2,626 research outputs found
Energy landscapes, supergraphs, and "folding funnels" in spin systems
Dynamical connectivity graphs, which describe dynamical transition rates
between local energy minima of a system, can be displayed against the
background of a disconnectivity graph which represents the energy landscape of
the system. The resulting supergraph describes both dynamics and statics of the
system in a unified coarse-grained sense. We give examples of the supergraphs
for several two dimensional spin and protein-related systems. We demonstrate
that disordered ferromagnets have supergraphs akin to those of model proteins
whereas spin glasses behave like random sequences of aminoacids which fold
badly.Comment: REVTeX, 9 pages, two-column, 13 EPS figures include
Statistical mechanics of the vertex-cover problem
We review recent progress in the study of the vertex-cover problem (VC). VC
belongs to the class of NP-complete graph theoretical problems, which plays a
central role in theoretical computer science. On ensembles of random graphs, VC
exhibits an coverable-uncoverable phase transition. Very close to this
transition, depending on the solution algorithm, easy-hard transitions in the
typical running time of the algorithms occur.
We explain a statistical mechanics approach, which works by mapping VC to a
hard-core lattice gas, and then applying techniques like the replica trick or
the cavity approach. Using these methods, the phase diagram of VC could be
obtained exactly for connectivities , where VC is replica symmetric.
Recently, this result could be confirmed using traditional mathematical
techniques. For , the solution of VC exhibits full replica symmetry
breaking.
The statistical mechanics approach can also be used to study analytically the
typical running time of simple complete and incomplete algorithms for VC.
Finally, we describe recent results for VC when studied on other ensembles of
finite- and infinite-dimensional graphs.Comment: review article, 26 pages, 9 figures, to appear in J. Phys. A: Math.
Ge
Dynamics of heuristic optimization algorithms on random graphs
In this paper, the dynamics of heuristic algorithms for constructing small
vertex covers (or independent sets) of finite-connectivity random graphs is
analysed. In every algorithmic step, a vertex is chosen with respect to its
vertex degree. This vertex, and some environment of it, is covered and removed
from the graph. This graph reduction process can be described as a Markovian
dynamics in the space of random graphs of arbitrary degree distribution. We
discuss some solvable cases, including algorithms already analysed using
different techniques, and develop approximation schemes for more complicated
cases. The approximations are corroborated by numerical simulations.Comment: 19 pages, 3 figures, version to app. in EPJ
Geometry and dimensionality reduction of feature spaces in primary visual cortex
Some geometric properties of the wavelet analysis performed by visual neurons
are discussed and compared with experimental data. In particular, several
relationships between the cortical morphologies and the parametric dependencies
of extracted features are formalized and considered from a harmonic analysis
point of view
Sampling properties of random graphs: the degree distribution
We discuss two sampling schemes for selecting random subnets from a network:
Random sampling and connectivity dependent sampling, and investigate how the
degree distribution of a node in the network is affected by the two types of
sampling. Here we derive a necessary and sufficient condition that guarantees
that the degree distribution of the subnet and the true network belong to the
same family of probability distributions. For completely random sampling of
nodes we find that this condition is fulfilled by classical random graphs; for
the vast majority of networks this condition will, however, not be met. We
furthermore discuss the case where the probability of sampling a node depends
on the degree of a node and we find that even classical random graphs are no
longer closed under this sampling regime. We conclude by relating the results
to real {\it E.coli} protein interaction network data.Comment: accepted for publication in Phys.Rev.
Coevolution of agents and networks: Opinion spreading and community disconnection
We study a stochastic model for the coevolution of a process of opinion
formation in a population of agents and the network which underlies their
interaction. Interaction links can break when agents fail to reach an opinion
agreement. The structure of the network and the distribution of opinions over
the population evolve towards a state where the population is divided into
disconnected communities whose agents share the same opinion. The statistical
properties of this final state vary considerably as the model parameters are
changed. Community sizes and their internal connectivity are the quantities
used to characterize such variations.Comment: To appear in Phys. Lett.
Stabilisers as a design tool for new forms of Lechner-Hauke-Zoller Annealer
In a recent paper Lechner, Hauke and Zoller (LHZ) described a means to
translate a Hamiltonian of spin- particles with 'all-to-all'
interactions into a larger physical lattice with only on-site energies and
local parity constraints. LHZ used this mapping to propose a novel form of
quantum annealing. Here we provide a stabiliser-based formulation within which
we can describe both this prior approach and a wide variety of variants.
Examples include a triangular array supporting all-to-all connectivity, and
moreover arrangements requiring only or spins but providing
interesting bespoke connectivities. Further examples show that arbitrarily high
order logical terms can be efficiently realised, even in a strictly 2D layout.
Our stabilisers can correspond to either even-parity constraints, as in the LHZ
proposal, or as odd-parity constraints. Considering the latter option applied
to the original LHZ layout, we note it may simplify the physical realisation
since the required ancillas are only spin- systems (i.e. qubits,
rather than qutrits) and moreover the interactions are very simple. We make a
preliminary assessment of the impact of this design choices by simulating small
(few-qubit) systems; we find some indications that the new variant may maintain
a larger minimum energy gap during the annealing process.Comment: A dramatically expanded revision: we now show how to use our
stabiliser formulation to construct a wide variety of new physical layouts,
including ones with fewer than Order N^2 spins but custom connectivities, and
a means to achieve higher order coupling even in 2
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