Calibration Regime¶

navi-fractal's sandbox method systematically underestimates fractal dimension on finite graphs. Rather than hiding this, we calibrate against networks with exact analytical dimensions and document the gap openly. This page explains why calibration matters, what the numbers show, and what they mean.

Why calibrate¶

Every measurement instrument has systematic error. A thermometer might read half a degree low; a bathroom scale might add a pound. The responsible thing is to characterize that error against known standards, not to pretend it doesn't exist.

The sandbox method's systematic underestimation has a clear geometric origin: on a finite graph, BFS balls near the boundary are truncated --- they run out of graph before they run out of radius. This pulls the average ball mass down at large radii, which flattens the log-log slope, which depresses the estimated dimension. The effect is worse on small graphs (more boundary relative to interior) and diminishes as graphs grow.

Calibrating against networks with known analytical dimensions lets us quantify this effect precisely. We use (u,v)-flower networks because their fractal dimension \( d_B = \frac{\log(u+v)}{\log u} \) is known exactly from the construction, and this formula has been formally proved in Lean 4 (see Theory Bridge).

The calibration table¶

All measurements use default parameters: seed=42, 256 centers, geometric mean, WLS, curvature guard on, slope stability guard off.

Convergence behavior¶

The gap generally shrinks with network size. This is consistent with the \( O(1/g) \) convergence rate established by the Lean squeeze bounds in fd-formalization. As the generation increases, the graph grows exponentially (each generation multiplies the edge count by \( u+v \)), and the boundary effects that cause underestimation become proportionally smaller.

Fitting \( \text{gap}(g) = a/g + b \) per flower family gives:

• Empirical rate constant: the fitted \( |a| \) values are 2--4x the theoretical bound from the Lean squeeze \( \frac{100 \log 2}{\log(u+v)} \).
• Amplification factor: this ratio \( |a_{\text{empirical}}| / a_{\text{theoretical}} \) is consistently greater than 1 across all families. This is expected because the theoretical bound applies to the global log-ratio (total vertex count divided by hub distance), while the sandbox measures a different geometric quantity (average local ball-mass scaling) that converges more slowly.

The amplification is not a deficiency of the sandbox method. It is a quantitative characterization of how much harder it is to estimate dimension from local measurements than from global counts.

The gen 7 to 8 reversal¶

The (2,2)-flower calibration shows a gap that goes from -6.0% at gen 7 to -9.5% at gen 8. This non-monotonic behavior deserves explanation because it looks like the measurement is getting worse even as the graph gets bigger.

This is a legitimate measurement characteristic, not a bug. Here is what happens:

1. At gen 7 (10,924 nodes), the window search finds a wide scaling window where the log-log plot is approximately linear. The curvature guard passes, the \( R^2 \) is high, and the slope lands at 1.881.

2. At gen 8 (43,692 nodes), the graph is larger and the potential radius range is wider. But the wider range also reveals curvature that wasn't visible at gen 7 --- the scaling behavior bends at large radii due to saturation effects that scale differently from the hub structure. The curvature guard correctly rejects the widest windows. The scoring function then selects the widest surviving window, which happens to have a slightly lower slope (1.812).

3. The scoring function's lexicographic preference for wide windows is working as designed. At gen 8, the widest curvature-clean window is different from the one that would minimize the gap to the analytical dimension. The algorithm doesn't know the analytical dimension --- it is optimizing for measurement quality (wide window, high \( R^2 \), low stderr), not for proximity to a number it doesn't have access to.

This reversal illustrates why the sandbox measures local ball-mass scaling, not the global log-ratio that fd-formalization proves. The two quantities converge in the limit but can diverge at any finite generation.

Convergence rate analysis¶

The scripts/convergence_analysis.py script performs a structured comparison of empirical convergence against the theoretical bound:

• Reads the calibration report (scripts/calibration-report.json)
• Groups results by flower family
• Fits \( \text{gap}(g) = a/g + b \) via least squares for each family
• Computes the theoretical rate constant: \( \frac{100 \log 2}{\log(u+v)} \)
• Reports the amplification factor and flags non-monotonic convergence

The theoretical bound comes from the Lean squeeze in FlowerLog.lean:

Dividing by \( g \log u \) and expressing as a percentage of \( d_B \) gives the rate constant. The empirical rate constant from the sandbox is larger because the sandbox gap includes contributions from window selection, center sampling, saturation filtering, and the fundamental geometric difference between global counts and local ball masses.

Non-fractal refusals¶

The calibration regime also validates the refusal behavior. Networks that should not have a finite fractal dimension are correctly refused:

• Barabási-Albert (preferential attachment): refused with no_valid_radii. Ball-mass growth saturates so quickly that too few non-degenerate radii survive filtering.
• Erdős-Rényi (random graphs): refused with no_valid_radii. The small-world property means ball mass saturates within a few hops, leaving insufficient radii for window construction.
• Complete graphs: refused with trivial_graph. Diameter is 1.
• (1,2)-flower (transfractal, \( u = 1 \)): refused with no_valid_radii. With \( u = 1 \), the hub distance is \( L_g = 1 \) for all generations --- the network grows but doesn't stretch. Ball-mass growth saturates too rapidly for any meaningful radius sequence.

These refusals are as important as the acceptances. A fractal dimension tool that reports \( D = 2.3 \) for a Barabási-Albert network is worse than useless --- it gives false confidence. The refusal is the feature.