76,633 research outputs found
Metric Representations Of Networks
The goal of this thesis is to analyze networks by first projecting them onto structured metric-like spaces -- governed by a generalized triangle inequality -- and then leveraging this structure to facilitate the analysis. Networks encode relationships between pairs of nodes, however, the relationship between two nodes can be independent of the other ones and need not be defined for every pair. This is not true for metric spaces, where the triangle inequality imposes conditions that must be satisfied by triads of distances and these must be defined for every pair of nodes. In general terms, this additional structure facilitates the analysis and algorithm design in metric spaces. In deriving metric projections for networks, an axiomatic approach is pursued where we encode as axioms intuitively desirable properties and then seek for admissible projections satisfying these axioms. Although small variations are introduced throughout the thesis, the axioms of projection -- a network that already has the desired metric structure must remain unchanged -- and transformation -- when reducing dissimilarities in a network the projected distances cannot increase -- shape all of the axiomatic constructions considered. Notwithstanding their apparent weakness, the aforementioned axioms serve as a solid foundation for the theory of metric representations of networks.
We begin by focusing on hierarchical clustering of asymmetric networks, which can be framed as a network projection problem onto ultrametric spaces. We show that the set of admissible methods is infinite but bounded in a well-defined sense and state additional desirable properties to further winnow the admissibility landscape. Algorithms for the clustering methods developed are also derived and implemented. We then shift focus to projections onto generalized q-metric spaces, a parametric family containing among others the (regular) metric and ultrametric spaces. A uniqueness result is shown for the projection of symmetric networks whereas for asymmetric networks we prove that all admissible projections are contained between two extreme methods. Furthermore, projections are illustrated via their implementation for efficient search and data visualization. Lastly, our analysis is extended to encompass projections of dioid spaces, natural algebraic generalizations of weighted networks
Spectral triples for noncommutative solenoidal spaces from self-coverings
Examples of noncommutative self-coverings are described, and spectral triples
on the base space are extended to spectral triples on the inductive family of
coverings, in such a way that the covering projections are locally isometric.
Such triples are shown to converge, in a suitable sense, to a semifinite
spectral triple on the direct limit of the tower of coverings, which we call
noncommutative solenoidal space. Some of the self-coverings described here are
given by the inclusion of the fixed point algebra in a C-algebra acted upon
by a finite abelian group. In all the examples treated here, the noncommutative
solenoidal spaces have the same metric dimension and volume as on the base
space, but are not quantum compact metric spaces, namely the pseudo-metric
induced by the spectral triple does not produce the weak topology on the
state space.Comment: v3: the paper will appear in the Journal of Mathematical Analysis and
Applications, 42 pages, no figure
Phases of Five-dimensional Theories, Monopole Walls, and Melting Crystals
Moduli spaces of doubly periodic monopoles, also called monopole walls or
monowalls, are hyperk\"ahler; thus, when four-dimensional, they are self-dual
gravitational instantons. We find all monowalls with lowest number of moduli.
Their moduli spaces can be identified, on the one hand, with Coulomb branches
of five-dimensional supersymmetric quantum field theories on
and, on the other hand, with moduli spaces of local
Calabi-Yau metrics on the canonical bundle of a del Pezzo surface. We explore
the asymptotic metric of these moduli spaces and compare our results with
Seiberg's low energy description of the five-dimensional quantum theories. We
also give a natural description of the phase structure of general monowall
moduli spaces in terms of triangulations of Newton polygons, secondary
polyhedra, and associahedral projections of secondary fans.Comment: 45 pages, 11 figure
Spectral convergence of non-compact quasi-one-dimensional spaces
We consider a family of non-compact manifolds X_\eps (``graph-like
manifolds'') approaching a metric graph and establish convergence results
of the related natural operators, namely the (Neumann) Laplacian \laplacian
{X_\eps} and the generalised Neumann (Kirchhoff) Laplacian \laplacian {X_0}
on the metric graph. In particular, we show the norm convergence of the
resolvents, spectral projections and eigenfunctions. As a consequence, the
essential and the discrete spectrum converge as well. Neither the manifolds nor
the metric graph need to be compact, we only need some natural uniformity
assumptions. We provide examples of manifolds having spectral gaps in the
essential spectrum, discrete eigenvalues in the gaps or even manifolds
approaching a fractal spectrum. The convergence results will be given in a
completely abstract setting dealing with operators acting in different spaces,
applicable also in other geometric situations.Comment: some references added, still 36 pages, 4 figure
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