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Colin Cooper

Publications and source records attributed to Colin Cooper.

At least 19 recordsLinked to original sources
arXiv2026

Discrete Incremental Voting: New Bounds for General Graphs and Expanders

We analyze the discrete incremental voting process (DIV) introduced by Cooper, Radzik, and Shiraga [OPODIS '23]. In this process, we consider a set $V$ of $n$ nodes connected in an undirected graph $G = (V, E)$ where each node has an integer opinion. In one step a randomly selected node interacts with its randomly selected neighbor and changes its opinion by $1$ in the direction of the neighbour's opinion. The process converges to a unique opinion that, in expectation, is the degree-weighted average of the initial opinions. We show that if the graph has conductance $\Phi(G)$, the ratio of the average to smallest degree is $\gamma(G)$, and the maximal difference between initial opinions is $K$, then the expected convergence time is ${O}\left({n\left(K\log (Kn)+\gamma(G) n \right)}/{\Phi(G)^2}\right)$. This bound is essentially optimal for a large class of graphs of bounded expansion. We also show that for regular graphs, if the second largest eigenvalue is $o(1/\log^2 n)$ and $K$ is $o\left({n}/{\log^2 n}\right)$, then w.h.p.\ DIV converges to the initial average opinion (rounded up or down).

cs.DC↗
arXiv2026

Edge disjoint Hamilton cycles in random digraphs of constant minimum degree

We study the existence of directed Hamilton cycles in random digraphs with $m$ edges where we condition on minimum in- and out-degree $\d \ge k+1$, where $k \ge 1$. Denote such a random graph by $D_{n,m}^{(\delta\geq k+1)}$. Let $m=cn$ and $c\ge c_k$, where $c_k$ is a sufficiently large constant. We prove that w.h.p. $D_{n,m}^{(\delta\geq k+1)}$ contains $k$ edge disjoint Hamilton cycles.

math.CO↗
arXiv2026

Undecided State Dynamics with Many Opinions

We study the Undecided-State Dynamics (USD), a fundamental consensus process in which each vertex holds one of $k$ decided opinions or the undecided state. We consider both the gossip model and the population protocol model. Prior work established tight bounds on the consensus time of this process only for the regime $k = O(\sqrt{n}/(\log n)^2)$ (for the population protocol model) and $k = O((n/\log n)^{1/3})$ (for the gossip model), often under restrictive assumptions on the initial configuration. In this paper, we obtain the first consensus-time guarantees for USD that hold for \emph{arbitrary} $2\le k\le n$ and for \emph{arbitrary} initial configurations in both the gossip model and the population protocol model. In the gossip model, USD reaches consensus within $\widetilde O(\min\{k,\sqrt n\})$ synchronous rounds with probability $1-p_{\bot}-n^{-c}$, where $p_{\bot}$ is the gossip-specific probability of collapsing to the all-undecided state in the first round. In the population protocol model, USD reaches consensus within $\widetilde O(\min\{kn,n^{3/2}\})$ asynchronous interactions with high probability. We also present lower bounds that match the upper bounds up to polylogarithmic factors for a specific initial configuration and show that our upper bounds are essentially optimal.

cs.DC↗
arXiv2025

Cover time of random subgraphs of the hypercube

$Q_{n,p}$, the random subgraph of the $n$-vertex hypercube $Q_n$, is obtained by independently retaining each edge of $Q_n$ with probability $p$. We give precise values for the cover time of $Q_{n,p}$ above the connectivity threshold.

math.CO↗
arXiv2025

Rainbow copies of spanning subgraphs

Let $G_{n,p}^{[\kappa]}$ denote the space of $n$-vertex edge coloured graphs, where each edge occurs independently with probability $p$. The colour of each existing edge is chosen independently and uniformly at random from the set $[\kappa]$. We consider the threshold for the existence of rainbow colored copies of a spanning subgraph $H$. We provide lower bounds on $p$ and $\kappa$ sufficient to prove the existence of such copies w.h.p.

math.CO↗
arXiv2024

WalkSAT is linear on random 2-SAT

In an influential article Papadimitriou [FOCS 1991] proved that a local search algorithm called WalkSAT finds a satisfying assignment of a satisfiable 2-CNF with $n$ variables in $O(n^2)$ expected time. Variants of the WalkSAT algorithm have become a mainstay of practical SAT solving (e.g., [Hoos and St\"utzle 2000]). In the present article we analyse the expected running time of WalkSAT on random 2-SAT instances. Answering a question raised by Alekhnovich and Ben-Sasson [SICOMP 2007], we show that WalkSAT runs in linear expected time for all clause/variable densities up to the random 2-SAT satisfiability threshold.

math.CO↗
arXiv2024

Asynchronous 3-Majority Dynamics with Many Opinions

We consider 3-Majority, a probabilistic consensus dynamics on a complete graph with $n$ vertices, each vertex starting with one of $k$ initial opinions. At each discrete time step, a vertex $u$ is chosen uniformly at random. The selected vertex $u$ chooses three neighbors $v_1,v_2,v_3$ uniformly at random with replacement and takes the majority opinion held by the three, where ties are broken in favor of the opinion of $v_3$. The main quantity of interest is the consensus time, the number of steps required for all vertices to hold the same opinion. This asynchronous version turns out to be considerably harder to analyze than the synchronous version and so far results have only been obtained for $k=2$. Even in the synchronous version the results for large $k$ are far from tight. In this paper we prove that the consensus time is $\tilde{\Theta}( \min(nk,n^{1.5}) )$ for all $k$. These are the first bounds for all $k$ that are tight up to a polylogarithmic factor.

cs.DC↗
arXiv2024

Discrete Incremental Voting on Expanders

Pull voting is a random process in which vertices of a connected graph have initial opinions chosen from a set of $k$ distinct opinions, and at each step a random vertex alters its opinion to that of a randomly chosen neighbour. If the system reaches a state where each vertex holds the same opinion, then this opinion will persist forthwith. In general the opinions are regarded as incommensurate, whereas in this paper we consider a type of pull voting suitable for integer opinions such as $\{1,2,\ldots,k\}$ which can be compared on a linear scale; for example, 1 ('disagree strongly'), 2 ('disagree'), $\ldots,$ 5 ('agree strongly'). On observing the opinion of a random neighbour, a vertex updates its opinion by a discrete change towards the value of the neighbour's opinion, if different. Discrete incremental voting is a pull voting process which mimics this behaviour. At each step a random vertex alters its opinion towards that of a randomly chosen neighbour; increasing its opinion by $+1$ if the opinion of the chosen neighbour is larger, or decreasing its opinion by $-1$, if the opinion of the neighbour is smaller. If initially there are only two adjacent integer opinions, for example $\{0,1\}$, incremental voting coincides with pull voting, but if initially there are more than two opinions this is not the case. For an $n$-vertex graph $G=(V,E)$, let $\lambda$ be the absolute second eigenvalue of the transition matrix $P$ of a simple random walk on $G$. Let the initial opinions of the vertices be chosen from $\{1,2,\ldots,k\}$. Let $c=\sum_{v \in V} \pi_v X_v$, where $X_v$ is the initial opinion of vertex $v$, and $\pi_v$ is the stationary distribution of the vertex. Then provided $\lambda k=o(1)$ and $k=o(n/\log n)$, with high probability the final opinion is the initial weighted average $c$ suitably rounded to $\lfloor c \rfloor$ or $\lceil c\rceil$.

cs.DM↗
arXiv2024

Random walks on edge colored random graphs

We consider random walks on edge coloured random graphs, where the colour of an edge reflects the cost of using it. In the simplest instance, the edges are coloured red or blue. Blue edges are free to use, whereas red edges incur a unit cost every time they are traversed.

math.CO↗
arXiv2023

Diffusion limited aggregation in the layers model

In the classical model of Diffusion Limited Aggregation (DLA), introduced by Witten and Sander, the process begins with a single particle cluster placed at the origin of a space, and then, one at a time, particles make a random walk from infinity until they collide with, and stick to, the existing cluster. We consider an analogous version of this process on large but finite graphs with a designated source and sink vertex. Initially the cluster of halted particles contains a single particle at the sink vertex. Starting one at a time from the source, each particle makes a random walk in the direction of the sink vertex. The particle halts at the last unoccupied vertex before the walk enters the cluster for the first time, thus increasing the size of the cluster. This continues until the source vertex becomes occupied, at which point the process ends. We study the DLA process on several classes of layered graphs, including Cayley trees of branching factor at least two with a sink vertex attached to the leaves. We determine the finish time of the process for a given class of graphs and show that the subcomponent of the final cluster linking source to sink is essentially a unique path.

math.PR↗
arXiv2023

Hamilton cycles in random digraphs with minimum degree at least one

We study the existence of a directed Hamilton cycle in random digraphs with $m$ edges where we condition on minimum in- and out-degree at least one. Denote such a random graph by $D_{n,m}^{(\delta\geq1)}$. We prove that if $m=\tfrac n2(\log n+2\log\log n+c_n)$ then \[ \lim_{n\to\infty}\Pr(D_{n,m}^{(\delta\geq1)}\text{ is Hamiltonian})=\begin{cases}0&c_n\to-\infty.\\e^{-e^{-c}/4}&c_n\to c.\\1&c_n\to\infty.\end{cases} \]

math.CO↗
arXiv2023

Rainbow Greedy Matching Algorithms

We consider the problem of finding a large rainbow matching in a random graph with randomly colored edges. In particular we analyze the performance of two greedy algorithms for this problem. The algorithms we study are colored versions of algorithms that were previously used to find large matchings in random graphs (i.e. the color-free version of our present problem).

math.CO↗
arXiv2023

Discrete Incremental Voting

We consider a type of pull voting suitable for discrete numeric opinions which can be compared on a linear scale, for example, 1 ('disagree strongly'), 2 ('disagree'), $\ldots,$ 5 ('agree strongly'). On observing the opinion of a random neighbour, a vertex changes its opinion incrementally towards the value of the neighbour's opinion, if different. For opinions drawn from a set $\{1,2,\ldots,k\}$, the opinion of the vertex would change by $+1$ if the opinion of the neighbour is larger, or by $-1$, if it is smaller. It is not clear how to predict the outcome of this process, but we observe that the total weight of the system, that is, the sum of the individual opinions of all vertices, is a martingale. This allows us analyse the outcome of the process on some classes of dense expanders such as clique graphs $K_n$ and random graphs $ G_{n,p}$ for suitably large $p$. If the average of the original opinions satisfies $i \le c \le i+1$ for some integer $i$, then the asymptotic probability that opinion $i$ wins is $i+1-c$, and the probability that opinion $i+1$ wins is $c-i$. With high probability, the winning opinion cannot be other than $i$ or $i+1$. To contrast this, we show that for a path and opinions $0,1,2$ arranged initially in non-decreasing order along the path, the outcome is very different. Any of the opinions can win with constant probability, provided that each of the two extreme opinions $0$ and $2$ is initially supported by a constant fraction of vertices.

math.PR↗
arXiv2023

A simple model of influence

We propose a simple model of influence in a network, based on edge density. In the model vertices (people) follow the opinion of the group they belong to. The opinion percolates down from an active vertex, the influencer, at the head of the group. Groups can merge, based on interactions between influencers (i.e., interactions along `active edges' of the network), so that the number of opinions is reduced. Eventually no active edges remain, and the groups and their opinions become static.

cs.DM↗
arXiv2023

Triangle processes on graphs with given degree sequence

The switch chain is a well-studied Markov chain which generates random graphs with a given degree sequence and has uniform stationary distribution. Motivated by the high number of triangles seen in some real-world networks, we study a variant of the switch chain which is more likely to produce graphs with higher numbers of triangles. Specifically, we apply a Metropolis scheme designed to have the following stationary distribution: graph $G$ has probability proportional to $\lambda^{\min\{t(G),\nu\}}$, where $t(G)$ is the number of triangles in $G$ and $\nu$ is a cut-off value introduced to moderate the impact of graphs with a very high number of triangles. We assume that the "activity" $\lambda$ satisfies $\lambda\geq 1$, and call the resulting chain the modified Metropolis switch chain. We prove that the modified Metropolis switch chain is rapidly mixing whenever the (standard) switch chain is rapidly mixing, provided that the activity and maximum degree are not too large. The triangle switch (or "$\triangle$-switch") chain is a restriction of the switch chain which only performs switches that change the set of triangles in the graph. We prove that the $\triangle$-switch chain is irreducible for any degree sequence with minimum degree at least 3, and prove a rapid mixing result for the modified Metropolis $\triangle$-switch chain. Finally, we investigate the distribution of triangles in random graphs with given degrees, under both the uniform distribution and the distribution in which graph $G$ has probability proportional to $\lambda^{t(G)}$. Our analysis implies that the imposition of the cut-off $\nu$ does not significantly impact the behaviour of these modified Metropolis chains over polynomially many steps

math.PR↗
arXiv2022

Distributed Averaging in Opinion Dynamics

We consider two simple asynchronous opinion dynamics on arbitrary graphs where every node $u$ has an initial value $\xi_u(0)$. In the first process, the NodeModel, at each time step $t\ge 0$, a random node $u$ and a random sample of $k$ of its neighbours $v_1,v_2,\cdots,v_k$ are selected. Then, $u$ updates its current value $\xi_u(t)$ to $\xi_u(t+1) = \alpha \xi_u(t) + \frac{(1-\alpha)}{k} \sum_{i=1}^k \xi_{v_i}(t)$, where $\alpha \in (0,1)$ and $k\ge 1$ are parameters of the process. In the second process, the EdgeModel, at each step a random pair of adjacent nodes $(u,v)$ is selected, and then node $u$ updates its value equivalently to the NodeModel with $k=1$ and $v$ as the selected neighbour. For both processes, the values of all nodes converge to $F$, a random variable depending on the random choices made in each step. For the NodeModel and regular graphs, and for the EdgeModel and arbitrary graphs, the expectation of $F$ is the average of the initial values $\frac{1}{n}\sum_{u\in V} \xi_u(0)$. For the NodeModel and non-regular graphs, the expectation of $F$ is the degree-weighted average of the initial values. Our results are two-fold. We consider the concentration of $F$ and show tight bounds on the variance of $F$ for regular graphs. We show that, when the initial values do not depend on the number of nodes, then the variance is negligible, hence the nodes are able to estimate the initial average of the node values. Interestingly, this variance does not depend on the graph structure. For the proof we introduce a duality between our processes and a process of two correlated random walks. We also analyse the convergence time for both models and for arbitrary graphs, showing bounds on the time $T_\varepsilon$ required to make all node values `$\varepsilon$-close' to each other. Our bounds are asymptotically tight under assumptions on the distribution of the initial values.

math.PR↗
arXiv2021

Rank of the vertex-edge incidence matrix of $r$-out hypergraphs

We consider a space of sparse Boolean matrices of size $n \times n$, which have finite co-rank over $GF(2)$ with high probability. In particular, the probability such a matrix has full rank, and is thus invertible, is a positive constant with value about $0.2574$ for large $n$. The matrices arise as the vertex-edge incidence matrix of 1-out 3-uniform hypergraphs The result that the null space is finite, can be contrasted with results for the usual models of sparse Boolean matrices, based on the vertex-edge incidence matrix of random $k$-uniform hypergraphs. For this latter model, the expected co-rank is linear in the number of vertices $n$, \cite{ACO}, \cite{CFP}. For fields of higher order, the co-rank is typically Poisson distributed.

math.CO↗
arXiv2020

A triangle process on regular graphs

Switches are operations which make local changes to the edges of a graph, usually with the aim of preserving the vertex degrees. We study a restricted set of switches, called triangle switches. Each triangle switch creates or deletes at least one triangle. Triangle switches can be used to define Markov chains which generate graphs with a given degree sequence and with many more triangles (3-cycles) than is typical in a uniformly random graph with the same degrees. We show that the set of triangle switches connects the set of all $d$-regular graphs on $n$ vertices, for all $d\geq 3$. Hence, any Markov chain which assigns positive probability to all triangle switches is irreducible on these graphs. We also investigate this question for 2-regular graphs.

math.CO↗