28 research outputs found
Evaluating links through spectral decomposition
Spectral decomposition has been rarely used to investigate complex networks.
In this work we apply this concept in order to define two types of
link-directed attacks while quantifying their respective effects on the
topology. Several other types of more traditional attacks are also adopted and
compared. These attacks had substantially diverse effects, depending on each
specific network (models and real-world structures). It is also showed that the
spectral-based attacks have special effect in affecting the transitivity of the
networks
Correlating sparse sensing for large-scale traffic speed estimation: A Laplacian-enhanced low-rank tensor kriging approach
Traffic speed is central to characterizing the fluidity of the road network.
Many transportation applications rely on it, such as real-time navigation,
dynamic route planning, and congestion management. Rapid advances in sensing
and communication techniques make traffic speed detection easier than ever.
However, due to sparse deployment of static sensors or low penetration of
mobile sensors, speeds detected are incomplete and far from network-wide use.
In addition, sensors are prone to error or missing data due to various kinds of
reasons, speeds from these sensors can become highly noisy. These drawbacks
call for effective techniques to recover credible estimates from the incomplete
data. In this work, we first identify the issue as a spatiotemporal kriging
problem and propose a Laplacian enhanced low-rank tensor completion (LETC)
framework featuring both lowrankness and multi-dimensional correlations for
large-scale traffic speed kriging under limited observations. To be specific,
three types of speed correlation including temporal continuity, temporal
periodicity, and spatial proximity are carefully chosen and simultaneously
modeled by three different forms of graph Laplacian, named temporal graph
Fourier transform, generalized temporal consistency regularization, and
diffusion graph regularization. We then design an efficient solution algorithm
via several effective numeric techniques to scale up the proposed model to
network-wide kriging. By performing experiments on two public million-level
traffic speed datasets, we finally draw the conclusion and find our proposed
LETC achieves the state-of-the-art kriging performance even under low
observation rates, while at the same time saving more than half computing time
compared with baseline methods. Some insights into spatiotemporal traffic data
modeling and kriging at the network level are provided as well
Joining forces for reconstruction inverse problems
A spectral inverse problem concerns the reconstruction of parameters of a parent graph from prescribed spectral data of subgraphs. Also referred to as the P–NP Isomorphism Problem, Reconstruction or Exact Graph Matching, the aim is to seek sets of parameters to determine a graph uniquely. Other related inverse problems, including the Polynomial Reconstruction Problem (PRP), involve the recovery of graph invariants. The PRP seeks to extract the spectrum of a graph from the deck of cards each showing the spectrum of a vertex-deleted subgraph. We show how various algebraic methods join forces to reconstruct a graph or its invariants from a minimal set of restricted eigenvalue-eigenvector information of the parent graph or its subgraphs. We show how functions of the entries of eigenvectors of the adjacency matrix A of a graph can be retrieved from the spectrum of eigenvalues of A. We establish that there are two subclasses of disconnected graphs with each card of the deck showing a common eigenvalue. These could occur as possible counter examples to the positive solution of the PRP.peer-reviewe
DCT and DST Filtering with Sparse Graph Operators
Graph filtering is a fundamental tool in graph signal processing. Polynomial
graph filters (PGFs), defined as polynomials of a fundamental graph operator,
can be implemented in the vertex domain, and usually have a lower complexity
than frequency domain filter implementations. In this paper, we focus on the
design of filters for graphs with graph Fourier transform (GFT) corresponding
to a discrete trigonometric transform (DTT), i.e., one of 8 types of discrete
cosine transforms (DCT) and 8 discrete sine transforms (DST). In this case, we
show that multiple sparse graph operators can be identified, which allows us to
propose a generalization of PGF design: multivariate polynomial graph filter
(MPGF). First, for the widely used DCT-II (type-2 DCT), we characterize a set
of sparse graph operators that share the DCT-II matrix as their common
eigenvector matrix. This set contains the well-known connected line graph.
These sparse operators can be viewed as graph filters operating in the DCT
domain, which allows us to approximate any DCT graph filter by a MPGF, leading
to a design with more degrees of freedom than the conventional PGF approach.
Then, we extend those results to all of the 16 DTTs as well as their 2D
versions, and show how their associated sets of multiple graph operators can be
determined. We demonstrate experimentally that ideal low-pass and exponential
DCT/DST filters can be approximated with higher accuracy with similar runtime
complexity. Finally, we apply our method to transform-type selection in a video
codec, AV1, where we demonstrate significant encoding time savings, with a
negligible compression loss.Comment: 16 pages, 11 figures, 5 table
Spectral Convergence of the connection Laplacian from random samples
Spectral methods that are based on eigenvectors and eigenvalues of discrete
graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used
for manifold learning and non-linear dimensionality reduction. It was
previously shown by Belkin and Niyogi \cite{belkin_niyogi:2007} that the
eigenvectors and eigenvalues of the graph Laplacian converge to the
eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold
in the limit of infinitely many data points sampled independently from the
uniform distribution over the manifold. Recently, we introduced Vector
Diffusion Maps and showed that the connection Laplacian of the tangent bundle
of the manifold can be approximated from random samples. In this paper, we
present a unified framework for approximating other connection Laplacians over
the manifold by considering its principle bundle structure. We prove that the
eigenvectors and eigenvalues of these Laplacians converge in the limit of
infinitely many independent random samples. We generalize the spectral
convergence results to the case where the data points are sampled from a
non-uniform distribution, and for manifolds with and without boundary