Limit of connected multigraph with fixed degree sequence

TL;DR

This paper investigates the limiting behavior of uniform connected multigraphs with fixed degree sequences and surplus, establishing convergence to inhomogeneous continuum random graphs under certain conditions.

Contribution

It introduces a framework for understanding the scaling limits of connected multigraphs with fixed degrees and surplus, extending previous models and utilizing novel algorithms.

Findings

Convergence of $( ext{D},k)$-graphs to $( ext{P},k)$-graphs or $( ext{Theta},k)$-ICRG.

Development of cycle-breaking and stick-breaking algorithms for these graphs.

Application of results to multiplicative graph models.

Abstract

Motivated by the scaling limits of the connected components of the configuration model, we study uniform connected multigraphs with fixed degree sequence $\mathcal{D}$ and with surplus $k$. We call those random graphs $(\mathcal{D},k)$-graphs. We prove that, for every $k\in \mathbb N$, under natural conditions of convergence of the degree sequence, ($\mathcal{D},k)$-graphs converge toward either $(\mathcal{P},k)$-graphs or $(\Theta,k)$-ICRG (inhomogeneous continuum random graphs). We prove similar results for $(\mathcal{P},k)$-graphs and $(\Theta,k)$-ICRG, which have applications to multiplicative graphs. Our approach relies on two algorithms, the cycle-breaking algorithm, and the stick-breaking construction of $\mathcal{D}$-tree that we introduced in a recent paper arXiv:2110.03378. From those algorithms we deduce a biased construction of $(\mathcal{D},k)$-graph, and we prove our results by studying this bias.