In this work, we generalize a normalized Laplacian matrix L on graphs to a normalized Hodge k-Laplacian matrix L k (i.e. L 0 = 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.do...
The normalized lazy random walk matrix is defined as: Zs = D1/2ZD−1/2 = αI + (1 − α)D−1/2SD−1/2 The matrices Zs and Z have the same eigenvalues and related eigenvectors. If Z has eigenvalue μi with an eigenvector fi, Zs has the same eigenvalue with eigenvector...
Following the random walk with restart model, in the paper, a new computational model based on Laplacian normalized random walk with restart algorithm in a heterogeneous network was proposed to predict the association between lncRNA and disease. Firstly, the disease semantic similarity (lncRNA functio...
denote the lazy walk matrix of the graph G, where W G def = (1/2) I +A G D −1 G . (8.4) This is the one assymetric matrix that we will deal with in this course. Fortunately, it is similar to a symmetric matrix we have studied, the normalized adjacency matrix. It is also...
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 ...
1.2Randomwalksondirectedgraphs...2 1.2.1Thetransitionmatrixandstationarydistribution2 1.2.2Perron-Frobeniustheoryandergodicity...6 1.3Spectralgraphtheory...9 1.3.1ThenormalizedLaplacian...9 1.3.2CirculationsandtheCheegerinequality...11 1.3.3Boundingthe...
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 ...
Thus, from any state q, the probability to jump to state q is proportional to the weight aqq of the edge from q to q (and then normalized). These transition probabilities are stored in an n × n transition matrix P = {pqq }q,q ∈N . We introduce discriminative random walks (D-...
It is easy to see that the spectrum of the symmetric normalized Laplacian matrix \({\hat{L}}=I_n-D^{-1/2}AD^{-1/2}\) equals the spectrum of the random walk normalized Laplacian matrix \(\tilde{L}:=I_n-D^{-1}A\). Moreover, for a graph with vertex set V, \(\tilde{L}...