is an example of this model used in mechanics to describe a system of springs of given stiffnesses and unit length, where the values of the stiffnesses play the role of the weights of the graph edges. We thus reuse the definition of the weightless incidence matrix B with element Bve for the vertex v and the edge e (connecting vertexes and , with i > j) defined by We now also define a diagonal matrix W containing the edge weights. Even though the edges in the definition of B are technically directed, their directions can be arbitrary, still resulting in the same symmetric Laplacian matrix L defined as where is the matrix transpose of B. The construction is illustrated in the following example, where every edge is assigned the weight value i, with Symmetric Laplacian for a directed graph Just like for simple graphs, the Laplacian matrix of a directed weighted graph is by definition generally non-symmetric. The symmetry can be enforced by turning the original directed graph into an undirected graph first before constructing the Laplacian. The adjacency matrix of the undirected graph could, e.g., be defined as a sum of the adjacency matrix of the original directed graph and its matrix transpose as in the following example: where the zero and one entries of are treated as numerical, rather than logical as for simple graphs, values, explaining the difference in the results - for simple graphs, the symmetrized graph still needs to be simple with its symmetrized adjacency matrix having only logical, not numerical values, e.g., the logical sum is 1 v 1 = 1, while the numeric sum is 1 + 1 = 2. Alternatively, the symmetric Laplacian matrix can be calculated from the two Laplacians using the indegree and outdegree, as in the following example: The sum of the out-degree Laplacian transposed and the in-degree Laplacian equals to the symmetric Laplacian matrix. Laplacian matrix normalization The goal of normalization is, like for simple graphs, to make the diagonal entries of the Laplacian matrix to be all unit, also scaling off-diagonal entries correspondingly. In a weighted graph, a vertex may have a large degree because of a small number of connected edges but with large weights just as well as due to a large number of connected edges with unit weights. Graph self-loops, i.e., non-zero entries on the main diagonal of the adjacency matrix, do not