Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2504.06493

Co-evolving vertex and edge dynamics in dense graphs

Siva Athreya·Frank den Hollander·Adrian Röllin
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider a random graph in which vertices can have one of two possible colours. Each vertex switches its colour at a rate that is proportional to the number of vertices of the other colour to which it is connected by an edge. Each edge turns on or off according to a rate that depends on whether the vertices at its two endpoints have the same colour or not. We prove that, in the limit as the graph size tends to infinity and the graph becomes dense, the graph process converges, in a suitable path topology, to a limiting Markov process that lives on a certain subset of the space of coloured graphons. In the limit, the density of each vertex colour evolves according to a Fisher-Wright diffusion driven by the density of the edges, while the underlying edge connectivity structure evolves according to a stochastic flow whose drift depends on the densities of the two vertex colours.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Siva Athreya, Frank den Hollander, Adrian Röllin. 2026-04-18. Co-evolving vertex and edge dynamics in dense graphs. https://arxiv.org/abs/2504.06493

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Breuer-Major Theorems for Hilbert Space-Valued Random Variables

Let $\{X_k\}_{k\in\mathbb Z}$ be a stationary Gaussian process with values in a separable Hilbert space $\mathcal H_1$, and let $G:\mathcal H_1\to\mathcal H_2$ be a measurable map into another separable Hilbert space $\mathcal H_2$. We derive a central limit theorem for the centered normalized partial sums of the Hilbert space-valued subordinated process $\{G[X_k]\}_{k\in\mathbb Z}$. Our result holds under either of two sets of sufficient conditions, formulated in terms of the transformation $G$ and the temporal and cross-sectional dependence structure of $\{X_k\}_{k\in\mathbb Z}$. These conditions coincide in finite dimensions but lead to genuinely different phenomena in the infinite-dimensional setting. The proof relies on the recently developed Fourth Moment Theorem on Hilbert spaces, leveraging tools from the infinite-dimensional Malliavin-Stein framework. We also provide continuous-time and quantitative versions of the central limit theorem. In a series of examples, we recover and strengthen limit theorems for a wide array of statistics relevant in functional data analysis, and present, as an application of our result, a novel limit theorem in the framework of neural operators.

math.PR↗
arXiv2026

The Fourth-Moment Theorem on Hilbert Spaces

In this work, we establish conditions ensuring convergence in distribution of a sequence admitting a Wiener-Itô chaos representation to a Gaussian measure on a separable Hilbert space. Our first main result shows that, assuming convergence of the associated covariance operators in the trace-class norm, a sequence lying in a fixed Wiener-Itô chaos converges in distribution if and only if its fourth weak moments converge to the corresponding Gaussian moments. For general sequences with infinite chaos expansions, we derive analogous sufficient conditions for convergence in distribution. A key ingredient in our approach is a Stein-Malliavin bound formulated with respect to a distance that metrizes weak convergence of probability measures on separable Hilbert spaces. The results are infinite-dimensional extensions of the classical real-valued Fourth-Moment Theorem of Nualart and Peccati [Ann. Probab. 33, 177-193 (2005)]. Our work builds upon the work by Bourguin and Campese [Electron. J. Probab. 25, 1-30 (2020)] who claimed a Fourth-Moment Theorem in separable Hilbert spaces. However, a recent work by Bassetti, Bourguin, Campese, and Peccati [Stat. Probab. Lett. 233, 110671, (2026)] showed that the distance employed in the former article does not metrize weak convergence of probability measures on separable Hilbert spaces. Consequently, the conditions stated in Bourguin and Campese are not sufficient to recover a valid Fourth-Moment Theorem in the Hilbert-space setting.

math.PR↗
arXiv2026

Collision types and times in interacting particle systems

We consider a system of stochastic interacting particles with general diffusion coefficient and drift functions and we study the types of collisions that arise in them. In particular, interactions between particles are inversely proportional to their separation, and the coupling function of interaction is also considered in great generality. Our main result shows that, under positivity and the stated variance compatibility conditions, no two distinct positive-root hyperplanes are reached simultaneously at a positive time. In type $A_{N-1}$, this excludes both collisions involving three or more particles and simultaneous collisions of disjoint pairs. In order to obtain our results we make use of symmetric polynomials in squared root projections; the degree of these polynomials indicates the type of collision, and by a locality argument we show that polynomials indicating a non-simple collision almost surely do not cancel. We use this result to obtain upper and lower bounds for the Hausdorff dimension of the set of collision times in terms of interaction-to-variance ratios. These bounds coincide in particular constant-ratio cases. Our results cover many of the most well-known particle systems, such as the Dyson model and Wishart processes and their extensions to non-constant diffusion coefficients and background drifts.

math.PR↗