Binomial Random Matroids

2026-03-11

AI Summary

This study investigates the conditions under which a randomly selected family of $k$-element subsets forms the collection of bases of a matroid, with an emphasis on its asymptotic existence and structural properties. Employing the binomial random model together with probabilistic methods, hypergraph matching estimates, and greedy algorithms, this work extends existing threshold results for the emergence of matroid bases to the regime where $k = k(n)$ grows slowly with $n$, thereby overcoming the prior restriction to fixed rank. The main contributions include a precise characterization of the phase transition threshold for such families to form matroid bases, the revelation that the resulting matroids are almost surely sparse paving, improved logarithmic estimates on the number of matroids and sparse paving matroids, and a new upper bound on the number of matchings in nearly regular low-degree hypergraphs.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Game Theory and Economic Paradigms: Other Foundations of Game Theory & Economic Paradigms

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

Let $\mathcal B=\mathcal B_{k,n,p}$ be a random collection of $k$-subsets of $[n]$ where each possible set is present independently with probability $p$. Let $\cal E_{\mathcal B}$ be the event that $\mathcal B$ defines the set of bases of a matroid. We prove that If $p= 1-\frac{c_n}{(k(n-k)\binom nk)^{1/2}}$ where $0\leq c_n\leq \infty$, then \[ \lim_{n\to\infty}\Pr[\cal E_{\cal B}\mid |\cal B|\geq2]=\begin{cases}1&c_n\to0.\\e^{-c^2}&c_n\to c.\\0&c_n\to \infty.\end{cases}\] In addition, we identify a condition preventing the occurence of $\cal E_{\cal B}$ and prove a hitting time version for the occurence of $\cal B$. We also prove that when $\cal E_{\mathcal B}$ occurs, $\mathcal B$ defines a sparse paving matroid w.h.p. In addition, study a greedy algorithm that produces a random matroid defined by a collection of hyperplanes. We use this to improve the estimates in \cite{HPV} on $\log m(n,k),\log p(n,k), \log s(n,k)$ where $ m(n, k), p(n, k), s(n, k)$ denote the number of matroids, paving matroids, and sparse paving matroids (respectively) of rank $k$ on $[n]$. Our improvement lies in that we can deal with $k$ growing slowly with $n$ as opposed to $k=O(1)$ in \cite{HPV}. More generally, we obtain estimates for the number of matchings in nearly-regular hypergraphs with small codegree, which may be of independent interest.

Problem

• random matroids
• binomial model
• sparse paving matroids
• threshold probability
• combinatorial structures

Innovation

• random matroids
• sparse paving matroids
• phase transition
• greedy algorithm
• hypergraph matchings