A CHEBYSHEV PROJECTION METHOD FOR SOLVING SHALLOW-WATER EQUATIONS

The aims of this paper is to propose a numerical approach to simulate water flows in a 2D shallow medium. We consider the 2D Shallow water equations following the velocity-denivelation formulation. We solve these equations by a projection technique using a $\mathbb{P}_{N,M}$-type Chebyshev spectral approach which uses the Chebyshev-Gauss-Lobatto collocation points.

Spectral estimation for detecting low-dimensional structure in networks using arbitrary null models

Discovering low-dimensional structure in real-world networks requires a suitable null model that defines the absence of meaningful structure. Here we introduce a spectral approach for detecting a network’s low-dimensional structure, and the nodes that participate in it, using any null model. We use generative models to estimate the expected eigenvalue distribution under a specified null model, and then detect where the data network’s eigenspectra exceed the estimated bounds. On synthetic networks, this spectral estimation approach cleanly detects transitions between random and community structure, recovers the number and membership of communities, and removes noise nodes. On real networks spectral estimation finds either a significant fraction of noise nodes or no departure from a null model, in stark contrast to traditional community detection methods. Across all analyses, we find the choice of null model can strongly alter conclusions about the presence of network structure. Our spectral estimation approach is therefore a promising basis for detecting low-dimensional structure in real-world networks, or lack thereof.