12,541 research outputs found
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
Pattern vectors from algebraic graph theory
Graphstructures have proven computationally cumbersome for pattern analysis. The reason for this is that, before graphs can be converted to pattern vectors, correspondences must be established between the nodes of structures which are potentially of different size. To overcome this problem, in this paper, we turn to the spectral decomposition of the Laplacian matrix. We show how the elements of the spectral matrix for the Laplacian can be used to construct symmetric polynomials that are permutation invariants. The coefficients of these polynomials can be used as graph features which can be encoded in a vectorial manner. We extend this representation to graphs in which there are unary attributes on the nodes and binary attributes on the edges by using the spectral decomposition of a Hermitian property matrix that can be viewed as a complex analogue of the Laplacian. To embed the graphs in a pattern space, we explore whether the vectors of invariants can be embedded in a low- dimensional space using a number of alternative strategies, including principal components analysis ( PCA), multidimensional scaling ( MDS), and locality preserving projection ( LPP). Experimentally, we demonstrate that the embeddings result in well- defined graph clusters. Our experiments with the spectral representation involve both synthetic and real- world data. The experiments with synthetic data demonstrate that the distances between spectral feature vectors can be used to discriminate between graphs on the basis of their structure. The real- world experiments show that the method can be used to locate clusters of graphs
Applications of Structural Balance in Signed Social Networks
We present measures, models and link prediction algorithms based on the
structural balance in signed social networks. Certain social networks contain,
in addition to the usual 'friend' links, 'enemy' links. These networks are
called signed social networks. A classical and major concept for signed social
networks is that of structural balance, i.e., the tendency of triangles to be
'balanced' towards including an even number of negative edges, such as
friend-friend-friend and friend-enemy-enemy triangles. In this article, we
introduce several new signed network analysis methods that exploit structural
balance for measuring partial balance, for finding communities of people based
on balance, for drawing signed social networks, and for solving the problem of
link prediction. Notably, the introduced methods are based on the signed graph
Laplacian and on the concept of signed resistance distances. We evaluate our
methods on a collection of four signed social network datasets.Comment: 37 page
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