Related papers: The grapheme-valued Wright-Fisher diffusion with m…
We study stochastic pairwise interaction network systems whereby a finite population of agents, identified with the nodes of a graph, update their states in response to both individual mutations and pairwise interactions with their…
In this paper we establish a diffusion limit for a multivariate continuous time Markov chain whose components are indexed by vertices of a finite graph. The components take values in a common finite set of non-negative integers and evolve…
Learning generative models for graph-structured data is challenging because graphs are discrete, combinatorial, and the underlying data distribution is invariant to the ordering of nodes. However, most of the existing generative models for…
Dynamical processes can be transformed into graphs through a family of mappings called visibility algorithms, enabling the possibility of (i) making empirical data analysis and signal processing and (ii) characterising classes of dynamical…
In a recent paper, Caron and Fox suggest a probabilistic model for sparse graphs which are exchangeable when associating each vertex with a time parameter in $\mathbb{R}_+$. Here we show that by generalizing the classical definition of…
Social networks have a small number of large hubs, and a large number of small dense communities. We propose a generative model that captures both hub and dense structures. Based on recent results about graphons on line graphs, our model is…
This paper uses the generator approach of Stein's method to analyze the gap between steady-state distributions of Markov chains and diffusion processes. Until now, the standard way to invoke Stein's method for this problem was to use the…
Graph generative models have broad applications in biology, chemistry and social science. However, modelling and understanding the generative process of graphs is challenging due to the discrete and high-dimensional nature of graphs, as…
We present a general approach to study the flooding time (a measure of how fast information spreads) in dynamic graphs (graphs whose topology changes with time according to a random process). We consider arbitrary converging Markovian…
We study the problem of generating connected random graphs with no self-loops or multiple edges and that, in addition, have a given degree sequence. The generation method we focus on is the edge-switching Markov-chain method, whose…
Diffusion models, as a novel generative paradigm, have achieved remarkable success in various image generation tasks such as image inpainting, image-to-text translation, and video generation. Graph generation is a crucial computational task…
Wright-Fisher diffusions and their dual ancestral graphs occupy a central role in the study of allele frequency change and genealogical structure, and they provide expressions, explicit in some special cases but generally implicit, for the…
For a directed graph $G(V_n, E_n)$ on the vertices $V_n = \{1,2, \dots, n\}$, we study the distribution of a Markov chain $\{ {\bf R}^{(k)}: k \geq 0\}$ on $\mathbb{R}^n$ such that the $i$th component of ${\bf R}^{(k)}$, denoted…
We consider a random model for directed graphs whereby an arc is placed from one vertex to another with a prescribed probability which may vary from arc to arc. Using perturbation bounds as well as Chernoff inequalities, we show that the…
In a (two-type) Wright-Fisher diffusion with directional selection and two-way mutation, let $x$ denote today's frequency of the beneficial type, and given $x$, let $h(x)$ be the probability that, among all individuals of today's…
Evolution in finite populations is often modelled using the classical Moran process. Over the last ten years this methodology has been extended to structured populations using evolutionary graph theory. An important question in any such…
Most real-world networks contain well-defined community structures where nodes are densely connected internally within communities. To learn from these networks, we develop MarkovGNN that captures the formation and evolution of communities…
Diffusion theory is a central tool of modern population genetics, yielding simple expressions for fixation probabilities and other quantities that are not easily derived from the underlying Wright-Fisher model. Unfortunately, the textbook…
We establish the representation of general regular diffusions on star-shaped graphs as time-changed Walsh Brownian motions. These are regular continuous Markov processes described locally by a family generalized second order differential…
We study branching Markov chains on a countable state space (space of types) $\mathscr{X}$, with the focus on the qualitative aspects of the limit behaviour of the evolving empirical population distributions. No conditions are imposed on…