On the properties of optimal weak links in consensus networks

We study consensus networks in which each node updates its state by taking a weighted average of the states of its neighbors. Our objective is to determine the optimal set of weak links whose addition to the network maximally improves the efficiency of reaching consensus. Allocating a small amount of resources to the entire network with which new links can be created, we employ a perturbation method to cast this problem as a linear program. We demonstrate that the set of optimal weak links is sparse and, based on extensive numerical experiments, conjecture that they are also long-range. Examples are provided to illustrate the utility of our developments.

[1]  Sandro Zampieri,et al.  Randomized consensus algorithms over large scale networks , 2007, 2007 Information Theory and Applications Workshop.

[2]  Mihailo R. Jovanovic,et al.  H2 norm of linear time-periodic systems: a perturbation analysis , 2006, 2006 American Control Conference.

[3]  Fu Lin,et al.  On the optimal design of structured feedback gains for interconnected systems , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[4]  Bassam Bamieh,et al.  Perturbation Methods in Stability and Norm Analysis of Spatially Periodic Systems , 2008, SIAM J. Control. Optim..

[5]  Mihailo R. Jovanovic.  H2 norm of linear time-periodic systems: a perturbation analysis , 2006 .

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

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

[8]  Fu Lin,et al.  Optimal Control of Vehicular Formations With Nearest Neighbor Interactions , 2011, IEEE Transactions on Automatic Control.

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

[10]  Rashad Moarref,et al.  Controlling the onset of turbulence by streamwise travelling waves. Part 2. Direct numerical simulation , 2010, Journal of Fluid Mechanics.

[11]  Stephen P. Boyd,et al.  Growing Well-connected Graphs , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

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

[13]  Fu Lin,et al.  Identification of sparse communication graphs in consensus networks , 2012, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[14]  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.

[15]  Rashad Moarref,et al.  Controlling the onset of turbulence by streamwise travelling waves. Part 1. Receptivity analysis , 2010, Journal of Fluid Mechanics.

[16]  Florian Dörfler,et al.  Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators , 2009, Proceedings of the 2010 American Control Conference.

[17]  Stephen P. Boyd,et al.  Distributed average consensus with least-mean-square deviation , 2007, J. Parallel Distributed Comput..

[18]  Jie Lin,et al.  Coordination of groups of mobile autonomous agents using nearest neighbor rules , 2003, IEEE Trans. Autom. Control..

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

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

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

[22]  Francesco Bullo,et al.  Distributed Control of Robotic Networks , 2009 .

[23]  Stephen P. Boyd,et al.  Randomized gossip algorithms , 2006, IEEE Transactions on Information Theory.

[24]  Fu Lin,et al.  Design of Optimal Sparse Interconnection Graphs for Synchronization of Oscillator Networks , 2013, IEEE Transactions on Automatic Control.

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