## Heatmap: Recurrent Iteration vs. Margin ### Overview The heatmap displays the relationship between recurrent iteration (t) and margin (M) for different hop lengths (2-hop, 5-hop, 10-hop, 13-hop, 16-hop, 30-hop) and a dynamic margin setting. The x-axis represents the recurrent iteration, while the y-axis represents the margin. ### Components/Axes - **X-axis**: Recurrent Iteration (t) - **Y-axis**: Margin (M) - **Legend**: - 2-hop - 5-hop - 10-hop - 13-hop - 16-hop - 30-hop - Avg. first-hit t* - Margin = 0 ### Detailed Analysis or Content Details The heatmap shows that as the recurrent iteration increases, the margin generally decreases for all hop lengths. The 2-hop and 5-hop lines are the most prominent, showing a steep decline in margin as t increases. The 10-hop, 13-hop, 16-hop, and 30-hop lines show a more gradual decline. The dynamic margin line (dashed line) remains relatively flat, indicating that the margin does not significantly change with increasing recurrent iteration. ### Key Observations - The 2-hop and 5-hop lines show the most significant decrease in margin as the recurrent iteration increases. - The dynamic margin line remains relatively flat, indicating minimal change in margin with increasing recurrent iteration. - The 10-hop, 13-hop, 16-hop, and 30-hop lines show a more gradual decline in margin. ### Interpretation The heatmap suggests that as the recurrent iteration increases, the margin decreases for all hop lengths. This could indicate that the system becomes less sensitive to changes in margin as the number of iterations increases. The dynamic margin line remaining relatively flat suggests that the system's margin is not significantly affected by changes in the recurrent iteration. This could be due to the system's ability to adapt to changes in the environment or the system's inherent stability.

## Line Graphs: Margin vs. Recurrent Iteration for Different R Values ### Overview The image contains four line graphs comparing the "Margin" metric across recurrent iterations (t) for different system configurations labeled R=6, R=7, R=8, and R=dynamic. Each graph includes multiple colored lines representing different "hop" configurations (2-hop, 5-hop, 10-hop, etc.), with a legend at the bottom. Vertical dashed lines mark "Avg. first-hit t*" thresholds, and a horizontal line at Margin=0 serves as a reference. ### Components/Axes - **X-axis**: Recurrent Iteration (t), ranging from 1 to 25. - **Y-axis**: Margin, with scales varying per graph (e.g., -25 to 50 for R=6, -10 to 20 for R=7, etc.). - **Legend**: Located at the bottom, mapping colors to hop configurations: - Blue: 2-hop - Orange: 5-hop - Green: 10-hop - Red: 13-hop - Purple: 16-hop - Brown: 30-hop - Dashed black line: Margin = 0 - Vertical dashed lines: Avg. first-hit t* (exact values vary per graph). ### Detailed Analysis #### R=6 Graph - **Y-axis range**: -25 to 50. - **Trends**: - All lines start near 0 and diverge sharply downward. - 2-hop (blue) drops steeply, crossing Margin=0 at t≈5. - Higher hops (e.g., 30-hop, brown) decline more gradually. - Vertical dashed lines (Avg. first-hit t*) appear near t=3–5. #### R=7 Graph - **Y-axis range**: -10 to 20. - **Trends**: - Lines start near 0 and decline, with 2-hop (blue) crossing Margin=0 at t≈4. - 30-hop (brown) remains above Margin=0 until t≈10. - Avg. first-hit t* marked near t=2–4. #### R=8 Graph - **Y-axis range**: -5 to 10. - **Trends**: - 2-hop (blue) crosses Margin=0 at t≈3. - 30-hop (brown) stays positive until t≈8. - Avg. first-hit t* marked near t=1–3. #### R=dynamic Graph - **Y-axis range**: 0 to 10. - **Trends**: - Lines converge sharply at t=10, with 30-hop (brown) dropping to Margin=0. - Avg. first-hit t* marked at t=10. ### Key Observations 1. **Margin Decline**: All configurations show a general decline in Margin with increasing t, but the rate varies by R and hop count. 2. **First-Hit Thresholds**: Avg. first-hit t* occurs earlier for lower R values (e.g., t≈3 for R=8) and later for higher R (e.g., t≈10 for R=dynamic). 3. **Hop Count Impact**: Lower-hop configurations (e.g., 2-hop) reach Margin=0 faster than higher-hop ones (e.g., 30-hop). 4. **Dynamic R Behavior**: The R=dynamic graph shows a sudden convergence at t=10, suggesting a threshold-based adjustment. ### Interpretation The data suggests that system performance (as measured by Margin) deteriorates with recurrent iterations, but the speed of decline depends on R and hop count. Lower R values and fewer hops lead to faster Margin collapse, while higher R or hop counts delay this effect. The "Avg. first-hit t*" thresholds likely represent critical points where system stability or efficiency thresholds are breached. The R=dynamic configuration’s abrupt change at t=10 implies a mechanism to reset or adjust parameters when Margin approaches zero, preventing further decline. This could indicate adaptive behavior in dynamic systems to maintain performance under varying conditions.