Analysis of Second Order Cluster Consensus for Directed Networks with Time Delay

Cluster consensus with first and higher-order antagonistic interaction dynamics

“…From Corollary 1, the signed Laplacian matrix L and the unsigned Laplacian matrix L have the same eigenvalues. From (23), L also has the same eigenvalues as L. Since the eigenvalues of C m and C m are functions of the eigenvalues of L and L, we conclude that the stabilities of S d in (32) and S d in (27) are equivalent for structurally balanced graphs.…”

“…This implies that L and L have the same eigenvalues. From (23), L also has the same eigenvalues with L. In the structurally balanced case, the stabilities of S c in (9) and S c in (1) are equivalent for m P 1.…”

“…(() Suppose S c is stable, i.e., L is a Laplacian matrix. Note from (23) that L has eigenvalues of L and L. As L is a Laplacian matrix by definition in (19) and S c is stable, it follows that S c is stable.…”

“…S c is stable. From (23), recall that the eigenvalues of L for S c are the union of the eigenvalues of L and L. By using this result, the eigenvalues of L m can be obtained with respect to the eigenvalues of L and L as…”

“…Lemma 2. [23] Suppose that c 1 -0 and the underlying unsigned digraph of the unsigned network consists of l p primary and l s secondary layer subgraphs. Then the system (27) achieves K ¼ l p þ l s cluster consensus if and only if the roots of the equation…”