Finding path and cycle counting formulae in graphs with Deep Reinforcement Learning

This paper presents Grammar Reinforcement Learning (GRL), a reinforcement learning algorithm that uses Monte Carlo Tree Search (MCTS) and a transformer architecture that models a Pushdown Automaton (PDA) within a context-free grammar (CFG) framework. Taking as use case the problem of efficiently counting paths and cycles in graphs, a key challenge in network analysis, computer science, biology, and social sciences, GRL discovers new matrix-based formulas for path/cycle counting that improve computational efficiency by factors of two to six w.r.t state-of-the-art approaches. Our contributions include: (i) a framework for generating gramformers that operate within a CFG, (ii) the development of GRL for optimizing formulas within grammatical structures, and (iii) the discovery of novel formulas for graph substructure counting, leading to significant computational improvements.

Technical report: Graph Neural Networks go Grammatical

This paper proposes a new GNN design strategy. This strategy relies on Context-Free Grammars (CFG) generating the matrix language MATLANG. It enables us to ensure both WL-expressive power, substructure counting abilities and spectral properties. Applying our strategy, we design Grammatical Graph Neural Network G$ ^2$N$^2$, a provably 3-WL GNN able to count at edge-level cycles of length up to 6 and able to reach band-pass filters. A large number of experiments covering these properties corroborate the presented theoretical results.