In an n-node connected graph A, each node i is with a real-valued state xi(t), i = 1, 2, …, n, and is able to communicate with certain other nodes. A periodic gossip sequence is able to drive all xi(t) to converge to equation exponentially fast. Different sequences are usually associated with different convergence rates for graphs with cycles. This paper mainly focuses on a type of optimal periodic gossip sequences for ring graphs. Explicit formulas to compute their convergence rates are given, which are determined by the adjacency matrix of the n over 2-node ring graph when n is even and Chebychev polynomials of the second kind when n is odd.
['Shaoshuai Mou', 'A. S. Morse', 'B. D. O. Anderson']
Convergence rate of optimal periodic gossiping on ring graphs