# On the Diffusion Geometry of Graph Laplacians and Applications

9 Nov 2016Xiuyuan ChengManas RachhStefan Steinerberger

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... (read more)

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