## Box Plot: Normalized Cut for Different Graphs ### Overview The image is a box plot visualizing the normalized cut values for 10 different graphs. Each graph was tested with 10 trials, and each graph has 15 nodes. The y-axis represents the normalized cut, ranging from 0.5 to 1. The x-axis represents the graph number, from 1 to 10. The box plots show the distribution of normalized cut values for each graph. ### Components/Axes * **Y-axis:** "Normalized cut", ranging from 0.5 to 1.0, with tick marks at 0.5, 0.6, 0.7, 0.8, 0.9, and 1. * **X-axis:** "Graph #", ranging from 1 to 10, with tick marks at each integer value. * **Box Plots:** Each box plot (in blue) represents the distribution of normalized cut values for a specific graph. The red line within each box represents the median. The top and bottom edges of the box represent the upper and lower quartiles. The red lines extending from the box represent the range of the data. Red dots represent outliers. * **Text:** "N = 15 nodes" and "10 trials per graph" are located in the bottom-right corner of the chart. ### Detailed Analysis Here's a breakdown of the box plots for each graph: * **Graph 1:** The box plot is a horizontal red line at approximately 0.99. * **Graph 2:** The box plot is a horizontal red line at approximately 0.99, with one outlier at approximately 0.95. * **Graph 3:** The box plot ranges from approximately 0.94 to 1.0, with a median around 0.98. * **Graph 4:** The box plot ranges from approximately 0.95 to 1.0, with a median around 0.98. * **Graph 5:** The box plot is a horizontal red line at approximately 0.99, with one outlier at approximately 0.95. * **Graph 6:** The box plot is a horizontal red line at approximately 0.99. * **Graph 7:** The box plot ranges from approximately 0.91 to 1.0, with a median around 0.98. * **Graph 8:** The box plot ranges from approximately 0.96 to 1.0, with a median around 0.98, and one outlier at approximately 0.92. * **Graph 9:** The box plot ranges from approximately 0.96 to 1.0, with a median around 0.98. * **Graph 10:** The box plot ranges from approximately 0.97 to 1.0, with a median around 0.98. ### Key Observations * The normalized cut values are generally high, clustered around 1.0 for most graphs. * Graphs 3, 4, 7, 8, 9, and 10 show more variability in their normalized cut values compared to the others. * Graphs 2, 5, and 8 have outliers, indicating some trials resulted in significantly lower normalized cut values. ### Interpretation The data suggests that for most of the tested graphs, the normalized cut is consistently high, indicating a strong separation of nodes within the graph. The variability observed in some graphs suggests that the normalized cut is more sensitive to the specific structure of those graphs. The outliers indicate that in some trials, the graph partitioning was less effective, resulting in a lower normalized cut. The fact that each graph has 15 nodes and was tested with 10 trials provides context for understanding the distribution of normalized cut values.

## Box Plot: Normalized Cut vs. Graph Number ### Overview The image presents a series of box plots illustrating the distribution of "Normalized Cut" values across ten different graphs. Each graph was generated using 15 nodes and 10 trials. The box plots display the median, quartiles, and outliers for each graph's normalized cut values. ### Components/Axes * **X-axis:** "Graph #" ranging from 1 to 10. * **Y-axis:** "Normalized cut" ranging from 0.5 to 1.0. * **Box Plot Elements:** Each box plot represents the distribution of normalized cut values for a single graph. The box represents the interquartile range (IQR), the line inside the box represents the median, and the whiskers extend to the most extreme data points within 1.5 times the IQR. Points beyond the whiskers are considered outliers and are plotted individually. * **Annotation:** "N = 15 nodes\n10 trials per graph" located in the bottom-right corner. ### Detailed Analysis The box plots are arranged horizontally, one for each graph number from 1 to 10. * **Graph 1:** The median normalized cut is approximately 0.98. The IQR extends from roughly 0.96 to 1.0. There is one outlier at approximately 0.94. * **Graph 2:** The median normalized cut is approximately 0.98. The IQR extends from roughly 0.96 to 1.0. There is one outlier at approximately 0.94. * **Graph 3:** The median normalized cut is approximately 0.98. The IQR extends from roughly 0.96 to 1.0. There is one outlier at approximately 0.94. * **Graph 4:** The median normalized cut is approximately 0.98. The IQR extends from roughly 0.96 to 1.0. There is one outlier at approximately 0.94. * **Graph 5:** The median normalized cut is approximately 0.98. The IQR extends from roughly 0.96 to 1.0. There is one outlier at approximately 0.94. * **Graph 6:** The median normalized cut is approximately 0.98. The IQR extends from roughly 0.96 to 1.0. * **Graph 7:** The median normalized cut is approximately 0.94. The IQR extends from roughly 0.92 to 0.97. * **Graph 8:** The median normalized cut is approximately 0.98. The IQR extends from roughly 0.96 to 1.0. There is one outlier at approximately 0.94. * **Graph 9:** The median normalized cut is approximately 0.98. The IQR extends from roughly 0.96 to 1.0. * **Graph 10:** The median normalized cut is approximately 0.98. The IQR extends from roughly 0.96 to 1.0. ### Key Observations * Graphs 1, 2, 3, 4, 5, 8, 9, and 10 exhibit very similar distributions of normalized cut values, with medians close to 1.0 and minimal variation. * Graph 7 shows a noticeably lower median normalized cut value (approximately 0.94) compared to the other graphs. * Outliers are present in Graphs 1, 2, 3, 4, 5, and 8, all falling below 0.95. * The normalized cut values are consistently high across most graphs, suggesting a strong degree of connectivity or separation within the graphs. ### Interpretation The data suggests that the normalized cut values are generally high and consistent across most of the graphs generated. The normalized cut is a measure of the "strength of cut" between two sets of nodes in a graph. A higher normalized cut indicates a stronger separation between the sets. The consistency across graphs 1, 2, 3, 4, 5, 8, 9, and 10 suggests that the underlying graph structures are similar in terms of their connectivity patterns. Graph 7 stands out as an anomaly, exhibiting a lower median normalized cut. This could indicate that the graph structure for Graph 7 is different from the others, potentially having weaker connections between node sets or a different overall topology. The outliers observed in some graphs may represent unusual configurations or edge cases within those specific graph instances. The annotation "N = 15 nodes, 10 trials per graph" indicates that each graph consists of 15 nodes and that the normalized cut was calculated based on 10 independent trials for each graph. This suggests that the observed distributions represent the variability in normalized cut values due to random factors or variations in the graph generation process.

## Box Plot: Normalized Cut Across 10 Graphs ### Overview This is a box-and-whisker plot displaying the distribution of "Normalized cut" values for 10 distinct graphs (labeled Graph #1 to #10). Annotations indicate each graph has 15 nodes, with 10 trials conducted per graph to generate the distribution data. ### Components/Axes - **Y-axis**: Labeled *Normalized cut*, with a linear scale ranging from 0.5 to 1.0, marked at intervals of 0.1 (0.5, 0.6, 0.7, 0.8, 0.9, 1.0). - **X-axis**: Labeled *Graph #*, with categorical markers for 10 discrete graphs (1 through 10). - **Box Plot Elements**: - Blue outlined boxes: Represent the interquartile range (IQR, 25th to 75th percentile) of normalized cut values for each graph. - Red horizontal lines inside boxes: Indicate the median value of the distribution. - Black whiskers: Extend to the minimum/maximum values within 1.5×IQR of the box. - Red dots: Mark outlier values outside the whisker range. - **Annotations**: Bottom-right corner contains text: *N = 15 nodes* and *10 trials per graph*. ### Detailed Analysis For each graph, the distribution of normalized cut values is as follows: 1. **Graph #1**: All data points cluster at 1.0; box, median, and whiskers are all at the top of the y-axis (1.0), no outliers. 2. **Graph #2**: Median and box at 1.0; one outlier (red dot) at ~0.95. 3. **Graph #3**: Box spans ~0.95 to 1.0; median at 1.0, no whiskers or outliers. 4. **Graph #4**: Box spans ~0.97 to 1.0; median at 1.0, no whiskers or outliers. 5. **Graph #5**: Median and box at 1.0; one outlier at ~0.95. 6. **Graph #6**: All data points cluster at 1.0; box, median, and whiskers at 1.0, no outliers. 7. **Graph #7**: Box spans ~0.92 to 1.0; median at 1.0, whisker extends down to ~0.9 (the lowest minimum value across all graphs), no outliers. 8. **Graph #8**: Box spans ~0.97 to 1.0; median at ~0.98, one outlier at ~0.92. 9. **Graph #9**: Box spans ~0.97 to 1.0; median at ~0.98, no whiskers or outliers. 10. **Graph #10**: All data points cluster at 1.0; box, median, and whiskers at 1.0, no outliers. ### Key Observations - 7 out of 10 graphs (1,2,3,4,5,6,10) have a median normalized cut of 1.0, the maximum value on the scale. - Only Graph #7 has a whisker extending below its box, indicating a wider spread of lower values. - Outliers are present in 4 graphs (2,5,8), all representing values below the main data cluster. - Graphs #8 and #9 are the only ones with medians slightly below 1.0 (~0.98). ### Interpretation Normalized cut is a metric for graph partitioning quality, where higher values (closer to 1.0) indicate better partitioning performance. The data suggests: - The partitioning algorithm performs extremely consistently for most 15-node graphs, achieving near-perfect normalized cut (1.0) across 10 trials. - Graphs #7, #8, and #9 show minor variability, with Graph #7 having the lowest minimum value, indicating some trials produced less optimal partitions. - Outliers represent rare trials where the algorithm underperformed relative to the majority of trials for that graph. - The consistent high performance across most graphs indicates the algorithm is reliable for small (15-node) graph partitioning tasks.

## Line Chart: Normalized Cut Values Across Graphs ### Overview The chart displays normalized cut values for 10 graphs (Graph #1 to Graph #10) with 15 nodes each, based on 10 trials per graph. The y-axis represents the normalized cut metric (0.5–1.0), while the x-axis enumerates graph instances. Red horizontal lines at y=1.0 serve as reference points, and blue boxes with red internal lines highlight specific graph ranges. Red dots indicate outlier or trial-specific values. ### Components/Axes - **X-axis (Graph #)**: Labeled "Graph #" with integer ticks from 1 to 10. - **Y-axis (Normalized cut)**: Labeled "Normalized cut" with increments of 0.1 from 0.5 to 1.0. - **Legend**: Located in the bottom-right corner, stating: - "N = 15 nodes" (graph size) - "10 trials per graph" (data granularity) - **Key Elements**: - **Red horizontal lines**: Fixed at y=1.0 across all graphs. - **Blue boxes**: Enclose data points for Graphs #3, #4, #7, #8, and #9. - **Red dots**: Scattered below the red lines (e.g., Graphs #2, #5, #8). - **Black crosshair**: Annotated at Graph #7, y≈0.9. ### Detailed Analysis 1. **Graph #1**: No visible data points; implied baseline at y=1.0 (red line). 2. **Graph #2**: Red dot at y≈0.95, below the red line. 3. **Graph #3**: Blue box spans y≈0.95–1.0, with a red line at y=1.0. 4. **Graph #4**: Blue box spans y≈0.95–1.0, with a red line at y=1.0. 5. **Graph #5**: Red dot at y≈0.95, below the red line. 6. **Graph #6**: No visible data points; implied baseline at y=1.0. 7. **Graph #7**: Blue box spans y≈0.9–1.0, with a red line at y=1.0. Black crosshair marks y≈0.9. 8. **Graph #8**: Blue box spans y≈0.95–1.0, with a red line at y=1.0. Red dot at y≈0.92. 9. **Graph #9**: Blue box spans y≈0.95–1.0, with a red line at y=1.0. 10. **Graph #10**: No visible data points; implied baseline at y=1.0. ### Key Observations - **Consistency**: Most graphs (1, 3, 4, 6, 9, 10) align with the red line at y=1.0, suggesting optimal normalized cut values. - **Variability**: Graphs #2, #5, and #8 show red dots below y=1.0, indicating lower cut values in specific trials. - **Confidence Intervals**: Blue boxes (Graphs #3, #4, #7, #8, #9) likely represent trial-based variability, with narrower ranges for #3, #4, and #9. - **Outliers**: The black crosshair at Graph #7 (y≈0.9) highlights a significant deviation from the baseline. ### Interpretation The data suggests that normalized cut values are generally high (close to 1.0), implying effective graph partitioning for most instances. However, the presence of red dots and variable blue box widths indicates trial-to-trial inconsistency. The red line at y=1.0 acts as a performance benchmark, while the blue boxes may reflect confidence intervals or interquartile ranges from the 10 trials. The outlier at Graph #7 (y≈0.9) warrants further investigation, as it deviates notably from the trend. The uniformity of Graphs #3, #4, and #9 (narrow blue boxes) suggests stable performance across trials, whereas Graphs #7 and #8 show greater variability. This could imply differences in graph structure or trial conditions affecting cut quality.