342 research outputs found
Spatial Mixing of Coloring Random Graphs
We study the strong spatial mixing (decay of correlation) property of proper
-colorings of random graph with a fixed . The strong spatial
mixing of coloring and related models have been extensively studied on graphs
with bounded maximum degree. However, for typical classes of graphs with
bounded average degree, such as , an easy counterexample shows that
colorings do not exhibit strong spatial mixing with high probability.
Nevertheless, we show that for with and
sufficiently large , with high probability proper -colorings of
random graph exhibit strong spatial mixing with respect to an
arbitrarily fixed vertex. This is the first strong spatial mixing result for
colorings of graphs with unbounded maximum degree. Our analysis of strong
spatial mixing establishes a block-wise correlation decay instead of the
standard point-wise decay, which may be of interest by itself, especially for
graphs with unbounded degree
Circular Networks from Distorted Metrics
Trees have long been used as a graphical representation of species
relationships. However complex evolutionary events, such as genetic
reassortments or hybrid speciations which occur commonly in viruses, bacteria
and plants, do not fit into this elementary framework. Alternatively, various
network representations have been developed. Circular networks are a natural
generalization of leaf-labeled trees interpreted as split systems, that is,
collections of bipartitions over leaf labels corresponding to current species.
Although such networks do not explicitly model specific evolutionary events of
interest, their straightforward visualization and fast reconstruction have made
them a popular exploratory tool to detect network-like evolution in genetic
datasets.
Standard reconstruction methods for circular networks, such as Neighbor-Net,
rely on an associated metric on the species set. Such a metric is first
estimated from DNA sequences, which leads to a key difficulty: distantly
related sequences produce statistically unreliable estimates. This is
problematic for Neighbor-Net as it is based on the popular tree reconstruction
method Neighbor-Joining, whose sensitivity to distance estimation errors is
well established theoretically. In the tree case, more robust reconstruction
methods have been developed using the notion of a distorted metric, which
captures the dependence of the error in the distance through a radius of
accuracy. Here we design the first circular network reconstruction method based
on distorted metrics. Our method is computationally efficient. Moreover, the
analysis of its radius of accuracy highlights the important role played by the
maximum incompatibility, a measure of the extent to which the network differs
from a tree.Comment: Submitte
Majority Dynamics and Aggregation of Information in Social Networks
Consider n individuals who, by popular vote, choose among q >= 2
alternatives, one of which is "better" than the others. Assume that each
individual votes independently at random, and that the probability of voting
for the better alternative is larger than the probability of voting for any
other. It follows from the law of large numbers that a plurality vote among the
n individuals would result in the correct outcome, with probability approaching
one exponentially quickly as n tends to infinity. Our interest in this paper is
in a variant of the process above where, after forming their initial opinions,
the voters update their decisions based on some interaction with their
neighbors in a social network. Our main example is "majority dynamics", in
which each voter adopts the most popular opinion among its friends. The
interaction repeats for some number of rounds and is then followed by a
population-wide plurality vote.
The question we tackle is that of "efficient aggregation of information": in
which cases is the better alternative chosen with probability approaching one
as n tends to infinity? Conversely, for which sequences of growing graphs does
aggregation fail, so that the wrong alternative gets chosen with probability
bounded away from zero? We construct a family of examples in which interaction
prevents efficient aggregation of information, and give a condition on the
social network which ensures that aggregation occurs. For the case of majority
dynamics we also investigate the question of unanimity in the limit. In
particular, if the voters' social network is an expander graph, we show that if
the initial population is sufficiently biased towards a particular alternative
then that alternative will eventually become the unanimous preference of the
entire population.Comment: 22 page
Constrained Non-Monotone Submodular Maximization: Offline and Secretary Algorithms
Constrained submodular maximization problems have long been studied, with
near-optimal results known under a variety of constraints when the submodular
function is monotone. The case of non-monotone submodular maximization is less
understood: the first approximation algorithms even for the unconstrainted
setting were given by Feige et al. (FOCS '07). More recently, Lee et al. (STOC
'09, APPROX '09) show how to approximately maximize non-monotone submodular
functions when the constraints are given by the intersection of p matroid
constraints; their algorithm is based on local-search procedures that consider
p-swaps, and hence the running time may be n^Omega(p), implying their algorithm
is polynomial-time only for constantly many matroids. In this paper, we give
algorithms that work for p-independence systems (which generalize constraints
given by the intersection of p matroids), where the running time is poly(n,p).
Our algorithm essentially reduces the non-monotone maximization problem to
multiple runs of the greedy algorithm previously used in the monotone case.
Our idea of using existing algorithms for monotone functions to solve the
non-monotone case also works for maximizing a submodular function with respect
to a knapsack constraint: we get a simple greedy-based constant-factor
approximation for this problem.
With these simpler algorithms, we are able to adapt our approach to
constrained non-monotone submodular maximization to the (online) secretary
setting, where elements arrive one at a time in random order, and the algorithm
must make irrevocable decisions about whether or not to select each element as
it arrives. We give constant approximations in this secretary setting when the
algorithm is constrained subject to a uniform matroid or a partition matroid,
and give an O(log k) approximation when it is constrained by a general matroid
of rank k.Comment: In the Proceedings of WINE 201
The theoretical capacity of the Parity Source Coder
The Parity Source Coder is a protocol for data compression which is based on
a set of parity checks organized in a sparse random network. We consider here
the case of memoryless unbiased binary sources. We show that the theoretical
capacity saturate the Shannon limit at large K. We also find that the first
corrections to the leading behavior are exponentially small, so that the
behavior at finite K is very close to the optimal one.Comment: Added references, minor change
Remarks on the notion of quantum integrability
We discuss the notion of integrability in quantum mechanics. Starting from a
review of some definitions commonly used in the literature, we propose a
different set of criteria, leading to a classification of models in terms of
different integrability classes. We end by highlighting some of the expected
physical properties associated to models fulfilling the proposed criteria.Comment: 22 pages, no figures, Proceedings of Statphys 2
Many-body localization and thermalization in the full probability distribution function of observables
We investigate the relation between thermalization following a quantum quench
and many-body localization in quasiparticle space in terms of the long-time
full distribution function of physical observables. In particular, expanding on
our recent work [E. Canovi {\em et al.}, Phys. Rev. B {\bf 83}, 094431 (2011)],
we focus on the long-time behavior of an integrable XXZ chain subject to an
integrability-breaking perturbation. After a characterization of the breaking
of integrability and the associated localization/delocalization transition
using the level spacing statistics and the properties of the eigenstates, we
study the effect of integrability-breaking on the asymptotic state after a
quantum quench of the anisotropy parameter, looking at the behavior of the full
probability distribution of the transverse and longitudinal magnetization of a
subsystem. We compare the resulting distributions with those obtained in
equilibrium at an effective temperature set by the initial energy. We find
that, while the long time distribution functions appear to always agree {\it
qualitatively} with the equilibrium ones, {\it quantitative} agreement is
obtained only when integrability is fully broken and the relevant eigenstates
are diffusive in quasi-particle space.Comment: 18 pages, 11 figure
Time Evolution within a Comoving Window: Scaling of signal fronts and magnetization plateaus after a local quench in quantum spin chains
We present a modification of Matrix Product State time evolution to simulate
the propagation of signal fronts on infinite one-dimensional systems. We
restrict the calculation to a window moving along with a signal, which by the
Lieb-Robinson bound is contained within a light cone. Signal fronts can be
studied unperturbed and with high precision for much longer times than on
finite systems. Entanglement inside the window is naturally small, greatly
lowering computational effort. We investigate the time evolution of the
transverse field Ising (TFI) model and of the S=1/2 XXZ antiferromagnet in
their symmetry broken phases after several different local quantum quenches.
In both models, we observe distinct magnetization plateaus at the signal
front for very large times, resembling those previously observed for the
particle density of tight binding (TB) fermions. We show that the normalized
difference to the magnetization of the ground state exhibits similar scaling
behaviour as the density of TB fermions. In the XXZ model there is an
additional internal structure of the signal front due to pairing, and wider
plateaus with tight binding scaling exponents for the normalized excess
magnetization. We also observe parameter dependent interaction effects between
individual plateaus, resulting in a slight spatial compression of the plateau
widths.
In the TFI model, we additionally find that for an initial Jordan-Wigner
domain wall state, the complete time evolution of the normalized excess
longitudinal magnetization agrees exactly with the particle density of TB
fermions.Comment: 10 pages with 5 figures. Appendix with 23 pages, 13 figures and 4
tables. Largely extended and improved versio
The two-spinon transverse structure factor of the gapped Heisenberg antiferromagnetic chain
We consider the transverse dynamical structure factor of the anisotropic
Heisenberg spin-1/2 chain (XXZ model) in the gapped antiferromagnetic regime
(). Specializing to the case of zero field, we use two independent
approaches based on integrability (one valid for finite size, the other for the
infinite lattice) to obtain the exact two-spinon part of this correlator. We
discuss in particular its asymmetry with respect to the momentum line,
its overall anisotropy dependence, and its contribution to sum rules.Comment: 19 pages, 6 figure
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