In this paper, we consider lightweight decentralised algorithms for achieving consensus in distributed systems. Each member of a distributed group has a private value from a fixed set consisting of, say, two elements, and the goal is for all members to reach consensus on the majority value. We explore variants of the voter model applied to this problem. In the voter model, each node polls a randomly chosen group member and adopts its value. The process is repeated until consensus is reached. We generalize this so that each member polls a (deterministic or random) number of other group members and changes opinion only if a suitably defined super-majority has a different opinion. We show that this modification greatly speeds up the convergence of the algorithm, as well as substantially reducing the probability of it reaching consensus on the incorrect value.
There is a well known duality between the voter model and coalescing random walks, which can be exploited to obtain bounds on the time to reach consensus in the voter model. On the complete graph, it shows that the expected time to reach consensus scales linearly in the number of nodes. Recent work in @cite_2 has used this approach to derive bounds on the consensus time in general graphs in terms of the number of nodes, the variance of node degrees, and the spectral gap of the transition probability matrix for the random walk associated with the voter model. Unfortunately, the duality with random walks does not extend to the generalised voter model considered in this paper. There also doesn't seem to be a natural martingale associated with it; in the voter model, this martingale is intimately related to the dual random walk. Hence, we need a quite different approach for our model, which doesn't extend to general graphs.
@cite_2
2090454980
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. We introduce and study the reverse voter model, a dynamics for spin variables similar to the well‐known voter dynamics. The difference is in the way neighbors influence each other: once a node is selected and one among its neighbors chosen, the neighbor is made equal to the selected node, while in the usual voter dynamics the update goes in the opposite direction. The reverse voter dynamics is studied analytically, showing that on networks with degree distribution decaying as k−v, the time to reach consensus is linear in the system size N for all v > 2. The consensus time for link‐update voter dynamics is computed as well. We verify the results numerically on a class of uncorrelated scale‐free graphs. We study simple interacting particle systems on heterogeneous networks, including the voter model and the invasion process. These are both two-state models in which in an update event an individual changes state to agree with a neighbor. For the voter model, an individual imports'' its state from a randomly chosen neighbor. Here the average time @math to reach consensus for a network of @math nodes with an uncorrelated degree distribution scales as @math , where @math is the @math moment of the degree distribution. Quick consensus thus arises on networks with broad degree distributions. We also identify the conservation law that characterizes the route by which consensus is reached. Parallel results are derived for the invasion process, in which the state of an agent is exported'' to a random neighbor. We further generalize to biased dynamics in which one state is favored. The probability for a single fitter mutant located at a node of degree @math to overspread the population the fixation probability is proportional to @math for the voter model and to @math for the invasion process. In this paper, we consider lightweight decentralised algorithms for achieving consensus in distributed systems. Each member of a distributed group has a private value from a fixed set consisting of, say, two elements, and the goal is for all members to reach consensus on the majority value. We explore variants of the voter model applied to this problem. In the voter model, each node polls a randomly chosen group member and adopts its value. The process is repeated until consensus is reached. We generalise this so that each member polls a (deterministic or random) number of other group members and changes opinion only if a suitably defined super-majority has a different opinion. We show that this modification greatly speeds up the convergence of the algorithm, as well as substantially reducing the probability of it reaching consensus on the incorrect value. We study the voter model on heterogeneous graphs. We exploit the nonconservation of the magnetization to characterize how consensus is reached. For a network of @math nodes with an arbitrary but uncorrelated degree distribution, the mean time to reach consensus @math scales as @math , where @math is the @math th moment of the degree distribution. For a power-law degree distribution @math , @math thus scales as @math for @math , as @math for @math , as @math for @math , as @math for @math , and as @math for @math . These results agree with simulation data for networks with both uncorrelated and correlated node degrees. Two of the simplest interacting particle systems are the coalescing random walks and the voter model. We are interested here in the asymptotic density and growth of these systems ast→∞. More specifically, let (ζ t Zd ) be a system of coalescing random walks with initial stateZd, and (ζ t O ) a voter model with a single individual originating atO. We analyse (p_t = P(0 _t^ Zd ) = .5em @math P ( _t^0 ) ), and show that (p_t 1 t t ) ast→∞ ford=2, andpt∼(γdt)−1 ast→∞ ford≧3 for someγd. As a consequence, conditioned on non-extinction ofζ t O , Pt¦ζ t O ¦ approaches an exponential distribution. Results of a recent paper by Sawyer are applied. We study properties of multiple random walks on a graph under various assumptions of interaction between the particles. To give precise results, we make the analysis for random regular graphs. The cover time of a random walk on a random @math -regular graph was studied in [C. Cooper and A. Frieze, SIAM J. Discrete Math., 18 (2005), pp. 728-740], where it was shown with high probability (whp) that for @math the cover time is asymptotic to @math , where @math . In this paper we prove the following (whp) results, arising from the study of multiple random walks on a random regular graph @math . For @math independent walks on @math , the cover time @math is asymptotic to @math , where @math is the cover time of a single walk. For most starting positions, the expected number of steps before any of the walks meet is @math . If the walks can communicate when meeting at a vertex, we show that, for most starting positions, the expected time for @math walks to broadcast a single piece of information to each other is asymptotic to @math as @math . We also establish properties of walks where there are two types of particles, predator and prey, or where particles interact when they meet at a vertex by coalescing or by annihilating each other. For example, the expected extinction time of @math explosive particles ( @math even) tends to @math as @math . The case of @math coalescing particles, where one particle is initially located at each vertex, corresponds to a voter model defined as follows: Initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbor. The expected time for a unique opinion to emerge is the same as the expected time for all the particles to coalesce, which is asymptotic to @math . Combining results from the predator-prey and multiple random walk models allows us to compare expected detection times of all prey in the following scenarios: Both the predator and the prey move randomly, the prey moves randomly and the predators stay fixed, and the predators move randomly and the prey stays fixed. In all cases, with @math predators and @math prey the expected detection time is @math , where @math is the @math th harmonic number. A voter sits on each vertex of an infinite tree of degree @math , and has to decide between two alternative opinions. At each time step, each voter switches to the opinion of the majority of her neighbors. We analyze this majority process when opinions are initialized to independent and identically distributed random variables. In particular, we bound the threshold value of the initial bias such that the process converges to consensus. In order to prove an upper bound, we characterize the process of a single node in the large @math -limit. This approach is inspired by the theory of mean field spin-glass and can potentially be generalized to a wider class of models. We also derive a lower bound that is nontrivial for small, odd values of @math . This work presents a novel procedure for computing (1) distances between nodes of a weighted, undirected, graph, called the Euclidean Commute Time Distance (ECTD), and (2) a subspace projection of the nodes of the graph that preserves as much variance as possible, in terms of the ECTD – a principal components analysis of the graph. It is based on a Markov-chain model of random walk through the graph. The model assigns transition probabilities to the links between nodes, so that a random walker can jump from node to node. A quantity, called the average commute time, computes the average time taken by a random walker for reaching node j for the first time when starting from node i, and coming back to node i. The square root of this quantity, the ECTD, is a distance measure between any two nodes, and has the nice property of decreasing when the number of paths connecting two nodes increases and when the "length" of any path decreases. The ECTD can be computed from the pseudoinverse of the Laplacian matrix of the graph, which is a kernel. We finally define the Principal Components Analysis (PCA) of a graph as the subspace projection that preserves as much variance as possible, in terms of the ECTD. This graph PCA has some interesting links with spectral graph theory, in particular spectral clustering. In a coalescing random walk, a set of particles make independent discrete-time random walks on a graph. Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues a random walk through the graph. Let @math be an undirected and connected graph with @math vertices and @math edges. The coalescence time, @math , is the expected time for all particles to coalesce, when initially one particle is located at each vertex. We study the problem of bounding the coalescence time for general connected graphs and prove that @math . Here @math is the second eigenvalue of the transition matrix of the random walk. To avoid problems arising from, e.g., lack of coalescence on bipartite graphs, we assume the random walk can be made lazy if required. The value of @math is given by @math , where @math is the degree of vertex @math , and @math is the average degree. The parame... This paper gives a lower bound on the convergence rate of a class of network consensus algorithms. Two different approaches using directed graphs as a main tool are introduced: one is to compute the "scrambling constants" of stochastic matrices associated with "neighbor shared graphs" and the other is to analyze random walks on a sequence of graphs. Both approaches prove that the time to reach consensus within a dynamic network is logarithmic in the relative error and is in worst case exponential in the size of the network. Networks are characterized by nodes and edges. While there has been a spate of recent work on estimating the number of nodes in a network, the edge-estimation question appears to be largely unaddressed. In this work we consider the problem of estimating the average degree of a large network using efficient random sampling, where the number of nodes is not known to the algorithm. We propose a new estimator for this problem that relies on access to node samples under a prescribed distribution. Next, we show how to efficiently realize this ideal estimator in a random walk setting. Our estimator has a natural and simple implementation using random walks; we bound its performance in terms of the mixing time of the underlying graph. We then show that our estimators are both provably and practically better than many natural estimators for the problem. Our work contrasts with existing theoretical work on estimating average degree, which assume that a uniform random sample of nodes is available and the number of nodes is known. We introduce the heterogeneous voter model (HVM), in which each agent has its own intrinsic rate to change state, reflective of the heterogeneity of real people, and the partisan voter model (PVM), in which each agent has an innate and fixed preference for one of two possible opinion states. For the HVM, the time until consensus is reached is much longer than in the classic voter model. For the PVM in the mean-field limit, a population evolves to a preference-based state, where each agent tends to be aligned with its internal preference. For finite populations, discrete fluctuations ultimately lead to consensus being reached in a time that scales exponentially with population size. The aim of this article is to discuss some applications of random processes in searching and reaching consensus on finite graphs. The topics covered are: Why random walks?, Speeding up random walks, Random and deterministic walks, Interacting particles and voting, Searching changing graphs. In this paper, we examine a spectral clustering algorithm for similarity graphs drawn from a simple random graph model, where nodes are allowed to have varying degrees, and we provide theoretical bounds on its performance. The random graph model we study is the Extended Planted Partition (EPP) model, a variant of the classical planted partition model. The standard approach to spectral clustering of graphs is to compute the bottom k singular vectors or eigenvectors of a suitable graph Laplacian, project the nodes of the graph onto these vectors, and then use an iterative clustering algorithm on the projected nodes. However a challenge with applying this approach to graphs generated from the EPP model is that unnormalized Laplacians do not work, and normalized Laplacians do not concentrate well when the graph has a number of low degree nodes. We resolve this issue by introducing the notion of a degree-corrected graph Laplacian. For graphs with many low degree nodes, degree correction has a regularizing eect on the Laplacian. Our spectral clustering algorithm projects the nodes in the graph onto the bottom k right singular vectors of the degree-corrected random-walk Laplacian, and clusters the nodes in this subspace. We show guarantees on the performance of this algorithm, demonstrating that it outputs the correct partition under a wide range of parameter values. Unlike some previous work, our algorithm does not require access to any generative parameters of the model. Random walk is an important tool in many graph mining applications including estimating graph parameters, sampling portions of the graph, and extracting dense communities. In this paper we consider the problem of sampling nodes from a large graph according to a prescribed distribution by using random walk as the basic primitive. Our goal is to obtain algorithms that make a small number of queries to the graph but output a node that is sampled according to the prescribed distribution. Focusing on the uniform distribution case, we study the query complexity of three algorithms and show a near-tight bound expressed in terms of the parameters of the graph such as average degree and the mixing time. Both theoretically and empirically, we show that some algorithms are preferable in practice than the others. We also extend our study to the problem of sampling nodes according to some polynomial function of their degrees; this has implications for designing efficient algorithms for applications such as triangle counting. We investigate network exploration by random walks defined via stationary and adaptive transition probabilities on large graphs. We derive an exact formula valid for arbitrary graphs and arbitrary walks with stationary transition probabilities (STP), for the average number of discovered edges as a function of time. We show that for STP walks site and edge exploration obey the same scaling nλ as a function of time n. Therefore, edge exploration on graphs with many loops is always lagging compared to site exploration, the revealed graph being sparse until almost all nodes have been discovered. We then introduce the edge explorer model (EEM), which presents a novel class of adaptive walks, that perform faithful network discovery even on dense networks. Consider k particles, 1 red and k - 1 white, chasing each other on the nodes of a graph G. If the red one catches one of the white, it "infects" it with its color. The newly red particles are now available to infect more white ones. When is it the case that all white will become red? It turns out that this simple question is an instance of information propagation between random walks and has important applications to mobile computing where a set of mobile hosts acts as an intermediary for the spread of information. In this paper we model this problem by k concurrent random walks, one corresponding to the red particle and k - 1 to the white ones. The infection time Tk of infecting all the white particles with red color is then a random variable that depends on k, the initial position of the particles, the number of nodes and edges of the graph, as well as on the structure of the graph. In this work we develop a set of probabilistic tools that we use to obtain upper bounds on the (worst case w.r.t. initial positions of particles) expected value of Tk for general graphs and important special cases. We easily get that an upper bound on the expected value of Tk is the worst case (over all initial positions) expected meeting time m* of two random walks multiplied by Θ(log k). We demonstrate that this is, indeed, a tight bound; i.e. there is a graph G (a special case of the "lollipop" graph), a range of values k < n (such that √n - k = Θ(√n)) and an initial position of particles achieving this bound. When G is a clique or has nice expansion properties, we prove much smaller bounds for Tk. We have evaluated and validated all our results by large scale experiments which we also present and discuss here. In particular, the experiments demonstrate that our analytical results for these expander graphs are tight. We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time - the expected time required to visit every node in a graph at least once - and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speed-up in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speed-up is sometimes possible, but that some natural graphs allow only a logarithmic speed-up. A problem related to ours (in which the walks start from some probablistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected s-t-connectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds. A general technique for proving lower bounds on expected covering times of random walks on graphs in terms of expected hitting times between vertices is given. This technique is used to prove (i) A tight bound of @math for the two-dimensional torus; (ii) A tight bound of @math for trees with maximum degree @math ; (iii) Tight bounds of @math for rapidly mixing walks on vertex transitive graphs, where @math denotes the maximum expected hitting time between vertices.In addition to these new results, the technique allows several known lower bounds on cover times to be systematically proved, often in a much simpler way.Finally, a different technique is used to prove an @math lower bound on the cover time, where @math is the second largest eigenvalue of the transition matrix. This was previously known only in the case where the walk starts in the stationary distribution [...