1,262 research outputs found
Quantum Operation Time Reversal
The dynamics of an open quantum system can be described by a quantum
operation, a linear, complete positive map of operators. Here, I exhibit a
compact expression for the time reversal of a quantum operation, which is
closely analogous to the time reversal of a classical Markov transition matrix.
Since open quantum dynamics are stochastic, and not, in general, deterministic,
the time reversal is not, in general, an inversion of the dynamics. Rather, the
system relaxes towards equilibrium in both the forward and reverse time
directions. The probability of a quantum trajectory and the conjugate, time
reversed trajectory are related by the heat exchanged with the environment.Comment: 4 page
Probability distribution of residence times of grains in models of ricepiles
We study the probability distribution of residence time of a grain at a site,
and its total residence time inside a pile, in different ricepile models. The
tails of these distributions are dominated by the grains that get deeply buried
in the pile. We show that, for a pile of size , the probabilities that the
residence time at a site or the total residence time is greater than , both
decay as for where
is an exponent , and values of and in the two
cases are different. In the Oslo ricepile model we find that the probability
that the residence time at a site being greater than or equal to ,
is a non-monotonic function of for a fixed and does not obey simple
scaling. For model in dimensions, we show that the probability of minimum
slope configuration in the steady state, for large , varies as where is a constant, and hence .Comment: 13 pages, 23 figures, Submitted to Phys. Rev.
Spectral coarse graining for random walk in bipartite networks
Many real-world networks display a natural bipartite structure, while
analyzing or visualizing large bipartite networks is one of the most
challenges. As a result, it is necessary to reduce the complexity of large
bipartite systems and preserve the functionality at the same time. We observe,
however, the existing coarse graining methods for binary networks fail to work
in the bipartite networks. In this paper, we use the spectral analysis to
design a coarse graining scheme specifically for bipartite networks and keep
their random walk properties unchanged. Numerical analysis on artificial and
real-world bipartite networks indicates that our coarse graining scheme could
obtain much smaller networks from large ones, keeping most of the relevant
spectral properties. Finally, we further validate the coarse graining method by
directly comparing the mean first passage time between the original network and
the reduced one.Comment: 7 pages, 3 figure
Excessive functions of continuous time Markov chains
AbstractWe consider transient continuous time Markov chains P(t) with P′ij(0)=qiΠij for i≠j and −qi for i=j. We assume 0<qi<∞ for all i. Then 1/qi is the mean time the process remains in state i, and Π is the transition matrix of the imbedded jump process. We let q be a diagonal matrix with diagonal entries qi.A non-negative function h is P(t)-excessive (invariant) if h≥P(t)h, (h=P(t) h) for all t. It is Π-superregular (regular) if h≥Πh (h=Πh). Our main results characterize the excessive functions of the minimal process in terms of q and Π. These results can also be used to characterize excessive functions of certain non-minimal processes
Computing the entropy of user navigation in the web
Navigation through the web, colloquially known as "surfing", is one of the main activities of users during web interaction. When users follow a navigation trail they often tend to get disoriented in terms of the goals of their original query and thus the discovery of typical user trails could be useful in providing navigation assistance. Herein, we give a theoretical underpinning of user navigation in terms of the entropy of an underlying Markov chain modelling the web topology. We present a novel method for online incremental computation of the entropy and a large deviation result regarding the length of a trail to realize the said entropy. We provide an error analysis for our estimation of the entropy in terms of the divergence between the empirical and actual probabilities. We then indicate applications of our algorithm in the area of web data mining. Finally, we present an extension of our technique to higher-order Markov chains by a suitable reduction of a higher-order Markov chain model to a first-order one
Maximal-entropy random walk unifies centrality measures
In this paper analogies between different (dis)similarity matrices are
derived. These matrices, which are connected to path enumeration and random
walks, are used in community detection methods or in computation of centrality
measures for complex networks. The focus is on a number of known centrality
measures, which inherit the connections established for similarity matrices.
These measures are based on the principal eigenvector of the adjacency matrix,
path enumeration, as well as on the stationary state, stochastic matrix or mean
first-passage times of a random walk. Particular attention is paid to the
maximal-entropy random walk, which serves as a very distinct alternative to the
ordinary random walk used in network analysis.
The various importance measures, defined both with the use of ordinary random
walk and the maximal-entropy random walk, are compared numerically on a set of
benchmark graphs. It is shown that groups of centrality measures defined with
the two random walks cluster into two separate families. In particular, the
group of centralities for the maximal-entropy random walk, connected to the
eigenvector centrality and path enumeration, is strongly distinct from all the
other measures and produces largely equivalent results.Comment: 7 pages, 2 figure
Micromagnetic understanding of stochastic resonance driven by spin-transfertorque
In this paper, we employ micromagnetic simulations to study non-adiabatic
stochastic resonance (NASR) excited by spin-transfer torque in a
super-paramagnetic free layer nanomagnet of a nanoscale spin valve. We find
that NASR dynamics involves thermally activated transitions among two static
states and a single dynamic state of the nanomagnet and can be well understood
in the framework of Markov chain rate theory. Our simulations show that a
direct voltage generated by the spin valve at the NASR frequency is at least
one order of magnitude greater than the dc voltage generated off the NASR
frequency. Our computations also reproduce the main experimentally observed
features of NASR such as the resonance frequency, the temperature dependence
and the current bias dependence of the resonance amplitude. We propose a simple
design of a microwave signal detector based on NASR driven by spin transfer
torque.Comment: 25 pages 8 figures, accepted for pubblication on Phys. Rev.
An average linear time algorithm for web data mining
In this paper, we study the complexity of a data mining algorithm for extracting patterns from user web navigation data that was proposed in previous work.3 The user web navigation sessions are inferred from log data and modeled as a Markov chain. The chain's higher probability trails correspond to the preferred trails on the web site. The algorithm implements a depth-first search that scans the Markov chain for the high probability trails. We show that the average behaviour of the algorithm is linear time in the number of web pages accessed
Improved coarse-graining of Markov state models via explicit consideration of statistical uncertainty
Markov state models (MSMs)---or discrete-time master equation models---are a
powerful way of modeling the structure and function of molecular systems like
proteins. Unfortunately, MSMs with sufficiently many states to make a
quantitative connection with experiments (often tens of thousands of states
even for small systems) are generally too complicated to understand. Here, I
present a Bayesian agglomerative clustering engine (BACE) for coarse-graining
such Markov models, thereby reducing their complexity and making them more
comprehensible. An important feature of this algorithm is its ability to
explicitly account for statistical uncertainty in model parameters that arises
from finite sampling. This advance builds on a number of recent works
highlighting the importance of accounting for uncertainty in the analysis of
MSMs and provides significant advantages over existing methods for
coarse-graining Markov state models. The closed-form expression I derive here
for determining which states to merge is equivalent to the generalized
Jensen-Shannon divergence, an important measure from information theory that is
related to the relative entropy. Therefore, the method has an appealing
information theoretic interpretation in terms of minimizing information loss.
The bottom-up nature of the algorithm likely makes it particularly well suited
for constructing mesoscale models. I also present an extremely efficient
expression for Bayesian model comparison that can be used to identify the most
meaningful levels of the hierarchy of models from BACE
Long-Range Navigation on Complex Networks using L\'evy Random Walks
We introduce a strategy of navigation in undirected networks, including
regular, random, and complex networks, that is inspired by L\'evy random walks,
generalizing previous navigation rules. We obtained exact expressions for the
stationary probability distribution, the occupation probability, the mean first
passage time, and the average time to reach a node on the network. We found
that the long-range navigation using the L\'evy random walk strategy, compared
with the normal random walk strategy, is more efficient at reducing the time to
cover the network. The dynamical effect of using the L\'evy walk strategy is to
transform a large-world network into a small world. Our exact results provide a
general framework that connects two important fields: L\'evy navigation
strategies and dynamics on complex networks.Comment: 6 pages, 3 figure
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