fixed coefficients yield the same spectral response for both graphon and graph convolutions

[[concept]]

Important

Given the same (fixed) coefficients $h_k$, define the graph convolution $H(S)x={\sum }_{k=0}^{K-1}{h}_k{S}^{k}x$. The spectral representation of that convolution is $h(\lambda )={\sum }_{k=0}^{K-1}{h}_k{\lambda }^k$

• this is the same as the spectral representation of graphon convolutions/function for the same coefficients.

• The only difference is where the function is evaluated.

  • For a graphon, we evaluate it at the graphon shift operator eigenvalues ${\lambda }_j(T_W)$ for graphon signal $T_{H_j}$
  • For a graph, this is the graph shift operator eigenvalues ${\lambda }_j(S)$ for the graph signal $H(S{)}_j$

Review

dsg

If we fix the coefficients $h_k$ for a graph convolution, the {ass||spectral representation||attribute} of the resulting graph filter {has||is the same||relationship} as the {ass||spectral representation||attribute} of {hha||the graphon convolution/filter||relation} with the same coefficients

Mentions

Mentions

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