22,888 research outputs found
PDEs level sets on weighted graphs
International audienceIn this paper we propose an adaptation of PDEs level sets over weighted graphs of arbitrary structure, based on PdEs and using a framework of discrete operators. A general PDEs level sets formulation is presented and an algorithm to solve such equation is described. Some transcriptions of well-known models under this formalism, as the mean-curvature-motion or active contours, are also provided. Then, we present several applications of our formalism, including image segmentation with active contours, using weighted graphs of arbitrary topologies
A Study on Topological Integer Additive Set-Labeling of Graphs
A set-labeling of a graph is an injective function , where is a finite set and a set-indexer of is a
set-labeling such that the induced function defined by
for every is also injective. Let be a graph and let be a
non-empty set. A set-indexer is called a topological
set-labeling of if is a topology of . An integer additive
set-labeling is an injective function ,
whose associated function is defined by
, where is the set of all
non-negative integers and is its power set. An
integer additive set-indexer is an integer additive set-labeling such that the
induced function defined by is also injective. In this paper, we extend the concepts of
topological set-labeling of graphs to topological integer additive set-labeling
of graphs.Comment: 16 pages, 7 figures, Accepted for publication. arXiv admin note: text
overlap with arXiv:1403.398
Ricci Curvature of the Internet Topology
Analysis of Internet topologies has shown that the Internet topology has
negative curvature, measured by Gromov's "thin triangle condition", which is
tightly related to core congestion and route reliability. In this work we
analyze the discrete Ricci curvature of the Internet, defined by Ollivier, Lin,
etc. Ricci curvature measures whether local distances diverge or converge. It
is a more local measure which allows us to understand the distribution of
curvatures in the network. We show by various Internet data sets that the
distribution of Ricci cuvature is spread out, suggesting the network topology
to be non-homogenous. We also show that the Ricci curvature has interesting
connections to both local measures such as node degree and clustering
coefficient, global measures such as betweenness centrality and network
connectivity, as well as auxilary attributes such as geographical distances.
These observations add to the richness of geometric structures in complex
network theory.Comment: 9 pages, 16 figures. To be appear on INFOCOM 201
Enumerations of finite topologies associated with a finite graph
The number of topologies and non-homeomorphic topologies on a fixed finite
set are now known up to , but still no complete formula yet
(Sloane). There are one to one correspondence among topologies, preorder and
digraphs. In this article, we enumerate topologies and non-homeomorphic
topologies whose underlying graph is a given finite graph.Comment: 2 figure
A semidefinite programming hierarchy for packing problems in discrete geometry
Packing problems in discrete geometry can be modeled as finding independent
sets in infinite graphs where one is interested in independent sets which are
as large as possible. For finite graphs one popular way to compute upper bounds
for the maximal size of an independent set is to use Lasserre's semidefinite
programming hierarchy. We generalize this approach to infinite graphs. For this
we introduce topological packing graphs as an abstraction for infinite graphs
coming from packing problems in discrete geometry. We show that our hierarchy
converges to the independence number.Comment: (v2) 25 pages, revision based on suggestions by referee, accepted in
Mathematical Programming Series B special issue on polynomial optimizatio
Topologies of nodal sets of random band limited functions
It is shown that the topologies and nestings of the zero and nodal sets of
random (Gaussian) band limited functions have universal laws of distribution.
Qualitative features of the supports of these distributions are determined. In
particular the results apply to random monochromatic waves and to random real
algebraic hyper-surfaces in projective space.Comment: An announcement of recent results. Includes an announcement of the
resolution of some open questions from the older version. 11 pages, 6 figure
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