Retrieving information from a noisy "knowledge network"

We address the problem of retrieving information from a noisy version of the ``knowledge networks'' introduced by Maslov and Zhang. We map this problem onto a disordered statistical mechanics model, which opens the door to many analytical and numerical approaches. We give the replica symmetric solution, compare with numerical simulations, and finally discuss an application to real datas from the United States Senate.Comment: 10 pages, 4 figures. Writing of the last section improved; version accepted in JSTA

Hierarchical models of rigidity percolation

We introduce models of generic rigidity percolation in two dimensions on hierarchical networks, and solve them exactly by means of a renormalization transformation. We then study how the possibility for the network to self organize in order to avoid stressed bonds may change the phase diagram. In contrast to what happens on random graphs and in some recent numerical studies at zero temperature, we do not find a true intermediate phase separating the usual rigid and floppy ones.Comment: 20 pages, 8 figures. Figures improved, references added, small modifications. Accepted in Phys. Rev.

Microcanonical Analysis of Exactness of the Mean-Field Theory in Long-Range Interacting Systems

Classical spin systems with nonadditive long-range interactions are studied in the microcanonical ensemble. It is expected that the entropy of such a system is identical to that of the corresponding mean-field model, which is called "exactness of the mean-field theory". It is found out that this expectation is not necessarily true if the microcanonical ensemble is not equivalent to the canonical ensemble in the mean-field model. Moreover, necessary and sufficient conditions for exactness of the mean-field theory are obtained. These conditions are investigated for two concrete models, the \alpha-Potts model with annealed vacancies and the \alpha-Potts model with invisible states.Comment: 23 pages, to appear in J. Stat. Phy

Oscillating elastic defects: competition and frustration

We consider a dynamical generalization of the Eshelby problem: the strain profile due to an inclusion or "defect" in an isotropic elastic medium. We show that the higher the oscillation frequency of the defect, the more localized is the strain field around the defect. We then demonstrate that the qualitative nature of the interaction between two defects is strongly dependent on separation, frequency and direction, changing from "ferromagnetic" to "antiferromagnetic" like behavior. We generalize to a finite density of defects and show that the interactions in assemblies of defects can be mapped to XY spin-like models, and describe implications for frustration and frequency-driven pattern transitions.Comment: 4 pages, 5 figure

Ensemble Inequivalence in Mean-field Models of Magnetism

Mean-field models, while they can be cast into an {\it extensive} thermodynamic formalism, are inherently {\it non additive}. This is the basic feature which leads to {\it ensemble inequivalence} in these models. In this paper we study the global phase diagram of the infinite range Blume-Emery-Griffiths model both in the {\it canonical} and in the {\it microcanonical} ensembles. The microcanonical solution is obtained both by direct state counting and by the application of large deviation theory. The canonical phase diagram has first order and continuous transition lines separated by a tricritical point. We find that below the tricritical point, when the canonical transition is first order, the phase diagrams of the two ensembles disagree. In this region the microcanonical ensemble exhibits energy ranges with negative specific heat and temperature jumps at transition energies. These two features are discussed in a general context and the appropriate Maxwell constructions are introduced. Some preliminary extensions of these results to weakly decaying nonintegrable interactions are presented.Comment: Chapter of the forthcoming "Lecture Notes in Physics" volume: ``Dynamics and Thermodynamics of Systems with Long Range Interactions'', T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens Eds., Lecture Notes in Physics Vol. 602, Springer (2002). (see http://link.springer.de/series/lnpp/

Exactly solvable models of adaptive networks

A satisfiability (SAT-UNSAT) transition takes place for many optimization problems when the number of constraints, graphically represented by links between variables nodes, is brought above some threshold. If the network of constraints is allowed to adapt by redistributing its links, the SAT-UNSAT transition may be delayed and preceded by an intermediate phase where the structure self-organizes to satisfy the constraints. We present an analytic approach, based on the recently introduced cavity method for large deviations, which exactly describes the two phase transitions delimiting this adaptive intermediate phase. We give explicit results for random bond models subject to the connectivity or rigidity percolation transitions, and compare them with numerical simulations.Comment: 4 pages, 4 figure

Lyapunov exponents as a dynamical indicator of a phase transition

We study analytically the behavior of the largest Lyapunov exponent $\lambda_1$ for a one-dimensional chain of coupled nonlinear oscillators, by combining the transfer integral method and a Riemannian geometry approach. We apply the results to a simple model, proposed for the DNA denaturation, which emphasizes a first order-like or second order phase transition depending on the ratio of two length scales: this is an excellent model to characterize $\lambda_1$ as a dynamical indicator close to a phase transition.Comment: 8 Pages, 3 Figure