6,860 research outputs found
k-connectivity of Random Graphs and Random Geometric Graphs in Node Fault Model
k-connectivity of random graphs is a fundamental property indicating
reliability of multi-hop wireless sensor networks (WSN). WSNs comprising of
sensor nodes with limited power resources are modeled by random graphs with
unreliable nodes, which is known as the node fault model. In this paper, we
investigate k-connectivity of random graphs in the node fault model by
evaluating the network breakdown probability, i.e., the disconnectivity
probability of random graphs after stochastic node removals. Using the notion
of a strongly typical set, we obtain universal asymptotic upper and lower
bounds of the network breakdown probability. The bounds are applicable both to
random graphs and to random geometric graphs. We then consider three
representative random graph ensembles: the Erdos-Renyi random graph as the
simplest case, the random intersection graph for WSNs with random key
predistribution schemes, and the random geometric graph as a model of WSNs
generated by random sensor node deployment. The bounds unveil the existence of
the phase transition of the network breakdown probability for those ensembles.Comment: 6 page
Economic Small-World Behavior in Weighted Networks
The small-world phenomenon has been already the subject of a huge variety of
papers, showing its appeareance in a variety of systems. However, some big
holes still remain to be filled, as the commonly adopted mathematical
formulation suffers from a variety of limitations, that make it unsuitable to
provide a general tool of analysis for real networks, and not just for
mathematical (topological) abstractions. In this paper we show where the major
problems arise, and how there is therefore the need for a new reformulation of
the small-world concept. Together with an analysis of the variables involved,
we then propose a new theory of small-world networks based on two leading
concepts: efficiency and cost. Efficiency measures how well information
propagates over the network, and cost measures how expensive it is to build a
network. The combination of these factors leads us to introduce the concept of
{\em economic small worlds}, that formalizes the idea of networks that are
"cheap" to build, and nevertheless efficient in propagating information, both
at global and local scale. This new concept is shown to overcome all the
limitations proper of the so-far commonly adopted formulation, and to provide
an adequate tool to quantitatively analyze the behaviour of complex networks in
the real world. Various complex systems are analyzed, ranging from the realm of
neural networks, to social sciences, to communication and transportation
networks. In each case, economic small worlds are found. Moreover, using the
economic small-world framework, the construction principles of these networks
can be quantitatively analyzed and compared, giving good insights on how
efficiency and economy principles combine up to shape all these systems.Comment: 17 pages, 10 figures, 4 table
Efficient Behavior of Small-World Networks
We introduce the concept of efficiency of a network, measuring how
efficiently it exchanges information. By using this simple measure small-world
networks are seen as systems that are both globally and locally efficient. This
allows to give a clear physical meaning to the concept of small-world, and also
to perform a precise quantitative a nalysis of both weighted and unweighted
networks. We study neural networks and man-made communication and
transportation systems and we show that the underlying general principle of
their construction is in fact a small-world principle of high efficiency.Comment: 1 figure, 2 tables. Revised version. Accepted for publication in
Phys. Rev. Let
Harmony in the Small-World
The Small-World phenomenon, popularly known as six degrees of separation, has
been mathematically formalized by Watts and Strogatz in a study of the
topological properties of a network. Small-worlds networks are defined in terms
of two quantities: they have a high clustering coefficient C like regular
lattices and a short characteristic path length L typical of random networks.
Physical distances are of fundamental importance in the applications to real
cases, nevertheless this basic ingredient is missing in the original
formulation. Here we introduce a new concept, the connectivity length D, that
gives harmony to the whole theory. D can be evaluated on a global and on a
local scale and plays in turn the role of L and 1/C. Moreover it can be
computed for any metrical network and not only for the topological cases. D has
a precise meaning in term of information propagation and describes in an
unified way both the structural and the dynamical aspects of a network:
small-worlds are defined by a small global and local D, i.e. by a high
efficiency in propagating information both on a local and on a global scale.
The neural system of the nematode C. elegans, the collaboration graph of film
actors, and the oldest U.S. subway system, can now be studied also as metrical
networks and are shown to be small-worlds.Comment: 16 pages, 3 figures, accepted for publication in Physica
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