Coalescing random walks and voting on graphs

Lecture Notes in Computer Science, 2014

Distributed voting is a fundamental topic in distributed computing. In the standard model of pull voting, in each step every vertex chooses a neighbour uniformly at random, and adopts the opinion of that neighbour. The voting is said to be completed when all vertices hold the same opinion. On many graph classes including regular graphs, irrespective of the expansion properties, pull voting requires Ω(n) expected time steps to complete, even if initially there are only two distinct opinions with the minority opinion being sufficiently large. In this paper we consider a related process which we call two-sample voting. In this process every vertex chooses two random neighbors in each step. If the opinions of these neighbors coincide, then the vertex revises its opinion according to the chosen sample. Otherwise, it keeps its own opinion. We consider the performance of this process in the case where two different opinions reside on vertices of some (arbitrary) sets A and B, respectively. Here, |A| + |B| = n is the number of vertices of the graph. We show that there is a constant K such that if the initial imbalance between the two opinions is ν 0 = (|A| − |B|)/n ≥ K (1/d) + (d/n), then with high probability two sample voting completes in a random d regular graph in O(log n) steps and the initial majority opinion wins. We also show the same performance for any regular graph, if ν 0 ≥ Kλ 2 , where λ 2 is the second largest eigenvalue of the transition matrix. In the graphs we consider, standard pull voting requires Ω(n) steps, and the minority can still win with probability |B|/n. Our results hold even if an adversary is able to rearrange the opinions in each step, and has complete knowledge of the graph structure.

Physical Review E, 2005

The majority-vote model with noise on Erdös-Rényi's random graphs has been studied. Monte Carlo simulations were performed to characterize the order-disorder phase transition appearing in the system. We found that the value of the critical noise parameter qc is an increasing function of the mean connectivity z of the random graph. The critical exponents β/ν, γ/ν and 1/ν were calculated for several values of z, and our analysis yielded critical exponents satisfying the hyperscaling relation with effective dimensionality equal to unity.

Journal of Physics A: Mathematical and Theoretical

The voter model is an archetypal stochastic process that represents opinion dynamics. In each update, one agent is chosen uniformly at random. The selected agent then copies the current opinion of a randomly selected neighbour. We investigate the voter model on a network with an exogenous community structure: two cliques (i.e. complete subgraphs) randomly linked by X interclique edges. We show that, counterintuitively, the mean consensus time is typically not a monotonically decreasing function of X. Cliques of fixed proportions with opposite initial opinions reach a consensus, on average, most quickly if X scales as N 3/2 , where N is the number of agents in the network. Hence, to accelerate a consensus between cliques, agents should connect to more members in the other clique as N increases but not to the extent that cliques lose their identity as distinct communities. We support our numerical results with an equation-based analysis. By interpolating between two asymptotic heterogeneous mean-field approximations, we obtain an equation for the mean consensus time that is in excellent agreement with simulations for all values of X.

The 31st ACM Symposium on Parallelism in Algorithms and Architectures

Given a graph G of n vertices, where each vertex is initially attached an opinion of either red or blue. We investigate a random process known as the Bestof-three voting. In this process, at each time step, every vertex chooses three neighbours at random and adopts the majority colour. We study this process for a class of graphs with minimum degree d = n α , where α = Ω (log log n) −1. We prove that if initially each vertex is red with probability greater than 1/2 + δ, and blue otherwise, where δ ≥ (log d) −C for some C > 0, then with high probability this dynamic reaches a final state where all vertices are red within O (log log n) + O log δ −1 steps .

Physical Review Letters, 2005

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 N nodes with an arbitrary but uncorrelated degree distribution, the mean time to reach consensus T N scales as N 2 1 = 2 , where k is the kth moment of the degree distribution. For a power-law degree distribution n k k ÿ , T N thus scales as N for > 3, as N= lnN for 3, as N 2ÿ4=ÿ1 for 2 < < 3, as lnN 2 for 2, and as O1 for < 2. These results agree with simulation data for networks with both uncorrelated and correlated node degrees.

SIAM Journal on Computing, 1991

This paper addresses the problem of distributively electing a leader in both synchronous and asynchronous complete networks. We present O(n log n) messages synchronous and asynchronous algorithms. The time complexity of the synchronous algorithm is O(log n), while that of the asynchronous algorithm is O(n). In the synchronous case, we prove a lower bound of (n log n) on the message complexity. We also prove that any message-optimal synchronous algorithm requires (log n) time. In proving these bounds we do not restrict the type of operations performed by nodes. The bounds thus apply to general algorithms and not just to comparison based algorithms.

Journal of Parallel and Distributed Computing, 2020

We present a self-stabilizing leader election algorithm for general networks, with space-complexity O(log ∆+log log n) bits per node in n-node networks with maximum degree ∆. This space complexity is sub-logarithmic in n as long as ∆ = n o(1). The best space-complexity known so far for general networks was O(log n) bits per node, and algorithms with sub-logarithmic space-complexities were known for the ring only. To our knowledge, our algorithm is the first algorithm for self-stabilizing leader election to break the Ω(log n) bound for silent algorithms in general networks. Breaking this bound was obtained via the design of a (non-silent) self-stabilizing algorithm using sophisticated tools such as solving the distance-2 coloring problem in a silent self-stabilizing manner, with space-complexity O(log ∆ + log log n) bits per node. Solving this latter coloring problem allows us to implement a sublogarithmic encoding of spanning trees-storing the IDs of the neighbors requires Ω(log n) bits per node, while we encode spanning trees using O(log ∆+log log n) bits per node. Moreover, we show how to construct such compactly encoded spanning trees without relying on variables encoding distances or number of nodes, as these two types of variables would also require Ω(log n) bits per node.

2011

We consider a consensus algorithm in which every node in a sequence of undirected, B-connected graphs assigns equal weight to each of its neighbors. Under the assumption that the degree of each node is fixed (except for times when the node has no connections to other nodes), we show that consensus is achieved within a given accuracy $\epsilon$ on n nodes in time $B+4n^3 B \ln(2n/\epsilon)$. Because there is a direct relation between consensus algorithms in time-varying environments and inhomogeneous random walks, our result also translates into a general statement on such random walks. Moreover, we give a simple proof of a result of Cao, Spielman, and Morse that the worst case convergence time becomes exponentially large in the number of nodes $n$ under slight relaxation of the degree constancy assumption.

Physical Review E, 2008

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 TN to reach consensus for a network of N nodes with an uncorrelated degree distribution scales as N µ 2 1 /µ2, where µ k is the k th 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 k to overspread the population-the fixation probability-is proportional to k for the voter model and to 1/k for the invasion process.

Discrete & Continuous Dynamical Systems - A

Building on recent work by Medvedev (2014) we establish new connections between a basic consensus model, called the voting model, and the theory of graph limits. We show that in the voting model if consensus is attained in the continuum limit then solutions to the finite model will eventually be close to a constant function, and a class of graph limits which guarantee consensus is identified. It is also proven that the dynamics in the continuum limit can be decomposed as a direct sum of dynamics on the connected components, using Janson's definition of connectivity for graph limits. This implies that without loss of generality it may be assumed that the continuum voting model occurs on a connected graph limit.