In this article, we show that magnitude homology and path are closely related, give some applications. We define differentials MH k ℓ ( G ) ⟶ − 1 $\operatorname{MH}^{\ell }_k(G) \longrightarrow \operatorname{MH}^{\ell -1}_{k-1}(G)$ between homologies of a digraph $G$ , which make them chain complexes. Then its $\mathcal {MH}^{\ell }_k(G)$ is non-trivial homotopy invariant in the context ‘homoto...