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Colin Cooper

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Papers by Colin Cooper

On the Length of a Random Minimum Spanning Tree

Combinatorics, Probability and Computing, 2015

We study the expected value of the lengthLnof the minimum spanning tree of the complete graphKnwh... more We study the expected value of the lengthLnof the minimum spanning tree of the complete graphKnwhen each edgeeis given an independent uniform [0, 1] edge weight. We sharpen the result of Frieze [6] that limn→∞$\mathbb{E}$(Ln) = ζ(3) and show that$$ \mathbb{E}(L_n)=\zeta(3)+\frac{c_1}{n}+\frac{c_2+o(1)}{n^{4/3}}, $$wherec1,c2are explicitly defined constants.

Cover time of a random graph with given degree sequence

Discrete Mathematics, 2012

In this paper we establish the cover time of a random graph G(d) chosen uniformly at random from ... more In this paper we establish the cover time of a random graph G(d) chosen uniformly at random from the set of graphs with vertex set [n] and degree sequence d. We show that under certain restrictions on d, the cover time of G(d) is with high probability asymptotic to d−1 d−2 θ d n log n. Here θ is the average degree and d is the effective minimum degree. The effective minimum degree is the first entry in the sorted degree sequence which occurs order n times.

Vacant Sets and Vacant Nets: Component Structures Induced by a Random Walk

SIAM Journal on Discrete Mathematics, 2016

Given a discrete random walk on a finite graph G, the vacant set and vacant net are, respectively... more 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. Let Γ(t) be the subgraph of G induced by the vacant set of the walk at step t. Similarly, let 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 Erdős-Renyi graphs G n,p. Thus, for r ≥ 3 there is an explicit value t * = t * (r) of the walk, such that for t ≤ (1−)t * , Γ(t) has a unique giant component, plus components of size O(log n), whereas for t ≥ (1 +)t * all the components of Γ(t) are of size O(log n). In this paper we establish the threshold value t for a phase transition in the graph 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 ≥ 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.

Multicoloured Hamilton cycles in random graphs; an anti-Ramsey threshold

The Electronic Journal of Combinatorics, 1995

Let the edges of a graph $G$ be coloured so that no colour is used more than $k$ times. We refer ... more Let the edges of a graph $G$ be coloured so that no colour is used more than $k$ times. We refer to this as a $k$-bounded colouring. We say that a subset of the edges of $G$ is multicoloured if each edge is of a different colour. We say that the colouring is $\cal H$-good, if a multicoloured Hamilton cycle exists i.e., one with a multicoloured edge-set. Let ${\cal AR}_k$ = $\{G :$ every $k$-bounded colouring of $G$ is $\cal H$-good$\}$. We establish the threshold for the random graph $G_{n,m}$ to be in ${\cal AR}_k$.

On the Cover Time of Dense Graphs

SIAM Journal on Discrete Mathematics, 2019

We consider arbitrary graphs G with n vertices and minimum degree at least δn where δ > 0 is cons... more 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 subgraphs with high conductance. Failing this we give a deterministic asymptotic (2+o(1))-approximation of C G .

Energy Efficient Randomized Communication in Unknown AdHoc Networks

Computing Research Repository, 2006

This paper studies broadcasting and gossiping algorithms in random and general AdHoc networks. Ou... more This paper studies broadcasting and gossiping algorithms in random and general AdHoc networks. Our goal is not only to minimise the broadcasting and gossiping time, but also to minimise the energy consumption, which is measured in terms of the total number of messages (or transmissions) sent. We assume that the nodes of the network do not know the network, and

Random walks in recommender systems

Proceedings of the 23rd International Conference on World Wide Web, 2014

A recommender system uses information about known associations between users and items to compute... more A recommender system uses information about known associations between users and items to compute for a given user an ordered recommendation list of items which this user might be interested in acquiring. We consider ordering rules based on various parameters of random walks on the graph representing associations between users and items. We experimentally compare the quality of recommendations and the required computational resources of two approaches: (i) calculate the exact values of the relevant random walk parameters using matrix algebra; (ii) estimate these values by simulating random walks. In our experiments we include methods proposed by Fouss et al. [8, 9] and Gori and Pucci [11], method P 3 , which is based on the distribution of the random walk after three steps, and method P 3 α , which generalises P 3. We show that the simple method P 3 can outperform previous methods and method P 3 α can offer further improvements. We show that the time-and memory-efficiency of direct simulation of random walks allows application of these methods to large datasets. We use in our experiments the three MovieLens datasets.

Rainbow Arborescence in Random Digraphs

Journal of Graph Theory, 2015

We consider the Erdős-Rényi random directed graph process, which is a stochastic process that sta... more 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 D(n, m) be a graph with m edges obtained after m steps of this process. Each edge e i (i = 1, 2,. .. , m) of 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: 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 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".

Coalescing random walks and voting on graphs

Proceedings of the 2012 ACM symposium on Principles of distributed computing - PODC '12, 2012

In a coalescing random walk, a set of particles make independent random walks on a graph. Wheneve... more 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 [11]. Let G = (V, E), be an undirected, connected n vertex graph. 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. For d-regular graphs with expansion parameterized by the eigenvalue gap 1 − λ2, where λ2 is the second eigenvalue of the transition matrix of the random walk, we establish that C(n) = O(n/(1 − λ2)). This result also extends of near regular graphs; where a graph is near regular if the ratio of the maximum and minimum degrees ∆/δ is constant. Our general result is, that C(n) = O(n/(ν(1 − λ2))), where ν = (d 2 (v))/(d 2 n), and d is the average node degree. The parameter ν is an indicator of the variability of node degrees: 1 ≤ ν = O(n), with ν = 1 for regular graphs. The result holds provided the maximum node degree is O(m 1−ǫ). A system of coalescing particles where initially one particle is located at each vertex, corresponds to a voter model. Initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbour. The voting process can be used for leader election in a distributed context. Let E(Cv) be the expected time for voting to complete, i.e. for a unique opinion to emerge. It is known that E(Cv) = C(n), so our results imply that E(Cv) = O(n/(ν(1 − λ2))). We also investigate how the voting time improves when a vertex elicits more than one opinion at each step. In a model which we call min-voting, each vertex initially holds an opinion from {1, 2, ..., n}. At each step each vertex takes the opinions of two random neighbours and keeps the smaller. We show that for regular graphs with very good expansion properties, voting is completed in O(log n) time with high probability. This result can be viewed as an example for the "power of two choices" in distributed voting.

Fast Low-Cost Estimation of Network Properties Using Random Walks

Lecture Notes in Computer Science, 2013

If citing, it is advised that you check and use the publisher's definitive version for pagination... more If citing, it is advised that you check and use the publisher's definitive version for pagination, volume/issue, and date of publication details. And where the final published version is provided on the Research Portal, if citing you are again advised to check the publisher's website for any subsequent corrections.

Searching for Black-Hole Faults in a Network Using Multiple Agents

Lecture Notes in Computer Science, 2006

We consider a fixed communication network where (software) agents can move freely from node to no... more We consider a fixed communication network where (software) agents can move freely from node to node along the edges. A black hole is a faulty or malicious node in the network such that if an agent enters this node, then it immediately "dies." We are interested in designing an efficient communication algorithm for the agents to identify all black holes. We assume that we have k agents starting from the same node s and knowing the topology of the whole network. The agents move through the network in synchronous steps and can communicate only when they meet in a node. At the end of the exploration of the network, at least one agent must survive and must know the exact locations of the black holes. If the network has n nodes and b black holes, then any exploration algorithm needs Ω(n/k + D b) steps in the worstcase, where D b is the worst case diameter of the network with at most b nodes deleted. We give a general algorithm which completes exploration in O((n/k) log n/ log log n+bD b) steps for arbitrary networks, if b ≤ k/2. In the case when b ≤ k/2, bD b = O(√ n) and k = O(√ n), we give a refined algorithm which completes exploration in asymptotically optimal O(n/k) steps.

Improved Duplication Models for Proteome Network Evolution

Lecture Notes in Computer Science

Protein-protein interaction networks, particularly that of the yeast S. Cerevisiae, have recently... more Protein-protein interaction networks, particularly that of the yeast S. Cerevisiae, have recently been studied extensively. These networks seem to satisfy the small world property and their (1-hop) degree distribution seems to form a power law. More recently, a number of duplication based random graph models have been proposed with the aim of emulating the evolution of protein-protein interaction networks and satisfying these two graph theoretical properties. In this paper, we show that the proposed model of Pastor-Satorras et al. does not generate the power law degree distribution with exponential cutoff as claimed and the more restrictive model by Chung et al. cannot be interpreted unconditionally. It is possible to slightly modify these models to ensure that they generate a power law degree distribution. However, even after this modification, the more general k-hop degree distribution achieved by these models, for k > 1, are very different from that of the yeast proteome network. We address this problem by introducing a new network growth model that takes into account the sequence similarity between pairs of proteins (as a binary relationship) as well as their interactions. The new model captures not only the k-hop degree distribution of the yeast protein interaction network for all k > 0, but it also captures the 1-hop degree distribution of the sequence similarity network, which again seems to form a power law.

Realistic Synthetic Data for Testing Association Rule Mining Algorithms for Market Basket Databases

Lecture Notes in Computer Science

We investigate the statistical properties of the databases generated by the IBM QUEST program. Mo... more We investigate the statistical properties of the databases generated by the IBM QUEST program. Motivated by the claim (also supported empirical evidence) that item occurrences in real life market basket databases follow a rather different pattern, we propose an alternative model for generating artificial data.

Martingales on Trees and the Empire Chromatic Number of Random Trees

Lecture Notes in Computer Science, 2009

We study the empire colouring problem (as defined by Percy Heawood in 1890) for maps whose dual p... more We study the empire colouring problem (as defined by Percy Heawood in 1890) for maps whose dual planar graph is a tree, with empires formed by exactly r countries. We prove that, for each fixed r > 1, with probability approaching one as the size of the graph grows to infinity, the minimum number of colours for which a feasible solution exists takes one of seven possible values.

Lower Bounds and Algorithms for Dominating Sets in Web Graphs

Internet Mathematics, 2005

In this paper we study the size of generalised dominating sets in two graph processes that are wi... more In this paper we study the size of generalised dominating sets in two graph processes that are widely used to model aspects of the World Wide Web. On the one hand, we show that graphs generated this way have fairly large dominating sets (i.e., linear in the size of the graph). On the other hand, we present efficient strategies to construct small dominating sets. The algorithmic results represent an application of a particular analysis technique which can be used to characterise the asymptotic behaviour of a number of dynamic processes related to the web.

Random walks which prefer unvisited edges

Proceedings of the 2012 ACM symposium on Principles of distributed computing - PODC '12, 2012

In this paper, we consider a modified random walk which uses unvisited edges whenever possible, a... more In this paper, we consider a modified random walk which uses unvisited edges whenever possible, and makes a simple random walk otherwise. We call such a walk an edge-process (or E-process). We assume there is a rule A, which tells the walk which unvisited edge to use whenever there are several unvisited edges. In the simplest case, A is a uniform random choice over unvisited edges incident with the current walk position. However we do not exclude arbitrary choices of rule A. For example, the rule could be determined on-line by an adversary, or could vary from vertex to vertex. For the class of connected, even degree graphs G of constant maximum degree, we characterize the vertex cover time of the E-process in terms of the edge expansion rate of G, as measured by eigenvalue gap 1 − λ max of the transition matrix of a simple random walk on G. A vertex v is-good, if any even degree subgraph containing all edges incident with v contains at least vertices. A graph G is-good, if every vertex has the-good property. In particular, for even degree expander graphs, of bounded maximum degree, we have the following result. Let G be an n vertex-good expander graph. Any E-process on G has cover time C G (E − process) = O n + n log n .

Randomized diffusion for indivisible loads

Journal of Computer and System Sciences, 2015

We present a new randomized diffusion-based algorithm for balancing indivisible tasks (tokens) on... more We present a new randomized diffusion-based algorithm for balancing indivisible tasks (tokens) on a network. Our aim is to minimize the discrepancy between the maximum and minimum load. The algorithm works as follows. Every vertex distributes its tokens as evenly as possible among its neighbors and itself. If this is not possible without splitting some tokens, the vertex redistributes its excess tokens among all its neighbors randomly (without replacement). In this paper we prove several upper bounds on the load discrepancy for general networks. These bounds depend on some expansion properties of the network, that is, the second largest eigenvalue, and a novel measure which we refer to as refined local divergence. We then apply these general bounds to obtain results for some specific networks. For constant-degree expanders and torus graphs, these yield exponential improvements on the discrepancy bounds. For hypercubes we obtain a polynomial improvement.

Some Typical Properties of the Spatial Preferred Attachment Model

Internet Mathematics, 2014

We investigate a stochastic model for complex networks, based on a spatial embedding of the nodes... more We investigate a stochastic model for complex networks, based on a spatial embedding of the nodes, called the Spatial Preferred Attachment (SPA) model. In the SPA model, nodes have spheres of influence of varying size, and new nodes may only link to a node if they fall within its influence region. The spatial embedding of the nodes models the background knowledge or identity of the node, which influences its link environment. In this paper, we focus on the (directed) diameter, small separators, and the (weak) giant component of the model.

The Cover Time of Cartesian Product Graphs

Lecture Notes in Computer Science, 2011

Let F = G2H be the Cartesian product of graphs G, H. We relate the cover time COV[F ] of F to the... more Let F = G2H be the Cartesian product of graphs G, H. We relate the cover time COV[F ] of F to the cover times of its factors. When one of the factors is in some sense larger than the other, its cover time dominates, and can become of the same order as the cover time of the product as a whole. Our main theorem effectively gives conditions for when this holds. The probabilistic technique which we introduce, based on the blanket time, is more general and may be of independent interest, as might some of our lemmas.

The degree distribution of the generalized duplication model

Theoretical Computer Science, 2006

We study the generalized duplication model of Pastor-Satorras et al. which generates a graph by i... more We study the generalized duplication model of Pastor-Satorras et al. which generates a graph by iteratively "duplicating" a randomly chosen node. The duplicate node v is connected to each neighbor of the original node with probability p; additionally v is connected to every other node with probability r/t where t is the iteration number. Unlike other copy based models, the degree of the duplicate node in this model is not fixed in advance; rather it strongly depends on the degree of the original node and thus the degree distribution of the generated network. Our main contributions are as follows. We show that (1) the generalized duplication model does not generate a truncated power law degree distribution as stated in [29] and (2) the special case where r = 0 does not generate a power law degree distribution as stated in [13]. In fact we show exactly the opposite: (3) apart from the special case considered in [13] the degree distribution of the generalized duplication model is a power law for p ≤ 0.58. (4) We also modify the model so that it achieves a power law degree distribution even for r = 0.