Parallel Tempering for the planted clique problem

The theoretical information threshold for the planted clique problem is $2\log_2(N)$, however no polynomial algorithm is known to recover a planted clique of size $O(N^{1/2-\epsilon})$, $\epsilon>0$. 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

Spin Glass in a Field: a New Zero-Temperature Fixed Point in Finite Dimensions

By using real space renormalisation group (RG) methods we show that spin-glasses in a field display a new kind of transition in high dimensions. The corresponding critical properties and the spin-glass phase are governed by two non-perturbative zero temperature fixed points of the RG flow. We compute the critical exponents, discuss the RG flow and its relevance for three dimensional systems. The new spin-glass phase we discovered has unusual properties, which are intermediate between the ones conjectured by droplet and full replica symmetry breaking theories. These results provide a new perspective on the long-standing debate about the behaviour of spin-glasses in a field.Comment: 5 pages, 3 figure

Real Space Renormalization Group Theory of Disordered Models of Glasses

We develop a real space renormalisation group analysis of disordered models of glasses, in particular of the spin models at the origin of the Random First Order Transition theory. We find three fixed points respectively associated to the liquid state, to the critical behavior and to the glass state. The latter two are zero-temperature ones; this provides a natural explanation of the growth of effective activation energy scale and the concomitant huge increase of relaxation time approaching the glass transition. The lower critical dimension depends on the nature of the interacting degrees of freedom and is higher than three for all models. This does not prevent three dimensional systems from being glassy. Indeed, we find that their renormalisation group flow is affected by the fixed points existing in higher dimension and in consequence is non-trivial. Within our theoretical framework the glass transition results to be an avoided phase transition.Comment: 6 pages, 3 figure

The Super-Potts glass: a new disordered model for glass-forming liquids

We introduce a new disordered system, the Super-Potts model, which is a more frustrated version of the Potts glass. Its elementary degrees of freedom are variables that can take M values and are coupled via pair-wise interactions. Its exact solution on a completely connected lattice demonstrates that for large enough M it belongs to the class of mean-field systems solved by a one step replica symmetry breaking Ansatz. Numerical simulations by the parallel tempering technique show that in three dimensions it displays a phenomenological behaviour similar to the one of glass-forming liquids. The Super-Potts glass is therefore the first long-sought disordered model allowing one to perform extensive and detailed studies of the Random First Order Transition in finite dimensions. We also discuss its behaviour for small values of M, which is similar to the one of spin-glasses in a field.Comment: 6 pages, 3 figure

Monte Carlo algorithms are very effective in finding the largest independent set in sparse random graphs

The effectiveness of stochastic algorithms based on Monte Carlo dynamics in solving hard optimization problems is mostly unknown. Beyond the basic statement that at a dynamical phase transition the ergodicity breaks and a Monte Carlo dynamics cannot sample correctly the probability distribution in times linear in the system size, there are almost no predictions nor intuitions on the behavior of this class of stochastic dynamics. The situation is particularly intricate because, when using a Monte Carlo based algorithm as an optimization algorithm, one is usually interested in the out of equilibrium behavior which is very hard to analyse. Here we focus on the use of Parallel Tempering in the search for the largest independent set in a sparse random graph, showing that it can find solutions well beyond the dynamical threshold. Comparison with state-of-the-art message passing algorithms reveals that parallel tempering is definitely the algorithm performing best, although a theory explaining its behavior is still lacking.Comment: 14 pages, 12 figure

Renormalization group and critical properties of Long Range models

We study a Renormalization Group transformation that can be used also for models with quenched disorder, like spin glasses, for which a commonly accepted and predictive Renormalization Group does not exist. We validate our method by applying it to a particular long-range model, the hierarchical one (both the diluted ferromagnetic version and the spin glass version), finding results in agreement with Monte Carlo simulations. In the second part we deeply analyze the connection between long-range and short-range models that still has some unclear aspects even for the ferromagnet. A systematic analysis is very important to understand if the use of long range models is justified to study properties of short range systems like spin-glasses

Renormalization group and critical properties of Long Range models

We study a Renormalization Group transformation that can be used also for models with quenched disorder, like spin glasses, for which a commonly accepted and predictive Renormalization Group does not exist. We validate our method by applying it to a particular long-range model, the hierarchical one (both the diluted ferromagnetic version and the spin glass version), finding results in agreement with Monte Carlo simulations. In the second part we deeply analyze the connection between long-range and short-range models that still has some unclear aspects even for the ferromagnet. A systematic analysis is very important to understand if the use of long range models is justified to study properties of short range systems like spin-glasses

One-loop topological expansion for spin glasses in the large connectivity limit

We apply for the first time a new one-loop topological expansion around the Bethe solution to the spin-glass model with field in the high connectivity limit, following the methodological scheme proposed in a recent work. The results are completely equivalent to the well known ones, found by standard field theoretical expansion around the fully connected model (Bray and Roberts 1980, and following works). However this method has the advantage that the starting point is the original Hamiltonian of the model, with no need to define an associated field theory, nor to know the initial values of the couplings, and the computations have a clear and simple physical meaning. Moreover this new method can also be applied in the case of zero temperature, when the Bethe model has a transition in field, contrary to the fully connected model that is always in the spin glass phase. Sharing with finite dimensional model the finite connectivity properties, the Bethe lattice is clearly a better starting point for an expansion with respect to the fully connected model. The present work is a first step towards the generalization of this new expansion to more difficult and interesting cases as the zero-temperature limit, where the expansion could lead to different results with respect to the standard one.Comment: 8 pages, 1 figur

Ensemble renormalization group for disordered systems

We propose and study a renormalization group transformation that can be used also for models with strong quenched disorder, like spin glasses. The method is based on a mapping between disorder distributions, chosen such as to keep some physical properties (e.g., the ratio of correlations averaged over the ensemble) invariant under the transformation. We validate this ensemble renormalization group by applying it to the hierarchical model (both the diluted ferromagnetic version and the spin glass version), finding results in agreement with Monte Carlo simulations.Comment: 7 pages, 10 figure

Spectral Detection on Sparse Hypergraphs

We consider the problem of the assignment of nodes into communities from a set of hyperedges, where every hyperedge is a noisy observation of the community assignment of the adjacent nodes. We focus in particular on the sparse regime where the number of edges is of the same order as the number of vertices. We propose a spectral method based on a generalization of the non-backtracking Hashimoto matrix into hypergraphs. We analyze its performance on a planted generative model and compare it with other spectral methods and with Bayesian belief propagation (which was conjectured to be asymptotically optimal for this model). We conclude that the proposed spectral method detects communities whenever belief propagation does, while having the important advantages to be simpler, entirely nonparametric, and to be able to learn the rule according to which the hyperedges were generated without prior information.Comment: 8 pages, 5 figure