Zipf Distribution

📊 Zipf Distribution in Python

The Zipf distribution is a discrete probability distribution that models ranked data, where the frequency of an item is inversely proportional to its rank.
It appears in linguistics, social networks, and city populations, following the 80/20 or power-law rule.

1. Characteristics of Zipf Distribution

Discrete distribution (integers: 1, 2, 3, …)
Parameter: a > 1 → exponent characterizing the distribution
Probability Mass Function (PMF):

$\frac{1/k^a}{\sum_{n=1}^{N} 1/n^a}, \quad k = 1,2,3,…$

Heavy-tailed distribution: a few items have very high frequency, many items have low frequency

Applications:

Word frequencies in text (most common words appear much more often)
City populations
Website traffic distribution

2. Generate Zipf Data Using NumPy

Output (example):

Mostly small integers (rank 1, 2, 3)
Few large values → heavy tail

3. Visualize Zipf Distribution

Most counts occur at low ranks
Long tail toward higher ranks

4. Compare Different Parameters

data1 = np.random.zipf(a=1.5, size=1000)
data2 = np.random.zipf(a=2.5, size=1000)

sns.histplot(data1, bins=50, color='red', label='a=1.5', alpha=0.5)
sns.histplot(data2, bins=50, color='blue', label='a=2.5', alpha=0.5)
plt.title("Zipf Distribution Comparison")
plt.legend()
plt.show()

data1 = np.random.zipf(a=1.5, size=1000)

data2 = np.random.zipf(a=2.5, size=1000)

sns.histplot(data1, bins=50, color='red', label='a=1.5', alpha=0.5)

sns.histplot(data2, bins=50, color='blue', label='a=2.5', alpha=0.5)

plt.title("Zipf Distribution Comparison")

plt.legend()

plt.show()

Smaller a → heavier tail (more high-rank values)
Larger a → concentrated at low ranks

5. Summary Table

Function	Parameters	Description
`np.random.zipf()`	a, size	Generates Zipf random integers
`a`	Exponent parameter	Controls heaviness of tail (>1)
`size`	Number of samples	Output array size

🎯 Practice Exercises

Generate 1000 Zipf random numbers with a=2 and plot histogram.
Compare Zipf distributions with a=1.5 vs a=3.
Analyze the proportion of top 10 ranks vs total data.

Zipf Distribution