  We introduce a modified non-linear heat equation $\partial_t u = \Delta u +
\Gamma u$ as a substitute of $\log P_t f$ where $P_t$ is the heat semigroup. We
prove an exponential decay of $\Gamma u$ under the Bakry Emery curvature
condition $CD(K,\infty)$ and prove the Li-Yau inequality $-\Delta u_t \leq
\frac{n}{2t}$ under the Bakry Emery curvature condition $CD(0,n)$. From this,
we deduce the volume doubling property which solves a major open problem in
discrete Ricci curvature. As an application, we show that there exist no
expander graphs satisfying $CD(0,n)$.
