$\L_{ij}$ and the scalar quantities $\u_{j}(t-1)$ also at nearest neighbors. We emphasize, again, that $\u_{i}(t)$ is a scalar quantity and `Random`($[0,1]$) is a random initial condition on the interval $[0,1]$. The vector $\v^{(j)}_{i}$ is the $i$-th component of the $j$-th eigenvector, $T_{max}$ is a positive integer derived in [@Cit:cluster1; @Cit:cluster2], $\texttt{FrequencyPeak(Y,j)}$ returns the frequency at which the $j$-th peak occurs and $\texttt{Coefficient}(\omega_{j})$ return the corresponding Fourier coefficient.

$\u_{i}(0) \leftarrow \texttt{Random}\:([0,1])$ $\u_{i}(-1) \leftarrow \u_{i}(0)$ $t\leftarrow 1$

  -----------------------------------------------------------------
  $\u_{i}(t) \leftarrow 2\u_{i}(t-1)-\u_{i}(t-2) -$
  $\qquad \qquad c^2 \sum_{j\in\mathcal{N}(i)}\L_{ij}\u_{j}(t-1)$
  -----------------------------------------------------------------

$t\leftarrow t+1$ $Y\leftarrow \texttt{FFT}\:(\left[\u_{i}(1),\dots\dots,\u_{i}(T_{max})\right])$ $\omega_{j} \leftarrow
        \texttt{FrequencyPeak}\:(Y,j)$ $\v^{(j)}_{i} \leftarrow \texttt{Coefficient}(\omega_{j})$ $A_j \leftarrow 1$ $A_j \leftarrow 0$ ClusterNumber $\leftarrow \sum_{j=1}^k A_j 2^{j-1}$

The clusters of graph $\mathcal{G}$, determined by the signs of the elements of the eigenvectors of $\L$, can be computed using the frequencies and coefficients obtained from the Fast Fourier Transform of $(\u_{i}(1),\dots,\u_{i}(T_{max}))$, for all $i$ and some $T_{max}>0$. Here $\u_i$ is governed by the wave equation on the graph (shown in Eqn. \[onenodewave\]) with the initial condition $\u(-1)=\u(0)$ and $0<c < \sqrt{2}$.

For the proofs see [@Cit:cluster1; @Cit:cluster2].

The above proof demonstrates that the approach is fundamentally *decentralized*. Moreover, it is shown in [@Cit:cluster1; @Cit:cluster2] that the convergence of the wave equation based 