On Optimality of Sparse Long-Range Links in Circulant Consensus Networks

We consider spatially invariant consensus networks in which the directed graph describing the interconnection topology, the link weights, and the temporal dynamics, are all characterized by circulant matrices. We seek the best new links, subject to budget constraints, whose addition to the network maximally improves its rate of convergence to consensus. Motivated by small-world networks, we apply the optimal link creation problem to circulant networks with local communication links. We observe that the optimal new links are sparse and long-range, and have an increasingly pronounced effect on the convergence rate of the network as its size grows. To further investigate the properties of optimal links analytically, we restrict attention to the creation of links with small weights, referred to as weak links. We employ perturbation methods to reformulate the problem of optimal weak link creation, and uncover conditions on the network architecture which guarantee sparse and long-range solutions to this optimization problem.

[1]  K. Seaton,et al.  Stations, trains and small-world networks , 2003, cond-mat/0311254.

[2]  M. Degroot.  Reaching a Consensus , 1974 .

[3]  Sandro Zampieri,et al.  Randomized consensus algorithms over large scale networks , 2007 .

[4]  O. Sporns,et al.  Organization, development and function of complex brain networks , 2004, Trends in Cognitive Sciences.

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

[6]  T. Prescott,et al.  Is there a brainstem substrate for action selection? , 2007, Philosophical Transactions of the Royal Society B: Biological Sciences.

[7]  Ran Dai,et al.  Optimal topology design for dynamic networks , 2011, IEEE Conference on Decision and Control and European Control Conference.

[8]  Fu Lin,et al.  On the properties of optimal weak links in consensus networks , 2014, 53rd IEEE Conference on Decision and Control.

[9]  Jon M. Kleinberg,et al.  Navigation in a small world , 2000, Nature.

[10]  Shan Yu,et al.  A Small World of Neuronal Synchrony , 2008, Cerebral cortex.

[11]  Makan Fardad,et al.  On the optimality of sparse long-range links in circulant consensus networks , 2017, 2015 American Control Conference (ACC).

[12]  Robert M. Gray,et al.  Toeplitz and Circulant Matrices: A Review , 2005, Found. Trends Commun. Inf. Theory.

[13]  Fu Lin,et al.  On optimal link creation for facilitation of consensus in social networks , 2014, 2014 American Control Conference.

[14]  Mark Newman,et al.  Networks: An Introduction , 2010 .

[15]  Stephen P. Boyd,et al.  Graph Implementations for Nonsmooth Convex Programs , 2008, Recent Advances in Learning and Control.

[16]  John S. Baras,et al.  Efficient and robust communication topologies for distributed decision making in networked systems , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[17]  Soummya Kar,et al.  Topology for Distributed Inference on Graphs , 2006, IEEE Transactions on Signal Processing.

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

[19]  Alireza Tahbaz-Salehi,et al.  Small world phenomenon, rapidly mixing Markov chains, and average consensus algorithms , 2007, 2007 46th IEEE Conference on Decision and Control.

[20]  Mark S. Granovetter.  The Strength of Weak Ties , 1973, American Journal of Sociology.

[21]  E. David,et al.  Networks, Crowds, and Markets: Reasoning about a Highly Connected World , 2010 .

[22]  John S. Baras,et al.  Average consensus over small world networks: A probabilistic framework , 2008, 2008 47th IEEE Conference on Decision and Control.

[23]  Stephen P. Boyd,et al.  Fastest Mixing Markov Chain on Graphs with Symmetries , 2007, SIAM J. Optim..

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

[25]  Massimo Marchiori,et al.  Economic small-world behavior in weighted networks , 2003 .

[26]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[27]  Frank Allgöwer,et al.  Performance and design of cycles in consensus networks , 2013, Syst. Control. Lett..