Notes on Gossip Dissemination and Averaging

Let P = [pij] be the probability probability transition matrix of Markov chain, so that pij is the probability that node i contacts node j. We assume that P is aperiodic, irreducible. It may or may not be symmetric. P is double–stochastic matrix, i.e. the sum of elements in each row or column is equal to 1. By the Perron–Frobenius theorem, P has a stationary distribution π = [πi] π = πP (1) where each πi positive, and ∑n i=1 πi = 1. Moreover, the rows of P k all tend to π as k → ∞. Moreover, P has n real eigenvalues λ 1 = λ1 > λ2 ≥ λ2 ≥ . . . ≥ λn ≥ −1. The conductance of P is defined as