Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Consensus of Complex Multi-agent Systems

  • Fabio Fagnani
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4471-5102-9_136-1


This entry provides a broad overview of the basic elements of consensus dynamics. It describes the classical Perron-Frobenius theorem that provides the main theoretical tool to study the convergence properties of such systems. Classes of consensus models that are treated include simple random walks on grid-like graphs and in graphs with a bottleneck, consensus on graphs with intermittently randomly appearing edges between nodes (gossip models), and models with nodes that do not modify their state over time (stubborn agent models). Application to cooperative control, sensor networks, and socioeconomic models are mentioned.

This is a preview of subscription content, log in to check access.


  1. Acemoglu D, Como G, Fagnani F, Ozdaglar A (2013) Opinion fluctuations and disagreement in social networks. Math Oper Res 38(1):1–27CrossRefMathSciNetGoogle Scholar
  2. Boyd S, Ghosh A, Prabhakar B, Shah D (2006) Randomized gossip algorithms. IEEE Trans Inf Theory 52(6):2508–2530CrossRefMATHMathSciNetGoogle Scholar
  3. Carli R, Fagnani F, Speranzon A, Zampieri S (2008) Communication constraints in the average consensus problem. Automatica 44(3):671–684CrossRefMATHMathSciNetGoogle Scholar
  4. Castellano C, Fortunato S, Loreto V (2009) Statistical physics of social dynamics. Rev Modern Phys 81:591–646CrossRefGoogle Scholar
  5. Fax JA, Murray RM (2004) Information flow and cooperative control of vehicle formations. IEEE Trans Autom Control 49(9):1465–1476CrossRefMathSciNetGoogle Scholar
  6. Galton F (1907) Vox populi. Nature 75:450–451CrossRefGoogle Scholar
  7. Gantmacher FR (1959) The theory of matrices. Chelsea Publishers, New YorkMATHGoogle Scholar
  8. Jadbabaie A, Lin J, Morse AS (2003) Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans Autom Control 48(6):988–1001CrossRefMathSciNetGoogle Scholar
  9. Levin DA, Peres Y, Wilmer EL (2008) Markov chains and mixing times. AMS, ProvidenceGoogle Scholar
  10. Strogatz SH (2003) Sync: the emerging science of spontaneous order. Hyperion, New YorkGoogle Scholar
  11. Surowiecki J (2004) The wisdom of crowds: why the many are smarter than the few and how collective wisdom shapes business, economies, societies and nations. Little, Brown. (Traduzione italiana: La saggezza della folla, Fusi Orari, 2007)Google Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  1. 1.Dipartimento di Scienze Matematiche ‘G.L. Lagrange’Politecnico di TorinoTorinoItaly