Graph Generation with $K^2$-trees

TL;DR

This work introduces a novel graph generation method leveraging $K^2-tree representation, originally designed for lossless graph compression, by introducing a Transformer-based architecture designed to generate the sequence by incorporating a specialized tree positional encoding scheme.

Abstract

Generating graphs from a target distribution is a significant challenge across many domains, including drug discovery and social network analysis. In this work, we introduce a novel graph generation method leveraging $K^2$-tree representation, originally designed for lossless graph compression. The $K^2$-tree representation {encompasses inherent hierarchy while enabling compact graph generation}. In addition, we make contributions by (1) presenting a sequential $K^2$-treerepresentation that incorporates pruning, flattening, and tokenization processes and (2) introducing a Transformer-based architecture designed to generate the sequence by incorporating a specialized tree positional encoding scheme. Finally, we extensively evaluate our algorithm on four general and two molecular graph datasets to confirm its superiority for graph generation.

Figures (16)

• Figure 1: (Left) Various representations used for graph generation.(Right) Comparing graph generative methods in terms of used graph representation. The comparison is made with respect to a method being hierarchical (H), able to handle attributed graphs (A), and domain-agnostic (DA).
• Figure 2: $K^{2}$--tree with $K=2$. The $K^{2}$--tree describes the hierarchy of the adjacency matrix iteratively being partitioned to $K\times K$ submatrices. It is compact due to summarizing any zero-filled submatrix with a size larger than $1\times 1$ (shaded in grey) by a leaf node $u$ with label $x_{u}=0$.
• Figure 3: Illustration of the sequential representation for $K^{2}$--tree. The shaded parts of the adjacency matrix $A$ and the $K^{2}$--tree $\mathcal{T}$ denote redundant parts, which are further pruned, while the purple-colored parts of $A$ and $\mathcal{T}$ denote non-redundant parts. Also, same-colored tree-nodes of pruned $K^{2}$--tree are grouped and tokenized into the same colored parts of the sequence $\bm{y}$.
• Figure 4: Illustration of the tree-node positions of $K^{2}$--tree. The shaded parts of the adjacency matrix denote redundant parts, e.g., $p_{u}<q_{u}$. Additionally, colored elements correspond to tree-nodes of the same color and the same-colored tree-edges signify the root-to-target downward path. Blue and red tuples denote the order in the first and second levels, respectively. The tree node $u$ is non-redundant as $p_u>q_u$ while $v$ is redundant as $p_{v}<q_{v}$.
• Figure 5: An example of featured $K^{2}$--tree representation. The shaded parts of the adjacency matrix and $K^{2}$--tree denote the redundant parts. The black-colored tree-nodes denote the normal tree-nodes with binary attributes while other-colored feature elements in the adjacency matrix $A$ denote the same-colored featured tree-nodes and sequence elements. The node features (i.e., C and N) and edge feature (i.e., single bond $-$) of the molecule are represented within the leaf nodes.