Alogithm 1 Modified diffusion wavelets.

Input: the number of sensor nodes $N$, communication radius $r$, decomposition level $\eta$, precision $\epsilon$ and MQR function.

Output: sparse basis $\mathsf{\Psi }$.

1 generate a graph $G(V,E)$

2 compute weight adjacency matrix $\mathsf{\Omega }=[w_i{}_j]$ according to the vertex degree/Equation (11)

3 calculate normalized Laplacian matrix $\mathsf{\Lambda }$ relying on Equation (12)/Equation (13)

4 generate diffusion operator $O=I-\mathsf{\Lambda }$

5 recursively raising ${\rm O}$ to power 2

5.1 for $\eta$= 0 to $\eta$ − 1

5.2 ${[x_{j+1}]}_{x_j}$, ${[O]}_{x_0}^{x_1}\leftarrow \mathrm{MQR}({[O^{2^{\eta }}]}_{x_j}^{x_j},\epsilon )$

5.3 $O_{j+1}:={[O^{2^{\eta +}{}^1}]}_{x_{j+}{}_1}^{x_{j+}{}_1}\leftarrow {[x_{j+1}]}_{x_j}{[O^{2^{\eta +}{}^1}]}_{x_j}^{x_j}{[x_{j+1}]}_{x_j}^{\ast }$

5.4 ${[{\mathsf{\Psi }}_{\eta }]}_{x_j}\leftarrow \mathrm{MQR}(I_{\langle x_j\rangle }-{[x_{j+1}]}_{x_j}{[x_{j+1}]}_{x_j}^{\ast },\epsilon )$

5.5 end for

6 concatenation of the scale functions and wavelet functions is regarded as the sparse basis $\mathsf{\Psi }$.

MQR Function:

$Q,R\leftarrow \mathrm{MQR}(B,\epsilon )$

Input: B: $N\times N$ sparse matrix, $\epsilon$

Output: $Q$, $R$ matrix, possibly sparse, such that $B{=}_{\epsilon }QR$

(1) $Q$ is orthogonal

(2) $R$ is upper triangular up to a permutation

(3) The columns of $Q$ $\epsilon$-span the space spanned by the columns of B