29,545 research outputs found
Sizing the length of complex networks
Among all characteristics exhibited by natural and man-made networks the
small-world phenomenon is surely the most relevant and popular. But despite its
significance, a reliable and comparable quantification of the question `how
small is a small-world network and how does it compare to others' has remained
a difficult challenge to answer. Here we establish a new synoptic
representation that allows for a complete and accurate interpretation of the
pathlength (and efficiency) of complex networks. We frame every network
individually, based on how its length deviates from the shortest and the
longest values it could possibly take. For that, we first had to uncover the
upper and the lower limits for the pathlength and efficiency, which indeed
depend on the specific number of nodes and links. These limits are given by
families of singular configurations that we name as ultra-short and ultra-long
networks. The representation here introduced frees network comparison from the
need to rely on the choice of reference graph models (e.g., random graphs and
ring lattices), a common practice that is prone to yield biased interpretations
as we show. Application to empirical examples of three categories (neural,
social and transportation) evidences that, while most real networks display a
pathlength comparable to that of random graphs, when contrasted against the
absolute boundaries, only the cortical connectomes prove to be ultra-short
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
Fractal and Transfractal Recursive Scale-Free Nets
We explore the concepts of self-similarity, dimensionality, and
(multi)scaling in a new family of recursive scale-free nets that yield
themselves to exact analysis through renormalization techniques. All nets in
this family are self-similar and some are fractals - possessing a finite
fractal dimension - while others are small world (their diameter grows
logarithmically with their size) and are infinite-dimensional. We show how a
useful measure of "transfinite" dimension may be defined and applied to the
small world nets. Concerning multiscaling, we show how first-passage time for
diffusion and resistance between hub (the most connected nodes) scale
differently than for other nodes. Despite the different scalings, the Einstein
relation between diffusion and conductivity holds separately for hubs and
nodes. The transfinite exponents of small world nets obey Einstein relations
analogous to those in fractal nets.Comment: Includes small revisions and references added as result of readers'
feedbac
Structures in supercritical scale-free percolation
Scale-free percolation is a percolation model on which can be
used to model real-world networks. We prove bounds for the graph distance in
the regime where vertices have infinite degrees. We fully characterize
transience vs. recurrence for dimension 1 and 2 and give sufficient conditions
for transience in dimension 3 and higher. Finally, we show the existence of a
hierarchical structure for parameters where vertices have degrees with infinite
variance and obtain bounds on the cluster density.Comment: Revised Definition 2.5 and an argument in Section 6, results are
unchanged. Correction of minor typos. 29 pages, 7 figure
Scale-Free Networks are Ultrasmall
We study the diameter, or the mean distance between sites, in a scale-free
network, having N sites and degree distribution p(k) ~ k^-a, i.e. the
probability of having k links outgoing from a site. In contrast to the diameter
of regular random networks or small world networks which is known to be d ~
lnN, we show, using analytical arguments, that scale free networks with 2<a<3
have a much smaller diameter, behaving as d ~ lnlnN. For a=3, our analysis
yields d ~ lnN/lnlnN, as obtained by Bollobas and Riordan, while for a>3, d ~
lnN. We also show that, for any a>2, one can construct a deterministic scale
free network with d ~ lnlnN, and this construction yields the lowest possible
diameter.Comment: Latex, 4 pages, 2 eps figures, small corrections, added explanation
Typical distances in a geometric model for complex networks
We study typical distances in a geometric random graph on the hyperbolic
plane. Introduced by Krioukov et al.~\cite{ar:Krioukov} as a model for complex
networks, vertices are drawn randomly within a bounded subset of the
hyperbolic plane and any two of them are joined if they are within a threshold
hyperbolic distance. With appropriately chosen parameters, the random graph is
sparse and exhibits power law degree distribution as well as local clustering.
In this paper we show a further property: the distance between two uniformly
chosen vertices that belong to the same component is doubly logarithmic in ,
i.e., the graph is an ~\emph{ultra-small world}. More precisely, we show that
the distance rescaled by converges in probability to a certain
constant that depends on the exponent of the power law. The same constant
emerges in an analogous setting with the well-known \emph{Chung-Lu} model for
which the degree distribution has a power law tail.Comment: 38 page
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