Given an adjacency matrix \(A\), the function builds the random walk (RW) normalised Laplacian \(I - D^{-1}A\) for a single-layer network. The RW normalised Laplacian is defined only for graphs without isolated nodes -- due to the inversion of the diagonal matrix of degrees \(D^{-1}\). Despite this, a (classical) random walk is still defined also in presence of isolates or nodes without out-going edges, simply setting the transition probability from those nodes outwards to be zero. Consequently we can extend the definition of the RW normalised Laplacian setting \(L_{ij} = 0\) if \(k_i = 0\) for all \(j\).
GetNormalizedLaplacianMatrix(AdjacencyMatrix)
| AdjacencyMatrix | the adjacency matrix characterising the network |
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Normalized Lapalacian Matrix