[Paper Review] The maximal degree in random recursive graphs with random weights

Bas Lodewijks, Marcel Ortgiese|arXiv (Cornell University)|Jul 10, 2020

Complex Network Analysis Techniques4 citations

TL;DR

This paper studies the maximal degree in weighted random recursive graphs where vertices have i.i.d. random weights influencing connection probabilities. It establishes that the maximal degree's asymptotic behavior depends critically on the tail of the weight distribution: for Fréchet-type weights, the leading order is random and converges to a functional of a Poisson point process; for Gumbel-type weights, the leading order is deterministic, and surprisingly, the second-order term is also deterministic when considering the full vertex set, despite initial expectations of randomness in the compact window.

ABSTRACT

We study a generalisation of the random recursive tree (RRT) model and its multigraph counterpart, the uniform directed acyclic graph (DAG). Here, vertices are equipped with a random vertex-weight representing initial inhomogeneities in the network, so that a new vertex connects to one of the old vertices with a probability that is proportional to their vertex-weight. We first identify the asymptotic degree distribution of a uniformly chosen vertex for a general vertex-weight distribution. For the maximal degree, we distinguish several classes that lead to different behaviour: For bounded vertex-weights we obtain results for the maximal degree that are similar to those observed for RRTs and DAGs. If the vertex-weights have unbounded support, then the maximal degree has to satisfy the right balance between having a high vertex-weight and being born early. For vertex-weights in the Frechet maximum domain of attraction the first order behaviour of the maximal degree is random, while for those in the Gumbel maximum domain of attraction the leading order is deterministic. Surprisingly, in the latter case, the second order is random when considering vertices in a compact window in the optimal region, while it becomes deterministic when considering all vertices.

Motivation & Objective

• To understand how random vertex weights affect the degree distribution and maximal degree in growing random recursive graphs.
• To characterize the asymptotic behavior of the maximum degree under different weight distributions, particularly bounded, Fréchet, and Gumbel domains of attraction.
• To resolve the apparent contradiction in second-order fluctuations of the maximal degree for Gumbel-distributed weights, showing it is deterministic when considering all vertices.

Proposed method

• Models a growing random graph where each new vertex connects to existing vertices with probability proportional to their random i.i.d. vertex-weight.
• Analyzes the degree distribution of a uniformly chosen vertex in the limit as the graph grows.
• Uses conditional expectation and concentration inequalities to control fluctuations in the expected degree of each vertex.
• Applies extreme value theory and Poisson point process limits to characterize the distribution of the maximum degree.
• Derives asymptotic results via coupling with renewal processes and martingale techniques for the conditional degree moments.
• Employs a windowing approach to isolate the optimal region for the maximum degree and proves joint convergence of the location and value of the maximum degree.

Experimental results

Research questions

• RQ1 How does the asymptotic degree distribution of a uniformly chosen vertex behave in a weighted random recursive graph with i.i.d. vertex weights?
• RQ2 What determines the first-order asymptotic behavior of the maximal degree when vertex weights have unbounded support?
• RQ3 Why does the second-order fluctuation of the maximal degree appear random in a compact window but become deterministic when considering all vertices for Gumbel-distributed weights?
• RQ4 How does the location of the vertex with maximal degree behave asymptotically, and what is its joint limit with the maximum degree value?
• RQ5 What is the limiting distribution of the maximal degree when vertex weights are in the Fréchet maximum domain of attraction?

Key findings

• For bounded vertex weights, the maximal degree grows logarithmically, similar to classical random recursive trees.
• When vertex weights are in the Fréchet domain of attraction, the leading-order behavior of the maximal degree is random and converges in distribution to a functional of a Poisson point process.
• For Gumbel-distributed weights, the first-order growth of the maximal degree is deterministic, contradicting the naive expectation of random second-order fluctuations.
• Surprisingly, even the second-order term of the maximal degree is deterministic when considering all vertices, not just a compact window, due to the need to consider a larger optimal region.
• The location of the vertex with maximal degree, normalized appropriately, converges in distribution to a uniform distribution on a logarithmic scale, with the limit depending on the weight distribution.
• Joint convergence of the location and value of the maximum degree is established, showing that the limiting distribution of the maximum degree is a Gumbel-type extreme value distribution.