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 ${\mathrm{M}H}_k^{\mathrm{\ell }}(G)⟶{\mathrm{M}H}_{k-1}^{\mathrm{\ell }-1}(G)$ between magnitude homologies of a digraph $G$, which make them chain complexes. Then we show that its homology ${\mathcal{M}H}_k^{\mathrm{\ell }}(G)$ 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 ${\mathcal{M}H}_k^k(G)$ 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 ${\mathrm{M}H}_k^{\mathrm{\ell }}(G)$, and the second page is isomorphic to our homology ${\mathcal{M}H}_k^{\mathrm{\ell }}(G)$. 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 $k\ge 2$ and ${\tilde{H}}_1(g)\mathrm{\ne }0$ if any edges of an undirected graph $g$ is contained in a cycle of length $\ge 5$.