70.4{\pm 6.8}$ & $60.2{\pm 2.4}$ & $72.2{\pm 1.5}$ & --- \\
DFA-GNN (random $R$, topo pseudo-error) & $68.1{\pm 5.9}$ & $60.0{\pm 2.2}$ & $70.5{\pm 2.0}$ & --- \\
VanillaGrAPE (learned $R$, $P\!=\!I$) & \best{77.3{\pm 1.0}} & $61.9{\pm 1.2}$ & \best{74.4{\pm 1.3}} & $82.0{\pm 0.6}$ \\
\textbf{GRAFT} (learned $R$, $P_\ell(\hat A)$) & \best{77.3{\pm 1.4}} & \best{62.8{\pm 1.6}} & $74.1{\pm 1.6}$ & \best{82.1{\pm 0.6}} \\
\bottomrule
\end{tabularx}
\end{table}

% =============================================================================
\section{Stackability (GRAFT $\times$ forward-side methods)}\label{sec:stack}
Source: \texttt{tab:stackability}, paper line 374. GCN $L=6$, 20 seeds.
\begin{table}[H]\centering\small
\begin{tabularx}{\textwidth}{l *{3}{>{\centering\arraybackslash}X}}
\toprule
Method & Cora & CiteSeer & DBLP \\
\midrule
BP                  & $68.8{\pm 4.6}$ & $54.0{\pm 4.1}$ & $80.5{\pm 1.0}$ \\
BP $+$ ResGCN       & $77.5{\pm 1.6}$ & $63.0{\pm 2.2}$ & $82.3{\pm 0.4}$ \\
BP $+$ JKNet        & $78.2{\pm 1.0}$ & \best{64.4{\pm 1.2}} & $79.9{\pm 0.8}$ \\
BP $+$ PairNorm     & $69.0{\pm 3.2}$ & $55.4{\pm 3.4}$ & $79.0{\pm 0.8}$ \\
BP $+$ DropEdge     & $74.8{\pm 1.8}$ & $64.0{\pm 1.6}$ & $81.6{\pm 0.5}$ \\
\midrule
GRAFT (backward only) & $76.7{\pm 1.8}$ & $62.4{\pm 1.9}$ & $82.1{\pm 0.4}$ \\
\midrule
GRAFT $+$ ResGCN    & $77.8{\pm 1.9}$ & $61.5{\pm 2.2}$ & \best{82.7{\pm 0.6}} \\
GRAFT $+$ JKNet     & \best{78.3{\pm 1.6}} & $61.8{\pm 2.2}$ & $82.4{\pm 0.4}$ \\
GRAFT $+$ PairNorm  & $75.8{\pm 1.5}$ & \best{64.3{\pm 2.0}} & $80.7{\pm 0.6}$ \\
GRAFT $+$ DropEdge  & $70.8{\pm 3.8}$ & $62.1{\pm 1.8}$ & $80.7{\pm 0.7}$ \\
\bottomrule
\end{tabularx}
\end{table}

\textbf{Notes.} GRAFT $+$ DropEdge is the one combination that fails to stack: forward--backward topology mismatch (forward drops edges, backward $P_\ell(\hat A)$ uses full $\hat A$). Synchronized variant recovers part of the gap but not all of it.

% =============================================================================
\section{Depth survival}\label{sec:depth}

\subsection{Cora / DBLP depth stress (T8)}
Source: \texttt{tab:depth-stress}, paper line 657. GCN, 3 seeds.
\begin{table}[H]\centering\small
\begin{tabularx}{\textwidth}{cl *{4}{>{\centering\arraybackslash}X}}
\toprule
Dataset & $L$ & BP & ResGCN & GRAFT & GRAFT $+$ ResGCN \\
\midrule
\multirow{6}{*}{Cora}
& 6  & $71.4{\pm 1.1}$ & $78.0{\pm 2.0}$ & $76.4{\pm 2.1}$ & \best{78.1{\pm 0.7}} \\
& 8  & $39.7{\pm 5.3}$ & \best{78.2{\pm 2.3}} & $63.8{\pm 5.0}$ & $51.7{\pm 11.0}$ \\
& 10 & $35.1{\pm 4.4}$ & \best{76.9{\pm 2.2}} & $54.5{\pm 4.7}$ & $47.3{\pm 5.3}$ \\
& 12 & $32.8{\pm 1.9}$ & \best{76.6{\pm 1.2}} & $45.7{\pm 1.8}$ & $42.3{\pm 1.3}$ \\
& 16 & $29.3{\pm 2.2}$ & \best{73.5{\pm 2.5}} & $35.4{\pm 2.6}$ & $31.6{\pm 0.5}$ \\
& 20 & $24.3{\pm 6.7}$ & \best{49.2{\pm 20.9}} & $38.3{\pm 5.0}$ & $34.1{\pm 3.1}$ \\
\midrule
\multirow{6}{*}{DBLP}
& 6  & $79.9{\pm 0.9}$ & $82.3{\pm 0.3}$ & $82.6{\pm 0.5}$ & \best{83.0{\pm 0.5}} \\
& 8  & $78.8{\pm 1.0}$ & $81.9{\pm 0.6}$ & \best{82.2{\pm 0.4}} & $81.6{\pm 1.1}$ \\
& 10 & $71.1{\pm 11.9}$ & \best{80.4{\pm 0.7}} & $78.1{\pm 1.0}$ & $69.4{\pm 0.9}$ \\
& 12 & $66.8{\pm 6.4}$ & \best{80.0{\pm 1.3}} & $73.4{\pm 3.2}$ & $64.8{\pm 8.1}$ \\
& 16 & $45.4{\pm 0.7}$ & $63.7{\pm 13.2}$ & \best{69.9{\pm 0.1}} & $60.3{\pm 11.3}$ \\
& 20 & $46.1{\pm 1.4}$ & $61.3{\pm 7.4}$ & \best{61.8{\pm 11.0}} & $46.8{\pm 3.0}$ \\
\bottomrule
\end{tabularx}
\caption{Three observations: (i) GRAFT sweet spot $L\!=\!5$--$8$. (ii) Cora $L\!\geq\!10$: ResGCN dominates. (iii) DBLP $L\!=\!16$: GRAFT \emph{overtakes} ResGCN (69.9 vs 63.7).}
\end{table}

\subsection{4 large real-world datasets, depth sweep (BP / DFA / DFA-GNN / GRAFT)}
Source: \texttt{gen\_realworld\_depth\_fig.py}, 3 seeds per cell. CitationFull-CiteSeer, CitationFull-DBLP, CitationFull-PubMed-biomed, Coauthor-Physics. $L\in\{3,5,8,10,12,14,16,18,20\}$.

\begin{figure}[H]\centering
\includegraphics[width=\textwidth]{graft_realworld_depth.pdf}
\caption{Real-world depth survival. Shallow ($L=3$) all methods tied $\geq 0.83$; from $L\!\geq\!10$ BP/DFA/DFA-GNN collapse, GRAFT descends gracefully and stays $\geq$10\,p.p.\ above the second-best at $L\!=\!20$ on every dataset.}
\end{figure}

\subsection{Real-world hero at $L=20$ (20 seeds)}
Source: \texttt{drafts/hero\_realworld\_L20.tex} + \texttt{realworld\_hero\_L20\_20seed.log}.
\begin{table}[H]\centering\small
\begin{tabularx}{\textwidth}{l *{4}{>{\centering\arraybackslash}X} c}
\toprule
Dataset & BP & DFA & DFA-GNN & GRAFT & $p$ (vs BP) \\
\midrule
CFull-CiteSeer       & $25.3{\pm 1.3}$  & $21.2{\pm 4.2}$ & $19.6{\pm 0.4}$ & \best{37.1{\pm 8.1}} & $5\!\times\!10^{-6}$ \\
CFull-DBLP           & $54.6{\pm 2.8}$  & $44.7{\pm 0.0}$ & $44.7{\pm 0.0}$ & \best{57.3{\pm 12.0}} & $0.34$ \\
CFull-PubMed (biomed)& $41.9{\pm 1.3}$  & $40.0{\pm 0.3}$ & $39.9{\pm 0.0}$ & \best{49.9{\pm 9.6}}  & $0.002$ \\
Coauthor-Physics     & $58.5{\pm 15.5}$ & $50.6{\pm 0.2}$ & $50.5{\pm 0.0}$ & \best{65.4{\pm 5.1}}  & $0.07$ \\
\bottomrule
\end{tabularx}
\caption{20-seed paired-$t$. GRAFT unique top performer everywhere; significant on CiteSeer (\,$p\!=\!5\!\times\!10^{-6}$\,) and PubMed (\,$p\!=\!0.002$\,), marginal on DBLP/Physics due to bimodal split-seed behaviour at $L\!=\!20$. DFA / DFA-GNN $\sigma\approx 0$ on 3 datasets = deterministic majority-class collapse.}
\end{table}

\subsection{Combined Fig 4-style depth panel}
\begin{figure}[H]\centering
\includegraphics[width=0.92\textwidth]{graft_fig4_combined.pdf}
\caption{Depth sweep across the four Planetoid-style datasets (Fig 4(a)) plus complementary panels.}
\end{figure}

\subsection{Original 4-dataset depth sweep}
\begin{figure}[H]\centering
\includegraphics[width=0.92\textwidth]{graft_depth_sweep.pdf}
\caption{Cora/CiteSeer/PubMed/DBLP, BP vs DFA-GNN vs GRAFT, $L\in\{4,8,10,12,16,20\}$, 20 seeds.}
\end{figure}

% =============================================================================
\section{Robustness}\label{sec:robustness}

\subsection{Wrong-topology causal control (T5)}\label{sec:wrong-topo}
Source: \texttt{tab:wrong-topo}, paper line 338. GCN $L=6$, 20 seeds. Forward uses true graph; only backward $P_\ell(\hat A)$ varies.
\begin{table}[H]\centering\small
\begin{tabularx}{\textwidth}{l *{3}{>{\centering\arraybackslash}X} c}
\toprule
Backward graph & Cora & CiteSeer & DBLP & vs.\ GRAFT \\
\midrule
GRAFT (correct $\hat A$) & $77.2{\pm 1.3}$ & \best{62.7{\pm 1.6}} & $81.9{\pm 0.8}$ & --- \\
VanillaGrAPE ($P=I$) & \best{77.5{\pm 1.7}} & $62.3{\pm 1.5}$ & \best{82.0{\pm 0.6}} & ns \\
\midrule
Rewired ($\tilde A$)         & \nega{32.3{\pm 1.3}} & \nega{29.6{\pm 8.0}} & \nega{46.1{\pm 5.1}} & $-35$ to $-45^{***}$ \\
Permuted ($\Pi\hat A\Pi^\top$) & \nega{32.5{\pm 2.0}} & \nega{48.1{\pm 6.5}} & \nega{75.8{\pm 3.9}} & $-6$ to $-45^{***}$ \\
Erd\H{o}s--R\'enyi           & \nega{31.9{\pm 0.0}} & \nega{27.4{\pm 5.8}} & \nega{44.8{\pm 0.3}} & $-37$ to $-45^{***}$ \\
\bottomrule
\end{tabularx}
\caption{Removing topology ($P=I$) is benign; \emph{wrong} topology is catastrophic. Forward--backward consistency is what the topology factor enforces.}
\end{table}

\subsection{Perturbation sweep (DFA-GNN-style Fig 4b/c/d)}
Source: \texttt{results/perturb\_20seeds/results.json} + \texttt{results/perturb\_extend/}. 3 attacks $\times$ 3 datasets (Cora, CiteSeer, PubMed) $\times$ 3 methods (BP, DFA-GNN, GRAFT) $\times$ rates $\{0,0.1,0.2,0.3,0.5,0.7\}$ $\times$ 20 seeds. Attacks: edge rewire, feature mask, label flip.

\begin{figure}[H]\centering
\includegraphics[width=\textwidth]{graft_perturb_sweep.pdf}
\caption{Perturbation robustness (DFA-GNN Fig 4b/c/d format). Top row: edge rewire; middle: feature mask; bottom: label flip. GRAFT keeps a positive margin over BP at most rates; both methods degrade symmetrically at extreme rates.}
\end{figure}

\textbf{Selected paired-$t$ from \texttt{perturb\_20seeds}} (CiteSeer edge-rewire example): rate$=$0\,$\Rightarrow$ BP 53.8/GRAFT 62.6 ($p\!=\!2.6e\text{-}8$); rate$=$0.1\,$\Rightarrow$ 36.4/42.7 ($p\!=\!1e\text{-}4$); rate$=$0.2\,$\Rightarrow$ 25.2/27.9 (ns); rate$=$0.3\,$\Rightarrow$ 21.6/20.4 (ns). The crossover is symmetric across attack types.

\subsection{Hyperparameter sensitivity (T9)}
Source: \texttt{tab:sensitivity}, paper line 689. Cora GCN $L=6$, 3 seeds. Default in \textbf{bold}.
\begin{table}[H]\centering\small
\begin{tabularx}{\textwidth}{l *{6}{>{\centering\arraybackslash}X}}
\toprule
\multicolumn{6}{l}{\textbf{(a) Probe count} (alignment every 10 steps, $K=3$)} \\
Probes & 16 & 32 & \textbf{64} & 128 & 256 \\
Acc (\%) & $74.6{\pm 0.8}$ & $76.1{\pm 1.1}$ & $\mathbf{77.5}{\pm 1.6}$ & $77.5{\pm 1.2}$ & $77.1{\pm 3.5}$ \\
\midrule
\multicolumn{6}{l}{\textbf{(b) Alignment frequency} (64 probes, $K=3$)} \\
Every $N$ steps & 1 & 5 & \textbf{10} & 20 & 50 \\
Acc (\%) & $77.1{\pm 1.8}$ & $76.9{\pm 0.3}$ & $\mathbf{78.2}{\pm 0.9}$ & $71.9{\pm 4.9}$ & $73.0{\pm 3.2}$ \\
\midrule
\multicolumn{6}{l}{\textbf{(c) Hop cap $K$} (64 probes, alignment every 10 steps)} \\
$K$ & 1 & 2 & \textbf{3} & 5 & --- \\
Acc (\%) & $77.1{\pm 1.9}$ & $76.0{\pm 0.9}$ & $\mathbf{78.3}{\pm 0.6}$ & $78.2{\pm 0.7}$ & --- \\
\bottomrule
\end{tabularx}
\caption{Variation $\leq 3\%$ across tested ranges; defaults at or near optimum on each axis.}
\end{table}

% =============================================================================
\section{Alignment analysis (per-layer cosine, gradient reach)}\label{sec:align}

\subsection{Per-layer cosine vs true BP gradient (T11)}
Source: \texttt{tab:per-layer-cos}, paper line 741. Cora GCN $L=6$, 200 epochs, 20 seeds.
\begin{table}[H]\centering\small
\begin{tabularx}{\textwidth}{l *{5}{>{\centering\arraybackslash}X}}
\toprule
Layer (input $\to$ output) & $\ell\!=\!0$ & $\ell\!=\!1$ & $\ell\!=\!2$ & $\ell\!=\!3$ & $\ell\!=\!4$ \\
\midrule
$\cos(\delta^{\text{GRAFT}},\nabla^{\text{BP}})$
& $0.33{\pm 0.12}$ & $0.36{\pm 0.15}$ & $0.39{\pm 0.16}$ & $0.42{\pm 0.16}$ & $0.59{\pm 0.19}$ \\
\bottomrule
\end{tabularx}
\caption{All five layers strictly positive (95\% CI $>$ 0); higher near loss, smooth degradation with depth (multi-probe variance $\uparrow$ as more matrices chain).}
\end{table}

\subsection{Gradient-reach summary (paper §5.1, prose)}
At GCN $L=10$, BP gradient norms $\|\partial\mathcal{L}/\partial Z_\ell\|_F < 10^{-38}$ across all 20 seeds and all hidden layers (single-precision underflow). Forward representations remain $\Theta(1)$. GRAFT $\|\delta_\ell\|_F\!\approx\!0.7$--$1.2$ across all layers with tight CI. Accuracy gap at $L=10$: GCN $\Delta=+16.3\%$ ($p=4\!\times\!10^{-4}$), APPNP $\Delta=+10.8\%$ ($p=8\!\times\!10^{-3}$). At $L=6$: BP norms $\sim 0.02$, GRAFT $\sim 0.17$ ($\sim 8\times$).

% =============================================================================
\section{Efficiency}\label{sec:efficient}

\subsection{Wall-clock (T4)}
Source: \texttt{tab:efficiency}, paper line 292. ms / training step, 5 timing runs, median reported.
\begin{table}[H]\centering\small
\begin{tabularx}{\textwidth}{ll *{3}{>{\centering\arraybackslash}X} >{\centering\arraybackslash}X}
\toprule
Dataset & $L$ & BP & ResGCN & GRAFT-Opt & Speedup vs BP \\
\midrule
Cora & 6  & 4.16 & 4.80 & \best{2.62} & $1.59\times$ \\
Cora & 10 & 7.03 & 6.40 & \best{4.07} & $1.73\times$ \\
DBLP & 6  & 5.51 & 5.35 & \best{5.34} & $1.03\times$ \\
DBLP & 10 & \best{7.13} & 7.42 & 7.33 & $0.97\times$ \\
\bottomrule
\end{tabularx}
\caption{Cora speedup driven by avoiding autograd + replacing $L$-step sequential backward with $O(1)$ batched kernels. DBLP speedup vanishes (large SpMM saturates GPU). Memory $1.2$--$1.4\times$ peak.}
\end{table}

\subsection{Reference vs Optimized accuracy parity (T12)}
Source: \texttt{tab:ref-vs-opt}, paper line 764. 9 settings, 5 seeds.
\begin{table}[H]\centering\small
\begin{tabularx}{\textwidth}{l *{3}{>{\centering\arraybackslash}X}}
\toprule
Setting (GCN/SAGE/APPNP $L=6$) & Cora & CiteSeer & DBLP \\
\midrule
GCN  & $76.9{\pm 2.2}$ & $61.6{\pm 2.7}$ & $82.5{\pm 0.3}$ \\
SAGE & $75.6{\pm 1.1}$ & $61.5{\pm 2.1}$ & $82.2{\pm 0.4}$ \\
APPNP& $76.1{\pm 1.7}$ & $59.4{\pm 1.7}$ & $82.8{\pm 0.3}$ \\
\bottomrule
\end{tabularx}
\caption{All within $\pm 2\%$ of reference; no setting significantly different at $p<0.05$.}
\end{table}

% =============================================================================
\section{Negative results / regime boundary}\label{sec:negative}

\subsection{Heterophily (T10)}
Source: \texttt{tab:hetero}, paper line 715. 3 seeds, GCN $L=6$.
\begin{table}[H]\centering\small
\begin{tabularx}{\textwidth}{l *{3}{>{\centering\arraybackslash}X} *{2}{>{\centering\arraybackslash}X}}
\toprule
Dataset & $N$ & deg & $h$ & BP & GRAFT \\
\midrule
Texas    & 183  & 1.8  & 0.108 & $47.4$ & $47.4$ \\
Cornell  & 183  & 1.6  & 0.131 & $39.5$ & $37.7$ \\
Chameleon& 2{,}277 & 15.9 & 0.235 & $52.3{\pm 1.2}$ & \nega{26.7{\pm 5.0}} \\
Squirrel & 5{,}201 & 41.7 & 0.224 & $28.1{\pm 3.5}$ & \nega{21.2{\pm 0.3}} \\
Actor    & 7{,}600 & 3.9  & 0.219 & $26.8{\pm 1.1}$ & $26.4{\pm 0.8}$ \\
\bottomrule
\end{tabularx}
\caption{GRAFT relies on homophily; useless when $h<0.3$. Edge-flow backward propagates supervision \emph{across} class boundaries.}
\end{table}

\subsection{Large dense graphs (paper-side prose)}
\begin{itemize}\setlength\itemsep{1pt}
\item \textbf{ogbn-arxiv} (169K nodes, 40 classes): GRAFT trails BP by 25--35\,pp at all class-counts (6/9/40). Identity-augmented kernel $(1{-}\beta)\hat A^k+\beta I$ at $\beta=0.5$ improves the 6-class case 48.6$\to$53.7\,\% but BP still 73.6\,\%.
\item \textbf{Flickr} (89K, deg $\sim$10, 7-cl, social): both BP and GRAFT collapse to majority at $L\!\geq\!10$ in paper setup.
\item \textbf{WikiCS} (11.7K, deg 36.9, 10-cl): GRAFT loses every depth $L\in\{3,5,10,14,20\}$, $\Delta\!=\!-9$ to $-20$\,pp. Confirms regime boundary: dense (deg $>$ 20) $\Rightarrow$ BP-stable, GRAFT collapses to majority (0.229) at deep $L$.
\end{itemize}

\subsection{Graph-level regression (Peptides-struct, PPI)}
\begin{itemize}\setlength\itemsep{1pt}
\item \textbf{Peptides-struct} (LRGB MAE): GRAFT carries an intrinsic $+0.11$ MAE offset from pool-transpose on graph-level regression; reuse of \texttt{src/trainers.GraphGrAPETrainer} v4 reproduces the same offset $\Rightarrow$ not a port bug. Failure mode of the framing.
\item \textbf{PPI} (multi-label F1): GRAFT loses $-0.04$ to $-0.12$ F1 vs BP at all depths (avg deg 18, dense).
\end{itemize}

\subsection{Other rejected candidates (triaged)}
ENZYMES (TUDataset, graph-level), Cora-Full ($\geq 70$ classes, both methods collapse $L\!\geq\!5$), Roman-empire / Chameleon / Squirrel / Texas / Cornell / Actor (heterophily, App N.1), Reddit2 (dense social), QM9 / ogbg-molhiv / MalNet-Tiny (graph-level regression / classification), CitationFull-Cora\_ML (3K, both methods saturate at $L=3$, similar profile to other CFull's). All triaged with rationale in \texttt{drafts/experiment\_queue.md}.

% =============================================================================
\section{BH multiple-comparisons correction}\label{sec:bh}
144 paired tests grouped: 96 BP-vs-GRAFT (full LR sweep), 12 ablation contrasts (DFA $\to$ DFA-GNN $\to$ VanillaGrAPE $\to$ GRAFT $\times$ 3 datasets), 12 wrong-topology, 12 stackability, 12 depth-stress at $L=8,10$. After BH at $q=0.05$: \textbf{117/144} significant; every test that survived unadjusted $p<0.05$ also survives BH. Non-significant residuals concentrated in GIN backbone, PubMed-SAGE, and high-perturbation feature-masking conditions.

% =============================================================================
\section{Additional running notes}\label{sec:notes}

\subsection{Identified GRAFT-win regime}
Sparse (deg $\leq$ 8) $\cap$ few-class ($\leq$ 10) $\cap$ node-level single-label $\cap$ homophilous ($h>0.5$) $\cap$ Planetoid-style 5\,\%/class semi-sup $\cap$ $L\geq 5$ where BP already starts to fail. Within this, GRAFT's edge of advantage grows with depth.

\subsection{Hyperparams that consistently work}
hidden=64, lr=0.01 (Adam, weight\_decay=$5e\text{-}4$), 200 epochs, no scheduler, no residual / BN / Dropout, 64 probes, alignment every 10 steps, $K=3$ hop cap, diffusion $\alpha=0.5$ for 10 iters.

\subsection{Failure-prone hyperparam choices we hit}
hidden=128, AdamW $+$ cosine LR, 20-per-class semi-sup. These broke the GRAFT port until we reverted to paper setup. Documented in commit history; flagged in CLAUDE memory as recurrent failure mode.

\subsection{Artifacts inventory}
\begin{itemize}\setlength\itemsep{1pt}
\item \texttt{neurips\_v4\_main.tex}: live paper, T1--T12 + appendix.
\item \texttt{drafts/hero\_table.tex}: wide backward-only hero, not in paper.
\item \texttt{drafts/hero\_realworld\_L20.tex}: deep real-world hero, not in paper.
\item \texttt{drafts/deep\_real\_world\_section.md}: prose for the new real-world section.
\item \texttt{graft\_depth\_sweep.\{pdf,png\}}, \texttt{graft\_perturb\_sweep.\{pdf,png\}}, \texttt{graft\_fig4\_combined.\{pdf,png\}}, \texttt{graft\_realworld\_depth.\{pdf,png\}}, \texttt{graft\_vs\_bp\_boxscatter.\{pdf,png\}}.
\item \texttt{results/}: per-experiment JSON dumps (\texttt{perturb\_20seeds/}, \texttt{ablation\_20seeds/}, \texttt{cafo\_baseline\_20seeds/}, \texttt{bp\_graft\_depth\_20seeds/}, \dots).
\item Logs: \texttt{realworld\_hero\_L20\_20seed.log}, \texttt{wikics\_paper\_setup.log}, \texttt{realworld\_10seed.log}, \texttt{realworld\_dfa\_10seed.log}, \texttt{cfull\_paper\_setup.log}, \texttt{dblpfull\_full\_depth.log}, \texttt{pubmedfull\_full\_depth.log}, \texttt{physics\_full\_depth.log}, \texttt{csfull\_full\_depth.log}, \texttt{perturb\_sweep.log}, \texttt{perturb\_extras.log}.
\end{itemize}

\end{document}