This is my first day to learn these stuff. I am confused about the following:
Suppose the underlying graph is just two nodes with edge value $1$.
So the adjacency matrix is: $$A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$
with the degree matrix $$D =\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $$
So the graph Laplacian: $L = D - A$. The consensus protocol should be $$\dot{x} = -Lx = \begin{bmatrix} x_2-x_1 \\ x_1-x_2 \end{bmatrix}$$
So let $x_0 = [0.25;-0.25]$, we have the following plot:
However, the phase plot is the following. Does this phase plot imply any consensus information? I cannot understand "consensus" from the following phase plot. What I mean is $x=0.25$ won't go to $0$ which contradicts the above plot. Please advise, thanks!
Your phase plot is correct. The states may not converge to zero since the Laplacian matrix has exactly 0 (zero) eigenvalue. According to your initial conditions, the states converge to the line $x=y$. Besides, consensus means that all states converge to the same value. It doesn't have to be zero.