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Motivated by questions in social networks, distributed computing and probabilistic combinatorics, the last few years have seen increasing interest in network evolution models where new vertices entering the system need to make decisions…
We define a class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favoring short distances and high degrees. The competition of preferential attachment and…
In the 2-choice allocation problem, $m$ balls are placed into $n$ bins, and each ball must choose between two random bins $i, j \in [n]$ that it has been assigned to. It has been known for more than two decades, that if each ball follows…
Species populations often modify their environment as they grow. When environmental feedback operates more slowly than population growth, the system can undergo boom-bust dynamics, where the population overshoots its carrying capacity and…
This paper describes a relatively simple way of allowing a brain model to self-organise its concept patterns through nested structures. Time is a key element and a simulator would be able to show how patterns may form and then fire in…
The planted partition model (also known as the stochastic blockmodel) is a classical cluster-exhibiting random graph model that has been extensively studied in statistics, physics, and computer science. In its simplest form, the planted…
To assess the number of life-bearing worlds in astrophysical environments, it is necessary to take the intertwined processes of abiogenesis (birth), extinction (death), and transfer of life (migration) into account. We construct a…
We consider a one-dimensional discrete-space birth process with a bounded number of particle per site. Under the assumptions of the finite range of interaction, translation invariance, and non-degeneracy, we prove a shape theorem. We also…
In this paper we investigate networks whose evolution is governed by the interaction of a random assembly process and an optimization process. In the first process, new nodes are added one at a time and form connections to randomly selected…
We investigate the problem of growing clusters, which is modeled by two dimensional disks and three dimensional droplets. In this model we place a number of seeds on random locations on a lattice with an initial occupation probability, $p$.…
The adoption of agroecological practices will be crucial to address the challenges of climate change and biodiversity loss. Such practices favor the cultivation of plants in complex mixtures with layouts differing from the monoculture…
An adult human body is made up of some 30 to 40 trillion cells, all of which stem from a single fertilized egg cell. The process by which the right cells appear to arrive in their right numbers at the right time at the right place --…
We consider a fitness-structured population model with competition and migration between nearest neighbors. Under a combination of large population and rare migration limits we are particularly interested in the asymptotic behavior of the…
Rooted, weighted continuum random trees are used to describe limits of sequences of random discrete trees. Formally, they are random quadruples $(\mathcal{T},d,r,p)$, where $(\mathcal{T},d)$ is a tree-like metric space, $r\in\mathcal{T}$ is…
The classical and extended occupancy distributions are useful for examining the number of occupied bins in problems involving random allocation of balls to bins. We examine the extended occupancy problem by framing it as a Markov chain and…
We present a systematic methodology to develop high order accurate numerical approaches for linear advection problems. These methods are based on evolving parts of the jet of the solution in time, and are thus called jet schemes. Through…
We introduce the concept of nested topological order in a class of exact quantum lattice Hamiltonian models with non-abelian discrete gauge symmetry. The topological order present in the models can be partially destroyed by introducing a…
This paper introduces a new type of covering process that covers the set of natural numbers using renewal processes as objects. Inspired by the behavior of prime numbers, the model in each step finds the smallest vacant point, $k$, and…
Cellular differentiation and evolution are stochastic processes that can involve multiple types (or states) of particles moving on a complex, high-dimensional state-space or "fitness" landscape. Cells of each specific type can thus be…
About 45,000 years ago, symbolic and technological complexity of human artefacts increased drastically. Computer simulations of Powell, Shennan and Thomas (2009) explained it through an increase of the population density, facilitating the…