we can write a graphon in the basis of its shift operator

[[concept]]

Theorem

We can write the graphon $W$ in the basis ${φ_{i}}$ for its graphon shift operator as

W (u, v) = \sum_{i = 0}^{1} λ_{i} φ_{i} (u) φ_{i} (v)

(compare this to $S = V Λ V^{*}$ for a spectral graph filter)

Proof

Since a graphon shift operator is a self-adjoint Hilbert-Schmidt integral operator, this is a direct consequence of the spectral theorem for self-adjoint compact operators on Hilbert spaces.

see graphon shift operator eigenvalues

Review

#flashcards/math/dsg

We can write the graphon $W$ in the basis of its {graphon shift operator}

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