Evolution of recursive trees with limited memory

TL;DR

The paper investigates recursive trees evolving under limited memory, introducing macroscopic ($j(n)=\lfloor \theta n\rfloor$) and mesoscopic ($j(n)=n-\lfloor n^{\beta}\rfloor$) information regimes and deriving detailed local and global asymptotics. It establishes local weak limits: in the macroscopic regime the tree converges to a sin-tree with fringe distribution $\boldsymbol{\varpi}_{\theta}$ derived from a continuous-time branching process $\mathrm{BP}_{\theta}$, while in the mesoscopic regime it converges to a sin-tree with fringe distribution $\boldsymbol{\varpi}_{{\sf Poisson},1}$ corresponding to a Poisson$(1)$ branching process. Global properties are characterized: in the macroscopic regime the height satisfies $H_n/\log n \to [\kappa(\theta)]^{-1}$, while in the mesoscopic regime $H_n/n^{1-\beta} \to 2/(1-\beta)$, with a phase transition at $\beta=1/2$ revealing line-like, star-like, or coalescing-fringe structures; an exploration algorithm is developed to reveal youngest-vertex ancestry paths. The work also connects to scaled-attachment random recursive trees (SARRTs) and extends local limit results to general SAARTs, offering potential universality classes for height and fractal scaling in memory-limited networks.

Abstract

Motivated by questions in social networks, distributed computing and probabilistic combinatorics, the last few years have seen increasing interest in network evolution models where new vertices entering the system need to make decisions based on a partial snapshot of the current state of the network. This paper considers a specific variant of the classical random recursive tree dynamics, where a vertex at time $n+1$ has information only on those vertices that have arrived in the interval $[j(n), n]$ for a sequence $j(n) \uparrow \infty$, and connects to vertices uniformly at random amongst this set. We consider two different regimes on the density information, termed macroscopic and mesoscopic regimes, which respectively correspond to $j(n)=θn$ for some $θ\in (0,1)$, and $j(n)=n-n^β$ for some $β\in (0,1)$. Our main interest is in studying asymptotics of various local and global functionals of the network. We show that in the macroscopic regime, the local limit is expressed in terms of an associated continuous time branching process that depends on the parameter $θ$, while it is a $\mathrm{Poisson}(1)$-branching process in the mesoscopic regime for any $β\in (0,1)$. Furthermore, the height of the macroscopic tree is logarithmic, which we prove exploiting a connection with scaled-attachment random recursive trees (SARRTs) as studied by Devroye, Fawzi and Fraiman (RSA 2011), while it is polynomial in the mesoscopic regime; our argument in this latter case relies on a differential equation approach to track the ancestor indices of late-coming vertices, together with a multiscale analysis. Further, we develop an exploration algorithm to simultaneously reveal the ancestral path of youngest vertices. Using this algorithm, we show that in the mesoscopic regime, the global structure experiences a phase transition at $β=1/2$.

Figures (3)

• Figure 1: A recursive tree with limited memory in the mesoscopic regime with 10,000 nodes and with $\beta=0.75$ (see Section \ref{['sec:model']} for relevant definitions). Section \ref{['sec:sims']} has more pictures.
• Figure 2: Pictures of the tree $\mathcal{T}_n$ in different mesoscopic regimes. As $\beta$ increases and passes through $0.5$, the tree from being more 'line-like', becomes more spread out and 'fatter', see Theorems \ref{['thm:ghp_meso']} and \ref{['thm:meso_star']}.
• Figure 3: Pictures of the tree $\mathcal{T}_n$ in different macroscopic regimes.

Theorems & Definitions (46)

• Remark 1.1
• Theorem 2.1: Local weak convergence in the macroscopic regime
• Corollary 2.2
• Theorem 2.3
• Remark 2.4
• Proposition 2.5
• Theorem 2.6
• Remark 2.7
• Theorem 2.8
• Theorem 2.9