TY - JOUR
T1 - Exponential convergence of the discrete- and continuous-time Altafini models
AU - Liu, Ji
AU - Chen, Xudong
AU - Başar, Tamer
AU - Belabbas, Mohamed Ali
N1 - Manuscript received January 23, 2017; accepted April 20, 2017. Date of publication May 2, 2017; date of current version December 1, 2017. This work was supported in part by AFOSR MURI Grant FA 9550-10-1-0573. This paper was presented in part at the 54th IEEE Conference on Decision and Control and the 2016 American Control Conference. Recommended by Associate Editor Z. Sun. (Corresponding author: Ji Liu.) J. Liu, T. Bas¸ar, and M.-A. Belabbas are with the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Champaign, IL 61820 USA (e-mail: [email protected]; [email protected]; belabbas@ illinois.edu).
PY - 2017
Y1 - 2017
N2 - This paper considers the discrete-time version of Altafini's model for opinion dynamics in which the interaction among a group of agents is described by a time-varying signed digraph. Prompted by an idea from [3], exponential convergence of the system is studied using a graphical approach. Necessary and sufficient conditions for exponential convergence with respect to each possible type of limit states are provided. Specifically, under the assumption of repeatedly jointly strong connectivity, it is shown that 1) a certain type of two-clustering will be reached exponentially fast for almost all initial conditions if, and only if, the sequence of signed digraphs is repeatedly jointly structurally balanced corresponding to that type of two-clustering; 2) the system will converge to zero exponentially fast for all initial conditions if, and only if, the sequence of signed digraphs is repeatedly jointly structurally unbalanced. An upper bound on the convergence rate is provided. The results are also extended to the continuous-time Altafini model.
AB - This paper considers the discrete-time version of Altafini's model for opinion dynamics in which the interaction among a group of agents is described by a time-varying signed digraph. Prompted by an idea from [3], exponential convergence of the system is studied using a graphical approach. Necessary and sufficient conditions for exponential convergence with respect to each possible type of limit states are provided. Specifically, under the assumption of repeatedly jointly strong connectivity, it is shown that 1) a certain type of two-clustering will be reached exponentially fast for almost all initial conditions if, and only if, the sequence of signed digraphs is repeatedly jointly structurally balanced corresponding to that type of two-clustering; 2) the system will converge to zero exponentially fast for all initial conditions if, and only if, the sequence of signed digraphs is repeatedly jointly structurally unbalanced. An upper bound on the convergence rate is provided. The results are also extended to the continuous-time Altafini model.
KW - Clustering
KW - Multi-agent systems
KW - Opinion dynamics
KW - Signed graphs
KW - Structural balance
UR - https://www.scopus.com/pages/publications/85040306828
UR - https://www.scopus.com/pages/publications/85040306828#tab=citedBy
U2 - 10.1109/TAC.2017.2700523
DO - 10.1109/TAC.2017.2700523
M3 - Article
AN - SCOPUS:85040306828
SN - 0018-9286
VL - 62
SP - 6168
EP - 6182
JO - IEEE Transactions on Automatic Control
JF - IEEE Transactions on Automatic Control
IS - 12
ER -