28,363 research outputs found
On the diameter of random planar graphs
We show that the diameter D(G_n) of a random labelled connected planar graph
with n vertices is equal to n^{1/4+o(1)}, in probability. More precisely there
exists a constant c>0 such that the probability that D(G_n) lies in the
interval (n^{1/4-\epsilon},n^{1/4+\epsilon}) is greater than
1-\exp(-n^{c\epsilon}) for {\epsilon} small enough and n>n_0(\epsilon). We
prove similar statements for 2-connected and 3-connected planar graphs and
maps.Comment: 24 pages, 7 figure
On the diameter of random planar graphs
International audienceWe show that the diameter of a random (unembedded) labelled connected planar graph with vertices is asymptotically almost surely of order , in the sense that there exists a constant such that for small enough and large enough . We prove similar statements for rooted -connected and -connected embedded (maps) and unembedded planar graphs
Geometry and Dynamics for Hierarchical Regular Networks
The recently introduced hierarchical regular networks HN3 and HN4 are
analyzed in detail. We use renormalization group arguments to show that HN3, a
3-regular planar graph, has a diameter growing as \sqrt{N} with the system
size, and random walks on HN3 exhibit super-diffusion with an anomalous
exponent d_w = 2 - \log_2\phi = 1.306..., where \phi = (\sqrt{5} + 1)/2 =
1.618... is the "golden ratio." In contrast, HN4, a non-planar 4-regular graph,
has a diameter that grows slower than any power of N, yet, fast than any power
of \ln N . In an annealed approximation we can show that diffusive transport on
HN4 occurs ballistically (d_w = 1). Walkers on both graphs possess a first-
return probability with a power law tail characterized by an exponent \mu = 2
-1/d_w . It is shown explicitly that recurrence properties on HN3 depend on the
starting site.Comment: 15 pages, revtex; published version; find related material at
http://www.physics.emory.edu/faculty/boettcher
Random graphs from a block-stable class
A class of graphs is called block-stable when a graph is in the class if and
only if each of its blocks is. We show that, as for trees, for most -vertex
graphs in such a class, each vertex is in at most blocks, and each path passes through at most blocks.
These results extend to `weakly block-stable' classes of graphs
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
Traffic Analysis in Random Delaunay Tessellations and Other Graphs
In this work we study the degree distribution, the maximum vertex and edge
flow in non-uniform random Delaunay triangulations when geodesic routing is
used. We also investigate the vertex and edge flow in Erd\"os-Renyi random
graphs, geometric random graphs, expanders and random -regular graphs.
Moreover we show that adding a random matching to the original graph can
considerably reduced the maximum vertex flow.Comment: Submitted to the Journal of Discrete Computational Geometr
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
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