[Paper Review] A playful note on spanning and surplus edges

Vlada Limic|arXiv (Cornell University)|Mar 7, 2017

Stochastic processes and statistical mechanics7 references3 citations

TL;DR

This paper extends the breadth-first-walk construction of the multiplicative coalescent to track both spanning and surplus edges in continuous-time random graphs, introducing a novel excursion mosaic framework that captures component sizes and surplus counts. It establishes that the canonical multi-graph of Bhamidi, Budhiraja, and Wang naturally emerges, and conjectures that this framework yields the scaling limit of near-critical random graphs with surplus edges, generalizing beyond the standard Aldous coalescent.

ABSTRACT

Consider a (not necessarily near-critical) random graph running in continuous time. A recent breadth-first-walk construction is extended in order to account for the surplus edge data in addition to the spanning edge data. Two different graph representations of the multiplicative coalescent, with different advantages and drawbacks, are discussed in detail. A canonical multi-graph of Bhamidi, Budhiraja and Wang (2014) naturally emerges. The presented framework should facilitate understanding of scaling limits with surplus edges for near-critical random graphs in the domain of attraction of general (not necessarily standard) eternal multiplicative coalescent.

Motivation & Objective

• To extend the breadth-first-walk construction of the multiplicative coalescent to include surplus edge data alongside spanning edge information.
• To provide a unified framework for modeling component sizes and surplus counts in continuous-time random graphs.
• To demonstrate that the canonical multi-graph of Bhamidi, Budhiraja, and Wang arises naturally within this framework.
• To lay the foundation for scaling limits of near-critical random graphs with surplus edges beyond the standard multiplicative coalescent.

Proposed method

• Introduces a graph representation using a continuous-time random graph process where edges appear at rates proportional to vertex masses.
• Defines a stochastic process $ Z^{\mathbf{x},q}(s) $ with unit negative drift and jumps at times $ \xi_{(i)}/q $, where $ \xi_{(i)} $ are order statistics of exponential random variables.
• Constructs an excursion mosaic from the process $ B^{\mathbf{x},q} $, where each component's surplus edge rate corresponds to the area of its $ \pi_l $-slice.
• Uses a Poisson point process $ N $ to mark excursions and define surplus edge counts $ Y_i(t) $ as the increase in $ N $-counts during each excursion.
• Derives that the cumulative rate of surplus edges from a vertex $ \pi_l $ is proportional to the area of its $ \pi_l $-slice at time $ q $, scaled by $ q $.
• Proposes a conjecture that the joint process $ (\mathbf{X}(t), \mathbf{Y}(t)) $, combining component sizes and surplus counts, converges to an eternal augmented multiplicative coalescent with parameters $ (\kappa, \tau, \mathbf{c}) $.

Experimental results

Research questions

• RQ1 How can the multiplicative coalescent framework be extended to simultaneously track spanning and surplus edges in random graphs?
• RQ2 What is the natural graph representation that captures both component structure and surplus edge counts in continuous time?
• RQ3 How do the surplus edge counts in a component relate to geometric features of the underlying stochastic process?
• RQ4 Can the excursion mosaic framework support scaling limits of near-critical random graphs with surplus edges beyond the standard Aldous coalescent?
• RQ5 Is there a canonical multi-graph representation that naturally emerges from the surplus edge construction and aligns with known scaling limits?

Key findings

• The canonical multi-graph of Bhamidi, Budhiraja, and Wang arises naturally from the proposed framework as a representation of the multiplicative coalescent with surplus edge data.
• The cumulative rate of surplus edges issued from a vertex $ \pi_l $ before time $ q $ is equal to the area of its $ \pi_l $-slice multiplied by $ q $, as formalized in Corollary 4(a).
• The total surplus edge rate in a component consisting of vertices $ \pi_h, \ldots, \pi_{h+r} $ is proportional to the area under the excursion of $ Z^{\mathbf{x},q} $ carrying those vertices, scaled by $ q $, as per Corollary 4(b).
• The framework supports a conjecture that the joint process $ (\mathbf{X}(t), \mathbf{Y}(t)) $, combining component sizes and surplus counts, converges to an eternal augmented multiplicative coalescent with parameters $ (\kappa, \tau, \mathbf{c}) $.
• The excursion mosaic and associated Poisson point process $ \zeta^{l;j-k} $ provide a richer geometric structure than component sizes alone, enabling a natural candidate for scaling limits in the near-critical regime.
• The framework generalizes beyond the standard multiplicative coalescent, offering a path to scaling limits for random graphs in the domain of attraction of general (non-standard) eternal multiplicative coalescents.