1,486 research outputs found
Adiabatic quantum algorithm for search engine ranking
We propose an adiabatic quantum algorithm for generating a quantum pure state
encoding of the PageRank vector, the most widely used tool in ranking the
relative importance of internet pages. We present extensive numerical
simulations which provide evidence that this algorithm can prepare the quantum
PageRank state in a time which, on average, scales polylogarithmically in the
number of webpages. We argue that the main topological feature of the
underlying web graph allowing for such a scaling is the out-degree
distribution. The top ranked entries of the quantum PageRank state
can then be estimated with a polynomial quantum speedup. Moreover, the quantum
PageRank state can be used in "q-sampling" protocols for testing properties of
distributions, which require exponentially fewer measurements than all
classical schemes designed for the same task. This can be used to decide
whether to run a classical update of the PageRank.Comment: 7 pages, 5 figures; closer to published versio
Quantum quenches and thermalization on scale-free graphs
We show that after a quantum quench of the parameter controlling the number
of particles in a Fermi-Hubbard model on scale free graphs, the distribution of
energy modes follows a power law dependent on the quenched parameter and the
connectivity of the graph. This paper contributes to the literature of quantum
quenches on lattices, in which, for many integrable lattice models the
distribution of modes after a quench thermalizes to a Generalized Gibbs
Ensemble; this paper provides another example of distribution which can arise
after relaxation. We argue that the main role is played by the symmetry of the
underlying lattice which, in the case we study, is scale free, and to the
distortion in the density of modes.Comment: 10 pages; 5 figures; accepted for publication in JTA
Scale-freeness for networks as a degenerate ground state: A Hamiltonian formulation
The origin of scale-free degree distributions in the context of networks is
addressed through an analogous non-network model in which the node degree
corresponds to the number of balls in a box and the rewiring of links to balls
moving between the boxes. A statistical mechanical formulation is presented and
the corresponding Hamiltonian is derived. The energy, the entropy, as well as
the degree distribution and its fluctuations are investigated at various
temperatures. The scale-free distribution is shown to correspond to the
degenerate ground state, which has small fluctuations in the degree
distribution and yet a large entropy. We suggest an implication of our results
from the viewpoint of the stability in evolution of networks.Comment: 7 pages, 3 figures. To appear in Europhysics lette
Nonextensive statistical mechanics and complex scale-free networks
One explanation for the impressive recent boom in network theory might be
that it provides a promising tool for an understanding of complex systems.
Network theory is mainly focusing on discrete large-scale topological
structures rather than on microscopic details of interactions of its elements.
This viewpoint allows to naturally treat collective phenomena which are often
an integral part of complex systems, such as biological or socio-economical
phenomena. Much of the attraction of network theory arises from the discovery
that many networks, natural or man-made, seem to exhibit some sort of
universality, meaning that most of them belong to one of three classes: {\it
random}, {\it scale-free} and {\it small-world} networks. Maybe most important
however for the physics community is, that due to its conceptually intuitive
nature, network theory seems to be within reach of a full and coherent
understanding from first principles ..
Scale free networks from a Hamiltonian dynamics
Contrary to many recent models of growing networks, we present a model with
fixed number of nodes and links, where it is introduced a dynamics favoring the
formation of links between nodes with degree of connectivity as different as
possible. By applying a local rewiring move, the network reaches equilibrium
states assuming broad degree distributions, which have a power law form in an
intermediate range of the parameters used. Interestingly, in the same range we
find non-trivial hierarchical clustering.Comment: 4 pages, revtex4, 5 figures. v2: corrected statements about
equilibriu
Multi-scaled analysis of the damped dynamics of an elastic rod with an essentially nonlinear end attachment
We study multi-frequency transitions in the transient dynamics of a viscously damped dispersive finite rod with an essentially nonlinear end attachment. The attachment consists of a small mass connected to the rod by means of an essentially nonlinear stiffness in parallel to a viscous damper. First, the periodic orbits of the underlying hamiltonian system with no damping are computed, and depicted in a frequency–energy plot (FEP). This representation enables one to clearly distinguish between the different types of periodic motions, forming back bone curves and subharmonic tongues. Then the damped dynamics of the system is computed; the rod and attachment responses are initially analyzed by the numerical Morlet wavelet transform (WT), and then by the empirical mode decomposition (EMD) or Hilbert–Huang transform (HTT), whereby, the time series are decomposed in terms of intrinsic mode functions (IMFs) at different characteristic time scales (or, equivalently, frequency scales). Comparisons of the evolutions of the instantaneous frequencies of the IMFs to the WT spectra of the time series enables one to identify the dominant IMFs of the signals, as well as, the time scales at which the dominant dynamics evolve at different time windows of the responses; hence, it is possible to reconstruct complex transient responses as superposition of the dominant IMFs involving different time scales of the dynamical response.
Moreover, by superimposing the WT spectra and the instantaneous frequencies of the IMFs to the FEPs of the underlying hamiltonian system, one is able to clearly identify the multi-scaled transitions that occur in the transient damped dynamics, and to interpret them as ‘jumps’ between different branches of periodic orbits of the underlying hamiltonian system. As a result, this work develops a physics-based, multi-scaled framework and provides the necessary computational tools for multi-scaled analysis of complex multi-frequency transitions of essentially nonlinear dynamical systems
Preferential attachment growth model and nonextensive statistical mechanics
We introduce a two-dimensional growth model where every new site is located,
at a distance from the barycenter of the pre-existing graph, according to
the probability law , and is attached to
(only) one pre-existing site with a probability ; is the number of links of the site of the
pre-existing graph, and its distance to the new site). Then we
numerically determine that the probability distribution for a site to have
links is asymptotically given, for all values of , by , where is the function
naturally emerging within nonextensive statistical mechanics. The entropic
index is numerically given (at least for not too large) by , and the characteristic number of links by . The particular case belongs to the same
universality class to which the Barabasi-Albert model belongs. In addition to
this, we have numerically studied the rate at which the average number of links
increases with the scaled time ; asymptotically, , the exponent being close to for , and zero otherwise.
The present results reinforce the conjecture that the microscopic dynamics of
nonextensive systems typically build (for instance, in Gibbs -space for
Hamiltonian systems) a scale-free network.Comment: 5 pages including 5 figures (the original colored figures 1 and 5a
can be asked directly to the authors
Randomness and Complexity in Networks
I start by reviewing some basic properties of random graphs. I then consider
the role of random walks in complex networks and show how they may be used to
explain why so many long tailed distributions are found in real data sets. The
key idea is that in many cases the process involves copying of properties of
near neighbours in the network and this is a type of short random walk which in
turn produce a natural preferential attachment mechanism. Applying this to
networks of fixed size I show that copying and innovation are processes with
special mathematical properties which include the ability to solve a simple
model exactly for any parameter values and at any time. I finish by looking at
variations of this basic model.Comment: Survey paper based on talk given at the workshop on ``Stochastic
Networks and Internet Technology'', Centro di Ricerca Matematica Ennio De
Giorgi, Matematica nelle Scienze Naturali e Sociali, Pisa, 17th - 21st
September 2007. To appear in proceeding
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