arXiv subjects

Colin Cooper

Publications and source records attributed to Colin Cooper.

At least 37 records · Page 2Linked to original sources

arXiv2020

A note on the rank of a sparse random matrix

Let $\mathbf{A}_{n,m;k}$ be a random $n \times m$ matrix with entries from some field $\mathbb{F}$ where there are exactly $k$ non-zero entries in each column, whose locations are chosen independently and uniformly at random from the set of all ${n \choose k}$ possibilities. In a previous paper (arXiv:1806.04988), we considered the rank of a random matrix in this model when the field is $\mathbb{F}=GF(2)$. In this note, we point out that with minimal modifications, the arguments from that paper actually allow analogous results when the field $\mathbb{F}$ is arbitrary. In particular, for any field $\mathbb{F}$ and any fixed $k\geq 3$, we determine an asymptotically correct estimate for the rank of $\mathbf{A}_{n,m;k}$ in terms of $c,n,k$ where $m=cn/k$, and $c$ is a constant. This formula works even when the values of the nonzero elements are adversarially chosen. When $\mathbb{F}$ is a finite field, we also determine the threshold for having full row rank, when the values of the nonzero elements are randomly chosen.

math.CO↗

arXiv2019

On the cover time of dense graphs

We consider arbitrary graphs $G$ with $n$ vertices and minimum degree at least $δn$ where $δ>0$ is constant. If the conductance of $G$ is sufficiently large then we obtain an asymptotic expression for the cover time $C_G$ of $G$ as the solution to an explicit transcendental equation. Failing this, if the mixing time of a random walk on $G$ is of a lesser magnitude than the cover time, then we can obtain an asymptotic deterministic estimate via a decomposition into a bounded number of dense sub-graphs with high conductance. Failing this we give a deterministic asymptotic (2+o(1))-approximation of $C_G$.

math.CO↗

arXiv2019

Triangle-creation processes on cubic graphs

An edge switch is an operation which makes a local change in a graph while maintaining the degree of every vertex. We introduce a switch move, called a triangle switch, which creates or deletes at least one triangle. Specifically, a make move is a triangle switch which chooses a path $zwvxy$ of length 4 and replaces it by a triangle $vxwv$ and an edge $yz$, while a break move performs the reverse operation. We consider various Markov chains which perform random triangle switches, and assume that every possible make or break move has positive probability of being performed. Our first result is that any such Markov chain is irreducible on the set of all 3-regular graphs with vertex set $\{1,2,\ldots, n\}$. For a particular, natural Markov chain of this type, we obtain a non-trivial linear upper and lower bounds on the number of triangles in the long run. These bounds are almost surely obtained in linear time, irrespective of the starting graph.

cs.DM↗

arXiv2019

Minors of a random binary matroid

Let ${\bf A}={\bf A}_{n,m,k}$ be a random $n\times m$ matrix over $\mathbf{GF}_2$ wher each column consists of $k$ randomly chosen ones. Let $M$ be an arbirary fixed binary matroid. We show that if $m/n$ and $k$ are sufficiently large then as $n\to\infty$ the binary matroid induced by {\bf A} contains $M$ as a minor.

math.CO↗

arXiv2018

The flip Markov chain for connected regular graphs

Mahlmann and Schindelhauer (2005) defined a Markov chain which they called $k$-Flipper, and showed that it is irreducible on the set of all connected regular graphs of a given degree (at least 3). We study the 1-Flipper chain, which we call the flip chain, and prove that the flip chain converges rapidly to the uniform distribution over connected $2r$-regular graphs with $n$ vertices, where $n\geq 8$ and $r = r(n)\geq 2$. Formally, we prove that the distribution of the flip chain will be within $\varepsilon$ of uniform in total variation distance after $\text{poly}(n,r,\log(\varepsilon^{-1}))$ steps. This polynomial upper bound on the mixing time is given explicitly, and improves markedly on a previous bound given by Feder et al.(2006). We achieve this improvement by using a direct two-stage canonical path construction, which we define in a general setting. This work has applications to decentralised networks based on random regular connected graphs of even degree, as a self-stabilising protocol in which nodes spontaneously perform random flips in order to repair the network.

cs.DM↗

arXiv2018

The cover time of a biased random walk on a random cubic graph

We study a random walk that prefers tou se unvisited edges in the context of random cubic graphs. We establish asymptotically correct estimates for the vertex and edge cover times, these being $\approx n\log n$ and $\approx \frac32n\log n$ respectively.

math.CO↗

arXiv2017

Dispersion processes

We study a synchronous dispersion process in which $M$ particles are initially placed at a distinguished origin vertex of a graph $G$. At each time step, at each vertex $v$ occupied by more than one particle at the beginning of this step, each of these particles moves to a neighbour of $v$ chosen independently and uniformly at random. The dispersion process ends once the particles have all stopped moving, i.e. at the first step at which each vertex is occupied by at most one particle. For the complete graph $K_n$ and star graph $S_n$, we show that for any constant $δ>1$, with high probability, if $M \le n/2(1-δ)$, then the process finishes in $O(\log n)$ steps, whereas if $M \ge n/2(1+δ)$, then the process needs $e^{Ω(n)}$ steps to complete (if ever). We also show that an analogous lazy variant of the process exhibits the same behaviour but for higher thresholds, allowing faster dispersion of more particles. For paths, trees, grids, hypercubes and Cayley graphs of large enough sizes (in terms of $M$) we give bounds on the time to finish and the maximum distance traveled from the origin as a function of the number of particles $M$.

cs.DM↗

arXiv2017

The covertime of a biased random walk on $G_{n,p}$

We analyze the covertime of a biased random walk on the random graph $G_{n,p}$. The walk is biased towards visiting vertices of low degree and this makes the covertime less than in the unbiased case

math.CO↗

arXiv2017

Population protocols for leader election and exact majority with O(log^2 n) states and O(log^2 n) convergence time

We consider the model of population protocols, which can be viewed as a sequence of random pairwise interactions of $n$ agents (nodes). We show population protocols for two problems: the leader election and the exact majority voting. The leader election starts with all agents in the same initial state and the goal is to converge to the (global) state when exactly one agent is in a distinct state $L$. The exact majority voting starts with each agent in one of the two distinct states $A$ or $B$ and the goal is to make all nodes know which of these two states was the initial majority state, even if that majority was just by a single vote. Alistarh and Gelashvili [ICALP 2015] showed a leader-election protocol which converges in $O(\log^3 n)$ time w.h.p. and in expectation and needs $Θ(\log^3 n)$ states per agent. We present a protocol which elects the leader in $O(\log^2 n)$ time w.h.p. and in expectation and uses $Θ(\log^2 n)$ states per agent. For the exact majority voting, we show a population protocol with the same asymptotic performance: $O(\log^2 n)$ time and $Θ(\log^2 n)$ states per agent. The exact-majority protocol proposed by Alistarh et al. [PODC 2015] achieves expected $O(\log^2 n)$ time, but requires a relatively high initial imbalance between $A$'s and $B$'s or a large number of states per agent. More recently, Alistarh et al. [SODA 2017] showed $O(\log^2 n)$-state protocols for both problems, with the exact majority protocol converging in time $O(\log^3 n)$, and the leader election protocol converging in time $O(\log^{6.3} n)$ w.h.p. and $O(\log^{5.3} n)$ in expectation. Our leader election and exact majority protocols are based on the idea of agents counting their local interactions and rely on the probabilistic fact that the uniform random selection would limit the divergence of the individual counts.

cs.DC↗

arXiv2017

Fast plurality consensus in regular expanders

Pull voting is a classic method to reach consensus among $n$ vertices with differing opinions in a distributed network: each vertex at each step takes on the opinion of a random neighbour. This method, however, suffers from two drawbacks. Even if there are only two opposing opinions, the time taken for a single opinion to emerge can be slow and the final opinion is not necessarily the initially held majority. We refer to a protocol where 2 neighbours are contacted at each step as a 2-sample voting protocol. In the two-sample protocol a vertex updates its opinion only if both sampled opinions are the same. Not much was known about the performance of two-sample voting on general expanders in the case of three or more opinions. In this paper we show that the following performance can be achieved on a $d$-regular expander using two-sample voting. We suppose there are $k \ge 3$ opinions, and that the initial size of the largest and second largest opinions is $A_1, A_2$ respectively. We prove that, if $A_1 - A_2 \ge C n \max\{\sqrt{(\log n)/A_1}, λ\}$, where $λ$ is the absolute second eigenvalue of matrix $P=Adj(G)/d$ and $C$ is a suitable constant, then the largest opinion wins in $O((n \log n)/A_1)$ steps with high probability. For almost all $d$-regular graphs, we have $λ=c/\sqrt{d}$ for some constant $c>0$. This means that as $d$ increases we can separate an opinion whose majority is $o(n)$, whereas $Θ(n)$ majority is required for $d$ constant. This work generalizes the results of Becchetti et. al (SPAA 2014) for the complete graph $K_n$.

cs.DM↗

arXiv2016

Coalescing random walks and voting on connected graphs

In a coalescing random walk, a set of particles make independent random walks on a graph. Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues the random walk through the graph. Coalescing random walks can be used to achieve consensus in distributed networks, and is the basis of the self-stabilizing mutual exclusion algorithm of Israeli and Jalfon. Let G=(V,E), be an undirected, connected n vertex graph with m edges. Let C(n) be the expected time for all particles to coalesce, when initially one particle is located at each vertex of an n vertex graph. We study the problem of bounding the coalescence time C(n) for general classes of graphs. Our main result is that C(n)= O(1/(1-lambda_2))*((log n)^4 +n/A)), where lambda_2 is the absolute value of the second largest eigenvalue of the transition matrix of the random walk, A= (sum d^2(v))/(d^2 n), d(v) is the degree of vertex v, and d is the average node degree. The parameter A is an indicator of the variability of node degrees. Thus 1 <= A =O(n), with A=1 for regular graphs.

cs.DS↗

arXiv2016

Discordant voting processes on finite graphs

We consider an asynchronous voting process on graphs which we call discordant voting, and which can be described as follows. Initially each vertex holds one of two opinions, red or blue say. Neighbouring vertices with different opinions interact pairwise. After an interaction both vertices have the same colour. The quantity of interest is T, the time to reach consensus , i.e. the number of interactions needed for all vertices have the same colour. An edge whose endpoint colours differ (i.e. one vertex is coloured red and the other one blue) is said to be discordant. A vertex is discordant if its is incident with a discordant edge. In discordant voting, all interactions are based on discordant edges. Because the voting process is asynchronous there are several ways to update the colours of the interacting vertices. Push: Pick a random discordant vertex and push its colour to a random discordant neighbour. Pull: Pick a random discordant vertex and pull the colour of a random discordant neighbour. Oblivious: Pick a random endpoint of a random discordant edge and push the colour to the other end point. We show that E(T), the expected time to reach consensus, depends strongly on the underlying graph and the update rule. For connected graphs on n vertices, and an initial half red, half blue colouring the following hold. For oblivious voting, E(T) =n^2/4 independent of the underlying graph. For the complete graph K_n, the push protocol has E(T)= Theta(n log(n)), whereas the pull protocol has E(T)= Theta(2^n). For the cycle C_n all three protocols have E(T) = Theta(n^2). For the star graph however, the pull protocol has E(T)=O(n^2), whereas the push protocol is slower with E(T) = Theta(n^2 log(n)). The wide variation in E(T) for the pull protocol is to be contrasted with the well known model of synchronous pull voting, for which E(T) = O(n) on many classes of expanders.

cs.DM↗

arXiv2016

The coalescing-branching random walk on expanders and the dual epidemic process

Information propagation on graphs is a fundamental topic in distributed computing. One of the simplest models of information propagation is the push protocol in which at each round each agent independently pushes the current knowledge to a random neighbour. In this paper we study the so-called coalescing-branching random walk (COBRA), in which each vertex pushes the information to $k$ randomly selected neighbours and then stops passing information until it receives the information again. The aim of COBRA is to propagate information fast but with a limited number of transmissions per vertex per step. In this paper we study the cover time of the COBRA process defined as the minimum time until each vertex has received the information at least once. Our main result says that if $G$ is an $n$-vertex $r$-regular graph whose transition matrix has second eigenvalue $λ$, then the COBRA cover time of $G$ is $\mathcal O(\log n )$, if $1-λ$ is greater than a positive constant, and $\mathcal O((\log n)/(1-λ)^3))$, if $1-λ\gg \sqrt{\log( n)/n}$. These bounds are independent of $r$ and hold for $3 \le r \le n-1$. They improve the previous bound of $O(\log^2 n)$ for expander graphs. Our main tool in analysing the COBRA process is a novel duality relation between this process and a discrete epidemic process, which we call a biased infection with persistent source (BIPS). A fixed vertex $v$ is the source of an infection and remains permanently infected. At each step each vertex $u$ other than $v$ selects $k$ neighbours, independently and uniformly, and $u$ is infected in this step if and only if at least one of the selected neighbours has been infected in the previous step. We show the duality between COBRA and BIPS which says that the time to infect the whole graph in the BIPS process is of the same order as the cover time of the COBRA process

cs.DC↗

arXiv2016

On randomly generated intersecting hypergraphs

Let $c$ be a positive constant. We show that if $r=\lfloor cn^{1/3}\rfloor$ and the members of ${[n]\choose r}$ are chosen sequentially at random to form an intersecting hypergraph then with limiting probability $(1+c^3)^{-1}$, as $n\to\infty$, the resulting family will be of maximum size ${n-1\choose r-1}$.

math.CO↗

arXiv2015

Vacant sets and vacant nets: Component structures induced by a random walk

Given a discrete random walk on a finite graph $G$, the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step $t$.%These sets induce subgraphs of the underlying graph. Let $Γ(t)$ be the subgraph of $G$ induced by the vacant set of the walk at step $t$. Similarly, let $\widehat Γ(t)$ be the subgraph of $G$ induced by the edges of the vacant net. For random $r$-regular graphs $G_r$, it was previously established that for a simple random walk, the graph $Γ(t)$ of the vacant set undergoes a phase transition in the sense of the phase transition on Erd\H{os}-Renyi graphs $G_{n,p}$. Thus, for $r \ge 3$ there is an explicit value $t^*=t^*(r)$ of the walk, such that for $t\leq (1-ε)t^*$, $Γ(t)$ has a unique giant component, plus components of size $O(\log n)$, whereas for $t\geq (1+ε)t^*$ all the components of $Γ(t)$ are of size $O(\log n)$. We establish the threshold value $\widehat t$ for a phase transition in the graph $\widehat Γ(t)$ of the vacant net of a simple random walk on a random $r$-regular graph. We obtain the corresponding threshold results for the vacant set and vacant net of two modified random walks. These are a non-backtracking random walk, and, for $r$ even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random $r$-regular graphs. The main findings are the following: As $r$ increases the threshold for the vacant set converges to $n \log r$ in all three walks. For the vacant net, the threshold converges to $rn/2 \; \log n$ for both the simple random walk and non-backtracking random walk. When $r\ge 4$ is even, the threshold for the vacant net of the unvisited edge process converges to $rn/2$, which is also the vertex cover time of the process.

math.CO↗

arXiv2015

Viral processes by random walks on random regular graphs

We study the SIR epidemic model with infections carried by $k$ particles making independent random walks on a random regular graph. Here we assume $k\leq n^ε$, where $n$ is the number of vertices in the random graph, and $ε$ is some sufficiently small constant. We give an edge-weighted graph reduction of the dynamics of the process that allows us to apply standard results of Erdős-Rényi random graphs on the particle set. In particular, we show how the parameters of the model give two thresholds: In the subcritical regime, $O(\ln k)$ particles are infected. In the supercritical regime, for a constant $β\in(0,1)$ determined by the parameters of the model, $βk$ get infected with probability $β$, and $O(\ln k)$ get infected with probability $(1-β)$. Finally, there is a regime in which all $k$ particles are infected. Furthermore, the edge weights give information about when a particle becomes infected. We exploit this to give a completion time of the process for the SI case.

math.PR↗

arXiv2014

Rainbow arborescence in random digraphs

We consider the Erdős-Rényi random directed graph process, which is a stochastic process that starts with $n$ vertices and no edges, and at each step adds one new directed edge chosen uniformly at random from the set of missing edges. Let $\mathcal{D}(n,m)$ be a graph with $m$ edges obtained after $m$ steps of this process. Each edge $e_i$ ($i=1,2,\ldots, m$) of $\mathcal{D}(n,m)$ independently chooses a colour, taken uniformly at random from a given set of $n(1 + O( \log \log n / \log n)) = n (1+o(1))$ colours. We stop the process prematurely at time $M$ when the following two events hold: $\mathcal{D}(n,M)$ has at most one vertex that has in-degree zero and there are at least $n-1$ distinct colours introduced ($M= n(n-1)$ if at the time when all edges are present there are still less than $n-1$ colours introduced; however, this does not happen asymptotically almost surely). The question addressed in this paper is whether $\mathcal{D}(n,M)$ has a rainbow arborescence (that is, a directed, rooted tree on $n$ vertices in which all edges point away from the root and all the edges are different colours). Clearly, both properties are necessary for the desired tree to exist and we show that, asymptotically almost surely, the answer to this question is "yes".

math.CO↗

arXiv2014

A note on the vacant set of random walks on the hypercube and other regular graphs of high degree

We consider a random walk on a $d$-regular graph $G$ where $d\to\infty$ and $G$ satisfies certain conditions. Our prime example is the $d$-dimensional hypercube, which has $n=2^d$ vertices. We explore the likely component structure of the vacant set, i.e. the set of unvisited vertices. Let $Λ(t)$ be the subgraph induced by the vacant set of the walk at step $t$. We show that if certain conditions are satisfied then the graph $Λ(t)$ undergoes a phase transition at around $t^*=n\log_ed$. Our results are that if $t\leq(1-ε)t^*$ then w.h.p. as the number vertices $n\to\infty$, the size $L_1(t)$ of the largest component satisfies $L_1\gg\log n$ whereas if $t\geq(1+\e)t^*$ then $L_1(t)=o(\log n)$.

math.CO↗