## 1 Introduction
Here, an Alternating Direction of Multipliers Method (ADMM)-based algorithm was employed to perform the factorization of the averaging matrix and to compute the average degree of the network concurrently. However, the main point in this work was the reformulation into the convex optimization problem, which is convenient to make use of the ADMM method to solve the problem. In addition to that, the impact of the Laplacian spectrum into power systems is expressed through energy management in smart grids [17] and the determination of the grid robustness against low frequency disturbance in [18]. In this work, in the framework of spectral graph theory, the authors reveal that the decomposition of frequency signal along scaled Laplacian spectrum when the damping-inertia ratios are uniform across buses not only makes the system respond faster but also helps lower the system nadir after a disturbance. In dynamic network systems, the spectrum of Laplacian matrix can also be utilized for locally checking the controllability and the observability [19]. From a short literature survey above, it is obvious that the Laplacian spectrum plays an important role in many fields. For instance, Laplacian spectrum can be used to design consensus matrices [11; 12], to compute these robustness indices [15; 16; 17; 18], or to check the controllability and the observability [19]. Hence, it is desirable to have an efficient method for monitoring the Laplacian spectrum of a dynamics network system. One thing to remark here is that if the global network topology is known in a-priori, the Laplacian matrix can be easily deduced. However, implementing a centralized structure is an expensive task due to the high computational cost, the heavy communication infrastructure aspect and the problem from large dimensionality. Additionally, if there is a failure problem from one point, it will affect the whole network. Therefore, our study is restricted to the assumption that the network topology (represented by the Laplacian Matrix) is unknown at the first glance. A dominant contribution of this paper is the possibility of implementing this monitoring scheme in a decentralized manner. In this paper, we present an Augmented Lagrangian based Alternating Direction Inexact Newton (ALADIN) method to estimate the Laplacian spectrum in decentralized scheme for dynamic controlled networks. The key feature of this paper is the direct solution to non-convex optimization for Laplacian spectrum estimation using ALADIN method. To simplify, the scope of this study is restricted to networks performing noise-free as well as the number of the agents \(N\) in the network in known in-priori by using random walk algorithm [20]. The network is modeled, then Laplacian eigenvalues and average consensus are retrieved respectively. Since the Laplacian spectrum matrix is not directly computable for undetermined network topology, the decentralized estimation of the Laplacian spectrum has been introduced with three main approaches in the recent literature: Fast Fourier Transform (FFT)-based methods [21; 22], local eigenvalue decomposition of given observability-based matrices [23], and distributed factorization of the averaging matrix \(\mathbf{J}_{N}=\frac{1}{N}\mathbf{1}\mathbf{1}^{T}\)[24]. FFT-based methods require a specific protocol. However, they do not make use of the available measurements coming from the consensus protocol. On the other hand, the method in [23] allows using the transient of the average consensus protocol but for several consecutive initial conditions. The distributed factorization of the averaging matrix in [24] yields the inverses of non-zero Laplacian eigenvalues and can be solved as a constrained consensus problem. The Laplacian eigenvalues can be deduced as the inverse of the stepsizes in each estimating factor, where these factors are constrained to be structured as Laplacian based consensus matrices.