## Diagram: Coalition Structure and Primary User Positioning ### Overview The left diagram illustrates a spatial distribution of coalitions and a primary user (PU) in a 2D coordinate system. Four coalitions are defined by dashed blue circles, each containing coalition heads (red circles) and members (green triangles). The PU is marked with a blue square. ### Components/Axes - **X-axis**: Position in x (km), ranging from -1.5 to 1.5. - **Y-axis**: Position in y (km), ranging from -1.5 to 1.5. - **Legend**: Located on the right side of the diagram. - **Red circles**: Coalition heads (e.g., SU₁, SU₂, SU₃, SU₄, SU₅, SU₆, SU₇, SU₈, SU₉, SU₁₀). - **Green triangles**: Coalition members (e.g., SU₇, SU₈, SU₉). - **Blue square**: Primary User (PU). - **Star symbols**: Coalition heads with special notation (e.g., SU₃, SU₅, SU₁₀). ### Detailed Analysis - **Coalition 1**: Centered at (-1, 0) with coalition head SU₁ and members SU₆, SU₅. - **Coalition 2**: Centered at (0, 1) with coalition head SU₂ and members SU₃, SU₄. - **Coalition 3**: Centered at (1, 0) with coalition head SU₉ and members SU₇, SU₈. - **Coalition 4**: Centered at (0, -1) with coalition head SU₆ and members SU₁, SU₂. - **PU**: Positioned at (0, 0.5). ### Key Observations - Coalition heads are marked with red circles, while members are green triangles. - The PU is centrally located but offset slightly upward. - Coalition 3 has the most members (2), while others have 1–2 members. ### Interpretation The diagram suggests a distributed network where coalitions are spatially separated, with the PU acting as a central reference point. The use of distinct symbols (red circles for heads, green triangles for members) implies hierarchical roles within coalitions. The PU’s position near Coalition 2 may indicate prioritized access or resource allocation. --- ## Line Graph: Average Coalition Size vs. Non-Cooperative False Alarm Probability ### Overview The right graph plots the relationship between non-cooperative false alarm probability (P_f) and average coalition size, with three data series: upper bound, average maximum, and actual average. ### Components/Axes - **X-axis**: Non-cooperative false alarm probability (P_f), logarithmic scale from 10⁻¹⁸ to 10⁻¹². - **Y-axis**: Average coalition size (number of SUs), ranging from 1 to 9. - **Legend**: Located at the top right. - **Green triangles**: Upper bound on number of SUs per coalition (M_max). - **Blue squares**: Average maximum number of SUs per coalition. - **Red circles**: Actual average number of SUs per coalition. ### Detailed Analysis - **Upper Bound (Green)**: - At P_f = 10⁻¹⁸: ~8.5 SUs. - At P_f = 10⁻¹²: ~1.5 SUs. - **Average Maximum (Blue)**: - At P_f = 10⁻¹⁸: ~4 SUs. - At P_f = 10⁻¹²: ~2 SUs. - **Actual Average (Red)**: - At P_f = 10⁻¹⁸: ~3 SUs. - At P_f = 10⁻¹²: ~1 SU. ### Key Observations - All three lines show a decreasing trend as P_f increases. - The upper bound (green) decreases most steeply, while the actual average (red) plateaus at lower P_f values. - The actual average (red) closely follows the average maximum (blue) until P_f ≈ 10⁻¹⁴, after which it drops sharply. ### Interpretation The graph demonstrates that as non-cooperative false alarms increase (lower P_f), coalition sizes shrink due to stricter resource constraints. The upper bound represents theoretical limits, while the actual average reflects real-world performance. The divergence between the actual average and upper bound at lower P_f suggests inefficiencies or practical limitations in coalition formation under high false alarm conditions. --- **Note**: No non-English text is present in the image. All labels and annotations are in English.