Fluctuation bounds for continuous time branching processes and nonparametric change point detection in growing networks
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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 τ(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 τ(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 τ(n) = Θ(n) as well as the quick big bang model when τ(n) = n^γ for some 0<γ <1. Proofs rely on a careful analysis of an associated 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 τ(n)→∞. 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.
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