## Line Graph: Utility Value vs Batch Number ### Overview The image depicts a line graph comparing utility values across 40 batches for four scenarios: "without distance adjustment" and three configurations with distance adjustment (m/n = 0.1, 0.5, 1.0). Utility values range from 0 to 0.8, with distinct trends observed for each configuration. ### Components/Axes - **X-axis (Batch number)**: Integers from 0 to 40, labeled "Batch number." - **Y-axis (Utility value)**: Scale from 0 to 0.8, labeled "Utility value." - **Legend**: Located at the bottom-left corner, with four entries: - Red stars: "without distance adjustment" - Blue circles: "with distance adjustment, m/n=0.1" - Magenta diamonds: "with distance adjustment, m/n=0.5" - Black squares: "with distance adjustment, m/n=1" - **Line styles**: Each configuration uses a unique color and marker type. ### Detailed Analysis 1. **Without distance adjustment (red stars)**: - Utility value remains stable (~0.55–0.6) until batch 25. - Sharp decline to ~0.05 by batch 30, then plateaus near 0. 2. **With distance adjustment, m/n=0.1 (blue circles)**: - Utility value starts at ~0.55, peaks at ~0.62 by batch 10. - Gradual decline to ~0.4 by batch 25, then steep drop to ~0.05 by batch 30. 3. **With distance adjustment, m/n=0.5 (magenta diamonds)**: - Utility value starts at ~0.58, peaks at ~0.68 by batch 10. - Steeper decline than m/n=0.1, reaching ~0.3 by batch 25 and ~0.05 by batch 30. 4. **With distance adjustment, m/n=1 (black squares)**: - Utility value starts at ~0.57, peaks at ~0.72 by batch 10. - Most pronounced decline, dropping to ~0.2 by batch 25 and ~0.05 by batch 30. ### Key Observations - **Peak utility**: All configurations with distance adjustment (m/n=0.1, 0.5, 1.0) exhibit higher initial utility values (~0.55–0.72) compared to the "without adjustment" baseline (~0.55–0.6). - **Decline pattern**: - The "without adjustment" line shows a delayed but abrupt collapse after batch 25. - Configurations with higher m/n values (e.g., m/n=1) experience steeper declines post-batch 25. - Lower m/n values (e.g., m/n=0.1) exhibit more gradual declines but still fall below the baseline by batch 30. - **Stability**: The "without adjustment" line remains stable until batch 25, suggesting distance adjustment introduces sensitivity to batch progression. ### Interpretation The data suggests that distance adjustment parameters (m/n) significantly impact utility value dynamics: - **Higher m/n values** (e.g., 1.0) correlate with higher peak utility but accelerated degradation over batches. - **Lower m/n values** (e.g., 0.1) balance peak utility and gradual decline, potentially indicating a trade-off between initial performance and long-term stability. - The "without adjustment" baseline demonstrates resilience until batch 25, after which utility collapses, implying that distance adjustment may introduce vulnerabilities in later stages. This pattern could reflect system optimization trade-offs, where distance adjustment improves early performance but destabilizes the system over time. Further investigation into the underlying mechanisms (e.g., parameter definitions, batch-specific stressors) is warranted to validate these hypotheses.