107 research outputs found
Sustaining the Internet with Hyperbolic Mapping
The Internet infrastructure is severely stressed. Rapidly growing overheads
associated with the primary function of the Internet---routing information
packets between any two computers in the world---cause concerns among Internet
experts that the existing Internet routing architecture may not sustain even
another decade. Here we present a method to map the Internet to a hyperbolic
space. Guided with the constructed map, which we release with this paper,
Internet routing exhibits scaling properties close to theoretically best
possible, thus resolving serious scaling limitations that the Internet faces
today. Besides this immediate practical viability, our network mapping method
can provide a different perspective on the community structure in complex
networks
Querying Probabilistic Neighborhoods in Spatial Data Sets Efficiently
In this paper we define the notion
of a probabilistic neighborhood in spatial data: Let a set of points in
, a query point , a distance metric \dist,
and a monotonically decreasing function be
given. Then a point belongs to the probabilistic neighborhood of with respect to with probability f(\dist(p,q)). We envision
applications in facility location, sensor networks, and other scenarios where a
connection between two entities becomes less likely with increasing distance. A
straightforward query algorithm would determine a probabilistic neighborhood in
time by probing each point in .
To answer the query in sublinear time for the planar case, we augment a
quadtree suitably and design a corresponding query algorithm. Our theoretical
analysis shows that -- for certain distributions of planar -- our algorithm
answers a query in time with high probability
(whp). This matches up to a logarithmic factor the cost induced by
quadtree-based algorithms for deterministic queries and is asymptotically
faster than the straightforward approach whenever .
As practical proofs of concept we use two applications, one in the Euclidean
and one in the hyperbolic plane. In particular, our results yield the first
generator for random hyperbolic graphs with arbitrary temperatures in
subquadratic time. Moreover, our experimental data show the usefulness of our
algorithm even if the point distribution is unknown or not uniform: The running
time savings over the pairwise probing approach constitute at least one order
of magnitude already for a modest number of points and queries.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-44543-4_3
Nearest-neighbour directed random hyperbolic graphs
Undirected hyperbolic graph models have been extensively used as models of
scale-free small-world networks with high clustering coefficient. Here we
presented a simple directed hyperbolic model, where nodes randomly distributed
on a hyperbolic disk are connected to a fixed number m of their nearest spatial
neighbours. We introduce also a canonical version of this network (which we
call "network with varied connection radius"), where maximal length of outgoing
bond is space-dependent and is determined by fixing the average out-degree to
m. We study local bond length, in-degree and reciprocity in these networks as a
function of spatial coordinates of the nodes, and show that the network has a
distinct core-periphery structure. We show that for small densities of nodes
the overall in-degree has a truncated power law distribution. We demonstrate
that reciprocity of the network can be regulated by adjusting an additional
temperature-like parameter without changing other global properties of the
network.Comment: 26 papers, 12 figure
Search in Complex Networks : a New Method of Naming
We suggest a method for routing when the source does not posses full
information about the shortest path to the destination. The method is
particularly useful for scale-free networks, and exploits its unique
characteristics. By assigning new (short) names to nodes (aka labelling) we are
able to reduce significantly the memory requirement at the routers, yet we
succeed in routing with high probability through paths very close in distance
to the shortest ones.Comment: 5 pages, 4 figure
Forman's Ricci curvature - From networks to hypernetworks
Networks and their higher order generalizations, such as hypernetworks or
multiplex networks are ever more popular models in the applied sciences.
However, methods developed for the study of their structural properties go
little beyond the common name and the heavy reliance of combinatorial tools. We
show that, in fact, a geometric unifying approach is possible, by viewing them
as polyhedral complexes endowed with a simple, yet, the powerful notion of
curvature - the Forman Ricci curvature. We systematically explore some aspects
related to the modeling of weighted and directed hypernetworks and present
expressive and natural choices involved in their definitions. A benefit of this
approach is a simple method of structure-preserving embedding of hypernetworks
in Euclidean N-space. Furthermore, we introduce a simple and efficient manner
of computing the well established Ollivier-Ricci curvature of a hypernetwork.Comment: to appear: Complex Networks '18 (oral presentation
Inferring periodic orbits from spectra of simple shaped micro-lasers
Dielectric micro-cavities are widely used as laser resonators and
characterizations of their spectra are of interest for various applications. We
experimentally investigate micro-lasers of simple shapes (Fabry-Perot, square,
pentagon, and disk). Their lasing spectra consist mainly of almost equidistant
peaks and the distance between peaks reveals the length of a quantized periodic
orbit. To measure this length with a good precision, it is necessary to take
into account different sources of refractive index dispersion. Our experimental
and numerical results agree with the superscar model describing the formation
of long-lived states in polygonal cavities. The limitations of the
two-dimensional approximation are briefly discussed in connection with
micro-disks.Comment: 13 pages, 19 figures, accepted for publication in Physical Review
Clustering and the hyperbolic geometry of complex networks
Clustering is a fundamental property of complex networks and it is the
mathematical expression of a ubiquitous phenomenon that arises in various types
of self-organized networks such as biological networks, computer networks or
social networks. In this paper, we consider what is called the global
clustering coefficient of random graphs on the hyperbolic plane. This model of
random graphs was proposed recently by Krioukov et al. as a mathematical model
of complex networks, under the fundamental assumption that hyperbolic geometry
underlies the structure of these networks. We give a rigorous analysis of
clustering and characterize the global clustering coefficient in terms of the
parameters of the model. We show how the global clustering coefficient can be
tuned by these parameters and we give an explicit formula for this function.Comment: 51 pages, 1 figur
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
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