Discrepancy Analysis of a New Randomized Diffusion Algorithm for Weighted Round Matrices

For an arbitrary initial configuration of indivisible (discrete) loads over vertices of a distributed network (graph), we consider the problem of minimizing the discrepancy between the maximum and minimum load among all vertices. For this problem, diffusion-based algorithms are well studied because of its simplicity. In diffusion-based algorithms, each vertex distributes its loads as evenly as possible among its neighbors in each synchronous round. This paper presents a new randomized diffusion-based algorithm inspired by multiple random walks. In multiple random walks, at each vertex, each token (load) generates a random variable in [0, 1), and move to a vertex corresponding to the given probability distribution (transition matrix). In our algorithm, at each vertex, each token k (k ∈ {0, 1, . . . , X−1}) generate a random number in [k/X, (k+1)/X), and moves to a vertex corresponding to the given probability distribution. Our algorithm is adaptive to any transition transition probabilities while almost all previous works are concerned with uniform transition probabilities. For this algorithm, we analyze the discrepancy between the token configuration and its expected value, and give an upper bound depending on the local 2-divergence of the transition matrix and √ log n, where n is the number of vertices. The local 2-divergence is a measure which often appeared in previous works. We also give an upper bound of the local-2 divergence for any reversible and lazy transition matrix. These yield the following specific results. First, our algorithm achieves O( √ d log n) discrepancy for any d regular graph, which matches the best result on previous works of diffusion model. Note that our algorithm does not need any assumption of the number of tokens such as negative loads which are often assumed in previous works. Second, for general graphs with maximum degree dmax, our algorithm achieves O( √ dmax log n) discrepancy using the transition matrix based on the metropolis hasting algorithm. Note that this algorithm does not need information of dmax while almost all previous works use it.