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
2237112965 1971118040 2533034710 1551454402
Distributed voting is a fundamental topic in distributed computing. In the standard model of pull voting, at each step every vertex chooses a neighbour uniformly at random and adopts its opinion. The voting is completed when all vertices hold the same opinion. In the simplest case, each vertex initially holds one of two different opinions. This partitions the vertices into arbitrary sets A and B. For many graphs, including regular graphs and irrespective of their expansion properties, if both A and B are sufficiently large sets, then pull voting requires @math expected steps, where n is the number of vertices of the graph. In this paper we consider a related class of voting processes based on sampling two opinions. In the simplest case, every vertex v chooses two random neighbours at each step. If both these neighbours have the same opinion, then v adopts this opinion. Otherwise, v keeps its own opinion. Let G be a connected graph with n vertices and m edges. Let P be the transition matrix of a simple random walk on G with second largest eigenvalue @math . We show that if the initial imbalance in degree between the two opinions satisfies @math , then with high probability voting completes in @math steps, and the opinion with the larger initial degree wins. The condition that @math , or only a bound on the conductance of the graph is known, the sampling process can be modified so that voting still provably completes in @math steps with high probability. The modification uses two sampling based on probing to a fixed depth @math from any vertex. In its most general form our voting process allows vertices to bias their sampling of opinions among their neighbours to achieve a desired outcome. This is done by allocating weights to edges. 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 . We study voting models on graphs. In the beginning, the vertices of a given graph have some initial opinion. Over time, the opinions on the vertices change by interactions between graph neighbours. Under suitable conditions the system evolves to a state in which all vertices have the same opinion. In this work, we consider a new model of voting, called the Linear Voting Model. This model can be seen as a generalization of several models of voting, including among others, pull voting and push voting. One advantage of our model is that, even though it is very general, it has a rich structure making the analysis tractable. In particular we are able to solve the basic question about voting, the probability that certain opinion wins the poll, and furthermore, given appropriate conditions, we are able to bound the expected time until some opinion wins. This article is concerned with a general class of stochastic spatial models for the dynamics of opinions. Like in the voter model, individuals are located on the vertex set of a connected graph and update their opinion at a constant rate based on the opinion of their neighbors. However, unlike in the voter model, the set of opinions is represented by the set of vertices of another connected graph that we call the opinion graph: when an individual interacts with a neighbor, she imitates this neighbor if and only if the distance between their opinions, defined as the graph distance induced by the opinion graph, does not exceed a certain confidence threshold. When the confidence threshold is at least equal to the radius of the opinion graph, we prove that the one-dimensional process fluctuates and clusters and give a universal lower bound for the probability of consensus of the process on finite connected graphs. We also establish a general sufficient condition for fixation of the infinite system based on the structure of the opinion graph, which we then significantly improve for opinion graphs which are distance-regular. Our general results are used to understand the dynamics of the system for various examples of opinion graphs: paths and stars, which are not distance-regular, and cycles, hypercubes and the five Platonic solids, which are distance-regular.