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Quadratic form and kernel of a weighted graph Laplacian

Statement

Let $V = {1, \dots, n}$ with $n \geq 1$ , let $W$ be symmetric, nonnegative, and zero-diagonal, let $D = diag (\sum_{j} w_{i j})$ , let $L = D - W$ , and let the support graph have edge ${i, j}$ exactly when $w_{i j} > 0$ . For every $f \in R^{n}$ , $f^{T} L f = \frac{1}{2} \sum_{i, j = 1}^{n} w_{i j} (f_{i} - f_{j})^{2} .$ Consequently $L$ is positive semidefinite and $\ker L = {f \in R^{n} : f is constant on each connected component of the support graph},$ including isolated singleton components.

Statement references

Ulrike von Luxburg, A Tutorial on Spectral Clustering, Statistics and Computing 17(4), 2007

Jean Gallier, Notes on Elementary Spectral Graph Theory: Applications to Graph Clustering Using Normalized Cuts, 2013

Facts & Assumptions

Given: A finite symmetric nonnegative zero-diagonal weight matrix $W$ , its degree matrix $D$ , Laplacian $L = D - W$ , support graph, and an arbitrary vector $f \in R^{n}$ .

[F1]

Finite weighted graph Laplacian: $d_{i} = \sum_{j} w_{i j}$ , $D = diag (d_{i})$ , $L = D - W$ , and the support graph has edge ${i, j}$ exactly when $w_{i j} > 0$ .

Proof

1.1givenF1algebra

By [F1], $f^{T} L f = \sum_{i} d_{i} f_{i}^{2} - \sum_{i, j} w_{i j} f_{i} f_{j}$ . Replacing $d_{i}$ by $\sum_{j} w_{i j}$ , using $w_{i j} = w_{j i}$ to identify the two square sums, and collecting terms gives $f^{T} L f = \frac{1}{2} \sum_{i, j} w_{i j} (f_{i}^{2} - 2 f_{i} f_{j} + f_{j}^{2}) = \frac{1}{2} \sum_{i, j} w_{i j} (f_{i} - f_{j})^{2}$ .

2.1givenF1step 1.1algebra

The displayed sum is nonnegative for every $f$ , and $L$ is symmetric because both $D$ and $W$ are symmetric. Therefore $L$ is positive semidefinite.

2.2givenF1step 1.1algebra

If $L f = 0$ , then $f^{T} L f = 0$ . By step 1.1 this is a finite sum of nonnegative terms, so every term with $w_{i j} > 0$ vanishes and $f_{i} = f_{j}$ on every support edge. Equality propagates along each finite path, hence $f$ is constant on every connected component, including an isolated singleton.

3.1givenF1step 2.2algebra∎

Conversely, suppose $f$ is constant on each support component. For each vertex $i$ , every $j$ with $w_{i j} > 0$ lies in the same component, so $f_{i} = f_{j}$ . Hence $(L f)_{i} = d_{i} f_{i} - \sum_{j} w_{i j} f_{j} = \sum_{j} w_{i j} (f_{i} - f_{j}) = 0$ for every $i$ , and $L f = 0$ . The two inclusions give the stated kernel equality.

Depends on

Finite weighted graph Laplacian

Used by

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Sources

Ulrike von Luxburg, A Tutorial on Spectral Clustering, Statistics and Computing 17(4), 2007 (standard reference, not scraped)
Jean Gallier, Notes on Elementary Spectral Graph Theory: Applications to Graph Clustering Using Normalized Cuts, 2013 (standard reference, not scraped)