The asymptotic rank of adjacency matrices of weighted configuration models over arbitrary fields

TL;DR

This paper investigates the asymptotic rank of adjacency matrices in weighted configuration models over arbitrary fields, revealing that the normalized rank's behavior is independent of edge weights and field choice.

Contribution

It introduces a novel adaptation of the component exploration method to analyze the asymptotic rank, extending previous combinatorial techniques to weighted models over arbitrary fields.

Findings

Normalized rank behavior is independent of edge weights.

Asymptotic rank does not depend on the field of weights.

New combinatorial approach for weighted adjacency matrices.

Abstract

We study the asymptotic rank of adjacency matrices of a large class of edge-weighted configuration models. Here, the weight of a (multi-)edge can be any fixed non-zero element from an arbitrary field, as long as it is independent of the (multi-)graph. Our main result demonstrates that the asymptotic behavior of the normalized rank of the adjacency matrix neither depends on the fixed edge-weights, nor on which field they are chosen from. Our approach relies on a novel adaptation of the component exploration method of \cite{janson2009new}, which enables the application of combinatorial techniques from \cite{coja2022rank, HofMul25}.