Boundary value problems in consensus networks

This paper studies the effect of boundary value conditions on consensus networks. Consider a network where some nodes keep their estimates constant while other nodes average their estimates with that of their neighbors. We analyze such networks and show that in contrast to standard consensus networks, the network estimate converges to a general harmonic function on the graph. Furthermore, the final value depends only on the value at the boundary nodes. This has important implications in consensus networks -- for example, we show that consensus networks are extremely sensitive to the existence of a single malicious node or consistent errors in a single node. We also discuss applications of this result in social and sensor networks. We investigate the existence of boundary nodes in human social networks via an experimental study involving human subjects. Finally, the paper is concluded with the numerical studies of the boundary value problems in consensus networks.

[1]  Ali H. Sayed,et al.  Diffusion Adaptation Strategies for Distributed Optimization and Learning Over Networks , 2011, IEEE Transactions on Signal Processing.

[2]  Peng Yang,et al.  Stability and Convergence Properties of Dynamic Average Consensus Estimators , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[3]  László Lovász,et al.  Harmonic and analytic functions on graphs , 2003 .

[4]  Reuven Cohen,et al.  Convergence Properties of the Gravitational Algorithm in Asynchronous Robot Systems , 2005, SIAM J. Comput..

[5]  Andrés M. Encinas,et al.  Solving Boundary Value Problems on Networks Using Equilibrium Measures , 2000 .

[6]  Reuven Cohen,et al.  Local spreading algorithms for autonomous robot systems , 2008, Theor. Comput. Sci..

[7]  Soummya Kar,et al.  Gossip Algorithms for Distributed Signal Processing , 2010, Proceedings of the IEEE.

[8]  Johannes Gehrke,et al.  Gossip-based computation of aggregate information , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[9]  John N. Tsitsiklis,et al.  Convergence Speed in Distributed Consensus and Averaging , 2009, SIAM J. Control. Optim..

[10]  Asuman E. Ozdaglar,et al.  Opinion Dynamics and Learning in Social Networks , 2010, Dyn. Games Appl..

[11]  Stephen P. Boyd,et al.  Gossip algorithms: design, analysis and applications , 2005, Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies..

[12]  Soummya Kar,et al.  Distributed Consensus Algorithms in Sensor Networks With Imperfect Communication: Link Failures and Channel Noise , 2007, IEEE Transactions on Signal Processing.

[13]  Reuven Cohen,et al.  Convergence of Autonomous Mobile Robots with Inaccurate Sensors and Movements , 2006, SIAM J. Comput..

[14]  R. Srikant,et al.  Quantized Consensus , 2006, 2006 IEEE International Symposium on Information Theory.

[15]  Jon M. Kleinberg,et al.  Spatial gossip and resource location protocols , 2001, JACM.

[16]  Nikolaos M. Freris,et al.  Randomized gossip algorithms for solving Laplacian systems , 2015, 2015 European Control Conference (ECC).

[17]  Hamid Krim,et al.  Analysis and Control of Beliefs in Social Networks , 2014, IEEE Transactions on Signal Processing.

[18]  Martin J. Wainwright,et al.  Network-Based Consensus Averaging With General Noisy Channels , 2008, IEEE Transactions on Signal Processing.

[19]  Sergio Barbarossa,et al.  Fast Distributed Average Consensus Algorithms Based on Advection-Diffusion Processes , 2010, IEEE Transactions on Signal Processing.

[20]  Anna Scaglione,et al.  Models for the Diffusion of Beliefs in Social Networks: An Overview , 2013, IEEE Signal Processing Magazine.

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

[22]  Reuven Cohen,et al.  Convergence of Autonomous Mobile Robots with Inaccurate Sensors and Movements , 2008, SIAM J. Comput..

[23]  Hamid Krim,et al.  Control and prediction of beliefs on social network , 2013, 2013 5th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).