11,180 research outputs found
Global Robustness vs. Local Vulnerabilities in Complex Synchronous Networks
In complex network-coupled dynamical systems, two questions of central
importance are how to identify the most vulnerable components and how to devise
a network making the overall system more robust to external perturbations. To
address these two questions, we investigate the response of complex networks of
coupled oscillators to local perturbations. We quantify the magnitude of the
resulting excursion away from the unperturbed synchronous state through
quadratic performance measures in the angle or frequency deviations. We find
that the most fragile oscillators in a given network are identified by
centralities constructed from network resistance distances. Further defining
the global robustness of the system from the average response over ensembles of
homogeneously distributed perturbations, we find that it is given by a family
of topological indices known as generalized Kirchhoff indices. Both resistance
centralities and Kirchhoff indices are obtained from a spectral decomposition
of the stability matrix of the unperturbed dynamics and can be expressed in
terms of resistance distances. We investigate the properties of these
topological indices in small-world and regular networks. In the case of
oscillators with homogeneous inertia and damping coefficients, we find that
inertia only has small effects on robustness of coupled oscillators. Numerical
results illustrate the validity of the theory.Comment: 11 pages, 9 figure
Randomly coupled minimal models
Using 1-loop renormalisation group equations, we analyze the effect of
randomness on multi-critical unitary minimal conformal models. We study the
case of two randomly coupled models and found that they flow in two
decoupled models, in the infra-red limit. This result is then extend
to the case with randomly coupled models, which will flow toward
decoupled .Comment: 12 pages, latex, 1 eps figures; new results adde
Learning Mixtures of Gaussians in High Dimensions
Efficiently learning mixture of Gaussians is a fundamental problem in
statistics and learning theory. Given samples coming from a random one out of k
Gaussian distributions in Rn, the learning problem asks to estimate the means
and the covariance matrices of these Gaussians. This learning problem arises in
many areas ranging from the natural sciences to the social sciences, and has
also found many machine learning applications. Unfortunately, learning mixture
of Gaussians is an information theoretically hard problem: in order to learn
the parameters up to a reasonable accuracy, the number of samples required is
exponential in the number of Gaussian components in the worst case. In this
work, we show that provided we are in high enough dimensions, the class of
Gaussian mixtures is learnable in its most general form under a smoothed
analysis framework, where the parameters are randomly perturbed from an
adversarial starting point. In particular, given samples from a mixture of
Gaussians with randomly perturbed parameters, when n > {\Omega}(k^2), we give
an algorithm that learns the parameters with polynomial running time and using
polynomial number of samples. The central algorithmic ideas consist of new ways
to decompose the moment tensor of the Gaussian mixture by exploiting its
structural properties. The symmetries of this tensor are derived from the
combinatorial structure of higher order moments of Gaussian distributions
(sometimes referred to as Isserlis' theorem or Wick's theorem). We also develop
new tools for bounding smallest singular values of structured random matrices,
which could be useful in other smoothed analysis settings
Genetic networks with canalyzing Boolean rules are always stable
We determine stability and attractor properties of random Boolean genetic
network models with canalyzing rules for a variety of architectures. For all
power law, exponential, and flat in-degree distributions, we find that the
networks are dynamically stable. Furthermore, for architectures with few inputs
per node, the dynamics of the networks is close to critical. In addition, the
fraction of genes that are active decreases with the number of inputs per node.
These results are based upon investigating ensembles of networks using
analytical methods. Also, for different in-degree distributions, the numbers of
fixed points and cycles are calculated, with results intuitively consistent
with stability analysis; fewer inputs per node implies more cycles, and vice
versa. There are hints that genetic networks acquire broader degree
distributions with evolution, and hence our results indicate that for single
cells, the dynamics should become more stable with evolution. However, such an
effect is very likely compensated for by multicellular dynamics, because one
expects less stability when interactions among cells are included. We verify
this by simulations of a simple model for interactions among cells.Comment: Final version available through PNAS open access at
http://www.pnas.org/cgi/content/abstract/0407783101v
- …