[Paper Review] Analysis of centrality in sublinear preferential attachment trees via the CMJ branching process

Varun S. Jog, Po‐Ling Loh|arXiv (Cornell University)|Jan 24, 2016

Stochastic processes and statistical mechanics32 references3 citations

TL;DR

This paper analyzes centrality and root inference in sublinear preferential attachment trees using the continuous-time Crump-Mode-Jagers (CMJ) branching process. It proves that a unique terminal tree centroid almost surely emerges, which becomes more central than any fixed vertex in the limit, and demonstrates that finite-sized confidence sets for the root node can be constructed in a subclass of sublinear models using centrality measures.

ABSTRACT

We investigate centrality and root-inference properties in a class of growing random graphs known as sublinear preferential attachment trees. We show that a continuous time branching processes called the Crump-Mode-Jagers (CMJ) branching process is well-suited to analyze such random trees, and prove that almost surely, a unique terminal tree centroid emerges, having the property that it becomes more central than any other fixed vertex in the limit of the random growth process. Our result generalizes and extends previous work establishing persistent centrality in uniform and linear preferential attachment trees. We also show that centrality may be utilized to generate a finite-sized $1-ε$ confidence set for the root node, for any $ε> 0$ in a certain subclass of sublinear preferential attachment trees.

Motivation & Objective

• To investigate the emergence of persistent centrality in sublinear preferential attachment trees, where attachment probability is proportional to a fractional power of vertex degree.
• To extend prior results on persistent hubs and centroids—previously established for linear and uniform attachment—to nonlinear, sublinear models.
• To develop a framework using the Crump-Mode-Jagers (CMJ) branching process to analyze tree growth dynamics and centrality in non-linear preferential attachment.
• To establish conditions under which a finite-sized confidence set for the root node can be constructed, even in sublinear models.
• To explore the relationship between node age and centrality, and to assess whether centrality can reliably infer the root in growing random trees.

Proposed method

• Model the growth of sublinear preferential attachment trees as a continuous-time Crump-Mode-Jagers (CMJ) branching process, leveraging its ability to track population growth and genealogical structure.
• Use the CMJ process to analyze the limiting behavior of vertex centrality, particularly focusing on the emergence of a unique terminal tree centroid.
• Define and apply a notion of 'balancedness centrality' to identify nodes that remain central over time, even as the network grows.
• Prove that the probability of one tree structure (e.g., line) outgrowing another (e.g., star) in population size converges to 1 as size increases, violating the 'irrelevance of structure' condition in linear models.
• Construct confidence sets for the root node by identifying nodes with high balancedness centrality, showing that such sets can be finite-sized for a subclass of sublinear models.
• Leverage the CMJ process to analyze the limiting random variable $ W $, which governs the long-term growth rate of subtrees, despite the lack of exact distributional knowledge.

Experimental results

Research questions

• RQ1 Does a unique terminal tree centroid emerge almost surely in sublinear preferential attachment trees, even when the attachment function is nonlinear?
• RQ2 Can finite-sized confidence sets for the root node be constructed in sublinear preferential attachment models using centrality measures?
• RQ3 How does the failure of the 'irrelevance of structure' condition in sublinear models affect the persistence of centrality?
• RQ4 To what extent is node age correlated with centrality in growing random trees under sublinear preferential attachment?
• RQ5 Can the CMJ branching process framework be used to generalize root inference and centrality results beyond linear and uniform attachment models?

Key findings

• A unique terminal tree centroid almost surely emerges in sublinear preferential attachment trees, becoming more central than any fixed vertex in the limit.
• The CMJ branching process provides a robust analytical framework for studying centrality and growth dynamics in sublinear preferential attachment trees.
• Finite-sized $1 - \epsilon$ confidence sets for the root node can be constructed in a subclass of sublinear preferential attachment trees using balancedness centrality.
• The irrelevance of structure condition—previously critical in linear models—does not hold in sublinear models due to structural biases in population growth (e.g., line trees grow faster than star trees).
• The results strengthen the intuition that node age and centrality are strongly correlated, making it extremely difficult for a node to hide its age in growing random trees.
• The lack of exact distributional knowledge of the limiting random variable $ W $ in CMJ processes remains a key obstacle to deriving tighter bounds on confidence set sizes or hub persistence.