## Histogram: Betweenness Centrality Distribution Across Iterations ### Overview The image presents four histograms displaying the distribution of betweenness centrality of nodes at different iterations (2, 100, 510, and 1024). The x-axis represents betweenness centrality, and the y-axis represents the number of nodes on a logarithmic scale. The histograms show how the distribution changes as the iterations progress. ### Components/Axes * **Titles:** * Top-Left: "Betweenness Centrality Distribution (Iteration 2)" * Top-Right: "Betweenness Centrality Distribution (Iteration 100)" * Bottom-Left: "Betweenness Centrality Distribution (Iteration 510)" * Bottom-Right: "Betweenness Centrality Distribution (Iteration 1024)" * **X-axis:** "Betweenness Centrality" * Top-Left: Scale from 0.0 to 0.7, with ticks at 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 * Top-Right: Scale from 0.00 to 0.40, with ticks at 0.00, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40 * Bottom-Left: Scale from 0.00 to 0.35, with ticks at 0.00, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35 * Bottom-Right: Scale from 0.00 to 0.25, with ticks at 0.00, 0.05, 0.10, 0.15, 0.20, 0.25 * **Y-axis:** "Number of Nodes (Log Scale)" * All plots: Logarithmic scale from 10⁰ (1) to 10³ (1000), with ticks at 10⁰, 10¹, 10², 10³. The top-right plot only goes to 10². * **Bars:** The histograms are represented by blue bars. ### Detailed Analysis * **Iteration 2 (Top-Left):** * A very high peak at approximately 0.0 betweenness centrality, reaching nearly 10¹ (10) nodes. * A smaller peak at approximately 0.08 betweenness centrality, reaching approximately 3 nodes. * Sparse nodes with betweenness centrality values around 0.2, 0.3, 0.4, 0.6, and 0.7, each with approximately 1 node. * **Iteration 100 (Top-Right):** * A high peak at approximately 0.00 betweenness centrality, reaching nearly 10³ (1000) nodes. * The number of nodes decreases rapidly as betweenness centrality increases. * A small number of nodes with betweenness centrality values around 0.10, 0.17, 0.25, and 0.40, each with approximately 1 node. * **Iteration 510 (Bottom-Left):** * A very high peak at approximately 0.00 betweenness centrality, exceeding 10³ (1000) nodes. * The number of nodes decreases rapidly as betweenness centrality increases. * A small number of nodes with betweenness centrality values around 0.06, 0.15, and 0.34, each with approximately 1 node. * **Iteration 1024 (Bottom-Right):** * A very high peak at approximately 0.00 betweenness centrality, exceeding 10³ (1000) nodes. * The number of nodes decreases rapidly as betweenness centrality increases. * A small number of nodes with betweenness centrality values around 0.06 and 0.22, each with approximately 1 node. ### Key Observations * The distribution of betweenness centrality is highly skewed towards 0.0 in all iterations. * As the iteration number increases, the peak at 0.0 becomes more pronounced. * The number of nodes with higher betweenness centrality values decreases significantly as the iteration number increases. * The x-axis scale decreases as the iteration number increases. ### Interpretation The histograms suggest that as the iterations progress, the network becomes increasingly centralized, with a large number of nodes having very low betweenness centrality. This indicates that a small number of nodes are becoming increasingly important in mediating connections between other nodes in the network. The decreasing x-axis scale and the increasing peak at 0.0 further support this interpretation. The network is evolving such that most nodes have very few connections to other nodes, while a few nodes act as hubs, connecting many other nodes.

## Grid of Histograms: Evolution of Betweenness Centrality Distribution ### Overview The image displays a 2x2 grid of histograms illustrating the "Betweenness Centrality Distribution" of a network across four distinct time steps or stages, labeled as "Iterations." The iterations shown are 2, 100, 510, and 1024. The charts demonstrate how the network's topology evolves, specifically showing a massive increase in the total number of nodes and a shift toward a highly skewed, power-law-like distribution where the vast majority of nodes have near-zero betweenness centrality. All text in the image is in English. ### Components/Axes The image is divided into four quadrants (Top-Left, Top-Right, Bottom-Left, Bottom-Right). All four charts share the following structural components: * **Chart Type:** Histogram (vertical bar charts). * **Visual Style:** Light blue bars with dark borders. Horizontal dashed gridlines correspond to major Y-axis ticks. * **X-axis Label:** "Betweenness Centrality" (Linear scale). Note: The maximum range of the X-axis changes across the iterations. * **Y-axis Label:** "Number of Nodes (Log Scale)" (Logarithmic scale: $10^0$, $10^1$, $10^2$, $10^3$). Note: The maximum range of the Y-axis changes across the iterations as the network grows. --- ### Detailed Analysis #### 1. Top-Left Chart: Iteration 2 * **Position:** Top-Left quadrant. * **Title:** Betweenness Centrality Distribution (Iteration 2) * **X-axis Range:** 0.0 to 0.7 (Markers at 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7). * **Y-axis Range:** $10^0$ (1) to $10^1$ (10). * **Visual Trend:** The distribution is sparse, indicating a very small network. There is a small cluster at 0.0, and a few individual nodes scattered across higher centrality values up to 0.7. * **Data Points (Approximate):** * ~0.00: ~9 nodes (just below the $10^1$ line) * ~0.02: 1 node ($10^0$ line) * ~0.06: ~2 nodes * ~0.23: 1 node * ~0.25: 1 node * ~0.32: 1 node * ~0.35: 1 node * ~0.70: 1 node #### 2. Top-Right Chart: Iteration 100 * **Position:** Top-Right quadrant. * **Title:** Betweenness Centrality Distribution (Iteration 100) * **X-axis Range:** 0.00 to 0.40 (Markers at 0.00, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40). *Note the scale reduction from Iteration 2.* * **Y-axis Range:** $10^0$ (1) to above $10^2$ (100). * **Visual Trend:** A heavy right-skewed distribution emerges. A massive spike occurs at 0.00, trailing off rapidly. The network has grown significantly. * **Data Points (Approximate):** * ~0.00: ~350 nodes (bar extends well above $10^2$) * ~0.01: ~40 nodes * ~0.02: ~15 nodes * ~0.03: ~15 nodes * ~0.04: ~9 nodes * ~0.05: ~6 nodes * ~0.06: ~4 nodes * ~0.07: ~2 nodes * ~0.09: ~3 nodes * ~0.14: 1 node * ~0.16: 1 node * ~0.25: 1 node * ~0.39: ~2 nodes * ~0.40: 1 node #### 3. Bottom-Left Chart: Iteration 510 * **Position:** Bottom-Left quadrant. * **Title:** Betweenness Centrality Distribution (Iteration 510) * **X-axis Range:** 0.00 to 0.35 (Markers at 0.00, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35). *Scale reduction continues.* * **Y-axis Range:** $10^0$ (1) to above $10^3$ (1000). * **Visual Trend:** Extreme right skew. The peak at zero centrality has grown exponentially compared to Iteration 100. * **Data Points (Approximate):** * ~0.00: ~3000 nodes (bar extends significantly above $10^3$) * ~0.01: ~60 nodes * ~0.02: ~20 nodes * ~0.03: ~10 nodes * ~0.04: ~5 nodes * ~0.05: ~6 nodes * ~0.06: 1 node * ~0.07: ~2 nodes * ~0.08: 1 node * ~0.11: 1 node * ~0.16: 1 node * ~0.35: 1 node #### 4. Bottom-Right Chart: Iteration 1024 * **Position:** Bottom-Right quadrant. * **Title:** Betweenness Centrality Distribution (Iteration 1024) * **X-axis Range:** 0.00 to 0.25 (Markers at 0.00, 0.05, 0.10, 0.15, 0.20, 0.25). *Maximum centrality is now much lower.* * **Y-axis Range:** $10^0$ (1) to above $10^3$ (1000). * **Visual Trend:** The distribution solidifies its extreme right-skewed shape. The vast majority of the network's nodes have near-zero betweenness. * **Data Points (Approximate):** * ~0.00: ~4500 nodes (highest peak across all charts) * ~0.01: ~80 nodes * ~0.02: ~25 nodes * ~0.03: ~18 nodes * ~0.04: ~3 nodes * ~0.05: ~3 nodes * ~0.06: 1 node * ~0.07: ~4 nodes * ~0.08: ~2 nodes * ~0.09: ~2 nodes * ~0.10: 1 node * ~0.13: 1 node * ~0.24: 1 node --- ### Key Observations 1. **Explosive Node Growth:** The Y-axis maximum increases from $10^1$ (Iteration 2) to well over $10^3$ (Iteration 1024). The network is actively growing over time, adding thousands of nodes. 2. **Decreasing Maximum Centrality:** In Iteration 2, the highest betweenness centrality is ~0.7. By Iteration 1024, the highest value has shrunk to ~0.24. The X-axis scale compresses accordingly in each subsequent chart. 3. **Solidification of Skew:** The distribution shifts from a sparse, somewhat random scattering (Iteration 2) to a highly defined, long-tail distribution (Iterations 510 and 1024). ### Interpretation Applying Peircean abductive reasoning to these visual trends reveals the underlying mechanics of the network being modeled: * **Network Growth Model:** The data strongly suggests a generative network model (such as the Barabási–Albert model of preferential attachment). We are watching a network grow from a tiny seed (Iteration 2) into a massive complex system (Iteration 1024). * **The Meaning of the Zero-Peak:** Betweenness centrality measures how often a node acts as a bridge along the shortest path between two other nodes. The massive, exponentially growing spike at 0.00 indicates that as the network grows, almost all *new* nodes are added to the periphery (leaf nodes). They do not act as bridges for any other nodes. * **Dilution of Centrality:** The fact that the *maximum* betweenness centrality drops from 0.7 to 0.24 is highly significant. In a tiny network (Iteration 2), a single central node might be the *only* bridge between two halves of the network, giving it a very high relative centrality score (0.7). As the network grows to thousands of nodes (Iteration 1024), it becomes denser. More alternative paths are created. Even the most critical "hub" nodes now share the bridging load with other pathways, diluting their individual betweenness score relative to the whole network, capping out at ~0.24. * **Conclusion:** The charts successfully visualize the emergence of a scale-free or complex network topology, characterized by a few highly connected hubs (the long tail on the right) and a massive majority of peripheral, low-influence nodes (the massive spike on the left).

## Histograms: Betweenness Centrality Distribution ### Overview The image presents four histograms, each depicting the distribution of Betweenness Centrality values for a network at different iterations (2, 100, 510, and 1024). The y-axis of each histogram represents the "Number of Nodes" on a logarithmic scale, while the x-axis represents the "Betweenness Centrality" value. All histograms share the same axes scales. ### Components/Axes * **Title:** "Betweenness Centrality Distribution (Iteration X)" where X is the iteration number (2, 100, 510, 1024). Positioned at the top-center of each individual histogram. * **X-axis Label:** "Betweenness Centrality". * **Y-axis Label:** "Number of Nodes (Log Scale)". * **Scale (Y-axis):** Logarithmic, ranging from 10¹ to 10³. * **Scale (X-axis):** Linear, ranging from 0.0 to approximately 0.7 (varies slightly between histograms). * **Histograms:** Each histogram is composed of a series of blue bars representing the frequency of nodes within specific Betweenness Centrality ranges. ### Detailed Analysis or Content Details **Histogram 1: Iteration 2** * The histogram is heavily skewed to the left. * The highest frequency of nodes occurs at very low Betweenness Centrality values (close to 0.0). * There are a few nodes with Betweenness Centrality values up to approximately 0.7. * Approximate counts (reading from the log scale): * Betweenness Centrality 0.0 - 0.1: ~500 nodes * Betweenness Centrality 0.1 - 0.2: ~100 nodes * Betweenness Centrality 0.2 - 0.3: ~30 nodes * Betweenness Centrality 0.3 - 0.4: ~10 nodes * Betweenness Centrality 0.4 - 0.5: ~5 nodes * Betweenness Centrality 0.5 - 0.6: ~2 nodes * Betweenness Centrality 0.6 - 0.7: ~1 node **Histogram 2: Iteration 100** * The histogram shows a more pronounced distribution compared to Iteration 2. * The peak frequency is still at low Betweenness Centrality values, but the distribution extends further to the right. * Approximate counts: * Betweenness Centrality 0.0 - 0.05: ~800 nodes * Betweenness Centrality 0.05 - 0.1: ~400 nodes * Betweenness Centrality 0.1 - 0.15: ~200 nodes * Betweenness Centrality 0.15 - 0.2: ~100 nodes * Betweenness Centrality 0.2 - 0.25: ~50 nodes * Betweenness Centrality 0.25 - 0.3: ~20 nodes * Betweenness Centrality 0.3 - 0.35: ~10 nodes * Betweenness Centrality 0.35 - 0.4: ~5 nodes **Histogram 3: Iteration 510** * The distribution continues to shift towards higher Betweenness Centrality values. * The peak is narrower and lower than in Iteration 100. * Approximate counts: * Betweenness Centrality 0.0 - 0.05: ~600 nodes * Betweenness Centrality 0.05 - 0.1: ~300 nodes * Betweenness Centrality 0.1 - 0.15: ~150 nodes * Betweenness Centrality 0.15 - 0.2: ~75 nodes * Betweenness Centrality 0.2 - 0.25: ~30 nodes * Betweenness Centrality 0.25 - 0.3: ~15 nodes * Betweenness Centrality 0.3 - 0.35: ~5 nodes **Histogram 4: Iteration 1024** * The distribution is further shifted to the right, with a more concentrated peak at lower values. * The maximum Betweenness Centrality value observed is approximately 0.25. * Approximate counts: * Betweenness Centrality 0.0 - 0.05: ~700 nodes * Betweenness Centrality 0.05 - 0.1: ~400 nodes * Betweenness Centrality 0.1 - 0.15: ~200 nodes * Betweenness Centrality 0.15 - 0.2: ~100 nodes * Betweenness Centrality 0.2 - 0.25: ~50 nodes ### Key Observations * The distribution of Betweenness Centrality values changes significantly over the iterations. * Initially (Iteration 2), most nodes have very low Betweenness Centrality. * As the iterations progress, the distribution shifts towards higher values, indicating that more nodes become central to the network's flow. * The peak of the distribution consistently remains at low Betweenness Centrality, suggesting that a large proportion of nodes still have relatively low centrality even after many iterations. * The range of Betweenness Centrality values decreases with increasing iterations, suggesting the network is converging. ### Interpretation The data suggests that the network is evolving over the iterations, with nodes gradually becoming more central. The shift in the distribution of Betweenness Centrality indicates that the network's structure is changing, potentially leading to the formation of more critical pathways and influential nodes. The convergence of the distribution (decreasing range of values) suggests that the network is stabilizing, and the centrality of nodes is becoming more defined. The logarithmic scale on the y-axis emphasizes the large number of nodes with low Betweenness Centrality, highlighting the hierarchical nature of the network. The decreasing number of nodes with high Betweenness Centrality as iterations increase suggests that the network is becoming more efficient, with fewer nodes controlling the flow of information or resources. This could be due to a process of network optimization or the emergence of a dominant core.

## Histograms: Betweenness Centrality Distribution Across Iterations ### Overview The image displays a 2x2 grid of four histograms, each titled "Betweenness Centrality Distribution" followed by an iteration number in parentheses. The charts visualize how the distribution of betweenness centrality values for nodes in a network evolves over time (iterations). All charts use a logarithmic scale for the vertical axis (Number of Nodes) and a linear scale for the horizontal axis (Betweenness Centrality). ### Components/Axes * **Chart Type:** Four separate histograms (bar charts). * **Titles:** * Top-Left: "Betweenness Centrality Distribution (Iteration 2)" * Top-Right: "Betweenness Centrality Distribution (Iteration 100)" * Bottom-Left: "Betweenness Centrality Distribution (Iteration 510)" * Bottom-Right: "Betweenness Centrality Distribution (Iteration 1024)" * **X-Axis (All Charts):** Label: "Betweenness Centrality". The scale range varies per chart. * **Y-Axis (All Charts):** Label: "Number of Nodes (Log Scale)". The scale is logarithmic (base 10), with major grid lines at powers of 10 (10⁰, 10¹, 10², 10³). * **Visual Elements:** Blue bars with black outlines represent the frequency (node count) for bins of betweenness centrality values. Dashed horizontal grid lines correspond to the logarithmic y-axis ticks. ### Detailed Analysis **1. Iteration 2 (Top-Left)** * **X-Axis Range:** 0.0 to 0.7. * **Distribution:** Extremely sparse. The vast majority of nodes have a betweenness centrality near 0.0. * **Key Data Points (Approximate):** * A dominant bar at ~0.0 centrality reaches just below 10¹ (≈8-9 nodes). * A smaller bar at ~0.06 centrality has a height of ~2-3 nodes. * Isolated, single-node bars (height = 10⁰) appear at approximately 0.23, 0.31, 0.34, and 0.69. * **Trend:** The distribution is highly skewed with a few outlier nodes possessing very high centrality. **2. Iteration 100 (Top-Right)** * **X-Axis Range:** 0.00 to 0.40. * **Distribution:** More nodes are present, forming a clearer, right-skewed distribution. The maximum centrality value has decreased. * **Key Data Points (Approximate):** * The tallest bar at ~0.00 centrality is between 10² and 10³ (≈300-400 nodes). * A rapid decay follows: bars at ~0.01 (≈40 nodes), ~0.02 (≈15 nodes), ~0.03 (≈10 nodes), ~0.04 (≈5 nodes), ~0.05 (≈3 nodes). * A small cluster appears around 0.09 (≈3 nodes). * Isolated single-node bars are at ~0.14, ~0.15, ~0.25, and ~0.39. * **Trend:** The network has grown, and centrality is more distributed, though still heavily concentrated near zero. **3. Iteration 510 (Bottom-Left)** * **X-Axis Range:** 0.00 to 0.35. * **Distribution:** The number of nodes has increased by an order of magnitude. The distribution is smoother but still strongly right-skewed. * **Key Data Points (Approximate):** * The peak at ~0.00 centrality is now above 10³ (≈1500-2000 nodes). * The decay is more gradual: bars at ~0.01 (≈60 nodes), ~0.02 (≈20 nodes), ~0.03 (≈10 nodes), ~0.04 (≈5 nodes), ~0.05 (≈2 nodes). * A small bump appears at ~0.06 (≈2 nodes). * Isolated single-node bars are at ~0.11, ~0.16, and ~0.35. * **Trend:** Continued network growth. The "tail" of the distribution (nodes with moderate centrality) is becoming more populated. **4. Iteration 1024 (Bottom-Right)** * **X-Axis Range:** 0.00 to 0.25. * **Distribution:** Similar in shape to Iteration 510, but the maximum centrality value has further decreased. The total node count appears similar or slightly higher. * **Key Data Points (Approximate):** * The peak at ~0.00 centrality is again above 10³ (≈2000-3000 nodes). * The decay pattern is consistent: bars at ~0.01 (≈80 nodes), ~0.02 (≈25 nodes), ~0.03 (≈15 nodes), ~0.04 (≈3 nodes), ~0.05 (≈4 nodes), ~0.06 (≈2 nodes). * Isolated single-node bars are at ~0.11 and ~0.24. * **Trend:** The distribution appears to be stabilizing. The network is large, and high betweenness centrality (above 0.1) is becoming increasingly rare for individual nodes. ### Key Observations 1. **Network Growth:** The total number of nodes (inferred from the area under the histogram and the y-axis peak) increases dramatically from Iteration 2 to Iteration 1024. 2. **Concentration at Zero:** In all iterations, the mode of the distribution is at or very near a betweenness centrality of 0.0, indicating most nodes lie on very few or no shortest paths between other nodes. 3. **Decreasing Maximum Centrality:** The highest observed betweenness centrality value decreases steadily across iterations: ~0.69 (It. 2) → ~0.39 (It. 100) → ~0.35 (It. 510) → ~0.24 (It. 1024). 4. **Distribution Shape Evolution:** The distribution evolves from a few extreme outliers (It. 2) to a smooth, rapidly decaying right-skewed distribution (It. 100 onwards), characteristic of many real-world networks. 5. **Sparse Tail:** Even in later iterations, the "tail" of the distribution (centrality > 0.1) is populated by only a handful of nodes, often just one. ### Interpretation This sequence of histograms likely depicts the evolution of a growing network model (e.g., a preferential attachment or spatial network model) where betweenness centrality is being recalculated at each major iteration. * **What the data suggests:** The data demonstrates a fundamental property of many evolving networks: as the network grows, the *relative* importance of any single node as a bridge (measured by betweenness centrality) tends to diminish for the vast majority of nodes. While new nodes are added, the "load" of being on shortest paths becomes more distributed. The decreasing maximum centrality suggests the network is becoming less reliant on a few critical "hub" nodes for global connectivity, potentially indicating increased robustness. * **How elements relate:** The four charts are snapshots in time. The shift in x-axis range and y-axis magnitude directly visualizes the network's growth and the changing statistical properties of its node centrality. The consistent log-scale y-axis allows for comparison of distribution shape across orders of magnitude in node count. * **Notable anomalies/trends:** The most striking trend is the compression of the centrality range. The outlier node with centrality ~0.69 in Iteration 2 is a significant anomaly compared to later stages. Its disappearance suggests a major structural change in the network between Iteration 2 and 100, where the initial "bottleneck" structure was broken or diluted by subsequent growth. The stabilization between Iterations 510 and 1024 suggests the network may be approaching a steady-state statistical distribution for betweenness centrality.

## Histograms: Betweenness Centrality Distribution Across Iterations ### Overview The image contains four histograms comparing the distribution of betweenness centrality values across network nodes at different computational iterations (Iteration 2, 100, 510, and 1024). All histograms use a logarithmic scale for the y-axis (Number of Nodes) and a linear scale for the x-axis (Betweenness Centrality). The data shows how node centrality distributions evolve as the algorithm progresses through iterations. ### Components/Axes - **X-axis**: Betweenness Centrality (ranges from 0.0 to 0.7 in all charts) - **Y-axis**: Number of Nodes (log scale: 10⁰ to 10³) - **Legend**: Not explicitly present, but colors are consistent across charts (blue bars) - **Titles**: Each subplot is labeled with "Betweenness Centrality Distribution (Iteration X)" where X = 2, 100, 510, 1024 ### Detailed Analysis #### Iteration 2 - **Peaks**: - 0.0: ~10 nodes - 0.05: ~5 nodes - **Trend**: Sharp drop after 0.05, with minimal values beyond 0.1 - **Spread**: Concentrated near 0.0 #### Iteration 100 - **Peaks**: - 0.0: ~100 nodes - 0.05: ~50 nodes - 0.1: ~10 nodes - **Trend**: Broader distribution than Iteration 2, with secondary peaks at 0.05 and 0.1 - **Spread**: Extends to 0.15 #### Iteration 510 - **Peaks**: - 0.0: ~1,000 nodes - 0.05: ~100 nodes - 0.1: ~50 nodes - **Trend**: Further concentration near 0.0, with smaller peaks at 0.05 and 0.1 - **Spread**: Extends to 0.15 #### Iteration 1024 - **Peaks**: - 0.0: ~1,000 nodes - 0.05: ~100 nodes - 0.075: ~50 nodes - 0.1: ~25 nodes - **Trend**: Highest concentration at 0.0, with gradual decay toward 0.1 - **Spread**: Extends to 0.25 ### Key Observations 1. **Logarithmic Scale Impact**: The y-axis scaling emphasizes differences in node counts across orders of magnitude. 2. **Iteration Progression**: - Early iterations (2, 100) show broader distributions. - Later iterations (510, 1024) exhibit sharper concentration near 0.0. 3. **Centrality Decay**: Nodes with higher betweenness centrality (>0.1) consistently represent smaller fractions of the network across all iterations. 4. **Stability**: The highest peak (0.0) remains dominant across all iterations, suggesting a core of highly central nodes persists. ### Interpretation The data demonstrates that as the algorithm iterates, the network's betweenness centrality distribution becomes increasingly polarized: - **Early iterations** (2, 100) suggest a more heterogeneous network with multiple moderately central nodes. - **Later iterations** (510, 1024) indicate convergence toward a network structure dominated by a small group of highly central nodes (near 0.0), with rapid decay in centrality values for most nodes. This pattern aligns with expectations for betweenness centrality in dynamic networks, where repeated computations may stabilize around a core-periphery structure. The logarithmic y-axis is critical for visualizing the exponential decay in node counts at higher centrality values, which would be less apparent on a linear scale.