429 research outputs found
Self-consistency and Symmetry in d-dimensions
Bethe approximation is shown to violate Bravais lattices translational
invariance. A new scheme is then presented which goes over the one-site Weiss
model yet preserving initial lattice symmetry. A mapping to a one-dimensional
finite closed chain in an external field is obtained. Lattice topology
determines the chain size. Using recent results in percolation, lattice
connectivity between chains is argued to be where is the
coordination number and is the space dimension. A new self-consistent
mean-field equation of state is derived. Critical temperatures are thus
calculated for a large variety of lattices and dimensions. Results are within a
few percent of exact estimates. Moreover onset of phase transitions is found to
occur in the range . For the Ising hypercube it yields the Golden
number limit .Comment: 16 pages, latex, Phys. Rev. B (in press
Expansion for -Core Percolation
The physics of -core percolation pertains to those systems whose
constituents require a minimum number of connections to each other in order
to participate in any clustering phenomenon. Examples of such a phenomenon
range from orientational ordering in solid ortho-para mixtures to
the onset of rigidity in bar-joint networks to dynamical arrest in
glass-forming liquids. Unlike ordinary () and biconnected ()
percolation, the mean field -core percolation transition is both
continuous and discontinuous, i.e. there is a jump in the order parameter
accompanied with a diverging length scale. To determine whether or not this
hybrid transition survives in finite dimensions, we present a expansion
for -core percolation on the -dimensional hypercubic lattice. We show
that to order the singularity in the order parameter and in the
susceptibility occur at the same value of the occupation probability. This
result suggests that the unusual hybrid nature of the mean field -core
transition survives in high dimensions.Comment: 47 pages, 26 figures, revtex
Critical phenomena in complex networks
The combination of the compactness of networks, featuring small diameters,
and their complex architectures results in a variety of critical effects
dramatically different from those in cooperative systems on lattices. In the
last few years, researchers have made important steps toward understanding the
qualitatively new critical phenomena in complex networks. We review the
results, concepts, and methods of this rapidly developing field. Here we mostly
consider two closely related classes of these critical phenomena, namely
structural phase transitions in the network architectures and transitions in
cooperative models on networks as substrates. We also discuss systems where a
network and interacting agents on it influence each other. We overview a wide
range of critical phenomena in equilibrium and growing networks including the
birth of the giant connected component, percolation, k-core percolation,
phenomena near epidemic thresholds, condensation transitions, critical
phenomena in spin models placed on networks, synchronization, and
self-organized criticality effects in interacting systems on networks. We also
discuss strong finite size effects in these systems and highlight open problems
and perspectives.Comment: Review article, 79 pages, 43 figures, 1 table, 508 references,
extende
A topological approximation of the nonlinear Anderson model
We study the phenomena of Anderson localization in the presence of nonlinear
interaction on a lattice. A class of nonlinear Schrodinger models with
arbitrary power nonlinearity is analyzed. We conceive the various regimes of
behavior, depending on the topology of resonance-overlap in phase space,
ranging from a fully developed chaos involving Levy flights to pseudochaotic
dynamics at the onset of delocalization. It is demonstrated that quadratic
nonlinearity plays a dynamically very distinguished role in that it is the only
type of power nonlinearity permitting an abrupt localization-delocalization
transition with unlimited spreading already at the delocalization border. We
describe this localization-delocalization transition as a percolation
transition on a Cayley tree. It is found in vicinity of the criticality that
the spreading of the wave field is subdiffusive in the limit
t\rightarrow+\infty. The second moment grows with time as a powerlaw t^\alpha,
with \alpha = 1/3. Also we find for superquadratic nonlinearity that the analog
pseudochaotic regime at the edge of chaos is self-controlling in that it has
feedback on the topology of the structure on which the transport processes
concentrate. Then the system automatically (without tuning of parameters)
develops its percolation point. We classify this type of behavior in terms of
self-organized criticality dynamics in Hilbert space. For subquadratic
nonlinearities, the behavior is shown to be sensitive to details of definition
of the nonlinear term. A transport model is proposed based on modified
nonlinearity, using the idea of stripes propagating the wave process to large
distances. Theoretical investigations, presented here, are the basis for
consistency analysis of the different localization-delocalization patterns in
systems with many coupled degrees of freedom in association with the asymptotic
properties of the transport.Comment: 20 pages, 2 figures; improved text with revisions; accepted for
publication in Physical Review
An improved Belief Propagation algorithm finds many Bethe states in the random field Ising model on random graphs
We first present an empirical study of the Belief Propagation (BP) algorithm,
when run on the random field Ising model defined on random regular graphs in
the zero temperature limit. We introduce the notion of maximal solutions for
the BP equations and we use them to fix a fraction of spins in their ground
state configuration. At the phase transition point the fraction of
unconstrained spins percolates and their number diverges with the system size.
This in turn makes the associated optimization problem highly non trivial in
the critical region. Using the bounds on the BP messages provided by the
maximal solutions we design a new and very easy to implement BP scheme which is
able to output a large number of stable fixed points. On one side this new
algorithm is able to provide the minimum energy configuration with high
probability in a competitive time. On the other side we found that the number
of fixed points of the BP algorithm grows with the system size in the critical
region. This unexpected feature poses new relevant questions on the physics of
this class of models.Comment: 20 pages, 8 figure
Universal features of the order-parameter fluctuations : reversible and irreversible aggregation
We discuss the universal scaling laws of order parameter fluctuations in any
system in which the second-order critical behaviour can be identified. These
scaling laws can be derived rigorously for equilibrium systems when combined
with the finite-size scaling analysis. The relation between order parameter,
criticality and scaling law of fluctuations has been established and the
connexion between the scaling function and the critical exponents has been
found. We give examples in out-of-equilibrium aggregation models such as the
Smoluchowski kinetic equations, or of at-equilibrium Ising and percolation
models.Comment: 19 pages, 10 figure
The Directed Dominating Set Problem: Generalized Leaf Removal and Belief Propagation
A minimum dominating set for a digraph (directed graph) is a smallest set of
vertices such that each vertex either belongs to this set or has at least one
parent vertex in this set. We solve this hard combinatorial optimization
problem approximately by a local algorithm of generalized leaf removal and by a
message-passing algorithm of belief propagation. These algorithms can construct
near-optimal dominating sets or even exact minimum dominating sets for random
digraphs and also for real-world digraph instances. We further develop a core
percolation theory and a replica-symmetric spin glass theory for this problem.
Our algorithmic and theoretical results may facilitate applications of
dominating sets to various network problems involving directed interactions.Comment: 11 pages, 3 figures in EPS forma
The Branched Polymer Growth Model Revisited
The Branched Polymer Growth Model (BPGM) has been employed to study the
kinetic growth of ramified polymers in the presence of impurities. In this
article, the BPGM is revisited on the square lattice and a subtle modification
in its dynamics is proposed in order to adapt it to a scenario closer to
reality and experimentation. This new version of the model is denominated the
Adapted Branched Polymer Growth Model (ABPGM). It is shown that the ABPGM
preserves the functionalities of the monomers and so recovers the branching
probability b as an input parameter which effectively controls the relative
incidence of bifurcations. The critical locus separating infinite from finite
growth regimes of the ABPGM is obtained in the (b,c) space (where c is the
impurity concentration). Unlike the original model, the phase diagram of the
ABPGM exhibits a peculiar reentrance.Comment: 8 pages, 10 figures. To be published in PHYSICA
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