This document presents the Geom-GCN model for graph neural networks. Geom-GCN maps graphs to continuous latent spaces to build structural neighborhoods for message passing aggregation. It uses a bi-level aggregator operating on these structural neighborhoods to update node representations while maintaining permutation invariance. Geom-GCN designs geometric relationships in Euclidean and hyperbolic spaces to define neighborhoods. It achieves state-of-the-art performance on benchmarks while addressing limitations of existing message passing approaches in capturing structure and long-range dependencies.

1. Nguyen Thanh Sang
Network Science Lab
Dept. of Artificial Intelligence
The Catholic University of Korea
E-mail: sang.ngt99@gmail.com
2023-03-24

2. 1
SOTA problems
• In a layer of MPNNs, each node sends its feature representation, a “message”, to the nodes in
its neighborhood; and then updates its feature representation by aggregating all “messages”
received from the neighborhood.
=> the aggregators lose the structural information of nodes in neighborhoods.

3. 2
SOTA problems
• In MPNNs,
• the neighborhood is defined as the set of all neighbors one hop away or all neighbors up
to r hops away.
• After n layers, all the embeddings feature of nodes are similar.
• They cannot capture the important features from distant but informative nodes.:
=> the aggregators lack the ability to capture long-range dependencies in disassortative
Graphs.

4. 3
SOTA problems
• Several existing works attempt to learn some implicit structure-like information to distinguish
different neighbors when aggregating features:
• Learn weights on “messages” from different neighbors by using attention mechanisms
and node and/or edge attributes
• CCN utilizes a covariance architecture to learn structure-aware representations.
 How about utilize geometry structure?

5. 4
Contributions
• Map a graph to a continuous latent, and use the geometric
relationships defined in the latent space to build structural
neighborhoods for aggregation.
• A bi-level aggregator operating on the structural
neighborhoods to update the feature representations of
nodes in graph neural networks, which are able to
guarantee permutation invariance for graph-structured
data.
 extracts more structural information of the graph and
aggregate feature representations from distant nodes via
mapping them to neighborhoods defined in the latent
space.
• Design particular geometric relationships to build the
structural neighborhood in Euclidean and hyperbolic
embedding space respectively.

6. 5
Geometric Aggregation
• Node embedding: maps the nodes in a graph to a latent continuous space:
• Structural neighborhood:
• Bi-level aggregation:

7. 6
Distinguish the non-isomorphic graphs
• Mean and maximum aggregators fail to distinguish the two different graphs.
• Structural neighborhood in aggregation.
• Different mapping function

8. 7
Geom-GCN
• The geometric operator τ as four relationships of the relative positions between two nodes
in a 2-D Euclidean or hyperbolic space.:
• Bi-level aggregation:
• Overall bi-level aggregation of Geom-GCN:
• Specify embedding methods to create suitable latent spaces where particular topology
patterns (e.g., hierarchy) are preserved:
• Isomap is a widely used isometry embedding method, by which distance patterns
(lengths of shortest paths) are preserved explicitly in the latent space.
• Poincare embedding and struc2vec can create particular latent spaces that preserve
hierarchies and local structures in a graph, respectively.

9. 8
Datasets

10. 9
Experiments
• Geom-GCN achieves state-of-the-art performance.

11. 10
Experiments
• Geom-GCN with two neighborhoods aggregate more irrelevant “messages” than Geom-GCN with
only one neighborhood, and the irrelevant “messages” adversely affect the performance

12. 11
Experiments
• One can observe that several combinations achieve better performance than Geom-GCN with
neighborhoods built by only one embedding space;
• There are also many combinations that have bad performance.

13. 12
Experiments
• Time complexity is very important for graph neural networks because real-world graphs are
always very large.
• All nodes distribute radially in the figure indicates the proposed model learn graph’s hierarchy by
Poincare embedding.

14. 13
Conclusions
• Tackling the two major weaknesses of existing message-passing neural networks over
graphs losses of :
• discriminative structures and
• long-range dependencies.
• Bridge a discrete graph to a continuous geometric space via graph embedding.
• Exploit the principle of convolution: spatial aggregation over a meaningful space.

15. 14