Small world phenomenon, rapidly mixing Markov chains, and average consensus algorithms

In this paper, we demonstrate the relationship between the diameter of a graph and the mixing time of a symmetric Markov chain defined on it. We use this relationship to show that graphs with the small world property have dramatically small mixing times. Based on this result, we conclude that addition of independent random edges with arbitrarily small probabilities to a cycle significantly increases the convergence speed of average consensus algorithms, meaning that small world networks reach consensus orders of magnitude faster than a cycle. Furthermore, this dramatic increase happens for any positive probability of random edges. The same argument is used to draw a similar conclusion for the case of addition of a random matching to the cycle.

[1]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

[2]  Jon M. Kleinberg,et al.  Small-World Phenomena and the Dynamics of Information , 2001, NIPS.

[3]  Stephen P. Boyd,et al.  Fastest Mixing Markov Chain on a Graph , 2004, SIAM Rev..

[4]  Alan M. Frieze,et al.  The diameter of randomly perturbed digraphs and some applications , 2007, International Workshop and International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques.

[5]  J.N. Tsitsiklis,et al.  Convergence in Multiagent Coordination, Consensus, and Flocking , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[6]  Stephen P. Boyd,et al.  The Fastest Mixing Markov Process on a Graph and a Connection to a Maximum Variance Unfolding Problem , 2006, SIAM Rev..

[7]  B. Mohar.  THE LAPLACIAN SPECTRUM OF GRAPHS y , 1991 .

[8]  Béla Bollobás,et al.  The Diameter of a Cycle Plus a Random Matching , 1988, SIAM J. Discret. Math..

[9]  Reza Olfati-Saber,et al.  Consensus and Cooperation in Networked Multi-Agent Systems , 2007, Proceedings of the IEEE.

[10]  S. Brenner,et al.  The structure of the nervous system of the nematode Caenorhabditis elegans. , 1986, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[11]  A. Sinclair.  Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow , 1992, Combinatorics, Probability and Computing.

[12]  Bojan Mohar,et al.  Eigenvalues, diameter, and mean distance in graphs , 1991, Graphs Comb..

[13]  A. Barabasi,et al.  Scale-free characteristics of random networks: the topology of the world-wide web , 2000 .

[14]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[15]  Fan Chung Graham,et al.  The Average Distance in a Random Graph with Given Expected Degrees , 2004, Internet Math..

[16]  R. Durrett.  Random Graph Dynamics: References , 2006 .

[17]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[18]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[19]  R. Olfati-Saber.  Ultrafast consensus in small-world networks , 2005, Proceedings of the 2005, American Control Conference, 2005..

[20]  M. Newman,et al.  Mean-field solution of the small-world network model. , 1999, Physical review letters.

[21]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[22]  H E Stanley,et al.  Classes of small-world networks. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[23]  M E Newman,et al.  Scientific collaboration networks. I. Network construction and fundamental results. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Luc Moreau,et al.  Stability of multiagent systems with time-dependent communication links , 2005, IEEE Transactions on Automatic Control.

[25]  Alistair Sinclair,et al.  Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow , 1992, Combinatorics, Probability and Computing.