7,278 research outputs found
On mixed radial Moore graphs of diameter 3
Radial Moore graphs and digraphs are extremal graphs related to the Moore
ones where the distance-preserving spanning tree is preserved for some
vertices. This leads to classify them according to their proximity to being a
Moore graph or digraph. In this paper we deal with mixed radial Moore graphs,
where the mixed setting allows edges and arcs as different elements. An
exhaustive computer search shows the top ranked graphs for an specific set of
parameters. Moreover, we study the problem of their existence by providing two
infinite families for different values of the degrees and diameter . One of
these families turns out to be optimal
Radially Moore graphs of radius three and large odd degree
Extremal graphs which are close related to Moore graphs have been defined in different ways. Radially Moore graphs are one of these examples of extremal graphs. Although it is proved that radially Moore graphs exist for radius two, the general problem remains open. Knor, and independently Exoo, gives
some constructions of these extremal graphs for radius three and small degrees. As far as we know, some few examples have been found for other small values of the degree and the radius.
Here, we consider the existence problem of radially Moore graphs of radius three. We use the generalized undirected de Bruijn
graphs to give a general construction of radially Moore graphs of radius three and large odd degree.Peer Reviewe
The boundary action of a sofic random subgroup of the free group
We prove that the boundary action of a sofic random subgroup of a finitely
generated free group is conservative. This addresses a question asked by
Grigorchuk, Kaimanovich, and Nagnibeda, who studied the boundary actions of
individual subgroups of the free group. Following their work, we also
investigate the cogrowth and various limit sets associated to sofic random
subgroups. We make heavy use of the correspondence between subgroups and their
Schreier graphs, and central to our approach is an investigation of the
asymptotic density of a given set inside of large neighborhoods of the root of
a sofic random Schreier graph.Comment: 21 pages, 2 figures, made minor corrections, to appear in Groups,
Geometry, and Dynamic
Hyperbolic Geometry of Complex Networks
We develop a geometric framework to study the structure and function of
complex networks. We assume that hyperbolic geometry underlies these networks,
and we show that with this assumption, heterogeneous degree distributions and
strong clustering in complex networks emerge naturally as simple reflections of
the negative curvature and metric property of the underlying hyperbolic
geometry. Conversely, we show that if a network has some metric structure, and
if the network degree distribution is heterogeneous, then the network has an
effective hyperbolic geometry underneath. We then establish a mapping between
our geometric framework and statistical mechanics of complex networks. This
mapping interprets edges in a network as non-interacting fermions whose
energies are hyperbolic distances between nodes, while the auxiliary fields
coupled to edges are linear functions of these energies or distances. The
geometric network ensemble subsumes the standard configuration model and
classical random graphs as two limiting cases with degenerate geometric
structures. Finally, we show that targeted transport processes without global
topology knowledge, made possible by our geometric framework, are maximally
efficient, according to all efficiency measures, in networks with strongest
heterogeneity and clustering, and that this efficiency is remarkably robust
with respect to even catastrophic disturbances and damages to the network
structure
Popularity versus Similarity in Growing Networks
Popularity is attractive -- this is the formula underlying preferential
attachment, a popular explanation for the emergence of scaling in growing
networks. If new connections are made preferentially to more popular nodes,
then the resulting distribution of the number of connections that nodes have
follows power laws observed in many real networks. Preferential attachment has
been directly validated for some real networks, including the Internet.
Preferential attachment can also be a consequence of different underlying
processes based on node fitness, ranking, optimization, random walks, or
duplication. Here we show that popularity is just one dimension of
attractiveness. Another dimension is similarity. We develop a framework where
new connections, instead of preferring popular nodes, optimize certain
trade-offs between popularity and similarity. The framework admits a geometric
interpretation, in which popularity preference emerges from local optimization.
As opposed to preferential attachment, the optimization framework accurately
describes large-scale evolution of technological (Internet), social (web of
trust), and biological (E.coli metabolic) networks, predicting the probability
of new links in them with a remarkable precision. The developed framework can
thus be used for predicting new links in evolving networks, and provides a
different perspective on preferential attachment as an emergent phenomenon
Random planar graphs and the London street network
In this paper we analyse the street network of London both in its primary and
dual representation. To understand its properties, we consider three idealised
models based on a grid, a static random planar graph and a growing random
planar graph. Comparing the models and the street network, we find that the
streets of London form a self-organising system whose growth is characterised
by a strict interaction between the metrical and informational space. In
particular, a principle of least effort appears to create a balance between the
physical and the mental effort required to navigate the city
Progress in extra-solar planet detection
Progress in extra-solar planet detection is reviewed. The following subject areas are covered: (1) the definition of a planet; (2) the weakness of planet signals; (3) direct techniques - imaging and spectral detection; and (4) indirect techniques - reflex motion and occultations
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