Optimal distributed consensus in Small World Network

The broad applications of multi-agent systems in many areas have stimulated a great deal of interests in studying consensus problems. This paper investigates an optimal distributed consensus and analysis the application in small world network by reducing the RH problem to 1-dimension and with the penalty F(x_i-x⁽ⁱ⁾) tracing the fast signal to reach consensus. We prove the algebraic connectivity of a graph is the second smallest eigenvalue of its Laplacian matrix and a measure of speed of solving consensus problems in networks. We find that the convergence speed of the consensus algorithm on a regular lattice can be greatly enhanced by just randomly rewiring a very small number of links in the network. By increasing the algebraic connectivity λ₂(G) and the Laplacian matrix λ₂(L), we can improve the convergent speed.