Glass models on Bethe lattices

We consider ``lattice glass models'' in which each site can be occupied by at most one particle, and any particle may have at most l occupied nearest neighbors. Using the cavity method for locally tree-like lattices, we derive the phase diagram, with a particular focus on the vitreous phase and the highest packing limit. We also study the energy landscape via the configurational entropy, and discuss different equilibrium glassy phases. Finally, we show that a kinetic freezing, depending on the particular dynamical rules chosen for the model, can prevent the equilibrium glass transitions.Comment: 24 pages, 11 figures; minor corrections + enlarged introduction and conclusio

A methodology for full-system power modeling in heterogeneous data centers

The need for energy-awareness in current data centers has encouraged the use of power modeling to estimate their power consumption. However, existing models present noticeable limitations, which make them application-dependent, platform-dependent, inaccurate, or computationally complex. In this paper, we propose a platform-and application-agnostic methodology for full-system power modeling in heterogeneous data centers that overcomes those limitations. It derives a single model per platform, which works with high accuracy for heterogeneous applications with different patterns of resource usage and energy consumption, by systematically selecting a minimum set of resource usage indicators and extracting complex relations among them that capture the impact on energy consumption of all the resources in the system. We demonstrate our methodology by generating power models for heterogeneous platforms with very different power consumption profiles. Our validation experiments with real Cloud applications show that such models provide high accuracy (around 5% of average estimation error).This work is supported by the Spanish Ministry of Economy and Competitiveness under contract TIN2015-65316-P, by the Gener- alitat de Catalunya under contract 2014-SGR-1051, and by the European Commission under FP7-SMARTCITIES-2013 contract 608679 (RenewIT) and FP7-ICT-2013-10 contracts 610874 (AS- CETiC) and 610456 (EuroServer).Peer ReviewedPostprint (author's final draft

Amorphous-amorphous transition and the two-step replica symmetry breaking phase

The nature of polyamorphism and amorphous-to-amorphous transition is investigated by means of an exactly solvable model with quenched disorder, the spherical s+p multi-spin interaction model. The analysis is carried out in the framework of Replica Symmetry Breaking theory and leads to the identification of low temperature glass phases of different kinds. Besides the usual `one-step' solution, known to reproduce all basic properties of structural glasses, also a physically consistent `two-step' solution arises. More complicated phases are found as well, as temperature is further decreased, expressing a complex variety of metastable states structures for amorphous systems.Comment: 8 pages, 7 figures, longer version, new references adde

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

Statistical mechanics of error exponents for error-correcting codes

Error exponents characterize the exponential decay, when increasing message length, of the probability of error of many error-correcting codes. To tackle the long standing problem of computing them exactly, we introduce a general, thermodynamic, formalism that we illustrate with maximum-likelihood decoding of low-density parity-check (LDPC) codes on the binary erasure channel (BEC) and the binary symmetric channel (BSC). In this formalism, we apply the cavity method for large deviations to derive expressions for both the average and typical error exponents, which differ by the procedure used to select the codes from specified ensembles. When decreasing the noise intensity, we find that two phase transitions take place, at two different levels: a glass to ferromagnetic transition in the space of codewords, and a paramagnetic to glass transition in the space of codes.Comment: 32 pages, 13 figure

The cavity method for large deviations

A method is introduced for studying large deviations in the context of statistical physics of disordered systems. The approach, based on an extension of the cavity method to atypical realizations of the quenched disorder, allows us to compute exponentially small probabilities (rate functions) over different classes of random graphs. It is illustrated with two combinatorial optimization problems, the vertex-cover and coloring problems, for which the presence of replica symmetry breaking phases is taken into account. Applications include the analysis of models on adaptive graph structures.Comment: 18 pages, 7 figure

Entropy landscape and non-Gibbs solutions in constraint satisfaction problems

We study the entropy landscape of solutions for the bicoloring problem in random graphs, a representative difficult constraint satisfaction problem. Our goal is to classify which type of clusters of solutions are addressed by different algorithms. In the first part of the study we use the cavity method to obtain the number of clusters with a given internal entropy and determine the phase diagram of the problem, e.g. dynamical, rigidity and SAT-UNSAT transitions. In the second part of the paper we analyze different algorithms and locate their behavior in the entropy landscape of the problem. For instance we show that a smoothed version of a decimation strategy based on Belief Propagation is able to find solutions belonging to sub-dominant clusters even beyond the so called rigidity transition where the thermodynamically relevant clusters become frozen. These non-equilibrium solutions belong to the most probable unfrozen clusters.Comment: 38 pages, 10 figure

Random multi-index matching problems

The multi-index matching problem (MIMP) generalizes the well known matching problem by going from pairs to d-uplets. We use the cavity method from statistical physics to analyze its properties when the costs of the d-uplets are random. At low temperatures we find for d>2 a frozen glassy phase with vanishing entropy. We also investigate some properties of small samples by enumerating the lowest cost matchings to compare with our theoretical predictions.Comment: 22 pages, 16 figure

An algorithm for counting circuits: application to real-world and random graphs

We introduce an algorithm which estimates the number of circuits in a graph as a function of their length. This approach provides analytical results for the typical entropy of circuits in sparse random graphs. When applied to real-world networks, it allows to estimate exponentially large numbers of circuits in polynomial time. We illustrate the method by studying a graph of the Internet structure.Comment: 7 pages, 3 figures, minor corrections, accepted versio