Normalized Hodge Laplacian Matrix and Application to Random Walk on Simplicial Complexes

Let X be a simplicial complex, \(\partial _k:C_k\rightarrow C_{k-1}\) ${\partial }_k:C_k\to C_{k-1}$ a boundary map between spaces of chain on X with respect to real coefficients and \(B_k\) $B_k$ a matrix representation of \(\partial _k\) ${\partial }_k$ . A Hodge k-Laplacian matrix on simplicial complexes is defined by \(L_k=B_{k+1}B_{k+1}^T+B_k^TB_k\) $L_k=B_{k+1}{B}_{k+1}^T+B_k^T{B}_k$ which is a generalization of a Laplacian matrix L on graphs. In this work, we generalize a normalized Laplacian matrix \(\mathcal {L}\) $\mathcal{L}$ on graphs to a normalized Hodge k-Laplacian matrix \(\mathcal {L}_k\) ${\mathcal{L}}_k$ (i.e. \(\mathcal {L}_0=\mathcal {L}\) ${\mathcal{L}}_0=\mathcal{L}$ ) on simplicial complexes. This matrix is also a Hodge Laplacian matrix and this fact leads some useful properties. We finally apply this matrix for random walks on simplicial complexes.