Zipf distribution

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The document describes how to generate combinations of items according to a Zipf distribution. It explains that the Zipf distribution assigns probabilities to ranks, with the highest ranked item having the greatest probability and each subsequent rank having less probability. It then shows how to calculate the Zipf probabilities for a set of 5 items and generate all possible combinations of those items weighted by their Zipf probabilities.

Technology

Zipf Distribution
Generate zipfian subscription spaces

Take all combinations C(5,i) ( 𝑖=0
5
𝐶 5, 𝑖 = 25
)
C(5,0) {}
C(5,1) {1} {2} {3} {4} {5}
C(5,2) {1,2} {1,3} {1,4} {1,5} {2,3} {2,4} {2,5} {3,4} {3,5} {4,5}
C(5,3) {1,2,3} {1,2,4} {1,2,5} {1,3,4} {1,3,5} {1,4,5} {2,3,4} {2,3,5} {2,4,5} {3,4,5}
C(5,4) {1,2,3,4} {1,2,3,5} {1,2,4,5} {1,3,4,5} {2,3,4,5}
C(5,5) {1,2,3,4,5}

C(5,0)
C(5,1) {1}
×
1/5
×
C(5,1) × 1
{2}
×
1/5
×
C(5,1) × 1
{3}
×
1/5
×
C(5,1) × 1
{4}
×
1/5
×
C(5,1) × 1
{5}
×
1/5
×
C(5,1) ×1
C(5,2) {1}
×
2/5
×
C(5,2) × 2
{2}
×
2/5
×
C(5,2) × 2
{3}
×
2/5
×
C(5,2) × 2
…. ….
C(5,3)
C(5,4)
C(5,5)

Normal Combinations
Prob(rank, C(5,i)) =
𝑖
5
Prob(rank, 𝑖=0
5
𝐶 5, 𝑖 ) =
𝑖=0
5
𝑃𝑟𝑜𝑏 𝑟𝑎𝑛𝑘,𝐶 5,𝑖 × 𝐶 5,𝑖 × 𝑖
𝑖=0
5
𝐶 5,𝑖 ×𝑖
= constant
Items have equal probability to appear because “no rank”

How to generate normal combinations?
• Take C(5,2), we have
• 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5
• (1,2) (1,3) (1,4) (1,5) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5)

Zipf Formula
𝑓 𝑘: 𝑠, 𝑁 =
1
𝑘 𝑠
𝑛=1
𝑁
(
1
𝑛 𝑠)
N - number of elements in distribution,
k - rank of element
s - value of exponent

Take N=5, rank ={1,….5}, s=1
Rank Zipf Prob
1 0.437956204379562062
2 0.218978102189781033
3 0.1459854014598544
4 0.109489051094890525
5 0.08759124087591241
CDF (5) 1.000000000000000004

Zipf Distribution
Prob(rank, C(5,i)) = 𝑧𝑖𝑝𝑓(𝑟𝑎𝑛𝑘)

Zipf Distribution
Prob(rank, C(5,i)) = 𝑧𝑖𝑝𝑓(𝑟𝑎𝑛𝑘)
Prob(rank, 𝑖=0
5
𝐶 5, 𝑖 ) =
𝑖=0
5
𝑃𝑟𝑜𝑏 𝑟𝑎𝑛𝑘,𝐶 5,𝑖 × 𝐶 5,𝑖 × 𝑖
𝑖=0
5
𝐶 5,𝑖 ×𝑖
Prob(rank,
𝑖=0
5
𝐶 5, 𝑖 ) ∝ 𝒛𝒊𝒑𝒇(𝒓𝒂𝒏𝒌)
Ranks appear according to zipf distribution

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