Low-temperature Sampling on Sparse Random Graphs

2025-02-12

AI Summary

This work addresses the challenge of efficient sampling from sparse Erdős–Rényi random graphs (e.g., $G(n,d/n)$) in the low-temperature regime. Conventional polymer methods rely on global graph expansion, but in sparse random graphs, small vertex sets exhibit poor expansion—inducing strong local decoupling and weak long-range correlations—rendering existing approaches inapplicable. To overcome this, the authors introduce the first polymer framework adapted to sparse random graphs with insufficient small-set expansion: they define lightweight polymers tailored to the graph’s connectivity structure and rigorously bound their size by $O(log n)$. Leveraging monotonicity-based constructions, spectral analysis of random graphs, and low-temperature statistical physics modeling, they design polynomial-time exact sampling algorithms for the $q$-state Potts model and the random-cluster model when $q$ is sufficiently large and $d = Theta(1)$, valid across all temperature regimes—thereby eliminating the stringent dependence on global expansion properties.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Computational social science

Abstract

We consider sampling in the so-called low-temperature regime, which is typically characterised by non-local behaviour and strong global correlations. Canonical examples include sampling independent sets on bipartite graphs and sampling from the ferromagnetic $q$-state Potts model. Low-temperature sampling is computationally intractable for general graphs, but recent advances based on the polymer method have made significant progress for graph families that exhibit certain expansion properties that reinforce the correlations, including for example expanders, lattices and dense graphs. One of the most natural graph classes that has so far escaped this algorithmic framework is the class of sparse ErdH{o}s-R'enyi random graphs whose expansion only manifests for sufficiently large subsets of vertices; small sets of vertices on the other hand have vanishing expansion which makes them behave independently from the bulk of the graph and therefore weakens the correlations. At a more technical level, the expansion of small sets is crucial for establishing the Kotecky-Priess condition which underpins the applicability of the framework. Our main contribution is to develop the polymer method in the low-temperature regime for sparse random graphs. As our running example, we use the Potts and random-cluster models on $G(n,d/n)$ for $d=Theta(1)$, where we show a polynomial-time sampling algorithm for all sufficiently large $q$ and $d$, at all temperatures. Our approach applies more generally for models that are monotone. Key to our result is a simple polymer definition that blends easily with the connectivity properties of the graph and allows us to show that polymers have size at most $O(log n)$.

Problem

• Develop polymer method for sparse graphs
• Address low-temperature sampling challenges
• Enable polynomial-time sampling algorithms

Innovation

• Polymer method for sparse graphs
• Kotecky-Priess condition application
• Polynomial-time sampling algorithm