1,131 research outputs found
Fastest mixing Markov chain on graphs with symmetries
We show how to exploit symmetries of a graph to efficiently compute the
fastest mixing Markov chain on the graph (i.e., find the transition
probabilities on the edges to minimize the second-largest eigenvalue modulus of
the transition probability matrix). Exploiting symmetry can lead to significant
reduction in both the number of variables and the size of matrices in the
corresponding semidefinite program, thus enable numerical solution of
large-scale instances that are otherwise computationally infeasible. We obtain
analytic or semi-analytic results for particular classes of graphs, such as
edge-transitive and distance-transitive graphs. We describe two general
approaches for symmetry exploitation, based on orbit theory and
block-diagonalization, respectively. We also establish the connection between
these two approaches.Comment: 39 pages, 15 figure
Graph Signal Processing: Overview, Challenges and Applications
Research in Graph Signal Processing (GSP) aims to develop tools for
processing data defined on irregular graph domains. In this paper we first
provide an overview of core ideas in GSP and their connection to conventional
digital signal processing. We then summarize recent developments in developing
basic GSP tools, including methods for sampling, filtering or graph learning.
Next, we review progress in several application areas using GSP, including
processing and analysis of sensor network data, biological data, and
applications to image processing and machine learning. We finish by providing a
brief historical perspective to highlight how concepts recently developed in
GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE
The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions
We consider a large class of self-adjoint elliptic problem associated with
the second derivative acting on a space of vector-valued functions. We present
two different approaches to the study of the associated eigenvalues problems.
The first, more general one allows to replace a secular equation (which is
well-known in some special cases) by an abstract rank condition. The latter
seems to apply particularly well to a specific boundary condition, sometimes
dubbed "anti-Kirchhoff" in the literature, that arise in the theory of
differential operators on graphs; it also permits to discuss interesting and
more direct connections between the spectrum of the differential operator and
some graph theoretical quantities. In either case our results yield, among
other, some results on the symmetry of the spectrum
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Relaxation-Based Coarsening for Multilevel Hypergraph Partitioning
Multilevel partitioning methods that are inspired by principles of
multiscaling are the most powerful practical hypergraph partitioning solvers.
Hypergraph partitioning has many applications in disciplines ranging from
scientific computing to data science. In this paper we introduce the concept of
algebraic distance on hypergraphs and demonstrate its use as an algorithmic
component in the coarsening stage of multilevel hypergraph partitioning
solvers. The algebraic distance is a vertex distance measure that extends
hyperedge weights for capturing the local connectivity of vertices which is
critical for hypergraph coarsening schemes. The practical effectiveness of the
proposed measure and corresponding coarsening scheme is demonstrated through
extensive computational experiments on a diverse set of problems. Finally, we
propose a benchmark of hypergraph partitioning problems to compare the quality
of other solvers
A modularity based spectral method for simultaneous community and anti-community detection
In a graph or complex network, communities and anti-communities are node sets
whose modularity attains extremely large values, positive and negative,
respectively. We consider the simultaneous detection of communities and
anti-communities, by looking at spectral methods based on various matrix-based
definitions of the modularity of a vertex set. Invariant subspaces associated
to extreme eigenvalues of these matrices provide indications on the presence of
both kinds of modular structure in the network. The localization of the
relevant invariant subspaces can be estimated by looking at particular matrix
angles based on Frobenius inner products
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