Spectral properties of the grounded Laplacian matrix with applications to consensus in the presence of stubborn agents

We study linear consensus and opinion dynamics in networks that contain stubborn agents. Previous work has shown that the convergence rate of such dynamics is given by the smallest eigenvalue of the grounded Laplacian induced by the stubborn agents. Building on this, we define a notion of centrality for each node in the network based upon the smallest eigenvalue obtained by removing that node from the network. We show that this centrality can deviate from other well known centralities. We then characterize certain properties of the smallest eigenvalue and corresponding eigenvector of the grounded Laplacian in terms of the graph structure and the expected absorption time of a random walk on the graph.

[1]  Fu Lin,et al.  Algorithms for leader selection in large dynamical networks: Noise-free leaders , 2011, IEEE Conference on Decision and Control and European Control Conference.

[2]  Aleksandar Ilic,et al.  Distance spectral radius of trees with fixed maximum degree , 2010 .

[3]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[4]  Maxi San Miguel,et al.  A measure of individual role in collective dynamics , 2010, Scientific Reports.

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

[6]  Martin G. Everett,et al.  A Graph-theoretic perspective on centrality , 2006, Soc. Networks.

[7]  Karl Henrik Johansson,et al.  A graph-theoretic approach on optimizing informed-node selection in multi-agent tracking control , 2014 .

[8]  Matthew O. Jackson,et al.  Naïve Learning in Social Networks and the Wisdom of Crowds , 2010 .

[9]  Shreyas Sundaram,et al.  Resilient Asymptotic Consensus in Robust Networks , 2013, IEEE Journal on Selected Areas in Communications.

[10]  R. Srikant,et al.  Opinion dynamics in social networks: A local interaction game with stubborn agents , 2012, 2013 American Control Conference.

[11]  Shreyas Sundaram,et al.  Distributed Function Calculation via Linear Iterative Strategies in the Presence of Malicious Agents , 2011, IEEE Transactions on Automatic Control.

[12]  Radha Poovendran,et al.  Leader selection for minimizing convergence error in leader-follower systems: A supermodular optimization approach , 2012, 2012 10th International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks (WiOpt).

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

[14]  P. Barooah,et al.  Graph Effective Resistance and Distributed Control: Spectral Properties and Applications , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[15]  Bassam Bamieh,et al.  Leader selection for optimal network coherence , 2010, 49th IEEE Conference on Decision and Control (CDC).

[16]  A. Ozdaglar,et al.  Discrete Opinion Dynamics with Stubborn Agents , 2011 .

[17]  John N. Tsitsiklis,et al.  Introduction to Probability , 2002 .

[18]  Charles M. Grinstead,et al.  Introduction to probability , 1986, Statistics for the Behavioural Sciences.

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

[20]  Ulla Miekkala,et al.  Graph properties for splitting with grounded Laplacian matrices , 1993 .

[21]  Magnus Egerstedt,et al.  Graph Theoretic Methods in Multiagent Networks , 2010, Princeton Series in Applied Mathematics.

[22]  Basel Alomair,et al.  Leader selection in multi-agent systems for smooth convergence via fast mixing , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[23]  Fu Lin,et al.  On new characterizations of social influence in social networks , 2013, 2013 American Control Conference.

[24]  Piet Van Mieghem,et al.  Graph Spectra for Complex Networks , 2010 .

[25]  Marco Dorigo,et al.  Swarm intelligence: from natural to artificial systems , 1999 .

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

[27]  Alireza Tahbaz-Salehi,et al.  A Necessary and Sufficient Condition for Consensus Over Random Networks , 2008, IEEE Transactions on Automatic Control.