The document discusses a preferential attachment model for hypergraphs called PAHG. PAHG adds new nodes and hyperedges to an initial hypergraph over time. New nodes preferentially attach to existing nodes based on their degree, while new hyperedges connect existing nodes. The model results in power law degree distributions, where the exponent depends on the rate of growth of hyperedge sizes. Many real-world networks are best modeled as hypergraphs, and PAHG provides a way to analyze hypergraph growth and properties directly. Open problems regarding properties of PAHG like the core size, expansion, influence, and diameter are mentioned.

1. Random
Preferential Attachment
Hypergraph.
Chen Avin, Zvi LoTkeR,
Yinon Nahum, David Peleg

2. Agenda
Why Hypergraphs
The model
Result
Conclusions
Open Problems

3. RUNDO 3
𝐺𝐺 𝑉𝑉, 𝐸𝐸 ,
𝐸𝐸 ⊆ 𝑉𝑉 × 𝑉𝑉
What is a
Graph
What is a
Hypergraphs
𝐺𝐺 𝑉𝑉, 𝐸𝐸 ,
𝐸𝐸 ⊆ 2𝑉𝑉

4. RUNDO 4
Many real-world networks are hypergraphs
1. Numerous people tagged in a picture
2. Numerous authors of a paper
3. Collaboration between many people
4. And many many others…
Harder to analyse?
HypergraphsWhy

5. RUNDO 5
Hypergraphs
Many real-world networks are hypergraphs
1. Numerous people tagged in a picture
2. Numerous authors of a paper
3. Collaboration between many people
4. And many many others…
Harder to analyse?
Why

6. RUNDO 6
P[x=k]~
𝟏𝟏
𝒌𝒌𝜷𝜷
Power Law Distributions
1. Observed in both network and non-network
structures
2. “On Power-Law Relationships of the Internet
Topology” (Faloutsos^3, 1999)
3. “Emergence of Scaling in Random Networks”
(Barabási and Albert, 1999)
4. “Networks of scientific papers” (de Solla Price,
1976).
5. Word frequencies, net worth, city populations,
etc.

7. RUNDO
In step 𝑡𝑡 vertex 𝑣𝑣𝑡𝑡 arrives,
and Pr[ 𝑣𝑣𝑡𝑡connects to 𝑣𝑣𝑖𝑖 ] =
𝑑𝑑𝑖𝑖
∑𝑗𝑗 𝑑𝑑𝑗𝑗
7
Preferential Attachment Process
In step 𝑡𝑡 vertex
Vertex event 𝑣𝑣𝑡𝑡 with probability 𝑝𝑝
Pr[ 𝑣𝑣𝑡𝑡connects to 𝑣𝑣𝑖𝑖 ] =
𝑑𝑑𝑖𝑖
∑𝑗𝑗 𝑑𝑑𝑗𝑗
Edge event 𝑒𝑒𝑡𝑡 with probability 1 − 𝑝𝑝
Pr[ 𝑣𝑣𝑘𝑘connects to 𝑣𝑣𝑖𝑖 ] =
𝑑𝑑𝑖𝑖
∑𝑗𝑗 𝑑𝑑𝑗𝑗
𝑑𝑑𝑘𝑘
∑𝑗𝑗 𝑑𝑑𝑗𝑗
HistoryChung and Lu
2006
Udny Yule1925, Price in 1976, Barabási, Albert in 1999

8. RUNDO
N
8
Our Model PAHG
Add a new node
and connect it to
𝑌𝑌𝑡𝑡 − 1 old nodes
Add a new edge
with 𝑌𝑌𝑡𝑡 old nodes
.
Start with
initial graph 𝐻𝐻𝐺𝐺0
At time 𝑡𝑡:
Choose a
edge size 𝑌𝑌𝑡𝑡
E
S
Existing nodes
are chosen w.p.
proportional
to their degree
with repetition.
p
1-p

9. RUNDO 9
Example
Chung and Lu
t=0
|E0|=3
Deg=(2,2,2)
Pr=(2,2,2)/6
|E0|=1
Deg=(1,1,1)
Pr=(1,1,1)/3
|E1|=6
Deg=(3,3,3,3)
Pr=(3,3,3,3)/12
|E1|=2
Deg=(1,2,2,1)
Pr=(1,2,2,1)/6
t=1 t=2
|E3|=9
Deg=(5,5,4,4)
Pr=(5,5,4,4)/18
|E2|=3
Deg=(3,3,2,2)
Pr=(2,3,2,2)/9
PTHG
Example
𝜕𝜕𝒅𝒅𝒊𝒊
𝜕𝜕𝒕𝒕
= 𝒑𝒑
𝟑𝟑𝟑𝟑𝒊𝒊
𝟔𝟔𝟔𝟔
+ (𝟏𝟏 − 𝒑𝒑)
𝒅𝒅𝒊𝒊
𝒕𝒕
𝑑𝑑𝑖𝑖 = 𝑡𝑡
2−𝑝𝑝
2 𝐜𝐜1
𝛃𝛃 = 𝟏𝟏 +
𝟐𝟐
𝟐𝟐 − 𝒑𝒑
𝜕𝜕𝒅𝒅𝒊𝒊
𝜕𝜕𝒕𝒕
= 𝒑𝒑
𝟐𝟐𝟐𝟐𝒊𝒊
𝟑𝟑𝟑𝟑
+ (𝟏𝟏 − 𝒑𝒑)
𝒅𝒅𝒊𝒊
𝒕𝒕
𝑑𝑑𝑖𝑖 = 𝑡𝑡
𝟑𝟑−𝑝𝑝
𝟑𝟑 𝐜𝐜1
𝛃𝛃 = 𝟏𝟏 +
𝟑𝟑
𝟑𝟑 − 𝒑𝒑

10. RUNDO 10
higher moments
Restrictions on 𝒀𝒀𝒕𝒕
Letting 𝑺𝑺𝒕𝒕 = ∑𝒊𝒊= 𝟏𝟏
𝒕𝒕
𝒀𝒀𝒊𝒊, Assume
𝜸𝜸 = 𝐥𝐥𝐥𝐥 𝐥𝐥
𝒕𝒕→∞
⁄𝑬𝑬 𝑺𝑺𝒕𝒕 𝒕𝒕
𝑬𝑬 𝒀𝒀𝒕𝒕+𝟏𝟏 −𝒑𝒑
>0
If 𝒀𝒀𝒕𝒕 = 𝒕𝒕𝜶𝜶
then 𝜸𝜸 =
𝟏𝟏
𝜶𝜶+𝟏𝟏
𝑬𝑬
𝟏𝟏
𝑺𝑺𝒕𝒕
−
𝟏𝟏
𝑬𝑬 𝑺𝑺𝒕𝒕
= 𝒐𝒐
𝟏𝟏
𝒕𝒕
𝑬𝑬
𝒀𝒀𝒕𝒕+𝟏𝟏
𝑺𝑺𝒕𝒕
𝟐𝟐
= 𝒐𝒐
𝟏𝟏
𝒕𝒕

11. RUNDO 11
TheoremUnder these assumptions,
PAHG follows a power law
with exponent 𝛽𝛽 = 1 + 𝜸𝜸
i.e., the expected percentage
of nodes of degree 𝑑𝑑
is proportional to 𝑑𝑑− 1+𝜸𝜸
If 𝒀𝒀𝒕𝒕 = 𝒕𝒕𝜶𝜶
then 𝛽𝛽 =
𝜶𝜶+𝟐𝟐
𝜶𝜶+𝟏𝟏

12. RUNDO 12
d-regular graphs Slowly increasing
hyperedge size
Rapidly increasing
hyperedge size:
.
Selected
Examples
𝑌𝑌𝑡𝑡 = 𝑑𝑑 always 𝐸𝐸 𝑌𝑌𝑡𝑡 = 𝑐𝑐 ⋅ log 𝑡𝑡
𝐸𝐸 𝑌𝑌𝑡𝑡
2
= 𝑜𝑜 𝑡𝑡
𝐸𝐸 𝑌𝑌𝑡𝑡 = 𝑐𝑐 ⋅ 𝑡𝑡 𝛼𝛼
𝐸𝐸 𝑌𝑌𝑡𝑡
2
= 𝑜𝑜 𝑡𝑡
𝛽𝛽 = 1 +
𝑑𝑑
𝑑𝑑 − 𝑝𝑝
𝛽𝛽 = 2 𝛽𝛽 = 1 +
1
1 + 𝛼𝛼

13. RUNDO 13
Many networks
are best modelled
as hypergraphs
Hypergraphs
should
be analyzed directly
In hypergraphs,
the power law
exponent can
< 2
Conclusions

14. RUNDO 14
Open Problems
What is the size of
The core in
PAHG(p) Expansion Property
Of PAHG(p)
Influence
In PAHG(p)
What is the
Diameter of
PAHG(p)