## Line Charts: Multi-Variable Analysis of Event Distributions and Elite Concentration ### Overview Three comparative line charts analyze different aspects of event distributions and elite concentration across agent-based systems. Panel (a) examines tail behavior, panel (b) shows extreme-value scaling, and panel (c) tracks elite concentration dynamics. All charts use logarithmic scales for quantitative precision. ### Components/Axes **Panel (a): Tail Behavior (TCE)** - X-axis: Event size (x) [log scale, 10¹ to 10³] - Y-axis: P(X ≥ x) [log scale, 10⁻⁴ to 10⁰] - Legend: - Pure PL (α=2.22) - dashed gray - Baseline - solid blue - DTI - solid green - Fixed-interval - dashed red - Excess tail mass - shaded blue - Spatial grounding: Legend positioned right, χ_c markers (48/57) annotated near curves **Panel (b): Extreme-value scaling** - X-axis: Number of agents (N) [log scale, 8 to 512] - Y-axis: Avg. max event size (X_max) [log scale, 10¹ to 10³] - Legend: - Baseline - blue circles - DTI - green circles - Spatial grounding: Legend right, crossover marker at N=32 **Panel (c): Elite concentration** - X-axis: Number of agents (N) [log scale, 8 to 512] - Y-axis: Top-10% share of TCE [linear scale, 0.1 to 0.35] - Legend: - Baseline - blue squares - DTI - green squares - Excess - shaded blue - Egalitarian (10%) - dashed gray - Spatial grounding: Legend right, excess annotations (+0.17/+0.21) near curves ### Detailed Analysis **Panel (a) Trends:** - Pure PL (dashed gray) shows steepest decay, with χ_c=57 - DTI (solid green) demonstrates faster tail decay (χ_c=48) - Baseline (solid blue) has α=2.22 vs DTI's α=2.21 - Fixed-interval (dashed red) intersects DTI at ~x=10² - Excess tail mass (shaded blue) spans x=10¹ to 10² **Panel (b) Trends:** - Both models show logarithmic growth in X_max with N - Baseline (blue) consistently outperforms DTI (green) - DTI crosses below Baseline at N=32 (24% excess) - γ values: Baseline=0.82 vs DTI=0.83 **Panel (c) Trends:** - Both models show linear increase in elite concentration - Baseline reaches 0.32 at N=512 (24% excess) - DTI reaches 0.27 at N=512 (17% excess) - Egalitarian line (10%) remains constant ### Key Observations 1. DTI demonstrates superior tail behavior (lower χ_c) but lags in extreme-value scaling after N=32 2. Baseline maintains higher elite concentration growth (+24% vs +17%) 3. Fixed-interval model shows critical intersection with DTI at mid-scale events 4. Excess tail mass dominates lower event sizes (<100) 5. Elite concentration grows logarithmically with agent count in both models ### Interpretation The data suggests: - DTI optimizes tail event management but becomes less effective at extreme scales - Baseline model exhibits stronger scaling properties but accumulates more elite concentration - Fixed-interval model's performance is context-dependent, excelling at mid-scale events - The 24% excess in Baseline's max event size at N=32 indicates critical threshold behavior - Elite concentration growth patterns suggest different resource distribution mechanisms between models The charts collectively demonstrate trade-offs between tail risk management and scaling efficiency, with DTI prioritizing tail behavior at the cost of extreme-scale performance, while Baseline maintains better scalability at the expense of tail risk mitigation.