Related papers: Path convergence of Markov chains on large graphs
We study various classes of random processes defined on the regular tree $T_d$ that are invariant under the automorphism group of $T_d$. Most important ones are factor of i.i.d. processes (randomized local algorithms), branching Markov…
This paper considers a distributed stochastic strongly convex optimization, where agents connected over a network aim to cooperatively minimize the average of all agents' local cost functions. Due to the stochasticity of gradient estimation…
We study the convergence of message passing graph neural networks on random graph models to their continuous counterpart as the number of nodes tends to infinity. Until now, this convergence was only known for architectures with aggregation…
In this work, we propose a first-order sampling method called the Metropolis-adjusted Preconditioned Langevin Algorithm for approximate sampling from a target distribution whose support is a proper convex subset of $\mathbb{R}^{d}$. Our…
Large continuous-time Markov chains with exponentially small transition rates arise in modeling complex systems in physics, chemistry and biology. We propose a constructive graph-algorithmic approach to determine the sequence of critical…
This paper introduces graphemes for constructing and analyzing stochastic processes that describe the evolution of large dynamic graphs. Unlike graphons, which capture the static properties of dense graphs via exchangeability or subgraph…
The Metropolis algorithm is arguably the most fundamental Markov chain Monte Carlo (MCMC) method. But the algorithm is not guaranteed to converge to the desired distribution in the case of multivariate binary distributions (e.g., Ising…
Convergence rate analyses of random walk Metropolis-Hastings Markov chains on general state spaces have largely focused on establishing sufficient conditions for geometric ergodicity or on analysis of mixing times. Geometric ergodicity is a…
Mathematical models of cellular physiological mechanisms often involve random walks on graphs representing transitions within networks of functional states. Schmandt and Gal\'{a}n recently introduced a novel stochastic shielding…
The configuration model is a standard tool for uniformly generating random graphs with a specified degree sequence, and is often used as a null model to evaluate how much of an observed network's structure can be explained by its degree…
Fitting neural networks often resorts to stochastic (or similar) gradient descent which is a noise-tolerant (and efficient) resolution of a gradient descent dynamics. It outputs a sequence of networks parameters, which sequence evolves…
We consider the problem of estimating the expected time to find a maximum degree node on a graph using a (parameterized) biased random walk. For assortative graphs the positive degree correlation serves as a local gradient for which a bias…
Various Markov chain Monte Carlo (MCMC) methods are studied to improve upon random walk Metropolis sampling, for simulation from complex distributions. Examples include Metropolis-adjusted Langevin algorithms, Hamiltonian Monte Carlo, and…
Motivated by robotic surveillance applications, this paper studies the novel problem of maximizing the return time entropy of a Markov chain, subject to a graph topology with travel times and stationary distribution. The return time entropy…
We study evolution equations on metric graphs with reservoirs, that is graphs where a one-dimensional interval is associated to each edge and, in addition, the vertices are able to store and exchange mass with these intervals. Focusing on…
We consider exploration algorithms of the random sequential adsorption type both for homogeneous random graphs and random geometric graphs based on spatial Poisson processes. At each step, a vertex of the graph becomes active and its…
Estimation in the deformable template model is a big challenge in image analysis. The issue is to estimate an atlas of a population. This atlas contains a template and the corresponding geometrical variability of the observed shapes. The…
Graph algorithms applied in many applications, including social networks, communication networks, VLSI design, graphics, and several others, require dynamic modifications -- addition and removal of vertices and/or edges -- in the graph.…
A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Originally discovered in the…
Our work is motivated by a desire to study the theoretical underpinning for the convergence of stochastic gradient type algorithms widely used for non-convex learning tasks such as training of neural networks. The key insight, already…