Network evolution with mesoscopic delay

TL;DR

The paper advances the theory of dynamic network models by incorporating mesoscopic delays in preferential attachment on growing trees. It develops stochastic-approximation tools to establish local weak convergence to a sin-tree with fringe distribution $\boldsymbol{\varpi}_{\mathrm{BP}_f}$, and derives explicit limiting degree distributions $p_k(f)$, as well as spectral convergence results. In the affine-linear case, it further proves second-order fluctuations of degree counts and provides detailed root-degree asymptotics, revealing regime-dependent scaling tied to the delay tail via $\beta$ and $\theta=1/(2+\alpha)$. The work connects local and global graph properties under delayed information, offering a robust framework that remains largely insensitive to mesoscopic delays and outlining avenues for future exploration, including non-tree settings and macroscopic-delay regimes handled in a companion paper.

Abstract

Owing to the influence of real-world networks both in science and society, numerous mathematical models have been developed to understand the structure and evolution of these systems, particularly in a temporal context. Recent advancements in fields like distributed cyber-security and social networks have spurred the creation of probabilistic models of evolution, where individuals make decisions based on only partial information about the network's current state. This paper seeks to explore models incorporating network delay, where new participants receive information from a time-lagged snapshot of the system. In the context of mesoscopic network delays, we develop probabilistic tools built on stochastic approximation to understand asymptotics of both local functionals, such as local neighborhoods and degree distributions, as well as global properties, such as the evolution of the degree of the network's initial founder. A companion paper explores the regime of macroscopic delays in the evolution of the network.

Figures (1)

• Figure 1: Simulated Linear Preferential Attachment trees in the mesoscopic regime with $\beta =1/2$, attachment function taken to be the pure linear preferential attachment function $f(k) = k$ with $\theta=1/2$ and different delay distributions arranged with increasing gradation of mass towards the right tail. Each simulation is on a network size of $n = 50000$ nodes with different delay distributions. All four settings satisfy the assumptions of Theorem \ref{['thm:meso-local']} (i.e.,\ref{['eqn:815']}) and thus have the same local limit as the no delay regime in a large network limit. Delay regime (D) does not satisfy the conditions of Theorem \ref{['thm:meso-max-degree']}--(i), and thus, for example, the root degree has a different scaling than the setting without delay.

Theorems & Definitions (31)

• Definition 1.1: Network evolution with delay
• Definition 2.2: Continuous time branching process (CTBP) jagers-ctbp-bookathreya1972nerman1981convergence
• Definition 2.3: Stable age distribution, jagers-ctbp-bookjagers-nerman-1jagers-nerman-2nerman1981convergence
• Lemma 2.5
• Remark 2.6
• Theorem 2.7: Local weak limit, mesoscopic regime
• Corollary 2.8
• Corollary 2.9
• Theorem 2.10: Second order fluctuations of degree counts
• Remark 2.11