2,980 research outputs found
Data driven approach to sparsification of reaction diffusion complex network systems
Graph sparsification is an area of interest in computer science and applied
mathematics. Sparsification of a graph, in general, aims to reduce the number
of edges in the network while preserving specific properties of the graph, like
cuts and subgraph counts. Computing the sparsest cuts of a graph is known to be
NP-hard, and sparsification routines exists for generating linear sized
sparsifiers in almost quadratic running time .
Consequently, obtaining a sparsifier can be a computationally demanding task
and the complexity varies based on the level of sparsity required. In this
study, we extend the concept of sparsification to the realm of
reaction-diffusion complex systems. We aim to address the challenge of reducing
the number of edges in the network while preserving the underlying flow
dynamics. To tackle this problem, we adopt a relaxed approach considering only
a subset of trajectories. We map the network sparsification problem to a data
assimilation problem on a Reduced Order Model (ROM) space with constraints
targeted at preserving the eigenmodes of the Laplacian matrix under
perturbations. The Laplacian matrix () is the difference between the
diagonal matrix of degrees () and the graph's adjacency matrix (). We
propose approximations to the eigenvalues and eigenvectors of the Laplacian
matrix subject to perturbations for computational feasibility and include a
custom function based on these approximations as a constraint on the data
assimilation framework. We demonstrate the extension of our framework to
achieve sparsity in parameter sets for Neural Ordinary Differential Equations
(neural ODEs)
Simple parallel and distributed algorithms for spectral graph sparsification
We describe a simple algorithm for spectral graph sparsification, based on
iterative computations of weighted spanners and uniform sampling. Leveraging
the algorithms of Baswana and Sen for computing spanners, we obtain the first
distributed spectral sparsification algorithm. We also obtain a parallel
algorithm with improved work and time guarantees. Combining this algorithm with
the parallel framework of Peng and Spielman for solving symmetric diagonally
dominant linear systems, we get a parallel solver which is much closer to being
practical and significantly more efficient in terms of the total work.Comment: replaces "A simple parallel and distributed algorithm for spectral
sparsification". Minor change
Similarity-Aware Spectral Sparsification by Edge Filtering
In recent years, spectral graph sparsification techniques that can compute
ultra-sparse graph proxies have been extensively studied for accelerating
various numerical and graph-related applications. Prior nearly-linear-time
spectral sparsification methods first extract low-stretch spanning tree from
the original graph to form the backbone of the sparsifier, and then recover
small portions of spectrally-critical off-tree edges to the spanning tree to
significantly improve the approximation quality. However, it is not clear how
many off-tree edges should be recovered for achieving a desired spectral
similarity level within the sparsifier. Motivated by recent graph signal
processing techniques, this paper proposes a similarity-aware spectral graph
sparsification framework that leverages efficient spectral off-tree edge
embedding and filtering schemes to construct spectral sparsifiers with
guaranteed spectral similarity (relative condition number) level. An iterative
graph densification scheme is introduced to facilitate efficient and effective
filtering of off-tree edges for highly ill-conditioned problems. The proposed
method has been validated using various kinds of graphs obtained from public
domain sparse matrix collections relevant to VLSI CAD, finite element analysis,
as well as social and data networks frequently studied in many machine learning
and data mining applications
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