Degree corrected stochastic block model: excursion representation

TL;DR

This paper develops a new excursion representation for analyzing the connected components of the degree-corrected stochastic block model, a complex network model with community structure and degree heterogeneity, extending existing methods to non-rank-one models.

Contribution

It introduces a novel random field encoding the component structure of DCSBM and a new composition operator, advancing the analysis of non-rank-one stochastic block models.

Findings

Constructed a random field encoding for DCSBM components

Reformulated a multidimensional minimization problem into a single process

Extended excursion representation techniques to non-rank-one models

Abstract

This is the first of two complementary works in which we analyze the connected components of the degree-corrected stochastic block model (DCSBM). Our model is a random graph with an underlying community structure and degree in-homogeneity. It belongs to a class of non-rank one models. The scaling limit of connected component sizes in the near-critical regime, obtained by Konarovskyi and Limic (2021) for a subfamily of DCSBM, is non-trivially different (although related to) the standard eternal multiplicative coalescent of Aldous (1997). The Aldous (1997) excursion representation combined with weak convergence approach to the scaling limits of connected components of random graphs proved to be much more difficult (and therefore rare) for non rank-one models. In this work we show how to build a random field encoding for the connected component structure of DCSBM, in part relying on the theory of Chaumont and Marolleau (2020). We then show how one can, under additional assumptions, reformulate the minimization problem stated in terms of multidimensional first hitting times into an equivalent minimization problem stated for a single real-valued stochastic process. This reformulation relies on a novel composition-like operator on pairs of compatible non-decreasing rcll functions, which might be of independent interest.