Limits of multi-relational graphs

• Published in: Machine learning Vol. 112; no. 1; pp. 177 - 216
• Main Authors: Alvarado, Juan, Wang, Yuyi, Ramon, Jan
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-01-2023; Springer Nature B.V; Springer Verlag
• Subjects: Artificial Intelligence; Computer Science; Control; Data Analysis, Statistics and Probability; Entropy; Graphs; Logic; Machine Learning; Mathematics; Maximum entropy; Mechatronics; Natural Language Processing (NLP); Physics; Robotics; Simulation and Modeling; Special issue on Learning and Reasoning; Statistics; Topology; Large deviation principle; Markov logic network; Large multigraph; Graphon theory; Large Deviation Principle; Markov Logic Network; Graphon Theory
• ISSN: 0885-6125; 1573-0565
• DOI: 10.1007/s10994-022-06281-x

Graphons are limits of large graphs. Motivated by a theoretical problem from statistical relational learning, we develop a generalization of basic results from graphon theory into the “multi-relational” setting. We show that their multi-relational counterparts, which we call multi-relational graphons, are analogically limits of large multi-relational graphs. We extend the cut-distance topology for graphons to multi-relational graphons and prove its compactness and the density of multi-relational graphs in this topology. In turn, compactness enables to prove the large deviation principle for Multi-Relational Graphs (LDP) which enables to prove the most typical random graphs constrained by marginal statistics converge asymptotically to constrained multi-relational graphons with maximum entropy. We show the equivalence between a restricted version of Markov Logic Network and Multi-Relational Graphons with maximum entropy.