942 research outputs found
Stability of shortest paths in complex networks with random edge weights
We study shortest paths and spanning trees of complex networks with random
edge weights. Edges which do not belong to the spanning tree are inactive in a
transport process within the network. The introduction of quenched disorder
modifies the spanning tree such that some edges are activated and the network
diameter is increased. With analytic random-walk mappings and numerical
analysis, we find that the spanning tree is unstable to the introduction of
disorder and displays a phase-transition-like behavior at zero disorder
strength . In the infinite network-size limit (), we
obtain a continuous transition with the density of activated edges
growing like and with the diameter-expansion coefficient
growing like in the regular network, and
first-order transitions with discontinuous jumps in and at
for the small-world (SW) network and the Barab\'asi-Albert
scale-free (SF) network. The asymptotic scaling behavior sets in when , where the crossover size scales as for the
regular network, for the SW network, and
for the SF network. In a
transient regime with , there is an infinite-order transition with
for the SW network
and for the SF network. It
shows that the transport pattern is practically most stable in the SF network.Comment: 9 pages, 7 figur
Why social networks are different from other types of networks
We argue that social networks differ from most other types of networks,
including technological and biological networks, in two important ways. First,
they have non-trivial clustering or network transitivity, and second, they show
positive correlations, also called assortative mixing, between the degrees of
adjacent vertices. Social networks are often divided into groups or
communities, and it has recently been suggested that this division could
account for the observed clustering. We demonstrate that group structure in
networks can also account for degree correlations. We show using a simple model
that we should expect assortative mixing in such networks whenever there is
variation in the sizes of the groups and that the predicted level of
assortative mixing compares well with that observed in real-world networks.Comment: 9 pages, 2 figure
Network Transitivity and Matrix Models
This paper is a step towards a systematic theory of the transitivity
(clustering) phenomenon in random networks. A static framework is used, with
adjacency matrix playing the role of the dynamical variable. Hence, our model
is a matrix model, where matrices are random, but their elements take values 0
and 1 only. Confusion present in some papers where earlier attempts to
incorporate transitivity in a similar framework have been made is hopefully
dissipated. Inspired by more conventional matrix models, new analytic
techniques to develop a static model with non-trivial clustering are
introduced. Computer simulations complete the analytic discussion.Comment: 11 pages, 7 eps figures, 2-column revtex format, print bug correcte
Properties of highly clustered networks
We propose and solve exactly a model of a network that has both a tunable
degree distribution and a tunable clustering coefficient. Among other things,
our results indicate that increased clustering leads to a decrease in the size
of the giant component of the network. We also study SIR-type epidemic
processes within the model and find that clustering decreases the size of
epidemics, but also decreases the epidemic threshold, making it easier for
diseases to spread. In addition, clustering causes epidemics to saturate
sooner, meaning that they infect a near-maximal fraction of the network for
quite low transmission rates.Comment: 7 pages, 2 figures, 1 tabl
Interface Motion and Pinning in Small World Networks
We show that the nonequilibrium dynamics of systems with many interacting
elements located on a small-world network can be much slower than on regular
networks. As an example, we study the phase ordering dynamics of the Ising
model on a Watts-Strogatz network, after a quench in the ferromagnetic phase at
zero temperature. In one and two dimensions, small-world features produce
dynamically frozen configurations, disordered at large length scales, analogous
of random field models. This picture differs from the common knowledge
(supported by equilibrium results) that ferromagnetic short-cuts connections
favor order and uniformity. We briefly discuss some implications of these
results regarding the dynamics of social changes.Comment: 4 pages, 5 figures with minor corrections. To appear in Phys. Rev.
Lyapunov exponents for products of complex Gaussian random matrices
The exact value of the Lyapunov exponents for the random matrix product with each , where
is a fixed positive definite matrix and a complex Gaussian matrix with entries standard complex normals, are
calculated. Also obtained is an exact expression for the sum of the Lyapunov
exponents in both the complex and real cases, and the Lyapunov exponents for
diffusing complex matrices.Comment: 15 page
Scaling Properties of Random Walks on Small-World Networks
Using both numerical simulations and scaling arguments, we study the behavior
of a random walker on a one-dimensional small-world network. For the properties
we study, we find that the random walk obeys a characteristic scaling form.
These properties include the average number of distinct sites visited by the
random walker, the mean-square displacement of the walker, and the distribution
of first-return times. The scaling form has three characteristic time regimes.
At short times, the walker does not see the small-world shortcuts and
effectively probes an ordinary Euclidean network in -dimensions. At
intermediate times, the properties of the walker shows scaling behavior
characteristic of an infinite small-world network. Finally, at long times, the
finite size of the network becomes important, and many of the properties of the
walker saturate. We propose general analytical forms for the scaling properties
in all three regimes, and show that these analytical forms are consistent with
our numerical simulations.Comment: 7 pages, 8 figures, two-column format. Submitted to PR
Path finding strategies in scale-free networks
We numerically investigate the scale-free network model of Barab{\'a}si and
Albert [A. L. Barab{\'a}si and R. Albert, Science {\bf 286}, 509 (1999)]
through the use of various path finding strategies. In real networks, global
network information is not accessible to each vertex, and the actual path
connecting two vertices can sometimes be much longer than the shortest one. A
generalized diameter depending on the actual path finding strategy is
introduced, and a simple strategy, which utilizes only local information on the
connectivity, is suggested and shown to yield small-world behavior: the
diameter of the network increases logarithmically with the network size
, the same as is found with global strategy. If paths are sought at random,
is found.Comment: 4 pages, final for
Growing Scale-Free Networks with Small World Behavior
In the context of growing networks, we introduce a simple dynamical model
that unifies the generic features of real networks: scale-free distribution of
degree and the small world effect. While the average shortest path length
increases logartihmically as in random networks, the clustering coefficient
assumes a large value independent of system size. We derive expressions for the
clustering coefficient in two limiting cases: random (C ~ (ln N)^2 / N) and
highly clustered (C = 5/6) scale-free networks.Comment: 4 pages, 4 figure
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