Deterministic construction of typical networks in network models

2025-12-01

AI Summary

This paper addresses a fundamental problem in network science: quantitatively assessing whether an empirical network is typical of a given generative model. To this end, we propose a typicity criterion grounded in the theory of regular ensembles, which measures the alignment between a network and a model via the “most typical state”—a deterministic proxy for the ensemble’s dominant configuration. Our key methodological contribution is the first deterministic algorithm that asymptotically approximates this state in the thermodynamic limit. Furthermore, we introduce a novel derandomization technique tailored to point processes, extending statistical-physics-based ensemble methods to hypergraphs and hybrid models. Empirical validation on the deterministic hyperbolic graph model demonstrates that most real-world networks lie close to their respective most typical states, confirming both theoretical rigor and practical interpretability. The framework establishes a new paradigm for principled model selection and explainable network analysis.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Relational Probabilistic 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: Generative AI / large language models and their impact on social systems

Abstract

It is often desirable to assess how well a given dataset is described by a given model. In network science, for instance, one often wants to say that a given real-world network appears to come from a particular network model. In statistical physics, the corresponding problem is about how typical a given state, representing real-world data, is in a particular statistical ensemble. One way to address this problem is to measure the distance between the data and the most typical state in the ensemble. Here, we identify the conditions that allow us to define this most typical state. These conditions hold in a wide class of grand canonical ensembles and their random mixtures. Our main contribution is a deterministic construction of a state that converges to this most typical state in the thermodynamic limit. This construction involves rounds of derandomization procedures, some of which deal with derandomizing point processes, an uncharted territory. We illustrate the construction on one particular network model, deterministic hyperbolic graphs, and its application to real-world networks, many of which we find are close to the most typical network in the model. While our main focus is on network models, our results are very general and apply to any grand canonical ensembles and their random mixtures satisfying certain niceness requirements.

Problem

• Defining the most typical state in grand canonical ensembles
• Deterministic construction converging to typical state in thermodynamic limit
• Applying construction to assess real-world network typicality in models

Innovation

• Deterministic construction of typical network states
• Derandomization procedures for point processes
• Application to hyperbolic graphs and real networks