2,038 research outputs found
Incremental eigenpair computation for graph Laplacian matrices: theory and applications
The smallest eigenvalues and the associated eigenvectors (i.e., eigenpairs) of a graph Laplacian matrix have been widely used for spectral clustering and community detection. However, in real-life applications, the number of clusters or communities (say, K) is generally unknown a priori. Consequently, the majority of the existing methods either choose K heuristically or they repeat the clustering method with different choices of K and accept the best clustering result. The first option, more often, yields suboptimal result, while the second option is computationally expensive. In this work, we propose an incremental method for constructing the eigenspectrum of the graph Laplacian matrix. This method leverages the eigenstructure of graph Laplacian matrix to obtain the Kth smallest eigenpair of the Laplacian matrix given a collection of all previously compute
Cascading Failures in Power Grids - Analysis and Algorithms
This paper focuses on cascading line failures in the transmission system of
the power grid. Recent large-scale power outages demonstrated the limitations
of percolation- and epid- emic-based tools in modeling cascades. Hence, we
study cascades by using computational tools and a linearized power flow model.
We first obtain results regarding the Moore-Penrose pseudo-inverse of the power
grid admittance matrix. Based on these results, we study the impact of a single
line failure on the flows on other lines. We also illustrate via simulation the
impact of the distance and resistance distance on the flow increase following a
failure, and discuss the difference from the epidemic models. We then study the
cascade properties, considering metrics such as the distance between failures
and the fraction of demand (load) satisfied after the cascade (yield). We use
the pseudo-inverse of admittance matrix to develop an efficient algorithm to
identify the cascading failure evolution, which can be a building block for
cascade mitigation. Finally, we show that finding the set of lines whose
removal has the most significant impact (under various metrics) is NP-Hard and
introduce a simple heuristic for the minimum yield problem. Overall, the
results demonstrate that using the resistance distance and the pseudo-inverse
of admittance matrix provides important insights and can support the
development of efficient algorithms
Subdivision Directional Fields
We present a novel linear subdivision scheme for face-based tangent
directional fields on triangle meshes. Our subdivision scheme is based on a
novel coordinate-free representation of directional fields as halfedge-based
scalar quantities, bridging the finite-element representation with discrete
exterior calculus. By commuting with differential operators, our subdivision is
structure-preserving: it reproduces curl-free fields precisely, and reproduces
divergence-free fields in the weak sense. Moreover, our subdivision scheme
directly extends to directional fields with several vectors per face by working
on the branched covering space. Finally, we demonstrate how our scheme can be
applied to directional-field design, advection, and robust earth mover's
distance computation, for efficient and robust computation
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