Limit of connected multigraph with fixed degree sequence
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 cyclebreaking algorithm, and the stickbreaking 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 arXiv:
 arXiv:2112.07725
 Bibcode:
 2021arXiv211207725B
 Keywords:

 Mathematics  Probability;
 60D05 60F05