518 research outputs found
On the properties of small-world network models
We study the small-world networks recently introduced by Watts and Strogatz
[Nature {\bf 393}, 440 (1998)], using analytical as well as numerical tools. We
characterize the geometrical properties resulting from the coexistence of a
local structure and random long-range connections, and we examine their
evolution with size and disorder strength. We show that any finite value of the
disorder is able to trigger a ``small-world'' behaviour as soon as the initial
lattice is big enough, and study the crossover between a regular lattice and a
``small-world'' one. These results are corroborated by the investigation of an
Ising model defined on the network, showing for every finite disorder fraction
a crossover from a high-temperature region dominated by the underlying
one-dimensional structure to a mean-field like low-temperature region. In
particular there exists a finite-temperature ferromagnetic phase transition as
soon as the disorder strength is finite.Comment: 19 pages including 15 figures, version accepted for publication in
EPJ
Solving satisfiability problems by fluctuations: The dynamics of stochastic local search algorithms
Stochastic local search algorithms are frequently used to numerically solve
hard combinatorial optimization or decision problems. We give numerical and
approximate analytical descriptions of the dynamics of such algorithms applied
to random satisfiability problems. We find two different dynamical regimes,
depending on the number of constraints per variable: For low constraintness,
the problems are solved efficiently, i.e. in linear time. For higher
constraintness, the solution times become exponential. We observe that the
dynamical behavior is characterized by a fast equilibration and fluctuations
around this equilibrium. If the algorithm runs long enough, an exponentially
rare fluctuation towards a solution appears.Comment: 21 pages, 18 figures, revised version, to app. in PRE (2003
Clustering by soft-constraint affinity propagation: Applications to gene-expression data
Motivation: Similarity-measure based clustering is a crucial problem
appearing throughout scientific data analysis. Recently, a powerful new
algorithm called Affinity Propagation (AP) based on message-passing techniques
was proposed by Frey and Dueck \cite{Frey07}. In AP, each cluster is identified
by a common exemplar all other data points of the same cluster refer to, and
exemplars have to refer to themselves. Albeit its proved power, AP in its
present form suffers from a number of drawbacks. The hard constraint of having
exactly one exemplar per cluster restricts AP to classes of regularly shaped
clusters, and leads to suboptimal performance, {\it e.g.}, in analyzing gene
expression data. Results: This limitation can be overcome by relaxing the AP
hard constraints. A new parameter controls the importance of the constraints
compared to the aim of maximizing the overall similarity, and allows to
interpolate between the simple case where each data point selects its closest
neighbor as an exemplar and the original AP. The resulting soft-constraint
affinity propagation (SCAP) becomes more informative, accurate and leads to
more stable clustering. Even though a new {\it a priori} free-parameter is
introduced, the overall dependence of the algorithm on external tuning is
reduced, as robustness is increased and an optimal strategy for parameter
selection emerges more naturally. SCAP is tested on biological benchmark data,
including in particular microarray data related to various cancer types. We
show that the algorithm efficiently unveils the hierarchical cluster structure
present in the data sets. Further on, it allows to extract sparse gene
expression signatures for each cluster.Comment: 11 pages, supplementary material:
http://isiosf.isi.it/~weigt/scap_supplement.pd
Boosting search by rare events
Randomized search algorithms for hard combinatorial problems exhibit a large
variability of performances. We study the different types of rare events which
occur in such out-of-equilibrium stochastic processes and we show how they
cooperate in determining the final distribution of running times. As a
byproduct of our analysis we show how search algorithms are optimized by random
restarts.Comment: 4 pages, 3 eps figures. References update
Ancient DNA from coral-hosted Symbiodinium reveal a static mutualism over the last 172 years.
Ancient DNA (aDNA) provides powerful evidence for detecting the genetic basis for adaptation to environmental change in many taxa. Among the greatest of changes in our biosphere within the last century is rapid anthropogenic ocean warming. This phenomenon threatens corals with extinction, evidenced by the increasing observation of widespread mortality following mass bleaching events. There is some evidence and conjecture that coral-dinoflagellate symbioses change partnerships in response to changing external conditions over ecological and evolutionary timescales. Until now, we have been unable to ascertain the genetic identity of Symbiodinium hosted by corals prior to the rapid global change of the last century. Here, we show that Symbiodinium cells recovered from dry, century old specimens of 6 host species of octocorals contain sufficient DNA for amplification of the ITS2 subregion of the nuclear ribosomal DNA, commonly used for genotyping within this genus. Through comparisons with modern specimens sampled from similar locales we show that symbiotic associations among several species have been static over the last century, thereby suggesting that adaptive shifts to novel symbiont types is not common among these gorgonians, and perhaps, symbiotic corals in general
Computational complexity arising from degree correlations in networks
We apply a Bethe-Peierls approach to statistical-mechanics models defined on
random networks of arbitrary degree distribution and arbitrary correlations
between the degrees of neighboring vertices. Using the NP-hard optimization
problem of finding minimal vertex covers on these graphs, we show that such
correlations may lead to a qualitatively different solution structure as
compared to uncorrelated networks. This results in a higher complexity of the
network in a computational sense: Simple heuristic algorithms fail to find a
minimal vertex cover in the highly correlated case, whereas uncorrelated
networks seem to be simple from the point of view of combinatorial
optimization.Comment: 4 pages, 1 figure, accepted in Phys. Rev.
Exactly solvable model with two conductor-insulator transitions driven by impurities
We present an exact analysis of two conductor-insulator transitions in the
random graph model. The average connectivity is related to the concentration of
impurities. The adjacency matrix of a large random graph is used as a hopping
Hamiltonian. Its spectrum has a delta peak at zero energy. Our analysis is
based on an explicit expression for the height of this peak, and a detailed
description of the localized eigenvectors and of their contribution to the
peak. Starting from the low connectivity (high impurity density) regime, one
encounters an insulator-conductor transition for average connectivity
1.421529... and a conductor-insulator transition for average connectivity
3.154985.... We explain the spectral singularity at average connectivity
e=2.718281... and relate it to another enumerative problem in random graph
theory, the minimal vertex cover problem.Comment: 4 pages revtex, 2 fig.eps [v2: new title, changed intro, reorganized
text
Multifractal analysis of perceptron learning with errors
Random input patterns induce a partition of the coupling space of a
perceptron into cells labeled by their output sequences. Learning some data
with a maximal error rate leads to clusters of neighboring cells. By analyzing
the internal structure of these clusters with the formalism of multifractals,
we can handle different storage and generalization tasks for lazy students and
absent-minded teachers within one unified approach. The results also allow some
conclusions on the spatial distribution of cells.Comment: 11 pages, RevTex, 3 eps figures, version to be published in Phys.
Rev. E 01Jan9
Random Graph Coloring - a Statistical Physics Approach
The problem of vertex coloring in random graphs is studied using methods of
statistical physics and probability. Our analytical results are compared to
those obtained by exact enumeration and Monte-Carlo simulations. We critically
discuss the merits and shortcomings of the various methods, and interpret the
results obtained. We present an exact analytical expression for the 2-coloring
problem as well as general replica symmetric approximated solutions for the
thermodynamics of the graph coloring problem with p colors and K-body edges.Comment: 17 pages, 9 figure
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