# Domain size asymptotics for Markov logic networks

arXiv: https://arxiv.org/abs/2509.04192 · HTML: https://mlsanity.com/paper/2509.04192/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 37 references that are in the map.
- First posted on arXiv 4 Sep 2025.

## Builds on (most cited first, 9 of 9)

- [An asymptotic analysis of probabilistic logic programming, with implications for expressing projective families of distributions](https://mlsanity.com/paper/2102.08777/) (2021)
- [Zero-One Laws of Graph Neural Networks](https://mlsanity.com/paper/2301.13060/) (2023)
- [Almost Surely Asymptotically Constant Graph Neural Networks](https://mlsanity.com/paper/2403.03880/) (2024)
- [Scaling the weight parameters in Markov logic networks and relational logistic regression models](https://mlsanity.com/paper/2103.15140/) (2021)
- [Asymptotic elimination of partially continuous aggregation functions in directed graphical models](https://mlsanity.com/paper/2106.10122/) (2021)
- [Probabilities of the Third Type: Statistical Relational Learning and Reasoning with Relative Frequencies](https://mlsanity.com/paper/2202.10367/) (2022)
- [On the relative asymptotic expressivity of inference frameworks](https://mlsanity.com/paper/2204.09457/) (2022)
- [Random expansions of trees with bounded height](https://mlsanity.com/paper/2410.11775/) (2024)
- [Convergence Laws for Extensions of First-Order Logic with Averaging](https://mlsanity.com/paper/2504.14270/) (2025)

## Built on by (most cited first, 3 of 3)

- [Random coloured digraphs defined by a Markov logic network](https://mlsanity.com/paper/2606.23715/) (2026)
- [A concentration result for multilayer feedforward neural networks](https://mlsanity.com/paper/2608.15335/) (2026)
- [Exponential random graph models with soft clique constraints](https://mlsanity.com/paper/2608.30869/) (2026)

## Abstract (arXiv)

A Markov logic network (MLN) mathbbM determines a probability distribution mathbbPn^mathbbM on the set Wn of structures, or ``possible worlds'', with domain 1, ldots, n. We study the properties of such distributions as n tends to infinity. We show that with mild assumptions on an MLN mathbbM with one soft constraint with an arbitrary positive weight the distribution mathbbPn^mathbbM will behave quite differently from the uniform distribution mathbbPnuni on Wn for all sufficiently large n. For a language with only one relation symbol R which has arity 1 we give an almost complete characterization of the possible asymptotic behaviours of mathbbPn^mathbbM as n → ∞, where mathbbM may be any MLN for this language. The asymptotic behaviour depends on the soft constraints and weights of the MLN. This characterization is used to show that if the language under consideration contains at least one relation symbol of arity 1 then the following holds: (a) There is an MLN mathbbM such that for every lifted Bayesian network (LBN) mathbbG there are infinitely many n such that mathbbM and mathbbG determine different distributions on Wn. (b) There is an LBN mathbbG such that for every MLN mathbbM there are infinitely many n such that mathbbG and mathbbM determine different distributions on Wn. We also show that, in the limit, the weight dimension and the domain size dimension may behave completely differently.