arXiv subjects

David Sivakoff

Publications and source records attributed to David Sivakoff.

At least 19 recordsLinked to original sources

arXiv2025

Disks, Surfaces, and Entanglement Percolation

We study the probability that a loop is null-homotopic -- that is, bounded by the continuous image of a disk -- in plaquette percolation on $\mathbb{Z}^3.$ Locally, the event that there is a ``horizontal disk crossing'' of a rectangular prism is dual to the event that there is a vertical crossing in entanglement percolation (with wired boundary conditions). However, the analysis of analogous events on the full lattice is complicated by the long-range nature of entanglement percolation. We show that the probability that a rectangular loop is contractible exhibits a phase transition from area law to perimeter law dual to the entanglement percolation threshold, conditional on a conjecture concerning the continuity of entanglement percolation thresholds with respect to truncation. We also show the continuity of a truncated entanglement percolation threshold in slabs and apply that to identify a regime where large plaquette surfaces exist but typically have many handles.

math.PR↗

arXiv2025

The effects of individual versus community-influenced isolation on SIS epidemic persistence on finite random graphs

The contact process, or the SIS epidemic model, is a continuous-time Markov process used to model the spread of a recurring infection on a graph. Each vertex is either healthy or infected, and each infected vertex independently infects each of its healthy neighbors at rate $\lambda$ and recovers at rate $1$. We study the contact process in the presence of additional intervention measures by introducing a third possible state for vertices, which we call isolated. Vertices may enter the isolated state either because of individual decisions or due to community-influenced decisions, which leads to two distinct models that we call the isolation model and the vigilance model, respectively. In the isolation model, infected vertices self-isolate at rate $\alpha$. In the vigilance model, each healthy vertex causes each of its infected neighbors to isolate at rate $\alpha$. Unlike the classical contact process, these models lack the key features of attractiveness and existence of a dual, which makes analyzing them more challenging. We study the persistence times of the infection on large, finite, random graphs with heavy-tailed degree distributions. We show that the infection in the isolation model persists for at least stretched exponential time in the size of the graph for all values of $\alpha$ and $\lambda$. By contrast, in the vigilance model, for every fixed $\alpha$ the persistence time of the infection exhibits a phase transition in $\lambda$: for small $\lambda$ the infection persists for at most a linear time in the size of the graph, while for large $\lambda$ the infection persists exponentially long. As a corollary of our main results we show that for any $\lambda, \alpha>0$ the persistence time of the SIRS model on the configuration model having $n$ vertices starting from all vertices infected is at least $\exp(n^{1-\eta})$ with high probability for any $\eta>0$.

math.PR↗

arXiv2025

On the Continuity of Enhancement Percolation

We study bond percolation in $\mathbb{Z}^d$ with an unbounded family of enhancements that enable additional bonds to act as open. A natural question is whether percolation occurs in this model if and only if percolation also occurs in the system with a finite subcollection of enhancements. We give an affirmative answer in dimension $d=2$ for symmetric families of connected enhancements, and in dimensions $d\ge 3$ we prove a partial result.

math.PR↗

arXiv2025

Two-stage Bootstrap Percolation

We introduce and study two variants of two-stage growth dynamics in $\mathbb{Z}^2$ with state space $\{0,1,2\}^{\mathbb{Z}^2}$. In each variant, vertices in state $0$ can be changed irreversibly to state $1$, and vertices in state $1$ can be changed permanently to state $2$. In the standard variant, a vertex flips from state $i$ to $i+1$ if it has at least two nearest-neighbors in state $i+1$. In the modified variant, a $0$ changes to a $1$ if it has both a north or south neighbor and an east or west neighbor in state $1$, and a $1$ changes to a $2$ if it has at least two nearest-neighbors in state $2$. We assume that the initial configuration is given by a product measure with small probabilities $p$ and $q$ of $1$s and $2$s. For both variants, as $p$ and $q$ tend to $0$, if $q$ is large compared to $p^{2+o(1)}$, then the final density of $0$s tends to $1$. When $q$ is small compared to $p^{2+o(1)}$, for standard variant the final density of $2$s tends to $1$, while for the modified variant the final density of $1$s tends to $1$. In fact, for the modified variant, the final density of $2$s approaches $0$ regardless of the relative size of $q$ versus $p$. These results remain unchanged if, in either variant, a $1$ changes to a $2$ only if it has both a north or south neighbor and an east or west neighbor in state $2$. An essential feature of these dynamics is that they are not monotone in the initial configuration.

math.PR↗

arXiv2025

Polluted Modified Bootstrap Percolation

In the polluted modified bootstrap percolation model, sites in the square lattice are independently initially occupied with probability $p$ or closed with probability $q$. A site becomes occupied at a subsequent step if it is not closed and has at least one occupied nearest neighbor in each of the two coordinates. We study the final density of occupied sites when $p$ and $q$ are both small. We show that this density approaches $0$ if $q\ge Cp^2/\log p^{-1}$ and $1$ if $q\le p^2/(\log p^{-1})^{1+o(1)}$. Thus we establish a logarithmic correction in the critical scaling, which is known not to be present in the standard model, settling a conjecture of Gravner and McDonald from 1997.

math.PR↗

arXiv2024

Phase transition in one-dimensional excitable media with variable interaction range

We investigate two discrete models of excitable media on a one-dimensional integer lattice $\mathbb{Z}$: the $κ$-color Cyclic Cellular Automaton (CCA) and the $κ$-color Firefly Cellular Automaton (FCA). In both models, sites are assigned uniformly random colors from $\mathbb{Z}/κ\mathbb{Z}$. Neighboring sites with colors within a specified interaction range $r$ tend to synchronize their colors upon a particular local event of 'excitation'. We establish that there are three phases of CCA/FCA on $\mathbb{Z}$ as we vary the interaction range $r$. First, if $r$ is too small (undercoupled), there are too many non-interacting pairs of colors, and the whole graph $\mathbb{Z}$ will be partitioned into non-interacting intervals of sites with no excitation within each interval. If $r$ is within a sweet spot (critical), then we show the system clusters into ever-growing monochromatic intervals. For the critical interaction range $r=\lfloor κ/2 \rfloor$, we show the density of edges of differing colors at time $t$ is $Θ(t^{-1/2})$ and each site excites $Θ(t^{1/2})$ times up to time $t$. Lastly, if $r$ is too large (overcoupled), then neighboring sites can excite each other and such 'defects' will generate waves of excitation at a constant rate so that each site will get excited at least at a linear rate. For the special case of FCA with $r=\lfloor 2/κ\rfloor+1$, we show that every site will become $(κ+1)$-periodic eventually.

math.PR↗

arXiv2024

Competing deterministic growth models in two dimensions

We consider three-state cellular automata in two dimensions in which two colored states, blue and red, compete for control of the empty background, starting from low initial densities $p$ and $q$. When the dynamics of both colored types are one-dimensional, the dynamics has three distinct phases, characterized by a power relationship between $p$ and $q$: two in which one of the colors is prevalent, and one when the colored types block each other and leave most of the space forever empty. When one of the colors spread in two dimensions and the other in one dimension, we also establish a power relation between $p$ and $q$ that characterizes which of the two colors eventually controls most of the space.

math.PR↗

arXiv2023

Particle density in diffusion-limited annihilating systems

Place an $A$-particle at each site of a graph independently with probability $p$ and otherwise place a $B$-particle. $A$- and $B$-particles perform independent continuous time random walks at rates $λ_A$ and $λ_B$, respectively, and annihilate upon colliding with a particle of opposite type. Bramson and Lebowitz studied the setting $λ_A = λ_B$ in the early 1990s. Despite recent progress, many basic questions remain unanswered for when $λ_A \neq λ_B$. For the critical case $p=1/2$ on low-dimensional integer lattices, we give a lower bound on the expected number of particles at the origin that matches physicists' predictions. For the process with $λ_B=0$ on the integers and the bidirected regular tree, we give sharp upper and lower bounds for the expected total occupation time of the root at and approaching criticality.

math.PR↗

arXiv2023

Two-dimensional supercritical growth dynamics with one-dimensional nucleation

We introduce a class of cellular automata growth models on the two-dimensional integer lattice with finite cross neighborhoods. These dynamics are determined by a Young diagram $\mathcal Z$ and the radius $ρ$ of the neighborhood, which we assume to be sufficiently large. A point becomes occupied if the pair of counts of currently occupied points on the horizontal and vertical parts of the neighborhood lies outside $\mathcal Z$. Starting with a small density $p$ of occupied points, we focus on the first time $T$ at which the origin is occupied. We show that $T$ scales as a power of $1/p$, and identify that power, when $\mathcal Z$ is the triangular set that gives threshold-$r$ bootstrap percolation, when $\mathcal Z$ is a rectangle, and when it is a union of a finite rectangle and an infinite strip. We give partial results when $\mathcal Z$ is a union of two finite rectangles. The distinguishing feature of these dynamics is nucleation of lines that grow to significant length before most of the space is covered.

math.PR↗

arXiv2023

Diffusion-limited annihilating-coalescing systems

We study a family of interacting particle systems with annihilating and coalescing reactions. Two types of particles are interspersed throughout a transitive unimodular graph. Both types diffuse as simple random walks with possibly different jump rates. Upon colliding, like particles coalesce up to some cap and unlike particles annihilate. We describe a phase transition as the initial particle density is varied and provide estimates for the expected occupation time of the root. For the symmetric setting with no cap on coalescence, we prove that the limiting occupation probability of the root is asymptotic to 2/3 the occupation probability for classical coalescing random walk. This addresses an open problem from Stephenson.

math.PR↗

arXiv2023

Sampling random graph homomorphisms and applications to network data analysis

A graph homomorphism is a map between two graphs that preserves adjacency relations. We consider the problem of sampling a random graph homomorphism from a graph into a large network. We propose two complementary MCMC algorithms for sampling random graph homomorphisms and establish bounds on their mixing times and the concentration of their time averages. Based on our sampling algorithms, we propose a novel framework for network data analysis that circumvents some of the drawbacks in methods based on independent and neighborhood sampling. Various time averages of the MCMC trajectory give us various computable observables, including well-known ones such as homomorphism density and average clustering coefficient and their generalizations. Furthermore, we show that these network observables are stable with respect to a suitably renormalized cut distance between networks. We provide various examples and simulations demonstrating our framework through synthetic networks. We also \commHL{demonstrate the performance of} our framework on the tasks of network clustering and subgraph classification on the Facebook100 dataset and on Word Adjacency Networks of a set of classic novels.

math.PR↗

arXiv2022

Dynamics of advantageous mutant spread in spatial death-birth and birth-death Moran models

The spread of an advantageous mutation through a population is of fundamental interest in population genetics. While the classical Moran model is formulated for a well-mixed population, it has long been recognized that in real-world applications, the population usually has an explicit spatial structure which can significantly influence the dynamics. In the context of cancer initiation in epithelial tissue, several recent works have analyzed the dynamics of advantageous mutant spread on integer lattices, using the biased voter model from particle systems theory. In this spatial version of the Moran model, individuals first reproduce according to their fitness and then replace a neighboring individual. From a biological standpoint, the opposite dynamics, where individuals first die and are then replaced by a neighboring individual according to its fitness, are equally relevant. Here, we investigate this death-birth analogue of the biased voter model. We construct the process mathematically, derive the associated dual process, establish bounds on the survival probability of a single mutant, and prove that the process has an asymptotic shape. We also briefly discuss alternative birth-death and death-birth dynamics, depending on how the mutant fitness advantage affects the dynamics. We show that birth-death and death-birth formulations of the biased voter model are equivalent when fitness affects the former event of each update of the model, whereas the birth-death model is fundamentally different from the death-birth model when fitness affects the latter event.

q-bio.PE↗

arXiv2021

Cyclic Cellular Automata and Greenberg-Hastings Models on Regular Trees

We study the cyclic cellular automaton (CCA) and the Greenberg-Hastings model (GHM) with $κ\ge 3$ colors and contact threshold $θ\ge 2$ on the infinite $(d+1)$-regular tree, $T_d$. When the initial state has the uniform product distribution, we show that these dynamical systems exhibit at least two distinct phases. For sufficiently large $d$, we show that if $κ(θ-1) \le d - O(\sqrt{dκ\ln(d)})$, then every vertex almost surely changes its color infinitely often, while if $κθ\ge d + O(κ\sqrt{d\ln(d)})$, then every vertex almost surely changes its color only finitely many times. Roughly, this implies that as $d\to \infty$, there is a phase transition where $κθ/d = 1$. For the GHM dynamics, in the scenario where every vertex changes color finitely many times, we moreover give an exponential tail bound for the distribution of the time of the last color change at a given vertex.

math.PR↗

arXiv2021

Two-type annihilating systems on the complete and star graph

Red and blue particles are placed in equal proportion through-out either the complete or star graph and iteratively sampled to take simple random walk steps. Mutual annihilation occurs when particles with different colors meet. We compare the time it takes to extinguish every particle to the analogous time in the (simple to analyze) one-type setting. Additionally, we study the effect of asymmetric particle speeds.

math.PR↗

arXiv2020

The effect of avoiding known infected neighbors on the persistence of a recurring infection process

We study a generalization of the classical contact process (SIS epidemic model) in a directed graph $G$. Our model is a continuous-time interacting particle system in which at every time, each vertex is either healthy or infected, and each oriented edge is either active or inactive. Infected vertices become healthy at rate $1$, and pass the infection along each active outgoing edge at rate $λ$. At rate $α$, healthy individuals deactivate each incoming edge from their infected neighbors. We study the persistence time of this epidemic model on the lattice $\mathbb{Z}$, the $n$-cycle $\mathbb{Z}_n$, and the $n$-star graph. We show that on $\mathbb{Z}$, for every $α>0$, there is a phase transition in $λ$ between almost sure extinction and positive probability of indefinite survival; on $\mathbb{Z}_n$ we show that there is a phase transition between poly-logarithmic and exponential survival time as the size of the graph increases. On the star graph, we show that the survival time is $n^{Δ+o(1)}$ for an explicit function $Δ(α,λ)$ whenever $α>0$ and $λ>0$. In the cases of $\mathbb{Z}$ and $\mathbb{Z}_n$, our results qualitatively match what has been shown for the classical contact process, while in the case of the star graph, the classical contact process exhibits exponential survival for all $λ> 0$, which is qualitatively different from our result. This model presents a challenge because, unlike the classical contact process, it has not been shown to be monotonic in the infection parameter $λ$ or the initial infected set.

math.PR↗

arXiv2020

Stretched exponential decay for subcritical parking times on $\mathbb{Z}^d$

In the parking model on $\mathbb{Z}^d$, each vertex is initially occupied by a car (with probability $p$) or by a vacant parking spot (with probability $1-p$). Cars perform independent random walks and when they enter a vacant spot, they park there, thereby rendering the spot occupied. Cars visiting occupied spots simply keep driving (continuing their random walk). It is known that $p=1/2$ is a critical value in the sense that the origin is a.s. visited by finitely many distinct cars when $p<1/2$, and by infinitely many distinct cars when $p\geq 1/2$. Furthermore, any given car a.s. eventually parks for $p \leq 1/2$ and with positive probability does not park for $p > 1/2$. We study the subcritical phase and prove that the tail of the parking time $τ$ of the car initially at the origin obeys the bounds \[ \exp\left( - C_1 t^{\frac{d}{d+2}}\right) \leq \mathbb{P}_p(τ> t) \leq \exp\left( - c_2 t^{\frac{d}{d+2}}\right) \] for $p>0$ sufficiently small. For $d=1$, we prove these inequalities for all $p \in [0,1/2)$. This result presents an asymmetry with the supercritical phase ($p>1/2$), where methods of Bramson--Lebowitz imply that for $d=1$ the corresponding tail of the parking time of the parking spot of the origin decays like $e^{-c\sqrt{t}}$. Our exponent $d/(d+2)$ also differs from those previously obtained in the case of moving obstacles.

math.PR↗

arXiv2018

A quest to unravel the metric structure behind perturbed networks

Graphs and network data are ubiquitous across a wide spectrum of scientific and application domains. Often in practice, an input graph can be considered as an observed snapshot of a (potentially continuous) hidden domain or process. Subsequent analysis, processing, and inferences are then performed on this observed graph. In this paper we advocate the perspective that an observed graph is often a noisy version of some discretized 1-skeleton of a hidden domain, and specifically we will consider the following natural network model: We assume that there is a true graph ${G^*}$ which is a certain proximity graph for points sampled from a hidden domain $\mathcal{X}$; while the observed graph $G$ is an Erd$ö$s-R$é$nyi type perturbed version of ${G^*}$. Our network model is related to, and slightly generalizes, the much-celebrated small-world network model originally proposed by Watts and Strogatz. However, the main question we aim to answer is orthogonal to the usual studies of network models (which often focuses on characterizing / predicting behaviors and properties of real-world networks). Specifically, we aim to recover the metric structure of ${G^*}$ (which reflects that of the hidden space $\mathcal{X}$ as we will show) from the observed graph $G$. Our main result is that a simple filtering process based on the \emph{Jaccard index} can recover this metric within a multiplicative factor of $2$ under our network model. Our work makes one step towards the general question of inferring structure of a hidden space from its observed noisy graph representation. In addition, our results also provide a theoretical understanding for Jaccard-Index-based denoising approaches.

cs.CG↗

arXiv2018

Parking on transitive unimodular graphs

Place a car independently with probability $p$ at each site of a graph. Each initially vacant site is a parking spot that can fit one car. Cars simultaneously perform independent random walks. When a car encounters an available parking spot it parks there. Other cars can still drive over the site, but cannot park there. For a large class of transitive and unimodular graphs, we show that the root is almost surely visited infinitely many times when $p \geq 1/2$, and only finitely many times otherwise.

math.PR↗