Connectedness matters: Construction and exact random sampling of connected graphs

TL;DR

This paper introduces a novel, efficient method for exactly sampling connected graphs with a specified degree sequence, applicable to both simple graphs and multigraphs, improving upon existing techniques.

Contribution

It extends a recent independent sampling approach to include connectedness constraints and provides a direct construction algorithm for connected graphs with given degrees.

Findings

Enforcing connectedness significantly changes network properties.

The method efficiently samples connected graphs with specified degrees.

Application to real-world networks demonstrates practical utility.

Abstract

We describe a new method for the random sampling of connected networks with a specified degree sequence. We consider both the case of simple graphs and that of loopless multigraphs. The constraints of fixed degrees and of connectedness are two of the most commonly needed ones when constructing null models for the practical analysis of physical or biological networks. Yet handling these constraints, let alone combining them, is non-trivial. Our method builds on a recently introduced novel sampling approach that constructs graphs with given degrees independently (unlike edge-switching Markov Chain Monte Carlo methods) and efficiently (unlike the configuration model), and extends it to incorporate the constraint of connectedness. Additionally, we present a simple and elegant algorithm for directly constructing a single connected realization of a degree sequence, either as a simple graph or a multigraph. Finally, we demonstrate our sampling method on a realistic scale-free example, as well as on degree sequences of connected real-world networks, and show that enforcing connectedness can significantly alter the properties of sampled networks.