## 3 Distributed Estimation of Laplacian Spectrum
### Distributed Estimation of Non-Zero Laplacian Eigenvalues  
For distributively carrying out the factorization of the average consensus matrix as factors of Laplacian based consensus matrices, the idea is to minimize the disagreement between neighbors on the value of \(\boldsymbol{\alpha}_{t}\) while ensuring that the factorization of the average consensus matrix is achieved. Such a factorization is assessed by constraining the values of the nodes after \(h\) iterations of the consensus algorithm to be equal to the average of the initial values:  
\[\min_{\boldsymbol{\alpha}_{t}\in R^{N\times 1},t=1,2,\ldots,h} \frac{1}{2}\sum_{t=1}^{h}\sum_{i\in V}\sum_{j\in\mathcal{N}_{i}} (\alpha_{t,j}-\alpha_{t,j})^{2}\] (5) subject to \[\mathbf{x}(h)=\tilde{\mathbf{x}}\]  
or, it can be rewritten in the following form:  
\[\min_{\boldsymbol{\alpha}_{t}\in R^{N\times 1}} \frac{1}{2}\sum_{t=1}^{h}\boldsymbol{\alpha}_{t}^{T}\mathbf{L} \boldsymbol{\alpha}_{t}.\] (6) subject to \[\mathbf{x}(h)=\tilde{\mathbf{x}}\]  
This optimization has been solved by applying Augmented Lagrange Method [1]. However, the disadvantage of this method is the slow convergence rate due to the fact that it is a non-convex optimization problem. To overcome this unexpected issue, the authors have suggested an interesting variant by converting the non-convex function into the convex one. By that, the optimization can be easily and effectively solved by Alternating Direction of Multipliers Methods(ADMM) [16; 27]. In this paper, we proposed a method that can solve a non-convex problem effectively by employing ALADIN method, which is described as following:  
1. **Step 1:** ALADIN solves in parallel a sequence of equality-constrained non-linear problems (NLP) by introducing an augmented variables \(\mathbf{y}\) as follows: \[\min_{\mathbf{\alpha}_{t}\in\mathbb{R}^{N\times 1},t=1,2,...,h}\frac{1}{2}\mathbf{ \alpha}_{t}^{T}\mathbf{L}\mathbf{\alpha}_{t}+\mathbf{\lambda}^{T}(\mathbf{\alpha}_{t}- \mathbf{y}_{t})+\frac{\rho}{2}\|(\mathbf{\alpha}_{t}-\mathbf{y}_{t})\|_{\mathbf{\Sigma} _{t}}^{2}\] (7) subject to \[\mathbf{x}(h)-\bar{\mathbf{x}}=0|\mathbf{\beta}\] \[\mathbf{y}_{t,i}=\mathbf{y}_{t,j}\;\;i=1,\ldots,N;j\in\mathcal{N }_{i}\quad(\mathcal{C})\] where \(\rho,\beta\) are penalty parameter and multiplier of the equality constraint, respectively. \(\mathcal{C}=\{\mathbf{y}_{t}:y_{t,i}=y_{t,j},i=1,\ldots,N;j\in\mathcal{N}_{i}\}\). One thing to note here is that with the given initial information \(x_{i}(0),i=1,2,\ldots,N\), running a standard consensus algorithm can determine the average value \(\bar{x}=\frac{1}{N}\sum_{i=1}^{N}x_{i}(0)\). In addition to that, the positive semi-definite scaling matrices \(\Sigma_{t}\) can be randomly initialized or even be an identity matrix \(\mathbf{I}_{N}\). In this optimization problem (7), augmented variables \(\mathbf{y}_{t}\) are introduced with respect to the constraint \(\mathcal{C}\). However, this NLP is to be solved to define the variables \(\mathbf{\alpha}_{t}\), hence, the constraint \(\mathcal{C}\) is going to be relaxed. The solution \(\mathbf{\alpha}_{t}[k+1],\mathbf{\beta}[k+1]\) obtained from (7) is then used to check the stopping criteria the the next steps of the ALADIN-based algorithm procedure with \(k\) being an iteration of the optimization process. Herein, if \(\frac{1}{2}\sum_{t=1}^{h}\sum_{i\in V}\sum_{j\in\mathcal{N}_{i}}(\alpha_{t,j} -\alpha_{t,i})^{2}<\epsilon\) and \(\rho\|\Sigma_{t}(\mathbf{\alpha}_{t}-\mathbf{y}_{t})\|_{1}\leq\epsilon\), then one can get \(\mathbf{\alpha}_{t}^{*}\) as well as the Algorithm stops. 2.