## 4 Simulation Results
### Case Study 2  
Let us consider a 6-node network described in Figure 6.  
It is known that for this topology, the Laplacian spectrum is \(sp(\mathbf{L})=\{0,1,2,3,3,5\}\).  
Let us define the same initial information state of each node at time \(t=0\): \(\mathbf{x}(0)=\{0.5832,0.74,0.2348,0.7350,0.9706,0.8669\}\) for both methods. By using a standard consensus algorithm [26], the consensus value \(\bar{x}=0.6884\) can be easily inferred.  
It can be seen in Figure 7 that the Lagrange-based method in [1] takes a long time to achieve the consensus first and then track the constraint to obtain the stepsizes \(\alpha_{t},t=1,...,N-1\).  
Now, with the same initial \(\mathbf{\alpha}_{t}(0),t=1,\ldots,5\), for the iterative procedure of the proposed method, the \(\mathbf{\alpha}_{t}\) after using the Algorithm 1 with the convergence trajectories of the \(\mathbf{\alpha}_{t}\) are illustrated in Figure 8.  
Figures 7 and 8 express the significant pros of our proposed method since the number of iterations is much less than that of Lagrange-based method. The consensus term is executed in advance from the start of the procedure and then try to reach the constraint term to figure out the expected values of the stepsizes \(\mathbf{\alpha}_{t}=\{1,0.5,0.1027,0.2,0.3333\}\). Obviously, since the authors operate the proposed algorithm with the number of \(\mathbf{\alpha}_{t}\) being \(N-1=5\), it is needed to run the next stage to eliminate the residual values. As can be seen clearly, the executive time for ALADIN-based method is significantly faster than the proposed method as showed in Figure 9.  
Figure 9 shows that the ALADIN-based method approaches the destined values earlier than Lagrange-based method. Furthermore, in order to get the non-zero Laplacian eigenvalues of the given network, the Algorithm 2 is carried out to obtain vector of stepsizes \(\mathbf{\alpha}_{t}=\{1,0.5,0.2,0.3333\}\).  
Finally, by constructing the Brand-and-Bound based method to solve the Problem 1, which has been proposed in [16], we achieve the vector of multiplicities \(\mathbf{m}=\{1,1,1,2\}\), hence deduce the Laplacian spectrum \(sp(\mathbf{L})=\{1,2,3,3,5\}\).  
\begin{table}
\begin{tabular}{l l} \hline \hline \(\mathbf{\mathsf{S}}_{1}\) & \(\{0.25,0.5919,0.5543\}\) \\ \(\mathbf{\mathsf{S}}_{2}\) & \(\{0.25\}\) \\ \(\mathbf{m}\) & 3 \\ Iterations & 130 \\ \(\mathcal{R}_{\mathbf{L}}\) & 3 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The achievement of the proposed procedure in the sense of 4-node topology.  
Figure 6: A new network constituted by 6 nodes.  
Figure 8: \(\mathbf{\alpha}_{t}\) convergence trajectories implemented by the proposed method.  
Figure 7: \(\mathbf{\alpha}_{t}\) convergence trajectories implemented by Lagrange-based method.  
Now, let us see how the proposed procedure works via the table below.  
Table 2 is obtained after executing the proposed procedure and the Lagrange-based method.  
Notice that the proposed method gives results much better than the Lagrange-based method. Let us see that at the iteration of 33,950, the Lagrange-based method gives the set of \(\mathbf{\mathcal{S}}_{1}\) that has still not satisfied the constraint.  
Recently, besides the Laplacian spectrum estimation basing on the optimization approaches, there are also some works that approximate the Laplacian spectrum via iterative dynamics process (taking random walk for an example). However, from our point of view, the dominant contribution of our proposed method is the possibility of implementation in a decentralized manner. Moreover, another method to faster the convergence rate of the Laplacian spectrum estimation procedure is the ADMM-based method, proposed in [16]. The important step in this work is to re-parameterize adequately the non-convex formulation into convex one. It is hard to compare the efficiency between ADMM-based method and the proposed method. Since, our study focuses on the non-convex formulation.