Related papers: Dynamics of Multidimensional Secession
We consider learning two layer neural networks using stochastic gradient descent. The mean-field description of this learning dynamics approximates the evolution of the network weights by an evolution in the space of probability…
We study a stochastic linear discrete metapolulation model to understand the effect of risk spreading by dispersion. We calculate analytically the stable distribution of populations that live in different habitats. The result shows that the…
In this paper, we consider the discrete membrane model in four dimensions. We confirm the existence of the scaling limit of the intermediate (i.e., a multiple of the expected maximum) level-sets of the model, and show that it is equal in…
Motivated by empirical research on bias and opinion formation, we formulate a multidimensional nonlinear opinion-dynamical model where agents have individual biases, which are fixed, as well as opinions, which evolve. The dimensions…
This contribution is concerned with mathematical models for the dynamics of the genetic composition of populations evolving under recombination. Recombination is the genetic mechanism by which two parent individuals create the mixed type of…
We consider supervised learning (regression/classification) problems with tensor-valued input. We derive multi-linear sufficient reductions for the regression or classification problem by modeling the conditional distribution of the…
Diffusion and flow matching approaches to generative modeling have shown promise in domains where the state space is continuous, such as image generation or protein folding & design, and discrete, exemplified by diffusion large language…
We study a multilayer SIR model with two levels of mixing, namely a global level which is uniformly mixing, and a local level with two layers distinguishing household and workplace contacts, respectively. We establish the large population…
We consider the multiparameter random simplicial complex on a vertex set $\{ 1,\dots,n \}$, which is parameterized by multiple connectivity probabilities. Our key results concern the topology of this complex of dimensions higher than the…
We describe a simple model of evolution which incorporates the branching and extinction of species lines, and also includes abiotic influences. A first principles approach is taken in which the probability for speciation and extinction are…
I consider a class of fitness landscapes, in which the fitness is a function of a finite number of phenotypic "traits", which are themselves linear functions of the genotype. I show that the stationary trait distribution in such a landscape…
Phase separation dynamics with an initially non-uniform concentration are studied. Critical and off-critical behavior is observed simultaneously. A mechanism for an expanding phase separated region is demonstrated and the time dependence of…
We study self-organization in a minimally nonlinear model of large random ecosystems. Populations evolve over time according to a piecewise linear system of ordinary differential equations subject to a non-negativity constraint resulting in…
Bayesian change-point detection, together with latent variable models, allows to perform segmentation over high-dimensional time-series. We assume that change-points lie on a lower-dimensional manifold where we aim to infer subsets of…
Fundamental properties of macroscopic gene-mating dynamic evolutionary systems are investigated. We focus on a single locus, any number of alleles in a two-gender dioecious population, for a large class of systems within population…
In many applications involving binary variables, only pairwise dependence measures, such as correlations, are available. However, for multi-way tables involving more than two variables, these quantities do not uniquely determine the joint…
Real networks exhibit nontrivial topological features such as heavy-tailed degree distribution, high clustering, and small-worldness. Researchers have developed several generative models for synthesizing artificial networks that are…
Mandelbrot multiplicative cascades provide a construction of a dynamical system on a set of probability measures defined by inequalities on moments. To be more specific, beyond the first iteration, the trajectories take values in the set of…
A hybrid stochastic individual-based model of proliferating cells with chemotaxis is presented. The model is expressed by a branching diffusion process coupled to a partial differential equation describing concentration of a chemotactic…
For a class of processes modeling the evolution of a spatially structured population with migration and a logistic local regulation of the reproduction dynamics, we show convergence to an upper invariant measure from a suitable class of…