Connectivity and Structure in Large Networks

TL;DR

This paper systematically compares various classes of random graph models, revealing fundamental incompatibilities between bounded expected degrees, near connectivity, and geometric constraints in large networks.

Contribution

It introduces generalized classes of random graph models and establishes relationships among them, highlighting key limitations and incompatibilities.

Findings

No random graph model can simultaneously have bounded expected degrees, be asymptotically almost connected, and satisfy locality and invariance.

The study extends the understanding of how different random graph models relate and differ.

It provides a framework for comparing the modeling strength of various large network models.

Abstract

Large real-life complex networks are often modeled by various random graph constructions and hundreds of further references therein. In many cases it is not at all clear how the modeling strength of differently generated random graph model classes relate to each other. We would like to systematically investigate such issues. Our approach was originally motivated to capture properties of the random network topology of wireless communication networks. We started some investigations, but here we elevate it to a more general level that makes it possible to compare the strength of different classes of random network models. Specially, we introduce various classes of random graph models that are significantly more general than the ones that are usually treated in the literature, and show relationships among them. One of our main results is that no random graph model can fall in the following three classes at the same time: (1) random graph models with bounded expected degrees; (2) random graph models that are asymptotically almost connected; (3) an abstracted version of geometric random graph models with two mild restrictions that we call locality and name invariance. In other words, in a mildly restricted, but still very general, class of generalized geometric-style models the requirements of bounded expected degrees and asymptotic almost connectivity are incompatible.