The paper introduces a general theoretical framework for fibrations on graphs labeled by a commutative monoid, extending the classic theory of graph fibrations to weighted and algebraically labeled graphs. It also accommodates approximate fibrations and demonstrates how this framework can be used to compress arbitrary neural networks, including CNNs, providing a solid theoretical basis for recent findings on fibration symmetries in geometric deep learning.
By Paolo Boldi
Posted by Ameya Velingker, Research Scientist, Google Research, and Balaji Venkatachalam, Software Engineer, Google Graphs , in which objects and their relations are represented as nodes (or vertices) and edges (or links) between pairs of nodes, are ubiquitous in computing and machine learning (ML). For example, social networks, road networks, and molecular structure and interactions are all domains in which underlying datasets have a natural graph structure.
By Google AI
The paper establishes a precise mathematical link between graph surgery and the do‑operator in deterministic acyclic structural causal models. It shows that deleting arrows in a graph corresponds exactly to replacing the associated mechanisms with constants, proving that “Graph(F^\iota)=Surg(Graph(F),T_\iota)”. The authors further characterize when this equality holds for the full graph, define the intervened model, and demonstrate how sequential interventions combine, concluding that an outcome depends only on interventions at its actual dependency ancestors.
By Satpreet Makhija
arXiv:2607. 01057v1 Announce Type: cross Abstract: We study a broad class of graphical models whose independencies correspond to vertex separation in mixed graphs with directed, undirected, and bidirected edges, that are capable of encoding independence structures arising from feedback, latent and selection mechanisms.
By Christopher Meek, Kayvan Sadeghi
arXiv:2603. 14846v3 Announce Type: replace Abstract: We define an information-complexity property for aggregation functions, capturing a vast range of practical aggregations, and prove that any Message-Passing Graph Neural Network (MP-GNN) model with such aggregations induces only a polynomial number of equivalence classes on all graphs - while the number of non-isomorphic graphs is super-exponential (in number of vertices).
By Eran Rosenbluth
arXiv:2606. 07728v1 Announce Type: new Abstract: It is well established that ReLU networks define continuous piecewise-linear functions, and that their linear regions are polyhedra in the input space.
By Blake B. Gaines, Jinbo Bi
arXiv:2607. 12026v1 Announce Type: cross Abstract: Finite groups are rigid algebraic objects, whose Cayley graphs expose a rich network geometry through which group-theoretic structure can be measured, compared, and learned.
By Rashid Barket, Enrico Grimaldi, Yacoub Hendi, Edward Hirst, Adam Onus, Harmeet Singh
arXiv:2605. 00725v2 Announce Type: replace Abstract: Topological neural networks have emerged as effective tools for modeling higher-order relational structures beyond pairwise graphs, including hypergraphs, simplicial complexes, and cell complexes.
By Jiawen Chen, Qi Shao, Zhiqiang Ge, Duxin Chen, Wenwu Yu
arXiv:2607. 23500v1 Announce Type: cross Abstract: Razborov's flag algebra method is a powerful tool for proving asymptotic inequalities in extremal graph theory, often reducing the task to finding a finite certificate by semidefinite programming.
By Gyeongwon Jeong, Seonghun Park, Jihoon Hyun, Sang-il Oum, Hongseok Yang
arXiv:2412. 03008v2 Announce Type: replace-cross Abstract: Local/seeded clustering aims to find a compact cluster near the given starting instances.
By Zihao Li, Dongqi Fu, Hengyu Liu, Jingrui He
Positional encodings (PEs) enhance the power of graph neural networks (GNNs), both theoretically and empirically. Two of the most popular families of PEs - spectral (e.