4 research outputs found
On large deviation properties of Erdos-Renyi random graphs
We show that large deviation properties of Erd\"os-R\'enyi random graphs can
be derived from the free energy of the -state Potts model of statistical
mechanics. More precisely the Legendre transform of the Potts free energy with
respect to is related to the component generating function of the graph
ensemble. This generalizes the well-known mapping between typical properties of
random graphs and the limit of the Potts free energy. For
exponentially rare graphs we explicitly calculate the number of components, the
size of the giant component, the degree distributions inside and outside the
giant component, and the distribution of small component sizes. We also perform
numerical simulations which are in very good agreement with our analytical
work. Finally we demonstrate how the same results can be derived by studying
the evolution of random graphs under the insertion of new vertices and edges,
without recourse to the thermodynamics of the Potts model.Comment: 38 pages, 9 figures, Latex2e, corrected and extended version
including numerical simulation result
Performance and service life in the Environmental Profiles Methodology and Green Guide to Specification
Heuristic methods for solution of problems in the NP-Complete class of
decision problems often reach exact solutions, but fail badly at "phase
boundaries", across which the decision to be reached changes from almost always
having one value to almost having a different value. We report an analytic
solution and experimental investigations of the phase transition that occurs in
the limit of very large problems in K-SAT. The nature of its "random
first-order" phase transition, seen at values of K large enough to make the
computational cost of solving typical instances increase exponenitally with
problem size, suggest a mechanism for the cost increase. There has been
evidence for features like the "backbone" of frozen inputs which characterizes
the UNSAT phase in K-SAT in the study of models of disordered materials, but
this feature and this transition are uniquely accessible to analysis in K-SAT.
The random first order transition combines properties of the 1st order
(discontinuous onset of order) and 2nd order (with power law scaling, e.g. of
the width of the the critical region in a finite system) transitions known in
the physics of pure solids. Such transitions should occur in other combinatoric
problems in the large N limit. Finally, improved search heuristics may be
developed when a "backbone" is known to exist.Comment: 25 pages, to appear in Random Structures and Algorithm
Thermodynamic Formalism Of Neural Computing
Neural networks are systems of interconnected processors mimicking some of the brain functions. After a rapid overview of neural computing, the thermodynamic formalism of the learning procedure is introduced. Besides its use in introducing efficient stochastic learning algorithms, it gives an insight in terms of information theory. Main emphasis is given in the information restitution process; stochastic evolution is used as the starting point for introducing statistical mechanics of associative memory. Instead of formulating problems in their most general setting, it is preferred stating precise results on specific models. In this report are mainly presented those features that are relevant when the neural net becomes very large. A survey of the most recent results is given and the main open problems are pointed out