175 research outputs found
Long-range frustration in T=0 first-step replica-symmetry-broken solutions of finite-connectivity spin glasses
In a finite-connectivity spin-glass at the zero-temperature limit, long-range
correlations exist among the unfrozen vertices (whose spin values being
non-fixed). Such long-range frustrations are partially removed through the
first-step replica-symmetry-broken (1RSB) cavity theory, but residual
long-range frustrations may still persist in this mean-field solution. By way
of population dynamics, here we perform a perturbation-percolation analysis to
calculate the magnitude of long-range frustrations in the 1RSB solution of a
given spin-glass system. We study two well-studied model systems, the minimal
vertex-cover problem and the maximal 2-satisfiability problem. This work points
to a possible way of improving the zero-temperature 1RSB mean-field theory of
spin-glasses.Comment: 5 pages, two figures. To be published in JSTA
Long Range Frustrations in a Spin Glass Model of the Vertex Cover Problem
In a spin glass system on a random graph, some vertices have their spins
changing among different configurations of a ground--state domain. Long range
frustrations may exist among these unfrozen vertices in the sense that certain
combinations of spin values for these vertices may never appear in any
configuration of this domain. We present a mean field theory to tackle such
long range frustrations and apply it to the NP-hard minimum vertex cover
(hard-core gas condensation) problem. Our analytical results on the
ground-state energy density and on the fraction of frozen vertices are in good
agreement with known numerical and mathematical results.Comment: An erratum is added to the main text. 5 pages, 5 figure
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
Optimal Location of Sources in Transportation Networks
We consider the problem of optimizing the locations of source nodes in
transportation networks. A reduction of the fraction of surplus nodes induces a
glassy transition. In contrast to most constraint satisfaction problems
involving discrete variables, our problem involves continuous variables which
lead to cavity fields in the form of functions. The one-step replica symmetry
breaking (1RSB) solution involves solving a stable distribution of functionals,
which is in general infeasible. In this paper, we obtain small closed sets of
functional cavity fields and demonstrate how functional recursions are
converted to simple recursions of probabilities, which make the 1RSB solution
feasible. The physical results in the replica symmetric (RS) and the 1RSB
frameworks are thus derived and the stability of the RS and 1RSB solutions are
examined.Comment: 38 pages, 18 figure
Message passing for vertex covers
Constructing a minimal vertex cover of a graph can be seen as a prototype for
a combinatorial optimization problem under hard constraints. In this paper, we
develop and analyze message passing techniques, namely warning and survey
propagation, which serve as efficient heuristic algorithms for solving these
computational hard problems. We show also, how previously obtained results on
the typical-case behavior of vertex covers of random graphs can be recovered
starting from the message passing equations, and how they can be extended.Comment: 25 pages, 9 figures - version accepted for publication in PR
Networking - A Statistical Physics Perspective
Efficient networking has a substantial economic and societal impact in a
broad range of areas including transportation systems, wired and wireless
communications and a range of Internet applications. As transportation and
communication networks become increasingly more complex, the ever increasing
demand for congestion control, higher traffic capacity, quality of service,
robustness and reduced energy consumption require new tools and methods to meet
these conflicting requirements. The new methodology should serve for gaining
better understanding of the properties of networking systems at the macroscopic
level, as well as for the development of new principled optimization and
management algorithms at the microscopic level. Methods of statistical physics
seem best placed to provide new approaches as they have been developed
specifically to deal with non-linear large scale systems. This paper aims at
presenting an overview of tools and methods that have been developed within the
statistical physics community and that can be readily applied to address the
emerging problems in networking. These include diffusion processes, methods
from disordered systems and polymer physics, probabilistic inference, which
have direct relevance to network routing, file and frequency distribution, the
exploration of network structures and vulnerability, and various other
practical networking applications.Comment: (Review article) 71 pages, 14 figure
Region graph partition function expansion and approximate free energy landscapes: Theory and some numerical results
Graphical models for finite-dimensional spin glasses and real-world
combinatorial optimization and satisfaction problems usually have an abundant
number of short loops. The cluster variation method and its extension, the
region graph method, are theoretical approaches for treating the complicated
short-loop-induced local correlations. For graphical models represented by
non-redundant or redundant region graphs, approximate free energy landscapes
are constructed in this paper through the mathematical framework of region
graph partition function expansion. Several free energy functionals are
obtained, each of which use a set of probability distribution functions or
functionals as order parameters. These probability distribution
function/functionals are required to satisfy the region graph
belief-propagation equation or the region graph survey-propagation equation to
ensure vanishing correction contributions of region subgraphs with dangling
edges. As a simple application of the general theory, we perform region graph
belief-propagation simulations on the square-lattice ferromagnetic Ising model
and the Edwards-Anderson model. Considerable improvements over the conventional
Bethe-Peierls approximation are achieved. Collective domains of different sizes
in the disordered and frustrated square lattice are identified by the
message-passing procedure. Such collective domains and the frustrations among
them are responsible for the low-temperature glass-like dynamical behaviors of
the system.Comment: 30 pages, 11 figures. More discussion on redundant region graphs. To
be published by Journal of Statistical Physic
- …