Normalized Hodge Laplacian Matrix and Application to Random Walk on Simplicial Complexesdoi:10.1007/s00373-024-02791-8Hodge LaplacianLaplacian matrixNormalized Laplacian matrixLetXbe a simplicial complex,k:Ck鈫扖k-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\use...
Let \({{{\Phi }}}^{(\alpha )}={({\phi }_{1}^{(\alpha )},\ldots ,{\phi }_{N}^{(\alpha )})}^{T}\) be the Laplacian normalized eigenvector associated the eigenvalue Λα for α = 1, …, N, such that $$\sum _{m}{{{\Delta }}}_{jm}{\phi }_{m}^...
Let be the N eigenvalues of matrix Γ for a network of size N, rearranged as and let denote the corresponding normalized and mutually orthogonal eigenvectors, where . Then, the FPT for a walker starting from node i to first arrive at node j can be expressed as56 For the particular ...
The graph correlation metric is a measure of similarity between a pair of graphs. This metric is defined herein to be the total number of different links present between the graphs, normalized by the number of edges in the graph containing the highest number of edges. Mathematically, letRp=(...
random walk with restart (RWR) on a LFS network [10]. Under the supposition that the more miRNAs two lncRNAs interacted, the more likely they are related to the analogous diseases, Zhou et al. proposed a LDA prediction model by implementing random walk on a heterogeneous network which ...
Frobeniustheoryandergodicity...61.3Spectralgraphtheory...91.3.1ThenormalizedLaplacian...91.3.2CirculationsandtheCheegerinequality...111.3.3Boundingtherateofconvergence...131.4Strongorientationsofgraphs...161.4.1PreliminariesandRobbins’Theorem...161.4.2Countingstrongorientations...171.5Overviewofmainresults....
it moves to a neighbor with probability proportional to the weight of the corresponding edge. Rather than tracking where some individual random walk goes, we will usually be interested in the probability distribution over vertices after a certain number of steps. We will let the vector p t ∈ ...
2 A Brief View of Personalized Random Walk On an undirected graph with edge weights W, let D be the diagonal matrix with D = diag(We), e = (1, . . . , 1)T , then P = (Pij ) is the transition probability from node i to node j, P = D−1W (1) Let fi be the ...
a normalized least square method (RLSMDA) was introduced by Chen and Yan11to identify the potential miRNA–disease associations. Shen et al.12presented the cooperative matrix decomposition (CMFMDA) algorithm in recommendation system to uncover potential associations. Xu et al.4designed a probability ...
(Related work on spectral theory) Similarly to the spectrum of the normalized Laplace operator, also the spectra of the non-normalized Laplacian matrix (defined in Sect.6) and the one of the adjacency matrix have been widely studied. We refer the reader to [8,44] for general references on...