243 research outputs found
Phase transitions for the cavity approach to the clique problem on random graphs
We give a rigorous proof of two phase transitions for a disordered system
designed to find large cliques inside Erdos random graphs. Such a system is
associated with a conservative probabilistic cellular automaton inspired by the
cavity method originally introduced in spin glass theory.Comment: 36 pages, 4 figure
Phase transitions in Pareto optimal complex networks
The organization of interactions in complex systems can be described by
networks connecting different units. These graphs are useful representations of
the local and global complexity of the underlying systems. The origin of their
topological structure can be diverse, resulting from different mechanisms
including multiplicative processes and optimization. In spatial networks or in
graphs where cost constraints are at work, as it occurs in a plethora of
situations from power grids to the wiring of neurons in the brain, optimization
plays an important part in shaping their organization. In this paper we study
network designs resulting from a Pareto optimization process, where different
simultaneous constraints are the targets of selection. We analyze three
variations on a problem finding phase transitions of different kinds. Distinct
phases are associated to different arrangements of the connections; but the
need of drastic topological changes does not determine the presence, nor the
nature of the phase transitions encountered. Instead, the functions under
optimization do play a determinant role. This reinforces the view that phase
transitions do not arise from intrinsic properties of a system alone, but from
the interplay of that system with its external constraints.Comment: 14 pages, 7 figure
Parallel Tempering for the planted clique problem
The theoretical information threshold for the planted clique problem is
, however no polynomial algorithm is known to recover a planted
clique of size , . In this paper we will apply
a standard method for the analysis of disordered models, the Parallel-Tempering
(PT) algorithm, to the clique problem, showing numerically that its
time-scaling in the hard region is indeed polynomial for the analyzed sizes. We
also apply PT to a different but connected model, the Sparse Planted
Independent Set problem. In this situation thresholds should be sharper and
finite size corrections should be less important. Also in this case PT shows a
polynomial scaling in the hard region for the recovery.Comment: 12 pages, 5 figure
A hard-sphere model on generalized Bethe lattices: Statics
We analyze the phase diagram of a model of hard spheres of chemical radius
one, which is defined over a generalized Bethe lattice containing short loops.
We find a liquid, two different crystalline, a glassy and an unusual
crystalline glassy phase. Special attention is also paid to the close-packing
limit in the glassy phase. All analytical results are cross-checked by
numerical Monte-Carlo simulations.Comment: 24 pages, revised versio
Glassy behavior induced by geometrical frustration in a hard-core lattice gas model
We introduce a hard-core lattice-gas model on generalized Bethe lattices and
investigate analytically and numerically its compaction behavior. If
compactified slowly, the system undergoes a first-order crystallization
transition. If compactified much faster, the system stays in a meta-stable
liquid state and undergoes a glass transition under further compaction. We show
that this behavior is induced by geometrical frustration which appears due to
the existence of short loops in the generalized Bethe lattices. We also compare
our results to numerical simulations of a three-dimensional analog of the
model.Comment: 7 pages, 4 figures, revised versio
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,
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