This document presents a Deep Hyper-Network Embedding (DHNE) model to learn low-dimensional representations of hypernetworks. The DHNE model uses an autoencoder and fully connected layer to preserve both local and global proximity in the embedding space. The DHNE model is tested on four datasets and outperforms other network embedding methods on tasks like network reconstruction, link prediction, and classification. The DHNE model can learn embeddings of hypernetworks while preserving the decomposability property of hyperedges using a nonlinear tuple similarity function.

1. Hyo Eun Lee
Network Science Lab
Dept. of Biotechnology
The Catholic University of Korea
E-mail: gydnsml@gmail.com
2023.07.17
AAAI 2018

2. 1
 Notations and Definitions
 Deep Hyper-Network Embedding
• Loss function
• Optimization
 Experiment
• Datasets and Parameter Settings
• Network Reconstruction
• Link Prediction and Classification
• Parameter Sensitivity
 Conclusion

3. 2
1. Notations and Definitions
Definitions
• Definitions 1. : 𝐺 = 𝑉 , 𝐸
𝑉 = 𝑉𝑡 𝑡=1
𝑇
, 𝐸 = 𝐸𝑖 = 𝑣1, … , 𝑣𝑛𝑖 𝑛𝑖 ≥ 2
 If the number of nodes is 2 for each hyperedge, the hyper-network degenerates to a network.
 If T ≥ 2, the hypernetwork is defined as a heterogeneous hyper-network.
• Definitions 2. (first-order proximity)
: First-order proximity in hypernetworks measures the N-tuple similarity between nodes.
 For 𝑁 nodes, if a hyperedge exists, their N-tuplewise similarity is defined as 1.
 No information about subset
• Definitions 3. (second-order proximity)
: Node-based similarity shared by hyperedge pairs
 Consider neighboring structures to discover hidden patterns or relationships.

4. 3
2. Deep Hyper-Network Embedding
Loss function
• If hyperedges exist between the 𝑁 vertices,
the N-tuple similarity(𝑆) must be large, otherwise it
must be small.
• Property 1.
 𝑖𝑓 𝑣1, 𝑣2, … , 𝑣𝑁 ∈ 𝐸, 𝑆 𝑋1, 𝑋2, … , 𝑋𝑁
𝑠. 𝑡. 𝑆 𝑙𝑎𝑟𝑔𝑒
 𝑖𝑓 𝑣1, 𝑣2, … , 𝑣𝑁 ∉ 𝐸, 𝑆 𝑋1, 𝑋2, … , 𝑋𝑁
𝑠. 𝑡. 𝑆 𝑙𝑎𝑟𝑔𝑒
+ In this paper, 𝑁 = 3
• Theorem 1.
𝐿𝑖𝑗𝑘 = 𝜎 𝑊
𝑎
2
∗ 𝑋𝑖
𝑎
+ 𝑊𝑏
2
∗ 𝑋𝑗
𝑏
+ 𝑊
𝑐
2
∗ 𝑋𝑘
𝑐
+ 𝑏2
𝑆𝑖𝑗𝑘 = 𝑆 𝑋𝑖
𝑎
, 𝑋𝑗
𝑏
, 𝑋𝑘
𝑐
= 𝜎 𝑊3
∗ 𝐿𝑖𝑗𝑘 + 𝑏3
Fill in this black

5. 4
2. Deep Hyper-Network Embedding
Loss function
• Preserve neighborhood structure using an
autoencoder as a model
𝐴 = 𝐻𝐻𝑇
− 𝐷𝑣
• Encoder ) 𝑋𝑖 = 𝜎(𝑊1
∗ 𝐴𝑖 + 𝑏1
)
• Decoder ) 𝐴𝑖 = 𝜎(𝑊1
∗ 𝑋𝑖 + 𝑏1
)
𝑠𝑖𝑔𝑛 𝐴𝑖 ⨀ 𝐴𝑖 − 𝐴𝑖
𝐹
2
∴ ℒ2 =
𝑡
𝑠𝑖𝑔𝑛 𝐴𝑖
𝑡
⨀ 𝐴𝑖
𝑡
− 𝐴𝑖
𝑡
𝐹
2
∴ ℒ = ℒ1 + 𝛼ℒ2
Fill in this black

6. 5
2. Deep Hyper-Network Embedding
Optimization
• Optimization : SGD
• Sampling multiple negatives based on a noisy
distribution and using them in a backpropagation
algorithm
• Out-of-sample extension
: Find the neighbor vector of the new arrival vertex
and feed it to the relevant autoencoder
𝑂(𝑑𝑣𝑑)
• Complexity analysis
: 𝑂 𝑛𝑑 + 𝑑𝑙 + 𝑙 𝑏𝐼
, Complexity is defined by the number of nodes
Fill in this black

7. 6
3. Experiment
Datasets
• GPS network, a social network, a medicine network and a semantic network
Methods
• Conventional pairwise network embedding methods) DeepWalk, LINE, node2vec
→ use clique expansion
• SHE : homogeneous hypernetwork embeddings, tensor method is a direct way for preserving
high-order relationship
• HEBE : heterogeneous event data

8. 7
Parameter Settings
• Represents the tuple similarity of hyperedges using the average or minimum of all pairwise
similarities for application to network reconstruction and link prediction in hypernetworks.
 DeepWalk and node2vec : window size – 10, walk length – 40, walks per vertex -10
 LINE : the number of negative samples – 5, 𝑝0 = 0.025
• DHNE : use one-layer autoencoder and fully connected layer
3. Experiment

9. 8
Network Reconstruction
3. Experiment

10. 9
Link Prediction
3. Experiment

11. 10
Link Prediction
3. Experiment

12. 11
Classification
3. Experiment

13. 12
Parameter Sensitivity
3. Experiment

14. 13
4. Contribution
Contribution
• Propose a DHNE model to learn low-dimensional representations of
hypernetworks with decomposable hyperedges
• Previous network embedding methods mainly focus on networks with pairwise relationships and
cannot preserve the decomposability property in hypernetworks using linear similarity metrics
in the embedding space.
• The DHNE model implements a nonlinear tuple similarity function that preserves both local and global
proximity in the formed embedding space.
• The DHNE model outperforms state of these algorithms in four types of experiments.

16. 15