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On the block size spectrum of a class of exchangeable dynamic random graphs

Frederic Alberti, Florin Boenkost, Fernando Cordero

TL;DR

This work develops a dynamic theory for block sizes in a class of exchangeable dynamic random graphs, focusing on the beta-dynamic case with Θ = $Beta(\alpha,\beta)$ × $\delta_1$ and $\theta=0$. It establishes a dynamic law of large numbers for the block-size spectrum, described by a system of ODEs for the limiting frequencies $\frak c_{t,i}$, and proves a nonstandard functional limit theorem for fluctuations, governed by a generalized Ornstein–Uhlenbeck process driven by a Lévy measure with density $u^{\alpha-3}$. A core methodological contribution is a Poisson integral representation that yields a tractable coupling between the drift and stochastic components, enabling precise control of both drift convergence and fluctuations. The results connect to the Beta-coalescent literature and reveal exponential decay of block frequencies under the slowing time $n^{\alpha-1}$, highlighting the momentum effect that larger blocks coalesce more rapidly in this dynamic setting.

Abstract

In this work we introduce the dynamic $Θ$-random graph and the associated $Θ$-coalescent with momentum. Dynamic $Θ$-random graphs are a subclass of exchangeable and consistent random graph processes, parametrised by a measure $Θ$ on $[0,1]\times (0,1]$, inspired by the classic $Λ$-coalescent from mathematical population genetics. The $Θ$-coalescent with momentum accounts for the small connected components of this graph; in contrast to the underlying random graph it is exchangeable but not consistent. Our main results specialise on the case where $Θ$ is the product of a beta measure and a Dirac mass at $1$. We prove a dynamic law of large numbers for the block size spectrum, which tracks the numbers of blocks containing $1,...,d$ elements. On top of that, we provide a functional limit theorem for the fluctuations. The limit process satisfies a stochastic differential equation of Ornstein-Uhlenbeck type.

On the block size spectrum of a class of exchangeable dynamic random graphs

TL;DR

This work develops a dynamic theory for block sizes in a class of exchangeable dynamic random graphs, focusing on the beta-dynamic case with Θ = × and . It establishes a dynamic law of large numbers for the block-size spectrum, described by a system of ODEs for the limiting frequencies , and proves a nonstandard functional limit theorem for fluctuations, governed by a generalized Ornstein–Uhlenbeck process driven by a Lévy measure with density . A core methodological contribution is a Poisson integral representation that yields a tractable coupling between the drift and stochastic components, enabling precise control of both drift convergence and fluctuations. The results connect to the Beta-coalescent literature and reveal exponential decay of block frequencies under the slowing time , highlighting the momentum effect that larger blocks coalesce more rapidly in this dynamic setting.

Abstract

In this work we introduce the dynamic -random graph and the associated -coalescent with momentum. Dynamic -random graphs are a subclass of exchangeable and consistent random graph processes, parametrised by a measure on , inspired by the classic -coalescent from mathematical population genetics. The -coalescent with momentum accounts for the small connected components of this graph; in contrast to the underlying random graph it is exchangeable but not consistent. Our main results specialise on the case where is the product of a beta measure and a Dirac mass at . We prove a dynamic law of large numbers for the block size spectrum, which tracks the numbers of blocks containing elements. On top of that, we provide a functional limit theorem for the fluctuations. The limit process satisfies a stochastic differential equation of Ornstein-Uhlenbeck type.
Paper Structure (13 sections, 12 theorems, 170 equations, 3 figures)

This paper contains 13 sections, 12 theorems, 170 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model and main results
  3. Two classes of exchangeable random graph processes
  4. A motivation: social bubbles
  5. $\Lambda$-dynamic random graphs and $\Lambda$-coalescents with momentum
  6. Results for the beta-dynamic random graph
  7. Proofs
  8. Integral/Poisson representation
  9. Structure of the proof / Heuristics
  10. Rigorous argument
  11. The functional limit theorem
  12. Some auxiliary results
  13. Derivation of identity \ref{['samplingformula1']}

Key Result

Theorem 2.5

For all $\varepsilon > 0$ and $T > 0$, where ${\mathfrak c}_t = ({\mathfrak c}_{t,1}^{}, \ldots, {\mathfrak c}_{t,d}^{})_{t \geqslant 0}$ solves the following system of ordinary differential equations with initial condition ${\mathfrak c}_{0,i} = \delta_{i,1}$.

Figures (3)

  • Figure 1: An illustration of two consecutive transitions of the $\Theta$-graph on $16$ vertices. Left: a given state of the graph. Independently of each other, vertices are coloured green with some probability $u$. Middle: each pair of coloured vertices is independently joined by an edge with some probability $q$. The same procedure is repeated to produce a second transition (middle to right). The top panel shows the special case $q = 1$ (which will be our focus in what follows). In the bottom panel, $q < 1$. In the case $q=1$, the associated $\Theta$ coalescent with momentum (Def. \ref{['def:multlambda']}) performs the transitions: $\text{ } \qquad \qquad \qquad \{ \{1\}, \{2,13\}, \{\textcolor{darkgreen}{3},5,6 \}, \{4 \}, \{7\}, \{8\}, \{9,14,15,\textcolor{darkgreen}{16}\}, \{10\}, \{\textcolor{darkgreen}{11}\}, \{\textcolor{darkgreen}{12}\}$$\text{ } \qquad \qquad \qquad \qquad \to \{ \{1\}, \{2,\textcolor{darkgreen}{13}\}, \{3,5,6,9,11,12,14,15,16\}, \{\textcolor{darkgreen}{4}\}, \{7\}, \{\textcolor{darkgreen}{8}\}, \{10\} \}$$\text{ } \qquad \qquad \qquad \qquad \qquad \to \{ \{1\}, \{2,4,8,13\}, \{3,5,6,9,11,12,14,15,16\}, \{7\}, \{10\} \}.$
  • Figure 2: Left: a graphical representation of the occurrence of social gatherings and the friendship relations established on their occasion in a community consisting of $9$ individuals; social gatherings occur at times $t_1, t_2, t_3$; participants are depicted as vertices of a graph; pairs of participants are connected by an edge if they become friends during the meeting. Right: the social network process associated to individual $3$ (resp. $7$) corresponding to the social gatherings in the left panel; the social gatherings are superposed on the diagram: at the times of a meeting, participants are depicted in dark red or dark blue; light red (resp. light blue) vertices represent individuals that belong to the social network of individual $3$ (resp. $7$), but that are not part of the meeting; light red and light blue edges represent connections established in previous meetings. At any time, vertices that are coloured red (resp. blue) belong to the social network of individual $3$ (resp. $7$).
  • Figure 3: A plot of ${\mathfrak c}_{t,i}$ for $i=1,...,4$ for $\alpha=0.5$. The right hand side is a scaled version of the first picture excluding the blocks which contain exactly one element.

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.1: Comparison to other random graph models
  • Definition 2.4
  • Theorem 2.5: A dynamic law of large numbers
  • Remark 2.2
  • Corollary 2.6
  • Remark 2.3
  • Theorem 2.7: Law of large numbers --- $L_2$-version
  • ...and 23 more