\section{Results}\label{sec:results}

\subsection{Family Results}

% Family Ontology
Figure~\ref{fig:results_family} summarizes the model performance on the \textit{Family} ontology under various noise types and levels, evaluated on both Membership (M) and Object Property Assertion (OPA) tasks. Across all models, OPA performance remains consistently low. This is expected since $98.4\%$ of the test triples correspond to object property assertion cases, many involving Pellet-generated inferences that require complex, non-local, multi-step reasoning—something non-symbolic reasoners models struggle to capture effectively.

For \textit{OWL2Vec*}, noise generally degrades performance. In the noise-free scenario, the model achieves membership and object property assertion MRRs of $0.562$ and $0.040$, respectively. As noise increases, performance declines steadily. Interestingly, a slight improvement in OPA occurs under GNN noise, likely because GNN-generated noisy triples preserve structural plausibility, reinforcing local relational patterns to some extent. The sharpest decline is observed at $100\%$ corruption, where membership MRR drops to $0.166$ (GNN noise) and OPA MRR to $0.028$ (random noise).

\textit{Box2EL} exhibits a similar downward trend. Starting from baseline MRRs of $0.299$ (M) and $0.018$ (OPA), its performance worsens with increasing noise, reaching a minimum membership MRR of $0.110$ under $100\%$ GNN noise and a near-zero OPA MRR of $0.001$ under $100\%$ logical noise.

In contrast, \textit{R-GCN} demonstrates greater robustness and even some performance gains under certain noise conditions. In the noise-free setting, \textit{R-GCN} scores MRRs of $0.323$ (M) and $0.040$ (OPA), with minimums of $0.295$ (M) at $75\%$ logical noise and $0.031$ (OPA) at $100\%$ logical noise. Because noise is introduced only in the test set, \textit{R-GCN}'s robustness stems from its reliance on neighborhood aggregation patterns learned during training. Its predictions depend on structural consistency in node embeddings, allowing it to effectively identify noisy triples that deviate from expected patterns.

\subsection{Pizza Results}

% Pizza Ontology (100 Triples)
Figure \ref{fig:results_pizza_100} shows the results for \textit{Pizza} ($100$ individuals). \textit{OWL2Vec*} starts with membership and object property assertion MRRs near $0.28$ but sharply declines to $0.040$ (M) and $0.004$ (OPA) under $100\%$ logical noise. \textit{Box2EL} follows a similar trend, dropping from $0.114$ (M) and $0.063$ (OPA) to nearly zero under 100\% logical noise. Once again, \textit{R-GCN} proves more resilient, maintaining relatively stable performance.

% Pizza Ontology (250 Triples)
Figure \ref{fig:results_pizza_250} shows the results for \textit{Pizza} ($250$ individuals) and reflects consistent patterns. \textit{OWL2Vec*} performance decreases with increasing logical noise, with membership MRR falling from $0.254$ to $0.040$, and OPA MRR dropping from $0.280$ to $0.003$ at $100\%$ logical noise. Again, the improvement in OPA observed under GNN noise during testing is likely because the GNN-generated noisy triples retain structural plausibility, which helps preserve local relational patterns to some extent. \textit{Box2EL} steadily declines from MRRs of $0.114$ (M) and $0.060$ (OPA) to near-zero under $100\%$ logical noise. \textit{R-GCN} remains robust across noise types, with performance largely unaffected—mirroring the resilience observed before.

\subsection{OWL2Bench Results}

% OWL2Bench Ontology
Figure \ref{fig:results_OWL2Bench} shows results on the \textit{OWL2Bench} ontology. In the noise-free setting, \textit{OWL2Vec*} achieves the highest object property assertion scores (MRR around $0.7$) and moderate membership MRR ($0.28$). While random and GNN noise degrade its performance gradually, logical noise has little impact on object property assertion accuracy and mainly affects membership predictions.

\textit{Box2EL} performs significantly lower overall and suffers substantial performance losses under all noise types, with logical noise particularly devastating object property assertion accuracy, which falls near zero at moderate noise levels.

\textit{R-GCN} offers a balanced and robust performance profile. Membership MRR remains relatively stable across noise types and levels, and object property assertion scores decline more gently than \textit{Box2EL}'s under logical noise. Notably, \textit{R-GCN} sometimes maintains or improves membership prediction under GNN noise, highlighting its effective use of graph structure to mitigate noise.

\subsection{Overall Analysis}

% Overall Analysis
Across all ontologies and noise types tested, \textit{R-GCN} consistently demonstrates superior robustness to noise compared to \textit{OWL2Vec*} and \textit{Box2EL}. While all models experience declines in performance as noise increases—particularly on complex object property assertion tasks—\textit{R-GCN}'s reliance on learned structural patterns and neighborhood aggregation enables it to better withstand noisy test data. \textit{OWL2Vec*} performs well in noise-free and logically noisy conditions for object property assertion but is more sensitive in membership tasks and under other noise types. Box2EL suffers the steepest performance drops, especially under logical noise, indicating limited noise tolerance.

\begin{figure}[!ht]
    \centering
    {\includegraphics[width=\textwidth]{figures/family_random.png}}\\[1em]
    {\includegraphics[width=\textwidth]{figures/family_gnn.png}}\\[1em]
    {\includegraphics[width=\textwidth]{figures/family_logical.png}}
    \caption{Results for \textit{Family} ontology. x\% noise indicates that the number of injected noisy assertions is x\% of the original assertions.}
    \label{fig:results_family}
\end{figure}

\begin{figure}[!ht]
    \centering
    {\includegraphics[width=\textwidth]{figures/pizza_100_random.png}}\\[1em]
    {\includegraphics[width=\textwidth]{figures/pizza_100_gnn.png}}\\[1em]
    {\includegraphics[width=\textwidth]{figures/pizza_100_logical.png}}
    \caption{Results for \textit{Pizza\_100} ontology. x\% noise indicates that the number of injected noisy assertions is x\% of the original assertions.}
    \label{fig:results_pizza_100}
\end{figure}

\begin{figure}[!ht]
    \centering
    {\includegraphics[width=\textwidth]{figures/pizza_250_random.png}}\\[1em]
    {\includegraphics[width=\textwidth]{figures/pizza_250_gnn.png}}\\[1em]
    {\includegraphics[width=\textwidth]{figures/pizza_250_logical.png}}
    \caption{Results for \textit{Pizza\_250} ontology. x\% noise indicates that the number of injected noisy assertions is x\% of the original assertions.}
    \label{fig:results_pizza_250}
\end{figure}

\begin{figure}[!ht]
    \centering
    {\includegraphics[width=\textwidth]{figures/OWL2DL-1_random.png}}\\[1em]
    {\includegraphics[width=\textwidth]{figures/OWL2DL-1_gnn.png}}\\[1em]
    {\includegraphics[width=\textwidth]{figures/OWL2DL-1_logical.png}}
    \caption{Results for \textit{OWL2Bench} ontology. x\% noise indicates that the number of injected noisy assertions is x\% of the original assertions.}
    \label{fig:results_OWL2Bench}
\end{figure}