Magnitude homology and Path homology

In this article, we show that magnitude homology and path homology are closely related, and we give some applications. We define differentials ${\mathrm MH}^{\ell}_k(G) \longrightarrow {\mathrm MH}^{\ell-1}_{k-1}(G)$ between magnitude homologies of a digraph $G$, which make them chain complexes. Then we show that its homology ${\mathcal MH}^{\ell}_k(G)$ is non-trivial and homotopy invariant in the context of `homotopy theory of digraphs' developed by Grigor'yan--Muranov--S.-T. Yau et al (G-M-Ys in the following). It is remarkable that the diagonal part of our homology ${\mathcal MH}^{k}_k(G)$ is isomorphic to the reduced path homology $\tilde{H}_k(G)$ also introduced by G-M-Ys. Further, we construct a spectral sequence whose first page is isomorphic to magnitude homology ${\mathrm MH}^{\ell}_k(G)$, and the second page is isomorphic to our homology ${\mathcal MH}^{\ell}_k(G)$. As an application, we show that the diagonality of magnitude homology implies triviality of reduced path homology. We also show that $\tilde{H}_k(g) = 0$ for $k \geq 2$ and $\tilde{H}_1(g) \neq 0$ if any edges of an undirected graph $g$ is contained in a cycle of length $\geq 5$.