Node isolation in large homogeneous binary multiplicative attribute graph models

TL;DR

This paper investigates the conditions under which isolated nodes appear or disappear in large homogeneous binary multiplicative attribute graph models, establishing a zero-one law that aligns with known connectivity results.

Contribution

It introduces a zero-one law for the absence of isolated nodes in MAG models, extending previous work on connectivity and degree distribution.

Findings

Zero-one law for isolated nodes in MAG models

Conditions under which isolated nodes almost surely appear or not

Alignment of zero-one law with graph connectivity results

Abstract

The multiplicative attribute graph (MAG) model was introduced by Kim and Leskovec as a mathematically tractable model of certain classes of real-world networks. It is an instance of hidden graph models, and implements the plausible idea that network structure is collectively shaped by attributes individually associated with nodes. These authors have studied several aspects of this model, including its connectivity, the existence of a giant component,its diameter and the degree distribution. This was done in the asymptotic regime when the number of nodes and the number of node attributes both grow unboundedly large, the latter scaling with the former under a natural admissibility condition. In the same setting, we explore the existence (or equivalently, absence) of isolated nodes, a property not discussed in the original paper. The main result of the paper is a {\em zero-one} law for the absence of isolated nodes; this zero-one law coincides with that obtained by Kim and Leskovec for graph connectivity (although under slightly weaker assumptions). We prove these results by applying the method of first and second moments in a non-standard way to multiple sets of counting random variables associated with the number of isolated nodes.