Depth First Exploration of a Configuration Model

TL;DR

This paper presents a novel algorithm for constructing random graphs with a given degree sequence and analyzes the depth first exploration process, revealing deterministic limiting behavior and properties of large paths in supercritical regimes.

Contribution

It introduces a new algorithm for uniform random graphs with prescribed degrees and characterizes the exploration process in the supercritical regime, including limiting profiles and path length bounds.

Findings

Deterministic limiting profile of the DFS exploration process in supercritical graphs.

Existence of a macroscopic simple path with a lower bound on its length.

Explicit solution to the differential equations governing degree distribution evolution.

Abstract

We introduce an algorithm that constructs a random uniform graph with prescribed degree sequence together with a depth first exploration of it. In the so-called supercritical regime where the graph contains a giant component, we prove that the renormalized contour process of the Depth First Search Tree has a deterministic limiting profile that we identify. The proof goes through a detailed analysis of the evolution of the empirical degree distribution of unexplored vertices. This evolution is driven by an infinite system of differential equations which has a unique and explicit solution. As a byproduct, we deduce the existence of a macroscopic simple path and get a lower bound on its length.