Mamoona Ghafoor

Mamoona Ghafoor, Tatsuya Akutsu

The generation of trees with a specified tree edit distance has significant applications across various fields, including computational biology, structured data analysis, and image processing. Recently, generative networks have been increasingly employed to synthesize new data that closely resembles the original datasets. However, the appropriate size and depth of generative networks required to generate data with a specified tree edit distance remain unclear. In this paper, we theoretically establish the existence and construction of generative networks capable of producing trees similar to a given tree with respect to the tree edit distance. Specifically, for a given rooted, ordered, and vertex-labeled tree T of size n + 1 with labels from an alphabet Σ, and a non-negative integer d, we prove that all rooted, ordered, and vertex-labeled trees over Σwith tree edit distance at most d from T can be generated using a ReLU-based generative network with size O(n^3 ) and constant depth. The proposed networks were implemented and evaluated for generating trees with up to 21 nodes. Due to their deterministic architecture, the networks successfully generated all valid trees within the specified tree edit distance. In contrast, state-of-the-art graph generative models GraphRNN and GraphGDP, which rely on non-deterministic mechanisms, produced significantly fewer valid trees, achieving validation rates of only up to 35% and 48%, respectively. These findings provide a theoretical foundation towards construction of compact generative models and open new directions for exact and valid tree-structured data generation. An implementation of the proposed networks is available at https://github.com/MGANN-KU/TreeGen_ReLUNetworks.

Mamoona Ghafoor, Tatsuya Akutsu

Generation of graphs constrained by a specified graph edit distance from a source graph is important in applications such as cheminformatics, network anomaly synthesis, and structured data augmentation. Despite the growing demand for such constrained generative models in areas including molecule design and network perturbation analysis, the neural architectures required to provably generate graphs within a bounded graph edit distance remain largely unexplored. In addition, existing graph generative models are predominantly data-driven and depend heavily on the availability and quality of training data, which may result in generated graphs that do not satisfy the desired edit distance constraints. In this paper, we address these challenges by theoretically characterizing ReLU neural networks capable of generating graphs within a prescribed graph edit distance from a given graph. In particular, we show the existence of constant depth and O(n^2 d) size ReLU networks that deterministically generate graphs within edit distance d from a given input graph with n vertices, eliminating reliance on training data while guaranteeing validity of the generated graphs. Experimental evaluations demonstrate that the proposed network successfully generates valid graphs for instances with up to 1400 vertices and edit distance bounds up to 140, whereas baseline generative models fail to generate graphs with the desired edit distance. These results provide a theoretical foundation for constructing compact generative models with guaranteed validity.