# Top Eigenpair Statistics for Weighted Sparse Graphs

## TL;DR

This paper introduces a formalism using cavity and replica methods to analytically compute the statistics of the top eigenpair in weighted sparse graphs, validated by numerical simulations.

## Contribution

It develops a novel analytical approach to determine top eigenpair statistics in weighted sparse graphs using cavity and replica techniques.

## Key findings

- Analytical expressions match numerical diagonalization results.
- Method applies to random regular graphs and sparse Markov transition matrices.
- Provides insights into eigenvector component contributions from nodes of different degrees.

## Abstract

We develop a formalism to compute the statistics of the top eigenpair of weighted sparse graphs with finite mean connectivity and bounded maximal degree. Framing the problem in terms of optimisation of a quadratic form on the sphere and introducing a fictitious temperature, we employ the cavity and replica methods to find the solution in terms of self-consistent equations for auxiliary probability density functions, which can be solved by population dynamics. This derivation allows us to identify and unpack the individual contributions to the top eigenvector's components coming from nodes of degree $k$. The analytical results are in perfect agreement with numerical diagonalisation of large (weighted) adjacency matrices, and are further cross-checked on the cases of random regular graphs and sparse Markov transition matrices for unbiased random walks.