On the Diffusion Geometry of Graph Laplacians and Applications

We study directed, weighted graphs $G=(V,E)$ and consider the (not necessarily symmetric) averaging operator $$ (\mathcal{L}u)(i) = -\sum{j \sim{} i}{p{ij} (u(j) - u(i))},$$ where $p{ij}$ are normalized edge weights. Given a vertex $i \in V$, we define the diffusion distance to a set $B \subset V$ as the smallest number of steps $d{B}(i) \in \mathbb{N}$ required for half of all random walks started in $i$ and moving randomly with respect to the weights $p{ij}$ to visit $B$ within $d{B}(i)$ steps. Our main result is that the eigenfunctions interact nicely with this notion of distance. In particular, if $u$ satisfies $\mathcal{L}u = \lambda u$ on $V$ and $$ B = \left{ i \in V: - \varepsilon \leq u(i) \leq \varepsilon \right} \neq \emptyset,$$ then, for all $i \in V$, $$ d{B}(i) \log{\left( \frac{1}{|1-\lambda|} \right) } \geq \log{\left( \frac{ |u(i)| }{|u|{L^{\infty}}} \right)} - \log{\left(\frac{1}{2} + \varepsilon\right)}.$$ $dB(i)$ is a remarkably good approximation of $|u|$ in the sense of having very high correlation. The result implies that the classical one-dimensional spectral embedding preserves particular aspects of geometry in the presence of clustered data. We also give a continuous variant of the result which has a connection to the hot spots conjecture.