[Paper Review] Fluctuation bounds for continuous time branching processes and nonparametric change point detection in growing networks

Sayan Banerjee, Shankar Bhamidi+1 more|arXiv (Cornell University)|Aug 7, 2018

Stochastic processes and statistical mechanics38 references4 citations

TL;DR

This paper develops continuous time branching processes (CTBP) to derive quantitative error bounds for network functionals in growing network models, enabling nonparametric change point detection. It establishes convergence rates for degree distributions under abrupt changes in attachment mechanisms, even in the 'quick big bang' regime where change occurs early (at $ n^\gamma $, $ 0<\gamma<1 $).

ABSTRACT

Motivated by applications, both for modeling real world systems as well as in the study of probabilistic systems such as recursive trees, the last few years have seen an explosion in models for dynamically evolving networks. The aim of this paper is two fold: (a) develop mathematical techniques based on continuous time branching processes (CTBP) to derive quantitative error bounds for functionals of a major class of these models about their large network limits; (b) develop general theory to understand the role of abrupt changes in the evolution dynamics of these models using which one can develop non-parametric change point detection estimators. In the context of the second aim, for fixed final network size $n$ and a change point $ au(n) < n$, we consider models of growing networks which evolve via new vertices attaching to the pre-existing network according to one attachment function $f$ till the system grows to size $ au(n)$ when new vertices switch their behavior to a different function $g$ till the system reaches size $n$. With general non-explosivity assumptions on the attachment functions $f,g$, we consider both the standard model where $ au(n) = \Theta(n)$ as well as the \emph{quick big bang model} when $ au(n) = n^\gamma$ for some $0<\gamma <1$. Proofs rely on a careful analysis of an associated \emph{inhomogeneous} continuous time branching process. Techniques developed in the paper are robust enough to understand the behavior of these models for any sequence of change points $ au(n) o\infty$. This paper derives rates of convergence for functionals such as the degree distribution; the same proof techniques should enable one to analyze more complicated functionals such as the associated fringe distributions.

Motivation & Objective

• To establish rigorous mathematical techniques based on continuous time branching processes (CTBP) for analyzing functionals of growing network models in their large-network limits.
• To address the challenge of detecting abrupt changes in network evolution dynamics without assuming parametric forms for attachment mechanisms.
• To analyze network models where attachment rules switch from function $ f $ to $ g $ at a change point $ \tau(n) $, under general non-explosivity assumptions.
• To extend theoretical understanding to the 'quick big bang' model where the change point scales as $ \tau(n) = n^\gamma $ for $ 0 < \gamma < 1 $, challenging standard asymptotic regimes.
• To provide a robust framework applicable to any diverging sequence of change points $ \tau(n) \to \infty $, enabling analysis of complex functionals like fringe distributions.

Proposed method

• Model network growth via an inhomogeneous continuous time branching process (CTBP), where edge formation and vertex attachment follow time-dependent rules.
• Use CTBP techniques to derive concentration and fluctuation bounds for key network functionals, such as degree distribution, under general attachment functions $ f $ and $ g $.
• Analyze the system in two phases: prior to $ \tau(n) $, attachment follows $ f $; after, it follows $ g $, with a discontinuity in dynamics at the change point.
• Establish convergence rates for functionals by controlling the fluctuation of the CTBP around its mean, leveraging moment and martingale methods.
• Generalize results to the 'quick big bang' model where $ \tau(n) = n^\gamma $, showing that convergence rates remain quantifiable despite early switching.
• Apply the framework to derive bounds for degree distribution and suggest extension to more complex functionals like fringe distributions.

Experimental results

Research questions

• RQ1 How can continuous time branching processes be used to derive quantitative error bounds for network functionals in growing network models?
• RQ2 What are the convergence rates of network functionals—such as the degree distribution—when the attachment mechanism abruptly changes at a time $ \tau(n) $?
• RQ3 How do the theoretical bounds behave in the 'quick big bang' regime where the change point occurs early, at $ \tau(n) = n^\gamma $ for $ 0 < \gamma < 1 $?
• RQ4 Can the proposed framework detect changes in network dynamics without assuming parametric forms for attachment functions?
• RQ5 To what extent are the derived techniques robust to arbitrary diverging sequences of change points $ \tau(n) \to \infty $?

Key findings

• The paper establishes explicit rates of convergence for the degree distribution in growing networks with abrupt changes in attachment rules.
• Convergence rates are derived under general non-explosivity assumptions on the attachment functions $ f $ and $ g $, ensuring applicability to a broad class of models.
• The framework remains valid even in the 'quick big bang' regime where $ \tau(n) = n^\gamma $, $ 0 < \gamma < 1 $, demonstrating robustness to early change points.
• The use of inhomogeneous continuous time branching processes enables precise fluctuation bounds for functionals, forming the basis for nonparametric change point detection.
• The same proof techniques are extendable to more complex functionals such as fringe distributions, suggesting broad theoretical applicability.
• The results provide a foundation for developing nonparametric estimators of change points in network evolution without assuming known parametric models.