2,724 research outputs found
Exploring complex networks via topological embedding on surfaces
We demonstrate that graphs embedded on surfaces are a powerful and practical
tool to generate, characterize and simulate networks with a broad range of
properties. Remarkably, the study of topologically embedded graphs is
non-restrictive because any network can be embedded on a surface with
sufficiently high genus. The local properties of the network are affected by
the surface genus which, for example, produces significant changes in the
degree distribution and in the clustering coefficient. The global properties of
the graph are also strongly affected by the surface genus which is constraining
the degree of interwoveness, changing the scaling properties from
large-world-kind (small genus) to small- and ultra-small-world-kind (large
genus). Two elementary moves allow the exploration of all networks embeddable
on a given surface and naturally introduce a tool to develop a statistical
mechanics description. Within such a framework, we study the properties of
topologically-embedded graphs at high and low `temperatures' observing the
formation of increasingly regular structures by cooling the system. We show
that the cooling dynamics is strongly affected by the surface genus with the
manifestation of a glassy-like freezing transitions occurring when the amount
of topological disorder is low.Comment: 18 pages, 7 figure
On projections of knots, links and spatial graphs
制度:新 ; 報告番号:甲3007号 ; 学位の種類:博士(理学) ; 授与年月日:2010/3/15 ; 早大学位記番号:新525
Refilling meridians in a genus 2 handlebody complement
Suppose a genus two handlebody is removed from a 3-manifold M and then a
single meridian of the handlebody is restored. The result is a knot or link
complement in M and it is natural to ask whether geometric properties of the
link complement say something about the meridian that was restored. Here we
consider what the relation must be between two not necessarily disjoint
meridians so that restoring each of them gives a trivial knot or a split link.Comment: This is the version published by Geometry & Topology Monographs on 29
April 200
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