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Muhan Zhang and Yixin Chen, NeurIPS 2018
Link Prediction Based on Graph Neural Networks
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Given an incomplete network, predict whether two nodes are likely to have a link.
Link Prediction (LP) Problem
Applications: • Friend recommendation in social networks• Product recommendation in ecommerce• Interaction prediction in biological networks• Knowledge graph completion• ...
Figures from the Internet.
social network Biological network
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Heuristic Methods for LPCalculate a proximity score for each pair of nodes.
• Good performance
• Easy to calculate
• Interpretable
• No training required
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First-Order HeuristicsNotations:! " is the neighbor set of node x in the graph
• The common neighbors (CN) heuristic: |! " ∩ ! $ |
x and y are likely to have a link if they have many common neighbors.yx
• First-order heuristic, need only 1-hop neighbors to compute.
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First-Order Heuristics• The preferential attachment (PA) heuristic: |! " |#|! $ |
yxx prefers to connect to y if y is popular.
• First-order heuristic, only involves 1-hop neighbors.
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Second-Order Heuristics• The Adamic-Adar (AA) heuristic:∑"∈$ % ∩' (
)*+, |$ " |
yx
Weighted common neighbors;
Popular common neighbors contribute less.
• Second-order heuristic. Involves 2-hop neighbors of x and y.
• First-order and second-order heuristics can be calculated from local subgraphs around links.
1log2
1log6a
b
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High-Order Heuristics• The Katz index heuristic:∑"#$% &"|walks(., 0) = 3|
yx
Sum all walks between x and y; each walk discounted by &".
& < 1 is the discount factor
3 is the length of a walk
Longer walks contribute less.
• High-order heuristic
• Need to search the entire network.
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High-Order Heuristics• The Rooted PageRank heuristic:
Let !" be the stationary distribution of a random walker starting from x who randomly moves to one of its current neighbors with probability # or returns to x with probability 1 − #.
yx
• High-order heuristic
• Need to know the entire network and iterate until convergence.
#4
#4
#4
#4
#3
#3
#3
Use [!"]* as the likelihood of link (x,y).
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Drawbacks of Heuristic Methods• Handcrafted graph structure features, not general.
• Have strong assumptions on link formation mechanisms.
• Only work well on certain networks.
• In our paper, we proposed SEAL:
1. Automatically learn general graph structure features.
2. No assumption on network properties at all.
3. New state-of-the-art link prediction performance based on a graph neural network.
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Proposed SEAL Framework
DC
AB
AB
DC
?
?
Extract enclosing subgraphs
common neighbors = 3Jaccard = 0.6preferential attachment = 16Katz ≈0.03……
Learn graph structure features
common neighbors = 0Jaccard = 0preferential attachment = 8Katz ≈0.001……
1 (link)
0 (non-link)
Predict links
Graph neural network
• Learn “heuristics” instead of using predefined ones.
• All first-order and second-order heuristics can be learned from local enclosing subgraphs.
• How about high-order heuristics?
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A !-decaying Heuristic Theory
1. A wide range of high-order heuristics can be unified into a !-decaying heuristic framework,including Katz index, rooted PageRank, SimRank etc. => They intrinsically have the same form!
2. Under mild assumptions, all !-decaying heuristics can be well approximated from local enclosing subgraphs. => We don’t need the entire network to learn them!
3. The approximation error decreases exponentially with the subgraph size. => A small subgraph is enough!
Main results:
Poster #121 Thurs 10:45 AM -- 12:45 PM @ Room 210 & 230 AB