Normalized Hodge Laplacian matrix and application to random walk on simplicial complexes (Q6544520)

One of the main purposes of this article is to generalize the notion of a normalized Laplacian matrix \(\mathcal{L}\) on graphs to a normalized Hodge \(k\)-Laplacian matrix \(\mathcal{L}_k\) on simplicial complexes. Let \(X\) be a simplicial complex and \(X^k\) denote the set of all \(k\)-simplices in \(X\). Two faces \(\sigma,\tau\in X^k\) are called upper adjacent if both \(\sigma\) and \(\tau\) are faces of the same \((k+1)\)-simplex of \(X\). The number of upper adjacent elements of \(\sigma\) in \((k+1)\)-simplex is denoted by \(\deg\sigma\). The finite-dimensional vector space with real coefficients whose basis elements are the oriented simplices in \(X^k\) is denoted by \(C_k\). Let \(B_k\) be a matrix representation of a boundary map \(\partial_k: C_k\rightarrow C_{k-1}\). Then the normalized Hodge \(k\)-Laplacian matrix \(\mathcal{L}_k\) is defined by\N\[\N\mathcal{L}_k=D_{k+1}^{-\frac{1}{2}}B_{k+1}B_{k+1}^TD_{k+1}^{-\frac{1}{2}}+D_{k+1}^{\frac{1}{2}}B_{k}^TB_{k}D_{k+1}^{\frac{1}{2}},\N\]\Nwhere \(D_{k+1}^{\frac{1}{2}}\) and \(D_{k+1}^{-\frac{1}{2}}\) are \(|X^k|\times |X^k|\) diagonal matrices defined by \((D_{k+1}^{\frac{1}{2}})_{\sigma\tau}=\max\{\sqrt{\deg\sigma},1\}\) if \(\sigma=\tau\), and \(0\) otherwise, and \(D_{k+1}^{-\frac{1}{2}}\) is the inverse of \(D_{k+1}^{\frac{1}{2}}\).\N\NIn Section \(4\) of the paper, the authors discuss some useful properties of the normalized Hodge \(k\)-Laplacian matrix \(\mathcal{L}_k\) defined above. In Section 5, they present the random walk normalized Hodge Laplacian matrix and explore the concept of simple random walk on simplicial complexes. The main results in this section are Theorem 5.9 and Theorem 5.10 where they provide expressions for the transition matrices of upper random \(k\)-walk and lower random \(k\)-walk on a simplicial complex \(X\), in terms of the random walk normalized Hodge \(k\)-Laplacian matrix.