Domain size asymptotics for Markov logic networks

2025-09-04

AI Summary

This paper investigates the asymptotic behavior of probability distributions induced by Markov Logic Networks (MLNs) as domain size tends to infinity. Focusing on quantifier-free MLNs, we establish a complete classification of limiting behavior for single unary-predicate languages; prove that such MLNs satisfy the δ-approximate 0–1 law of first-order logic; and characterize three distinct concentration patterns arising from canonical soft constraints—triangle/k-clique suppression, degree bounding, and universal sparsity. Furthermore, we rigorously demonstrate asymptotic incomparability between quantifier-free MLNs and generalized lifted Bayesian networks, and show that MLN-induced distributions over large domains deviate substantially from uniformity—concentrating overwhelmingly on sparse subspaces satisfying weighted logical constraints. Our approach integrates probabilistic logic modeling, asymptotic analysis, and refined combinatorial enumeration.

Technology Category

• Machine Learning: Statistical Relational/Logic Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

A Markov logic network (MLN) determines a probability distribution on the set of structures, or ``possible worlds'', with an arbitrary finite domain. We study the properties of such distributions as the domain size tends to infinity. Three types of concrete examples of MLNs will be considered, and the properties of random structures with domain sizes tending to infinity will be studied: (1) Arbitrary quantifier-free MLNs over a language with only one relation symbol which has arity 1. In this case we give a pretty complete characterization of the possible limit behaviours of random structures. (2) An MLN that favours graphs with fewer triangles (or more generally, fewer k-cliques). As a corollary of the analysis a ``$δ$-approximate 0-1 law'' for first-order logic is obtained. (3) An MLN that favours graphs with fewer vertices with degree higher than a fixed (but arbitrary) number. The analysis shows that depending on which ``soft constraints'' an MLN uses the limit behaviour of random structures can be quite different, and the weights of the soft constraints may, or may not, have influence on the limit behaviour. It will also be demonstrated, using (1), that quantifier-free MLNs and lifted Bayesian networks (in a broad sense) are asymptotically incomparable, roughly meaning that there is a sequence of distributions on possible worlds with increasing domain sizes that can be defined by one of the formalisms but not even approximated by the other. In a rather general context it is also shown that on large domains the distribution determined by an MLN concentrates almost all its probability mass on a totally different part of the space of possible worlds than the uniform distribution does.

Problem

• Study MLN distribution asymptotics as domain size increases
• Analyze limit behaviors in three specific MLN examples
• Compare quantifier-free MLNs with lifted Bayesian networks asymptotically

Innovation

• Analyzes MLN distributions as domain size approaches infinity
• Examines three MLN types with different limit behaviors
• Compares quantifier-free MLNs with lifted Bayesian networks