[Paper Review] Spectra of large random trees

Shankar Bhamidi, Steven N. Evans+1 more|arXiv (Cornell University)|Mar 20, 2009

Random Matrices and Applications34 references4 citations

TL;DR

This paper establishes the almost sure convergence of the empirical spectral distribution (ESD) of adjacency matrices of various large random trees to a deterministic limit using probability fringe convergence and interlacing inequalities. It shows that for models like preferential attachment, recursive trees, and uniform trees, the limiting spectral measure has a dense set of atoms on the real line and derives precise asymptotics for the number of zero eigenvalues via maximal matchings, extending results on weighted matchings and isospectrality.

ABSTRACT

We analyze the eigenvalues of the adjacency matrices of a wide variety of random trees. Using general, broadly applicable arguments based on the interlacing inequalities for the eigenvalues of a principal submatrix of a Hermitian matrix and a suitable notion of local weak convergence for an ensemble of random trees, we show that the empirical spectral distributions for each of a number of random tree models converge to a deterministic (model dependent) limit as the number of vertices goes to infinity. We conclude for ensembles such as the linear preferential attachment models, random recursive trees, and the uniform random trees that the limiting spectral distribution has a set of atoms that is dense in the real line. We obtain precise asymptotics on the mass assigned to zero by the empirical spectral measures via the connection with the cardinality of a maximal matching. Moreover, we show that the total weight of a weighted matching is asymptotically equivalent to a constant multiple of the number of vertices when the edge weights are independent, identically distributed, non-negative random variables with finite expected value. We greatly extend a celebrated result obtained by Schwenk for the uniform random trees by showing that, under mild conditions, with probability converging to one, the spectrum of a realization is shared by at least one other tree. For the the linear preferential attachment model with parameter $a > -1$, we show that the suitably rescaled $k$ largest eigenvalues converge jointly.

Motivation & Objective

• To establish the weak convergence of the empirical spectral distribution (ESD) of adjacency matrices of large random trees to a deterministic limit across diverse models.
• To analyze the limiting behavior of the number of zero eigenvalues in random trees using combinatorial properties of maximal matchings.
• To extend results on weighted matchings in random trees, showing asymptotic equivalence to a constant multiple of the number of vertices under i.i.d. edge weights.
• To investigate isospectrality in random tree ensembles, proving that with high probability, a tree's spectrum is shared by a non-isomorphic tree.
• To characterize the joint limiting distribution of the largest eigenvalues in the linear preferential attachment model, rescaled by $ n^{1/2\gamma_a} $.

Proposed method

• Utilizes probability fringe convergence—a form of local weak convergence—for random tree ensembles to analyze spectral limits.
• Applies interlacing inequalities for eigenvalues of principal submatrices of Hermitian matrices to relate local tree structure to spectral behavior.
• Employs a simplified version of the Karp-Sipser algorithm to construct and analyze maximal matchings in random trees.
• Connects the number of zero eigenvalues of the adjacency matrix to the size of a maximal matching via combinatorial identities.
• Uses generating functions and matrix powers on the lamplighter group's Cayley graph to compute moments of the limiting spectral measure.
• Applies representation theory of the lamplighter group to analyze matrix powers and derive spectral moment expressions.

Experimental results

Research questions

• RQ1 Does the empirical spectral distribution of the adjacency matrix of a large random tree converge weakly to a deterministic limit as the number of vertices tends to infinity?
• RQ2 What is the limiting behavior of the number of zero eigenvalues in random trees, and how is it related to the size of a maximal matching?
• RQ3 Can the total weight of a maximal weighted matching in a random tree with i.i.d. non-negative edge weights be asymptotically characterized as a constant multiple of the number of vertices?
• RQ4 With what probability does a random tree have a non-isomorphic sibling with the same spectrum (isospectrality) in large random tree models?
• RQ5 How do the largest eigenvalues of the adjacency matrix in the linear preferential attachment model behave asymptotically, and what is their joint limiting distribution after appropriate rescaling?

Key findings

• The empirical spectral distribution of adjacency matrices of many random tree models converges weakly to a deterministic limit as the number of vertices tends to infinity.
• The limiting spectral measure for models such as linear preferential attachment, recursive trees, and uniform trees has a set of atoms that is dense in the real line.
• The proportion of zero eigenvalues in the adjacency matrix converges in distribution to a constant, with the limit determined by the size of a maximal matching in the tree.
• For i.i.d. non-negative edge weights with finite mean, the total weight of a maximal weighted matching is asymptotically equivalent to a constant multiple of the number of vertices.
• With probability tending to one, a random tree from the linear preferential attachment model with parameter $ a > -1 $ shares its spectrum with a non-isomorphic tree.
• The $ k $ largest eigenvalues of the linear preferential attachment model, when rescaled by $ n^{1/2\gamma_a} $ with $ \gamma_a = a+2 $, jointly converge in distribution to a non-degenerate limit.