Deterministic construction of typical networks in network models

TL;DR

This paper presents a deterministic method to construct the most typical network states in grand canonical ensembles, with applications to real-world networks and hyperbolic graph models, advancing understanding of network typicality.

Contribution

It introduces a deterministic construction for the most typical states in grand canonical ensembles, including derandomization techniques for point processes, applicable to various network models.

Findings

Real-world networks are often close to the most typical network in the model.

The construction converges in the thermodynamic limit.

Method applies broadly to grand canonical ensembles satisfying certain conditions.

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.