490 research outputs found
Spreading and shortest paths in systems with sparse long-range connections
Spreading according to simple rules (e.g. of fire or diseases), and
shortest-path distances are studied on d-dimensional systems with a small
density p per site of long-range connections (``Small-World'' lattices). The
volume V(t) covered by the spreading quantity on an infinite system is exactly
calculated in all dimensions. We find that V(t) grows initially as t^d/d for
t>t^*$,
generalizing a previous result in one dimension. Using the properties of V(t),
the average shortest-path distance \ell(r) can be calculated as a function of
Euclidean distance r. It is found that
\ell(r) = r for r<r_c=(2p \Gamma_d (d-1)!)^{-1/d} log(2p \Gamma_d L^d), and
\ell(r) = r_c for r>r_c.
The characteristic length r_c, which governs the behavior of shortest-path
lengths, diverges with system size for all p>0. Therefore the mean separation s
\sim p^{-1/d} between shortcut-ends is not a relevant internal length-scale for
shortest-path lengths. We notice however that the globally averaged
shortest-path length, divided by L, is a function of L/s only.Comment: 4 pages, 1 eps fig. Uses psfi
Scaling for the Percolation Backbone
We study the backbone connecting two given sites of a two-dimensional lattice
separated by an arbitrary distance in a system of size . We find a
scaling form for the average backbone mass: , where
can be well approximated by a power law for : with . This result implies that for the entire range . We also propose a scaling
form for the probability distribution of backbone mass for a given
. For is peaked around , whereas for decreases as a power law, , with . The exponents and satisfy the relation
, and is the codimension of the backbone,
.Comment: 3 pages, 5 postscript figures, Latex/Revtex/multicols/eps
Mean-field solution of the small-world network model
The small-world network model is a simple model of the structure of social
networks, which simultaneously possesses characteristics of both regular
lattices and random graphs. The model consists of a one-dimensional lattice
with a low density of shortcuts added between randomly selected pairs of
points. These shortcuts greatly reduce the typical path length between any two
points on the lattice. We present a mean-field solution for the average path
length and for the distribution of path lengths in the model. This solution is
exact in the limit of large system size and either large or small number of
shortcuts.Comment: 14 pages, 2 postscript figure
Ising model in small-world networks
The Ising model in small-world networks generated from two- and
three-dimensional regular lattices has been studied. Monte Carlo simulations
were carried out to characterize the ferromagnetic transition appearing in
these systems. In the thermodynamic limit, the phase transition has a
mean-field character for any finite value of the rewiring probability p, which
measures the disorder strength of a given network. For small values of p, both
the transition temperature and critical energy change with p as a power law. In
the limit p -> 0, the heat capacity at the transition temperature diverges
logarithmically in two-dimensional (2D) networks and as a power law in 3D.Comment: 6 pages, 7 figure
Structure of Growing Networks: Exact Solution of the Barabasi--Albert's Model
We generalize the Barab\'{a}si--Albert's model of growing networks accounting
for initial properties of sites and find exactly the distribution of
connectivities of the network and the averaged connectivity
of a site in the instant (one site is added per unit of
time). At long times at and
at , where the exponent
varies from 2 to depending on the initial attractiveness of sites. We
show that the relation between the exponents is universal.Comment: 4 pages revtex (twocolumn, psfig), 1 figur
Spin-dependent tunneling through high-k LaAlO3
We report on the use of the LaAlO3 (LAO) high-k dielectric as a tunnel
barrier in magnetic tunnel junctions. From tunnel magnetoresistance (TMR)
measurements on epitaxial La2/3Sr1/3MnO3/LAO/La2/3Sr1/3MnO3 junctions, we
estimate a spin polarization of 77% at low temperature for the
La2/3Sr1/3MnO3/LAO interface. Remarkably, the TMR of La2/3Sr1/3MnO3/LAO/Co
junctions at low bias is negative, evidencing a negative spin polarization of
Co at the interface with LAO, and its bias dependence is very similar to that
of La2/3Sr1/3MnO3/STO/Co junctions. We discuss possible reasons for this
behaviour.Comment: 14 pages, 3 figure
Polynomial kernels for 3-leaf power graph modification problems
A graph G=(V,E) is a 3-leaf power iff there exists a tree T whose leaves are
V and such that (u,v) is an edge iff u and v are at distance at most 3 in T.
The 3-leaf power graph edge modification problems, i.e. edition (also known as
the closest 3-leaf power), completion and edge-deletion, are FTP when
parameterized by the size of the edge set modification. However polynomial
kernel was known for none of these three problems. For each of them, we provide
cubic kernels that can be computed in linear time for each of these problems.
We thereby answer an open problem first mentioned by Dom, Guo, Huffner and
Niedermeier (2005).Comment: Submitte
Small-world networks: Evidence for a crossover picture
Watts and Strogatz [Nature 393, 440 (1998)] have recently introduced a model
for disordered networks and reported that, even for very small values of the
disorder in the links, the network behaves as a small-world. Here, we test
the hypothesis that the appearance of small-world behavior is not a
phase-transition but a crossover phenomenon which depends both on the network
size and on the degree of disorder . We propose that the average
distance between any two vertices of the network is a scaling function
of . The crossover size above which the network behaves as a
small-world is shown to scale as with .Comment: 5 pages, 5 postscript figures (1 in color),
Latex/Revtex/multicols/epsf. Accepted for publication in Physical Review
Letter
Small-world properties of the Indian Railway network
Structural properties of the Indian Railway network is studied in the light
of recent investigations of the scaling properties of different complex
networks. Stations are considered as `nodes' and an arbitrary pair of stations
is said to be connected by a `link' when at least one train stops at both
stations. Rigorous analysis of the existing data shows that the Indian Railway
network displays small-world properties. We define and estimate several other
quantities associated with this network.Comment: 5 pages, 7 figures. To be published in Phys. Rev.
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