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The notion of the Laplacian of weighted graphs will be introduced, the eigenvectors belonging to k consecutive eigen-values of which define optimal k-dimensional Euclidean representation of the vertices. By means of these spectral techniques some combinatorial problems concerning minimal (k+ 1)-cuts of weighted graphs can be handled easily with linear(More)
Two penalized-balanced and normalized-versions of the Newman-Girvan modularity are introduced and estimated by the non-negative eigenvalues of the modularity and normalized modularity matrix, respectively. In this way, the partition of the vertices that maximizes the modularity can be obtained by applying the k-means algorithm for the representatives of the(More)
Asymptotic behavior of the singular value decomposition (SVD) of blown up matrices and normalized blown up contingency tables exposed to Wigner-noise is investigated. It is proved that such an m × n matrix almost surely has a constant number of large singular values (of order √ mn), while the rest of the singular values are of order √ m + n as m, n → ∞.(More)
The role of the normalized modularity matrix in finding homogeneous cuts will be presented. We also discuss the testability of the structural eigenvalues and that of the subspace spanned by the corresponding eigenvectors of this matrix. In the presence of a spectral gap between the k − 1 largest absolute value eigenvalues and the remainder of the spectrum,(More)
Expander graphs are widely used in communication problems and construction of error correcting codes. In such graphs, information gets through very quickly. Typically, it is not true for social or biological networks, though we may find a partition of the vertices such that the induced subgraphs on them and the bipartite subgraphs between any pair of them(More)
Testable weighted graph parameters and equivalent notions of testability – proved in [4] for simple graphs – are generalized for vertex-and edge-weighted graphs with no dominant vertex-weights. We prove that certain balanced minimum multiway cut densities are testable. Using this fact, quadratic programming techniques are applied to approximate some of(More)
Preliminaries Examples Results Motivation To recover the structure of large edge-weighted graphs, for example: metabolic, social, economic, or communication networks. To find a clustering (partition) of the vertices such that the induced subgraphs on them and the bipartite subgraphs between any pair of them exhibit regular behavior of information flow(More)