Time estimation for heat diffusion on graphs

This paper studies the estimation of the starting time of a diffusion process from its noisy measurements when there is a single point source located on a known vertex of a graph with unknown starting time. The diffusion process is assumed to be governed by the heat equation. In particular, the Cramer-Rao lower bound (CRLB) for the problem is derived. It is shown that the problem has a larger CRLB for graphs with higher connectivity. Closed form expression of the bound is derived for some graphs. The ML estimator is numerically verified to be unbiased, and achieve the CRLB for some graphs.