2,090 research outputs found
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.
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Quasiperiodic graphs at the onset of chaos
We examine the connectivity fluctuations across networks obtained when the
horizontal visibility (HV) algorithm is used on trajectories generated by
nonlinear circle maps at the quasiperiodic transition to chaos. The resultant
HV graph is highly anomalous as the degrees fluctuate at all scales with
amplitude that increases with the size of the network. We determine families of
Pesin-like identities between entropy growth rates and generalized
graph-theoretical Lyapunov exponents. An irrational winding number with pure
periodic continued fraction characterizes each family. We illustrate our
results for the so-called golden, silver and bronze numbers.Comment: arXiv admin note: text overlap with arXiv:1205.190
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
Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs
We consider the problem of coloring Erdos-Renyi and regular random graphs of
finite connectivity using q colors. It has been studied so far using the cavity
approach within the so-called one-step replica symmetry breaking (1RSB) ansatz.
We derive a general criterion for the validity of this ansatz and, applying it
to the ground state, we provide evidence that the 1RSB solution gives exact
threshold values c_q for the q-COL/UNCOL phase transition. We also study the
asymptotic thresholds for q >> 1 finding c_q = 2qlog(q)-log(q)-1+o(1) in
perfect agreement with rigorous mathematical bounds, as well as the nature of
excited states, and give a global phase diagram of the problem.Comment: 23 pages, 10 figures. Replaced with accepted versio
Communication and correlation among communities
Given a network and a partition in communities, we consider the issues "how
communities influence each other" and "when two given communities do
communicate". Specifically, we address these questions in the context of
small-world networks, where an arbitrary quenched graph is given and long range
connections are randomly added. We prove that, among the communities, a
superposition principle applies and gives rise to a natural generalization of
the effective field theory already presented in [Phys. Rev. E 78, 031102]
(n=1), which here (n>1) consists in a sort of effective TAP (Thouless, Anderson
and Palmer) equations in which each community plays the role of a microscopic
spin. The relative susceptibilities derived from these equations calculated at
finite or zero temperature, where the method provides an effective percolation
theory, give us the answers to the above issues. Unlike the case n=1,
asymmetries among the communities may lead, via the TAP-like structure of the
equations, to many metastable states whose number, in the case of negative
short-cuts among the communities, may grow exponentially fast with n. As
examples we consider the n Viana-Bray communities model and the n
one-dimensional small-world communities model. Despite being the simplest ones,
the relevance of these models in network theory, as e.g. in social networks, is
crucial and no analytic solution were known until now. Connections between
percolation and the fractal dimension of a network are also discussed. Finally,
as an inverse problem, we show how, from the relative susceptibilities, a
natural and efficient method to detect the community structure of a generic
network arises.
For a short presentation of the main result see arXiv:0812.0608.Comment: 29 pages, 5 figure
Potts q-color field theory and scaling random cluster model
We study structural properties of the q-color Potts field theory which, for
real values of q, describes the scaling limit of the random cluster model. We
show that the number of independent n-point Potts spin correlators coincides
with that of independent n-point cluster connectivities and is given by
generalized Bell numbers. Only a subset of these spin correlators enters the
determination of the Potts magnetic properties for q integer. The structure of
the operator product expansion of the spin fields for generic q is also
identified. For the two-dimensional case, we analyze the duality relation
between spin and kink field correlators, both for the bulk and boundary cases,
obtaining in particular a sum rule for the kink-kink elastic scattering
amplitudes.Comment: 27 pages; 6 figures. Published version, some comments and references
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