arXiv subjects

David Sivakoff

Publications and source records attributed to David Sivakoff.

33 records · Page 2Linked to original sources

arXiv2018

Bootstrap percolation on the product of the two-dimensional lattice with a Hamming square

Bootstrap percolation on a graph is a deterministic process that iteratively enlarges a set of occupied sites by adjoining points with at least $θ$ occupied neighbors. The initially occupied set is random, given by a uniform product measure with a low density $p$. Our main focus is on this process on the product graph $\mathbb{Z}^2\times K_n^2$, where $K_n$ is a complete graph. We investigate how $p$ scales with $n$ so that a typical site is eventually occupied. Under critical scaling, the dynamics with even $θ$ exhibits a sharp phase transition, while odd $θ$ yields a gradual percolation transition. We also establish a gradual transition for bootstrap percolation on $\mathbb{Z}^2\times K_n$. The main tool is heterogeneous bootstrap percolation on $\mathbb{Z}^2$.

math.PR↗

arXiv2018

Coarsening model on $\mathbb{Z}^d$ with biased zero-energy flips and an exponential large deviation bound for ASEP

We study the coarsening model (zero-temperature Ising Glauber dynamics) on $\mathbb{Z}^d$ (for $d \geq 2$) with an asymmetric tie-breaking rule. This is a Markov process on the state space $\{-1,+1\}^{\mathbb{Z}^d}$ of "spin configurations" in which each vertex updates its spin to agree with a majority of its neighbors at the arrival times of a Poisson process. If a vertex has equally many $+1$ and $-1$ neighbors, then it updates its spin value to $+1$ with probability $q \in [0,1]$ and to $-1$ with probability $1-q$. The initial state of this Markov chain is distributed according to a product measure with probability $p$ for a spin to be $+1$. In this paper, we show that for any given $p>0$, there exist $q$ close enough to 1 such that a.s. every spin has a limit of $+1$. This is of particular interest for small values of $p$, for which it is known that if $q=1/2$, a.s. all spins have a limit of $-1$. For dimension $d=2$, we also obtain near-exponential convergence rates for $q$ sufficiently large, and for general $d$, we obtain stretched exponential rates independent of $d$. Two important ingredients in our proofs are refinements of block arguments of Fontes-Schonmann-Sidoravicius and a novel exponential large deviation bound for the Asymmetric Simple Exclusion Process.

math.PR↗

arXiv2017

Persistence of sums of correlated increments and clustering in cellular automata

We consider sums of increments given by a functional of a stationary Markov chain. Letting $T$ be the first return time of the partial sums process to $(-\infty,0]$, under general assumptions, we determine the asymptotic behavior of the survival probability, $\mathbb{P}(T\ge t)\sim Ct^{-1/2}$ for an explicit constant $C$. Our analysis is based on a connection between the survival probability and the running maximum of the time-reversed process, and relies on a functional central limit theorem for Markov chains. Our result extends the classic theorem of Sparre Anderson on sums of mean zero and independent increments to the case of correlated increments. As applications, we recover known clustering results for the 3-color cyclic cellular automaton and the Greenberg-Hastings model in one dimension, and we prove a new clustering result for the 3-color firefly cellular automaton.

math.PR↗

arXiv2017

Polluted Bootstrap Percolation in Three Dimensions

In the polluted bootstrap percolation model, vertices of the cubic lattice $\mathbb{Z}^3$ are independently declared initially occupied with probability $p$ or closed with probability $q$. Under the standard (respectively, modified) bootstrap rule, a vertex becomes occupied at a subsequent step if it is not closed and it has at least $3$ occupied neighbors (respectively, an occupied neighbor in each coordinate). We study the final density of occupied vertices as $p,q\to 0$. We show that this density converges to $1$ if $q \ll p^3(\log p^{-1})^{-3}$ for both standard and modified rules. Our principal result is a complementary bound with a matching power for the modified model: there exists $C$ such that the final density converges to $0$ if $q > Cp^3$. For the standard model, we establish convergence to $0$ under the stronger condition $q>Cp^2$.

math.PR↗

arXiv2017

Hierarchical Change Point Detection on Dynamic Networks

This paper studies change point detection on networks with community structures. It proposes a framework that can detect both local and global changes in networks efficiently. Importantly, it can clearly distinguish the two types of changes. The framework design is generic and as such several state-of-the-art change point detection algorithms can fit in this design. Experiments on both synthetic and real-world networks show that this framework can accurately detect changes while achieving up to 800X speedup.

cs.SI↗

arXiv2017

Fast Change Point Detection on Dynamic Social Networks

A number of real world problems in many domains (e.g. sociology, biology, political science and communication networks) can be modeled as dynamic networks with nodes representing entities of interest and edges representing interactions among the entities at different points in time. A common representation for such models is the snapshot model - where a network is defined at logical time-stamps. An important problem under this model is change point detection. In this work we devise an effective and efficient three-step-approach for detecting change points in dynamic networks under the snapshot model. Our algorithm achieves up to 9X speedup over the state-of-the-art while improving quality on both synthetic and real world networks.

cs.SI↗

arXiv2016

Limiting behavior of 3-color excitable media on arbitrary graphs

Fix a simple graph $G=(V,E)$ and choose a random initial 3-coloring of vertices drawn from a uniform product measure. The 3-color cycle cellular automaton is a process in which at each discrete time step in parallel, every vertex with color $i$ advances to the successor color $(i+1)$ mod 3 if in contact with a neighbor with the successor color, and otherwise retains the same color. In the Greenberg-Hastings Model, the same update rule applies only to color 0, while other two colors automatically advance. The limiting behavior of these processes has been studied mainly on the integer lattices. In this paper, we introduce a monotone comparison process defined on the universal covering space of the underlying graph, and characterize the limiting behavior of these processes on arbitrary connected graphs. In particular, we establish a phase transition on the Erdös-Rényi random graph. On infinite trees, we connect the rate of color change to the cloud speed of an associated tree-indexed walk. We give estimates of the cloud speed by generalizing known results to trees with leaves.

math.PR↗

arXiv2016

Neighborhood growth dynamics on the Hamming plane

We initiate the study of general neighborhood growth dynamics on two dimensional Hamming graphs. The decision to add a point is made by counting the currently occupied points on the horizontal and the vertical line through it, and checking whether the pair of counts lies outside a fixed Young diagram. We focus on two related extremal quantities. The first is the size of the smallest set that eventually occupies the entire plane. The second is the minimum of an energy-entropy functional that comes from the scaling of the probability of eventual full occupation versus the density of the initial product measure within a rectangle. We demonstrate the existence of this scaling and study these quantities for large Young diagrams.

math.CO↗

arXiv2015

Jigsaw percolation: What social networks can collaboratively solve a puzzle?

We introduce a new kind of percolation on finite graphs called jigsaw percolation. This model attempts to capture networks of people who innovate by merging ideas and who solve problems by piecing together solutions. Each person in a social network has a unique piece of a jigsaw puzzle. Acquainted people with compatible puzzle pieces merge their puzzle pieces. More generally, groups of people with merged puzzle pieces merge if the groups know one another and have a pair of compatible puzzle pieces. The social network solves the puzzle if it eventually merges all the puzzle pieces. For an Erdős-Rényi social network with $n$ vertices and edge probability $p_n$, we define the critical value $p_c(n)$ for a connected puzzle graph to be the $p_n$ for which the chance of solving the puzzle equals $1/2$. We prove that for the $n$-cycle (ring) puzzle, $p_c(n)=Θ(1/\log n)$, and for an arbitrary connected puzzle graph with bounded maximum degree, $p_c(n)=O(1/\log n)$ and $ω(1/n^b)$ for any $b>0$. Surprisingly, with probability tending to 1 as the network size increases to infinity, social networks with a power-law degree distribution cannot solve any bounded-degree puzzle. This model suggests a mechanism for recent empirical claims that innovation increases with social density, and it might begin to show what social networks stifle creativity and what networks collectively innovate.

math.PR↗

arXiv2015

Bootstrap percolation on products of cycles and complete graphs

Bootstrap percolation on a graph iteratively enlarges a set of occupied sites by adjoining points with at least $θ$ occupied neighbors. The initially occupied set is random, given by a uniform product measure, and we say that spanning occurs if every point eventually becomes occupied. The main question concerns the critical probability, that is, the minimal initial density that makes spanning likely. The graphs we consider are products of cycles of $m$ points and complete graphs of $n$ points. The major part of the paper focuses on the case when two factors are complete graphs and one factor is a cycle. We identify the asymptotic behavior of the critical probability and show that, when $θ$ is odd, there are two qualitatively distinct phases: the transition from low to high probability of spanning as the initial density increases is sharp or gradual, depending on the size of $m$.

math.PR↗

arXiv2015

Bootstrap percolation on the Hamming torus

The Hamming torus of dimension $d$ is the graph with vertices $\{1,\dots,n\}^d$ and an edge between any two vertices that differ in a single coordinate. Bootstrap percolation with threshold $θ$ starts with a random set of open vertices, to which every vertex belongs independently with probability $p$, and at each time step the open set grows by adjoining every vertex with at least $θ$ open neighbors. We assume that $n$ is large and that $p$ scales as $n^{-α}$ for some $α>1$, and study the probability that an $i$-dimensional subgraph ever becomes open. For large $θ$, we prove that the critical exponent $α$ is about $1+d/θ$ for $i=1$, and about $1+2/θ+Θ(θ^{-3/2})$ for $i\ge2$. Our small $θ$ results are mostly limited to $d=3$, where we identify the critical $α$ in many cases and, when $θ=3$, compute exactly the critical probability that the entire graph is eventually open.

math.PR↗

arXiv2014

Deterministic walk in an excited random environment

Deterministic walk in an excited random environment is a non-Markov integer-valued process $(X_n)_{n=0}^{\infty}$, whose jump at time $n$ depends on the number of visits to the site $X_n$. The environment can be understood as stacks of cookies on each site of $\mathbb Z$. Once all cookies are consumed at a given site, every subsequent visit will result in a walk taking a step according to the direction prescribed by the last consumed cookie. If each site has exactly one cookie, then the walk ends in a loop if it ever visits the same site twice. If the number of cookies per site is increased to two, the walk can visit a site infinitely many times and still not end in a loop. Nevertheless the moments of $X_n$ are sub-linear in $n$ and we establish monotonicity results on the environment that imply large deviations.

math.PR↗

arXiv2014

Nucleation scaling in jigsaw percolation

Jigsaw percolation is a nonlocal process that iteratively merges connected clusters in a deterministic "puzzle graph" by using connectivity properties of a random "people graph" on the same set of vertices. We presume the Erdos--Renyi people graph with edge probability p and investigate the probability that the puzzle is solved, that is, that the process eventually produces a single cluster. In some generality, for puzzle graphs with N vertices of degrees about D (in the appropriate sense), this probability is close to 1 or small depending on whether pD(log N) is large or small. The one dimensional ring and two dimensional torus puzzles are studied in more detail and in many cases the exact scaling of the critical probability is obtained. The paper settles several conjectures posed by Brummitt, Chatterjee, Dey, and Sivakoff who introduced this model.

math.PR↗

arXiv2012

Contact process on a graph with communities

We are interested in the spread of an epidemic between two communities that have higher connectivity within than between them. We model the two communities as independent Erdos-Renyi random graphs, each with n vertices and edge probability p = n^{a-1} (0 1 then the contact process on the Erdos-Renyi random graph is supercritical, and we show that it survives for exponentially long. Further, let τbe the time to infect a positive fraction of vertices in the second community when the infection starts from a single vertex in the first community. We show that on the event that the contact process survives exponentially long, τ|B|/(np) converges in distribution to an exponential random variable with a specified rate. These results generalize to a graph with N communities.

math.PR↗

arXiv2010

Emergence of a Giant Component in Random Site Subgraphs of a d-Dimensional Hamming Torus

The d-dimensional Hamming torus is the graph whose vertices are all of the integer points inside an a_1 n X a_2 n X ... X a_d n box in R^d (for constants a_1, ..., a_d > 0), and whose edges connect all vertices within Hamming distance one. We study the size of the largest connected component of the subgraph generated by independently removing each vertex of the Hamming torus with probability 1-p. We show that if p=λ/ n, then there exists λ_c > 0, which is the positive root of a degree d polynomial whose coefficients depend on a_1, ..., a_d, such that for λ< λ_c the largest component has O(log n) vertices (a.a.s. as n \to \infty), and for λ> λ_c the largest component has (1-q) λ(\prod_i a_i) n^{d-1} + o(n^{d-1}) vertices and the second largest component has O(log n) vertices (a.a.s.). An implicit formula for q < 1 is also given. Surprisingly, the value of λ_c that we find is distinct from the critical value for the emergence of a giant component in the random edge subgraph of the Hamming torus. Additionally, we show that if p = c log n / n, then when c < (d-1) / (\sum a_i) the site subgraph of the Hamming torus is not connected, and when c > (d-1) / (\sum a_i) the subgraph is connected (a.a.s.). We also show that the subgraph is connected precisely when it contains no isolated vertices.

math.PR↗