Leader selection for coherence in symmetric and asymmetric trees

Abstract

This paper studies consensus dynamics in symmetric and asymmetric trees with leader selection under additive noise environment, which is characterized as leader-follower coherence determined by the spectrum. In order to compare the effect of the leader's positions on the coherence, we choose a treelike network model and employ three different ways to assign the leaders. Based on the tree construction, we obtain exact solutions for the leader-follower coherence depending on the number of leaders and network parameters. We then show that the consensus in symmetric and asymmetric trees becomes better with an increasing number of leaders. When the number of leaders and network parameters satisfy certain conditions, the leader's positions and topological structures have a profound impact on the coherence.

Consensus-Related Performance of Triplex MASs Based on Partial Complete Graph Structure

“…(Refs. [ 11 , 12 , 13 , 14 , 15 ]) The first-order system with disturbance is with and where is a vector of uncorrelated noise. is the Laplacian matrix.…”

“…As a significant interdisciplinary field, consensus-related research has received much attention from both engineers and scholars in the past two decades, and there exist many applications or research works on the consensus problem, for instances, formation control, sensor networks, decision making, and so on. Scholars have conducted significant valuable research on consensus from various aspects, such as the order of system (first or second order [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 ]), ways of communication (continuous or intermittent [ 4 , 7 ], control methods (adaptive control [ 5 ], intermittent control [ 7 ], sliding mode control [ 19 ]), convergence time (finite time or fixed time [ 6 , 8 ]) and so on.…”

“…For solving consensus problems, the connecting relations among nodes are characterised by the graph of the network, and the convergence index of the dynamics model, such as consensus speed [ 1 , 9 ] and network coherence [ 11 , 12 , 13 , 14 , 15 , 16 ], can be determined by the second smallest Laplacian eigenvalue and the Laplacian spectrum. Similarly, as a branch of coordination problems, synchronization problems are always connected with the notion of system topologies [ 20 , 21 , 22 , 23 , 24 , 25 ].…”

“…There exists significant coordination-related research on the Laplacian eigenvalues [ 1 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 ] of the network graph. In these enlightening works, Ref.…”

“…[ 14 ], an analytical expression for the leader–follower coherence is determined depending on the number of leader nodes and network parameters. In [ 15 ], based on the tree construction, the authors show that the consensus in symmetric and asymmetric trees becomes better with an increasing number of leader nodes.…”