11,541 research outputs found
Geometric Graph Properties of the Spatial Preferred Attachment model
The spatial preferred attachment (SPA) model is a model for networked
information spaces such as domains of the World Wide Web, citation graphs, and
on-line social networks. It uses a metric space to model the hidden attributes
of the vertices. Thus, vertices are elements of a metric space, and link
formation depends on the metric distance between vertices. We show, through
theoretical analysis and simulation, that for graphs formed according to the
SPA model it is possible to infer the metric distance between vertices from the
link structure of the graph. Precisely, the estimate is based on the number of
common neighbours of a pair of vertices, a measure known as {\sl co-citation}.
To be able to calculate this estimate, we derive a precise relation between the
number of common neighbours and metric distance. We also analyze the
distribution of {\sl edge lengths}, where the length of an edge is the metric
distance between its end points. We show that this distribution has three
different regimes, and that the tail of this distribution follows a power law
Spatial preferential attachment networks: Power laws and clustering coefficients
We define a class of growing networks in which new nodes are given a spatial
position and are connected to existing nodes with a probability mechanism
favoring short distances and high degrees. The competition of preferential
attachment and spatial clustering gives this model a range of interesting
properties. Empirical degree distributions converge to a limit law, which can
be a power law with any exponent . The average clustering coefficient
of the networks converges to a positive limit. Finally, a phase transition
occurs in the global clustering coefficients and empirical distribution of edge
lengths when the power-law exponent crosses the critical value . Our
main tool in the proof of these results is a general weak law of large numbers
in the spirit of Penrose and Yukich.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1006 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Robustness of scale-free spatial networks
A growing family of random graphs is called robust if it retains a giant
component after percolation with arbitrary positive retention probability. We
study robustness for graphs, in which new vertices are given a spatial position
on the -dimensional torus and are connected to existing vertices with a
probability favouring short spatial distances and high degrees. In this model
of a scale-free network with clustering we can independently tune the power law
exponent of the degree distribution and the rate at which the
connection probability decreases with the distance of two vertices. We show
that the network is robust if , but fails to be robust if
. In the case of one-dimensional space we also show that the network is
not robust if . This implies that robustness of a
scale-free network depends not only on its power-law exponent but also on its
clustering features. Other than the classical models of scale-free networks our
model is not locally tree-like, and hence we need to develop novel methods for
its study, including, for example, a surprising application of the
BK-inequality.Comment: 34 pages, 4 figure
STEPS - an approach for human mobility modeling
In this paper we introduce Spatio-TEmporal Parametric Stepping (STEPS) - a simple parametric mobility model which can cover a large spectrum of human mobility patterns. STEPS makes abstraction of spatio-temporal preferences in human mobility by using a power law to rule the nodes movement. Nodes in STEPS have preferential attachment to favorite locations where they spend most of their time. Via simulations, we show that STEPS is able, not only to express the peer to peer properties such as inter-ontact/contact time and to reflect accurately realistic routing performance, but also to express the structural properties of the underlying interaction graph such as small-world phenomenon. Moreover, STEPS is easy to implement, exible to configure and also theoretically tractable
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