A dynamical approach to spanning and surplus edges of random graphs

Abstract

Consider a finite inhomogeneous random graph evolving in continuous time, where each vertex is assigned a mass, and an edge between any pair of vertices appears at a rate proportional to the product of their masses. The process tracking the evolution of component sizes evolves according to the multiplicative coalescent dynamic and can be encoded using the simultaneous breadth-first walk introduced by Limic (2019). We extend this encoding to incorporate surplus edge data within each connected component. Two distinct graph-based representations of the multiplicative coalescent, each with its own advantages and limitations, are analyzed in detail. In particular, a canonical multigraph introduced by Bhamidi, Budhiraja and Wang (2014), which is naturally connected to the augmented multiplicative coalescent, emerges from our framework. We demonstrate that a transformation of the simultaneous breadth-first walk, supplemented with an additional and independent source of randomness, encodes the full dynamics of the augmented multiplicative coalescent.