What Is a Power Law? Discovering Scale Invariance with Real Examples
Learn what power laws are, how log-log plots reveal them, and why phenomena like earthquakes, city sizes, and metabolic scaling follow these surprising patterns.
🚧 This post is under construction 🚧
TL;DR
- Point 1
- Point 2
Table of Contents
- Introduction
- Example 1: Kleiber’s Law (Metabolic Rate vs Body Mass)
- Example 2: Gutenberg-Richter Law (Earthquake Magnitudes)
- The General Recipe
- So, what is a Power Law?
- Starting Simple: Concrete Calculations
- The Signature Property: Scale Invariance
- Where Power Laws Appear?
- How to Distinguish a Power Law from an Exponential?
- Fitting: How to Determine That a Phenomenon Follows a Power Law
- Can We Convert Between Power Law and Exponential?
- Summary: When to Use a Power Law
- Conclusion
- Webliography
Introduction
Lorem
Example 1: Kleiber’s Law (Metabolic Rate vs Body Mass)
The Setup
In the 1930s, Max Kleiber was measuring how much oxygen different animals consume per day (from mice to cows). He had a simple, practical question: if I double an animal’s body mass, how much more food does it need?
The naive hypothesis (the “obvious” one) was linear: twice the mass \(\rightarrow\) twice the energy. Some biologists argued for a \(2/3\) exponent, based on surface-area reasoning (heat loss scales with surface, surface scales as \(M^{2/3}\)).
Kleiber just… measured things.
The Raw Data
| Animal | Body mass \(M\) (kg) | Metabolic rate \(B\) (watts) |
|---|---|---|
| Mouse | 0.025 | 0.24 |
| Rat | 0.3 | 1.45 |
| Cat | 3 | 8.0 |
| Dog | 11 | 22.7 |
| Human | 70 | 82 |
| Cow | 400 | 270 |
| Elephant | 3000 | 1100 |
Step 1: Plot linearly. It’s a mess.
On linear axes, the elephant dominates everything and we can’t see structure in the small animals at all. We notice it’s “some kind of curve” but can’t say more.
import matplotlib.pyplot as plt
# Data: Kleiber's metabolic scaling
animals = ["Mouse", "Rat", "Cat", "Dog", "Human", "Cow", "Elephant"]
M = [0.025, 0.3, 3, 11, 70, 400, 3000] # body mass (kg)
B = [0.24, 1.45, 8.0, 22.7, 82, 270, 1100] # metabolic rate (W)
fig, ax = plt.subplots(figsize=(9, 6))
ax.plot(M, B, 'o', markersize=8, color='#2a6496')
for name, m, b in zip(animals, M, B):
ax.annotate(name, (m, b), textcoords="offset points", xytext=(8, 8), fontsize=10)
ax.set_xlabel("Body mass M (kg)", fontsize=13)
ax.set_ylabel("Metabolic rate B (W)", fontsize=13)
ax.set_title("Metabolic Rate vs Body Mass", fontsize=14)
ax.grid(True, alpha=0.3)
fig.tight_layout()
plt.show()
With the help of the graph, we immediately see the problem: Mouse, Rat, Cat, and Dog are all crushed into the bottom-left corner while Elephant dominates. That’s exactly the visual motivation for switching to another scaling.
Step 2: Try log-linear. Does it straighten?
Plot \(\log B\) vs \(M\). Still curved. Not exponential.
import numpy as np
import matplotlib.pyplot as plt
# Data: Kleiber's metabolic scaling
animals = ["Mouse", "Rat", "Cat", "Dog", "Human", "Cow", "Elephant"]
M = np.array([0.025, 0.3, 3, 11, 70, 400, 3000]) # body mass (kg)
B = np.array([0.24, 1.45, 8.0, 22.7, 82, 270, 1100]) # metabolic rate (W)
fig, ax = plt.subplots(figsize=(9, 6))
ax.plot(M, np.log10(B), 'o', markersize=8, color='#2a6496')
for name, m, b in zip(animals, M, B):
ax.annotate(name, (m, np.log10(b)), textcoords="offset points", xytext=(8, 8), fontsize=10)
ax.set_xlabel("Body mass M (kg)", fontsize=13)
ax.set_ylabel("log₁₀(B) [B in watts]", fontsize=13)
ax.set_title("log(Metabolic Rate) vs Body Mass — semi-log", fontsize=14)
ax.grid(True, alpha=0.3)
fig.tight_layout()
plt.show()
The small animals are a bit more visible now, but the points still curve. This confirms this isn’t an exponential relationship (which would appear as a straight line here)
Step 3: Try log-log.
Plot \(\log B\) vs \(\log M\):
| \(\log_{10} M\) | \(\log_{10} B\) |
|---|---|
| -1.60 | -0.62 |
| -0.52 | 0.16 |
| 0.48 | 0.90 |
| 1.04 | 1.36 |
| 1.85 | 1.91 |
| 2.60 | 2.43 |
| 3.48 | 3.04 |
import numpy as np
import matplotlib.pyplot as plt
# Data: Kleiber's metabolic scaling
animals = ["Mouse", "Rat", "Cat", "Dog", "Human", "Cow", "Elephant"]
M = np.array([0.025, 0.3, 3, 11, 70, 400, 3000]) # body mass (kg)
B = np.array([0.24, 1.45, 8.0, 22.7, 82, 270, 1100]) # metabolic rate (W)
# Log-log linear fit: log(B) = a * log(M) + b
coeffs = np.polyfit(np.log10(M), np.log10(B), 1)
exponent, log_prefactor = coeffs
prefactor = 10 ** log_prefactor
# Fit line over a smooth range
M_fit = np.logspace(np.log10(M.min()) - 0.3, np.log10(M.max()) + 0.3, 200)
B_fit = prefactor * M_fit ** exponent
# Plot
fig, ax = plt.subplots(figsize=(9, 6))
ax.loglog(M, B, 'o', markersize=9, color='#2a6496', zorder=5)
ax.loglog(M_fit, B_fit, '--', color='#cc4444', linewidth=1.5,
label = "Fit: $B = %.2f \\, M^{%.2f}$" % (prefactor, exponent))
# Annotate each animal
offsets = {
"Mouse": (10, 8), "Rat": (10, -12), "Cat": (10, 8),
"Dog": (-15, -18), "Human": (10, 8), "Cow": (-15, -18), "Elephant": (10, 8),
}
for name, m, b in zip(animals, M, B):
dx, dy = offsets[name]
ax.annotate(name, (m, b), textcoords="offset points", xytext=(dx, dy), fontsize=10)
ax.set_xlabel("Body mass $M$ (kg)", fontsize=13)
ax.set_ylabel("Metabolic rate $B$ (W)", fontsize=13)
ax.set_title("Kleiber's Law — Metabolic Scaling", fontsize=14)
ax.legend(fontsize=12)
ax.grid(True, which="both", alpha=0.3)
fig.tight_layout()
plt.show()
Now the points fall on a straight line. This is the moment of discovery.
Step 4: Read off the slope
From mouse to elephant: \(\Delta \log M = 3.48 - (-1.60) = 5.08\), \(\Delta \log B = 3.04 - (-0.62) = 3.66\).
\[\hat{\alpha} = \frac{3.66}{5.08} \approx 0.72\]Linear regression on all 7 log-log points gives \(\hat{\alpha} \approx 0.74\), intercept \(\approx -0.10\), so:
\[\boxed{B \approx 0.8 \cdot M^{3/4}}\]Step 5: Reject the competing hypotheses
| Hypothesis | Exponent | Prediction for elephant vs mouse ratio | Observed |
|---|---|---|---|
| Linear | 1.0 | \((3000/0.025)^{1.0} = 120{,}000\times\) | — |
| Surface area | 0.67 | \((120{,}000)^{0.67} = 3{,}800\times\) | — |
| Kleiber | 0.75 | \((120{,}000)^{0.75} = 18{,}600\times\) | \(1100/0.24 \approx 4{,}600\times\) |
None is perfect (real biology is noisy), but \(3/4\) fits dramatically better than \(1\) or \(2/3\), and it holds across 27 orders of magnitude of body mass when we include bacteria and whales. That robustness is the signature that we’ve found something real.
Why 3/4 and not 2/3? The physical story came after the data.
The \(\frac{3}{4}\) exponent was a mystery for 60 years. In 1997, West, Brown & Enquist showed it emerges from the fractal geometry of vascular networks: the network must fill a 3D volume but itself has fractal dimension and the math forces \(\alpha = \frac{3}{4}\). The law was discovered empirically decades before it was understood theoretically. Data first, theory second.
Example 2: Gutenberg-Richter Law (Earthquake Magnitudes)
The Setup
In the 1940s, Beno Gutenberg and Charles Richter were cataloguing California earthquakes. They had counts of earthquakes per year at each magnitude level. The question: is there a pattern in how often large vs small quakes occur?
The engineering hope was that large earthquakes were rare “anomalies” — exponentially rare, like failures in most engineered systems, so we could define a “maximum credible earthquake.” Spoiler: that hope was wrong.
The Raw Data (simplified, order-of-magnitude realistic)
| Magnitude \(M\) | Quakes/year in California |
|---|---|
| 2 | ~1000 |
| 3 | ~300 |
| 4 | ~100 |
| 5 | ~30 |
| 6 | ~10 |
| 7 | ~3 |
| 8 | ~1 |
import matplotlib.pyplot as plt
# Gutenberg-Richter: earthquake frequency in California
magnitude = [2, 3, 4, 5, 6, 7, 8]
quakes_per_year = [1000, 300, 100, 30, 10, 3, 1]
fig, ax = plt.subplots(figsize=(9, 6))
ax.plot(magnitude, quakes_per_year, 'o', markersize=8, color='#2a6496')
ax.set_xlabel("Magnitude M", fontsize=13)
ax.set_ylabel("Earthquakes per year", fontsize=13)
ax.set_title("Earthquake Frequency vs Magnitude (California)", fontsize=14)
ax.grid(True, alpha=0.3)
fig.tight_layout()
plt.show()
Step 1: First look: is it exponential?
Plot \(\log N\) vs \(M\) (log-linear):
| \(M\) | \(\log_{10} N\) |
|---|---|
| 2 | 3.0 |
| 3 | 2.5 |
| 4 | 2.0 |
| 5 | 1.5 |
| 6 | 1.0 |
| 7 | 0.5 |
| 8 | 0.0 |
import numpy as np
import matplotlib.pyplot as plt
# Gutenberg-Richter: earthquake frequency in California
magnitude = np.array([2, 3, 4, 5, 6, 7, 8])
quakes_per_year = np.array([1000, 300, 100, 30, 10, 3, 1])
fig, ax = plt.subplots(figsize=(9, 6))
ax.plot(magnitude, np.log10(quakes_per_year), 'o', markersize=8, color='#2a6496')
ax.set_xlabel("Magnitude M", fontsize=13)
ax.set_ylabel("log₁₀(N)", fontsize=13)
ax.set_title("log₁₀(Earthquakes per year) vs Magnitude", fontsize=14)
ax.grid(True, alpha=0.3)
fig.tight_layout()
plt.show()
This is a perfect straight line. Slope = \(-0.5\) per unit magnitude. So:
\[\log_{10} N = 3 - 0.5 \cdot M \implies N = 10^{3 - 0.5M} = 10^3 \cdot 10^{-0.5M}\]Wait, that looks like an exponential in \(M\), not a power law! Case closed?
Step 2: The key insight: magnitude is already a log scale
Richter magnitude is defined as:
\[M = \log_{10}\left(\frac{E}{E_0}\right) \cdot \frac{2}{3} \quad \text{(approximately)}\]So \(M \propto \log E\), meaning energy \(E \propto 10^{1.5M}\).
Substitute \(M = \frac{2}{3}\log_{10}(E/E_0)\) into the Gutenberg-Richter relation:
\[\log_{10} N = a - b \cdot M = a - b \cdot \frac{2}{3}\log_{10} E + \text{const}\] \[\log_{10} N = \text{const} - \frac{2b}{3} \log_{10} E\] \[\boxed{N(E) \propto E^{-2b/3}}\]It’s a power law in energy, with exponent \(\approx -2b/3 \approx -1.0\) for \(b \approx 1.5\).
Step 3: What this means concretely
\[N(>E) \propto E^{-1}\]A quake releasing \(10\times\) more energy is \(10\times\) rarer. A quake releasing \(1000\times\) more energy is \(1000\times\) rarer. There is no characteristic earthquake size. The ratio is always the same. That’s scale invariance in action.
The exponential-in-magnitude formula and the power-law-in-energy formula are the same statement, just in different variables. The discovery was realizing that magnitude was already a compressed (logarithmic) representation, and that uncompressing it reveals the true power law structure.
The engineering consequence
If seismic energy followed a true exponential distribution (with a characteristic scale), there would be a natural maximum earthquake and risk would be bounded. The power law means there is no such maximum. The distribution has a heavy tail. Every time we think we’ve seen the biggest possible earthquake, the math says there’s a finite probability of something larger. This fundamentally changed how nuclear plants, dams, and bridges are designed.
The General Recipe
Both examples follow the same narrative, the same story:
- Measure without assumptions. Just collect \((x_i, y_i)\) pairs
- Plot linearly. See a curve, gain intuition about scale
- Try log-linear. Test the exponential hypothesis explicitly
- Try log-log. If this straightens, we have a power law candidate
- Fit the slope. Get \(\hat{\alpha}\) with error bars
- Check competing models. Don’t just confirm, try to falsify
- Ask what the exponent means. The number \(\alpha\) often encodes deep geometry or mechanism
- Watch for hidden variables. Like Richter’s magnitude being already logarithmic
The moment of discovery is always step 4: the log-log plot snapping into a straight line when everything else was a curve. That’s the “aha” moment we will remember.
So, what is a Power Law?
A power law is a relationship of the form:
\[y = C \cdot x^{\alpha}\]where \(C\) is a constant and \(\alpha\) is the exponent (the key parameter). That’s it. Deceptively simple.
Compare immediately with an exponential:
\[y = C \cdot e^{\lambda x}\]The crucial structural difference: in a power law, \(x\) is the base. In an exponential, \(x\) is the exponent. This one difference has enormous consequences.
Starting Simple: Concrete Calculations
Power law examples we already know
Area of a circle: \(A = \pi r^2\) \rightarrow exponent \(\alpha = 2\)
Volume of a sphere: \(V = \frac{4}{3}\pi r^3\) \rightarrow exponent \(\alpha = 3\)
Gravitational force: \(F = G \frac{m_1 m_2}{r^2}\) \rightarrow exponent \(\alpha = -2\) (decreasing)
Now let’s feel the difference between power law and exponential growth numerically.
Take \(y = x^2\) vs \(y = e^x\), starting at \(x = 1\):
| x | \(x^2\) | \(e^x\) |
|---|---|---|
| 1 | 1 | 2.7 |
| 2 | 4 | 7.4 |
| 5 | 25 | 148 |
| 10 | 100 | 22,026 |
| 20 | 400 | 485,165,195 |
import numpy as np
import matplotlib.pyplot as plt
# Comparison: power law vs exponential
x = np.array([1, 2, 5, 10, 20])
power = x ** 2
expo = np.exp(x)
fig, ax = plt.subplots(figsize=(9, 6))
ax.semilogy(x, power, 'o-', markersize=8, color='#2a6496', label='$x^2$ (power law)')
ax.semilogy(x, expo, 's-', markersize=8, color='#cc4444', label='$e^x$ (exponential)')
ax.set_xlabel("x", fontsize=13)
ax.set_ylabel("y (log scale)", fontsize=13)
ax.set_title("Power Law vs Exponential: the exponential always wins", fontsize=14)
ax.legend(fontsize=12)
ax.grid(True, which="both", alpha=0.3)
fig.tight_layout()
plt.show()
\(e^x\) shoots up as a straight line while \(x^2\) barely curves upward visually hammering home that exponentials always dominate power laws at large \(x\).
The exponential annihilates the power law for large \(x\). This is the fundamental theorem of comparing these two families: exponentials always eventually dominate power laws, no matter how large \(\alpha\) is.
But for small \(x\), or when \(\alpha\) is large, power laws can look like exponentials locally and this is a source of much confusion.
The Signature Property: Scale Invariance
Power laws have a remarkable property that exponentials do not share: scale invariance (also called self-similarity).
Watch what happens when we multiply \(x\) by a constant factor \(k\):
\[y(kx) = C(kx)^{\alpha} = k^{\alpha} \cdot Cx^{\alpha} = k^{\alpha} \cdot y(x)\]Doubling \(x\) always multiplies \(y\) by exactly \(2^{\alpha}\), regardless of where we are on the curve. The ratio depends only on the factor \(k\), not on the starting value of \(x\).
With an exponential: \(y(kx) = Ce^{\lambda kx} = e^{\lambda(k-1)x} \cdot Ce^{\lambda x}\). The multiplier depends on \(x\), it’s not constant. Scale invariance is broken.
Practical meaning: For a power law with \(\alpha = 2\), a city 10× bigger than another will always have roughly \(10^2 = 100\)× more of whatever is being measured (road surface, number of gas stations…), regardless of whether we are comparing 10k vs 100k cities, or 100k vs 1M cities. The ratio is always the same.
import numpy as np
import matplotlib.pyplot as plt
# Scale invariance: power law vs exponential
x = np.linspace(1, 10, 200)
alpha = 2
lam = 0.5
y_power = x ** alpha
y_expo = np.exp(lam * x)
# Pick two starting points, double each
x1, x2 = 2, 5
k = 2
fig, axes = plt.subplots(1, 2, figsize=(14, 6))
# --- Left panel: Power law ---
ax = axes[0]
ax.plot(x, y_power, '-', color='#2a6496', linewidth=2)
for xi in [x1, x2]:
y_lo = xi ** alpha
y_hi = (k * xi) ** alpha
ratio = y_hi / y_lo
ax.plot([xi, k * xi], [y_lo, y_hi], 'o', markersize=9, color='#cc4444')
ax.annotate('', xy=(k * xi, y_hi), xytext=(k * xi, y_lo),
arrowprops=dict(arrowstyle='<->', color='#cc4444', lw=1.5))
ax.text(k * xi + 0.3, (y_lo + y_hi) / 2, f'×{ratio:.0f}',
color='#cc4444', fontsize=12, va='center')
ax.set_xlabel('x', fontsize=13)
ax.set_ylabel('y', fontsize=13)
ax.set_title(f'Power law $y = x^{{{alpha}}}$\nDoubling x → always ×{k**alpha}', fontsize=13)
ax.grid(True, alpha=0.3)
# --- Right panel: Exponential ---
ax = axes[1]
ax.plot(x, y_expo, '-', color='#2a6496', linewidth=2)
for xi in [x1, x2]:
y_lo = np.exp(lam * xi)
y_hi = np.exp(lam * k * xi)
ratio = y_hi / y_lo
ax.plot([xi, k * xi], [y_lo, y_hi], 'o', markersize=9, color='#cc4444')
ax.annotate('', xy=(k * xi, y_hi), xytext=(k * xi, y_lo),
arrowprops=dict(arrowstyle='<->', color='#cc4444', lw=1.5))
ax.text(k * xi + 0.3, (y_lo + y_hi) / 2, f'×{ratio:.1f}',
color='#cc4444', fontsize=12, va='center')
ax.set_xlabel('x', fontsize=13)
ax.set_ylabel('y', fontsize=13)
ax.set_title(f'Exponential $y = e^{{{lam}x}}$\nDoubling x → ratio depends on x!', fontsize=13)
ax.grid(True, alpha=0.3)
fig.tight_layout()
plt.show()
On the left, both red arrows show the same ×4 ratio (since \(2^2 = 4\)) whether we start at \(x=2\) or \(x=5\). That’s scale invariance. On the right, the same doubling produces very different ratios depending on where we start. The exponential breaks scale invariance.
Where Power Laws Appear?
Economics & Society
- Zipf’s law: The \(n\)-th most common word in any large corpus appears with frequency \(\propto 1/n\). The most common word (“the”) appears roughly twice as often as the second most common, three times the third, etc.
- City sizes: In most countries, city population follows \(P(rank) \propto rank^{-1}\). Paris is roughly 2× Lyon, 3× Marseille as a metropolitan area.
- Wealth distribution: Top 20% hold ~80% of wealth (Pareto’s 80/20 rule). This is a power law with \(\alpha \approx 1.16\).
- Company sizes: Number of companies with \(n\) employees \(\propto n^{-\alpha}\).
Internet & Networks
- Web links: Most pages have few inbound links; a tiny number (Google, Wikipedia) have billions. Distribution: \(P(k) \propto k^{-\gamma}\), with \(\gamma \approx 2.1\) for the web.
- Degree distribution in social networks (Facebook friends, Twitter followers).
Nature & Physics
- Earthquake magnitudes (Gutenberg-Richter law): \(\log N = a - b \cdot M\), i.e., \(N \propto 10^{-bM}\). The number of earthquakes of magnitude \(\geq M\) follows a power law in energy.
- Avalanche sizes, solar flares, forest fire extents.
- Metabolic rate vs body mass: \(B \propto M^{3/4}\) (Kleiber’s law). An elephant uses ~\(10^4\) times more energy than a mouse, yet its mass is \(\sim 10^{4.5}\) times greater. The \(3/4\) exponent reflects fractal-like vascular network efficiency.
- Coastline length vs measurement scale (Mandelbrot’s fractal work).
Information & Language
- File size distributions on the internet.
- Number of citations a scientific paper receives.
How to Distinguish a Power Law from an Exponential?
The log-log trick
Take \(\log\) of both sides of \(y = Cx^{\alpha}\):
\[\log y = \log C + \alpha \log x\]On a log-log plot, a power law becomes a straight line with slope \(\alpha\). This is the primary diagnostic tool.
For an exponential \(y = Ce^{\lambda x}\), take \(\log\):
\[\log y = \log C + \lambda x\]On a log-linear plot (log \(y\), linear \(x\)), an exponential is a straight line. On a log-log plot, it curves upward.
import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(1, 50, 200)
y_power = 2 * x ** 1.5 # power law: C=2, alpha=1.5
y_expo = 2 * np.exp(0.15 * x) # exponential: C=2, lambda=0.15
fig, axes = plt.subplots(1, 2, figsize=(14, 6))
# --- Left: log-log plot ---
ax = axes[0]
ax.loglog(x, y_power, '-', color='#2a6496', linewidth=2, label='Power law $2x^{1.5}$')
ax.loglog(x, y_expo, '-', color='#cc4444', linewidth=2, label='Exponential $2e^{0.15x}$')
ax.set_xlabel('x', fontsize=13)
ax.set_ylabel('y', fontsize=13)
ax.set_title('Log-log plot\nPower law → straight line', fontsize=13)
ax.legend(fontsize=11)
ax.grid(True, which='both', alpha=0.3)
# --- Right: log-linear plot ---
ax = axes[1]
ax.semilogy(x, y_power, '-', color='#2a6496', linewidth=2, label='Power law $2x^{1.5}$')
ax.semilogy(x, y_expo, '-', color='#cc4444', linewidth=2, label='Exponential $2e^{0.15x}$')
ax.set_xlabel('x', fontsize=13)
ax.set_ylabel('y', fontsize=13)
ax.set_title('Log-linear plot\nExponential → straight line', fontsize=13)
ax.legend(fontsize=11)
ax.grid(True, which='both', alpha=0.3)
fig.tight_layout()
plt.show()
Left panel: on a log-log plot the power law is a perfect straight line while the exponential curves upward.
Right panel: on a log-linear plot the roles swap. The exponential becomes the straight line. That’s the diagnostic trick in one glance.
Decision table
| Plot type | Power law | Exponential |
|---|---|---|
| Linear-linear | Curve | Curve |
| Log-log | Straight line ✓ | Curves |
| Log-linear | Curves | Straight line ✓ |
Worked example
Suppose we observe this data:
| x | y |
|---|---|
| 1 | 3 |
| 2 | 12 |
| 4 | 48 |
| 8 | 192 |
| 16 | 768 |
import matplotlib.pyplot as plt
x = [1, 2, 4, 8, 16]
y = [3, 12, 48, 192, 768]
fig, ax = plt.subplots(figsize=(9, 6))
ax.plot(x, y, 'o', markersize=8, color='#2a6496')
ax.set_xlabel("x", fontsize=13)
ax.set_ylabel("y", fontsize=13)
ax.set_title("Raw Data", fontsize=14)
ax.grid(True, alpha=0.3)
fig.tight_layout()
plt.show()
Check ratios: when \(x\) doubles, \(y\) quadruples. That’s consistent every time \(\rightarrow\) power law candidate with \(\alpha = 2\).
Check: \(y = 3x^2\). Indeed \(3(1)^2=3\), \(3(2)^2=12\), \(3(4)^2=48\).
On a log-log plot: \(\log y = \log 3 + 2 \log x\) \(\rightarrow\) slope = 2, intercept = \(\log 3 \approx 0.48\).
import numpy as np
import matplotlib.pyplot as plt
x = np.array([1, 2, 4, 8, 16])
y = np.array([3, 12, 48, 192, 768])
# Linear fit in log-log space
coeffs = np.polyfit(np.log10(x), np.log10(y), 1)
slope, intercept = coeffs
# Fit line
x_fit = np.logspace(0, np.log10(20), 100)
y_fit = 10 ** (slope * np.log10(x_fit) + intercept)
fig, ax = plt.subplots(figsize=(9, 6))
ax.loglog(x, y, 'o', markersize=9, color='#2a6496', zorder=5, label='Data')
ax.loglog(x_fit, y_fit, '--', color='#cc4444', linewidth=1.5,
label=f'Fit: slope = {slope:.1f}, intercept = {intercept:.2f}')
ax.set_xlabel("x", fontsize=13)
ax.set_ylabel("y", fontsize=13)
ax.set_title("Log-log plot: perfect straight line confirms $y = 3x^2$", fontsize=14)
ax.legend(fontsize=12)
ax.grid(True, which='both', alpha=0.3)
fig.tight_layout()
plt.show()
Now contrast with exponential data: \(y = 2 \cdot 3^x\)
| x | y |
|---|---|
| 1 | 6 |
| 2 | 18 |
| 3 | 54 |
| 4 | 162 |
Here when \(x\) increases by 1 (additive), \(y\) multiplies by 3 (constant ratio). For a power law, we need a multiplicative step in \(x\) to get a constant ratio in \(y\). That’s the operational test.
The “ratio test” in practice
- Power law: \(y(2x)/y(x) =\) constant (depends only on the factor, not on \(x\))
- Exponential: \(y(x+1)/y(x) =\) constant (constant ratio per additive step)
Fitting: How to Determine That a Phenomenon Follows a Power Law
Step 1: Log-log linearization
Let’s plot our data on log-log axes. If it looks linear, we have a candidate.
Step 2: Linear regression on log-transformed data
\[\log y = \alpha \log x + \log C\]Run OLS on \((\log x_i, \log y_i)\) pairs. The slope gives \(\hat{\alpha}\), the intercept gives \(\log \hat{C}\).
Caution: This minimizes error in log-space, which implicitly assumes multiplicative (relative) errors, not additive ones. Fine for many natural phenomena, but check our assumptions.
Step 3: The Clauset-Shalizi-Newman warning (2009)
A landmark paper showed that “looks linear on log-log” is necessary but not sufficient. Many distributions that look like power laws on a log-log plot are actually log-normal or stretched exponential.
More rigorous approach:
- Estimate \(x_{min}\) (the lower cutoff, below which the power law doesn’t hold)
- Use Maximum Likelihood Estimation to estimate \(\alpha\):
- Use a Kolmogorov-Smirnov test to measure goodness-of-fit
- Compare with alternative distributions (log-normal, exponential) via likelihood ratio tests
Step 4: Watch for the heavy tail
Power law distributions with \(\alpha \leq 2\) have infinite variance. With \(\alpha \leq 1\), even the mean is infinite. This has practical consequences: our sample mean will never converge, and “average” behavior is meaningless. This is why financial risk models built on Gaussian assumptions catastrophically failed in 2008 — asset return tails are closer to power laws (\(\alpha \approx 3\)) than Gaussians.
Can We Convert Between Power Law and Exponential?
This is a subtle and interesting question.
Mathematically: yes, via a change of variable
Start with an exponential: \(y = Ce^{\lambda x}\). Substitute \(x = \log t\) (so \(t = e^x\)):
\[y = Ce^{\lambda \log t} = C \cdot t^{\lambda}\]That’s a power law in \(t\)! So: an exponential in \(x\) is a power law in \(e^x\), and conversely a power law in \(x\) is an exponential in \(\log x\).
This means log-normal data will appear as a power law if we accidentally plot it on the wrong axes — a very common mistake.
Practically: when does this matter?
In network growth models, preferential attachment (new nodes connect to existing nodes proportionally to their degree) generates power-law degree distributions. If instead growth were random (each new node equally likely to connect to any existing node), we would get an exponential distribution. The mechanism determines the law.
In machine learning: learning curves often follow a power law in the number of training samples: \(\text{error} \propto n^{-\alpha}\). This has become crucial for scaling laws in large language models (Chinchilla, GPT-4 scaling). The implication is that we can extrapolate performance from small experiments — if the log-log relationship is linear, we can predict what a 10× larger model will do.
Summary: When to Use a Power Law
We can use a power law when:
- Scale invariance is physically plausible. There’s no natural characteristic scale in the system (vs. radioactive decay, which has a half-life: a characteristic time scale \(\rightarrow\) exponential)
- We observe “the rich get richer” dynamics: multiplicative feedback, preferential attachment, cumulative advantage
- Our data has a heavy tail — extreme events are far more common than a Gaussian or exponential would predict
- The phenomenon involves geometric/fractal self-similarity (coastlines, vascular networks, drainage basins)
- We are dealing with rank-frequency relationships (words, cities, firms)
We should prefer an exponential when:
- There’s a constant per-unit-time probability of an event (Poisson process \(\rightarrow\) exponential waiting times)
- There’s a characteristic scale (half-life, mean free path, time constant)
- The system has a maximum or saturation (logistic growth, not pure exponential, but related)
The deepest intuition: power laws emerge from multiplicative processes without a characteristic scale; exponentials emerge from additive or constant-rate processes with a characteristic scale.
Conclusion
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