Tuned preconditioners for the eigensolution of large SPD matrices arising in engineering problems

In this paper, we study a class of tuned preconditioners that will be designed to accelerate both the DACG-Newton method and the implicitly restarted Lanczos method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices arising in large-scale scientific computations. These tuning strategies are based on low-rank modifications of a given initial preconditioner. We present some theoretical properties of the preconditioned matrix. We experimentally show how the aforementioned methods benefit from the acceleration provided by these tuned/deflated preconditioners. Comparisons are carried out with the Jacobi-Davidson method onto matrices arising from various large realistic problems arising from finite element discretization of PDEs modeling either groundwater flow in porous media or geomechanical processes in reservoirs. The numerical results show that the Newton-based methods (which includes also the Jacobi-Davidson method) are to be preferred to the - yet efficiently implemented - implicitly restarted Lanczos method whenever a small to moderate number of eigenpairs is required. \ua9 2016 John Wiley & Sons, Ltd

Distributed Current Flow Betweeness Centrality

—The computation of nodes centrality is of great importance for the analysis of graphs. The current flow betweenness is an interesting centrality index that is computed by considering how the information travels along all the possible paths of a graph. The current flow betweenness exploits basic results from electrical circuits, i.e. Kirchhoff’s laws, to evaluate the centrality of vertices. The computation of the current flow betweenness may exceed the computational capability of a single machine for very large graphs composed by millions of nodes. In this paper we propose a solution that estimates the current flow betweenness in a distributed setting, by defining a vertex-centric, gossip-based algorithm. Each node, relying on its local information, in a selfadaptive way generates new flows to improve the betweenness of all the nodes of the graph. Our experimental evaluation shows that our proposal achieves high correlation with the exact current flow betweenness, and provides a good centrality measure for large graphs

A variant of the current flow betweenness centrality and its application in urban networks

The current flow betweenness centrality is a useful tool to estimate traffic status in spatial networks and, in general, to measure the intermediation of nodes in networks where the transition between them takes place in a random way. The main drawback of this centrality is its high computational cost, especially for very large networks, as it is the case of urban networks. In this paper, a new approach to the current flow betweenness centrality for its practical application in urban networks with data is presented and discussed. The new centrality measure allows the estimation of pedestrian flow developed in urban networks, taking into account both the network topology and its associated data. In addition, its computational cost makes it suitable for application in networks with a large number of nodes. Some examples are studied in order to better understand the characteristics and behaviour of the proposed centrality in the context of the city.Partially supported by the Spanish Government, Ministerio de Economía y Competividad, grant number TIN2017-84821-P

Structure of complex networks: Quantifying edge-to-edge relations by failure-induced flow redistribution

The analysis of complex networks has so far revolved mainly around the role of nodes and communities of nodes. However, the dynamics of interconnected systems is commonly focalised on edge processes, and a dual edge-centric perspective can often prove more natural. Here we present graph-theoretical measures to quantify edge-to-edge relations inspired by the notion of flow redistribution induced by edge failures. Our measures, which are related to the pseudo-inverse of the Laplacian of the network, are global and reveal the dynamical interplay between the edges of a network, including potentially non-local interactions. Our framework also allows us to define the embeddedness of an edge, a measure of how strongly an edge features in the weighted cuts of the network. We showcase the general applicability of our edge-centric framework through analyses of the Iberian Power grid, traffic flow in road networks, and the C. elegans neuronal network.Comment: 24 pages, 6 figure

A Fast Hierarchically Preconditioned Eigensolver Based on Multiresolution Matrix Decomposition

In this paper we propose a new iterative method to hierarchically compute a relatively large number of leftmost eigenpairs of a sparse symmetric positive matrix under the multiresolution operator compression framework. We exploit the well-conditioned property of every decomposition component by integrating the multiresolution framework into the implicitly restarted Lanczos method. We achieve this combination by proposing an extension-refinement iterative scheme, in which the intrinsic idea is to decompose the target spectrum into several segments such that the corresponding eigenproblem in each segment is well-conditioned. Theoretical analysis and numerical illustration are also reported to illustrate the efficiency and effectiveness of this algorithm