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

Finding long cycles in graphs

We analyze the problem of discovering long cycles inside a graph. We propose and test two algorithms for this task. The first one is based on recent advances in statistical mechanics and relies on a message passing procedure. The second follows a more standard Monte Carlo Markov Chain strategy. Special attention is devoted to Hamiltonian cycles of (non-regular) random graphs of minimal connectivity equal to three

On the path integral representation for quantum spin models and its application to the quantum cavity method and to Monte Carlo simulations

The cavity method is a well established technique for solving classical spin models on sparse random graphs (mean-field models with finite connectivity). Laumann et al. [arXiv:0706.4391] proposed recently an extension of this method to quantum spin-1/2 models in a transverse field, using a discretized Suzuki-Trotter imaginary time formalism. Here we show how to take analytically the continuous imaginary time limit. Our main technical contribution is an explicit procedure to generate the spin trajectories in a path integral representation of the imaginary time dynamics. As a side result we also show how this procedure can be used in simple heat-bath like Monte Carlo simulations of generic quantum spin models. The replica symmetric continuous time quantum cavity method is formulated for a wide class of models, and applied as a simple example on the Bethe lattice ferromagnet in a transverse field. The results of the methods are confronted with various approximation schemes in this particular case. On this system we performed quantum Monte Carlo simulations that confirm the exactness of the cavity method in the thermodynamic limit.Comment: 25 pages, 15 figures, typos correcte

Non-linear susceptibilities of spherical models

The static and dynamic susceptibilities for a general class of mean field random orthogonal spherical spin glass models are studied. We show how the static and dynamical properties of the linear and nonlinear susceptibilities depend on the behaviour of the density of states of the two body interaction matrix in the neighbourhood of the largest eigenvalue. Our results are compared with experimental results and also with those of the droplet theory of spin glasses.Comment: 20 pages, 2 fig

The number of matchings in random graphs

We study matchings on sparse random graphs by means of the cavity method. We first show how the method reproduces several known results about maximum and perfect matchings in regular and Erdos-Renyi random graphs. Our main new result is the computation of the entropy, i.e. the leading order of the logarithm of the number of solutions, of matchings with a given size. We derive both an algorithm to compute this entropy for an arbitrary graph with a girth that diverges in the large size limit, and an analytic result for the entropy in regular and Erdos-Renyi random graph ensembles.Comment: 17 pages, 6 figures, to be published in Journal of Statistical Mechanic

Fluctuations in the coarsening dynamics of the O(N) model: are they similar to those in glassy systems?

We study spatio-temporal fluctuations in the non-equilibrium dynamics of the d dimensional O(N) in the large N limit. We analyse the invariance of the dynamic equations for the global correlation and response in the slow ageing regime under transformations of time. We find that these equations are invariant under scale transformations. We extend this study to the action in the dynamic generating functional finding similar results. This model therefore falls into a different category from glassy problems in which full time-reparametrisation invariance, a larger symmetry that emcompasses time scale invariance, is expected to be realised asymptotically. Consequently, the spatio-temporal fluctuations of the large N O(N) model should follow a different pattern from that of glassy systems. We compute the fluctuations of local, as well as spatially separated, two-field composite operators and responses, and we confront our results with the ones found numerically for the 3d Edwards-Anderson model and kinetically constrained lattice gases. We analyse the dependence of the fluctuations of the composite operators on the growing domain length and we compare to what has been found in super-cooled liquids and glasses. Finally, we show that the development of time-reparametrisation invariance in glassy systems is intimately related to a well-defined and finite effective temperature, specified from the modification of the fluctuation-dissipation theorem out of equilibrium. We then conjecture that the global asymptotic time-reparametrisation invariance is broken down to time scale invariance in all coarsening systems.Comment: 57 pages, 5 figure

Spatially heterogeneous ages in glassy dynamics

We construct a framework for the study of fluctuations in the nonequilibrium relaxation of glassy systems with and without quenched disorder. We study two types of two-time local correlators with the aim of characterizing the heterogeneous evolution: in one case we average the local correlators over histories of the thermal noise, in the other case we simply coarse-grain the local correlators. We explain why the former describe the fingerprint of quenched disorder when it exists, while the latter are linked to noise-induced mesoscopic fluctuations. We predict constraints on the pdfs of the fluctuations of the coarse-grained quantities. We show that locally defined correlations and responses are connected by a generalized local out-of-equilibrium fluctuation-dissipation relation. We argue that large-size heterogeneities in the age of the system survive in the long-time limit. The invariance of the theory under reparametrizations of time underlies these results. We relate the pdfs of local coarse-grained quantities and the theory of dynamic random manifolds. We define a two-time dependent correlation length from the spatial decay of the fluctuations in the two-time local functions. We present numerical tests performed on disordered spin models in finite and infinite dimensions. Finally, we explain how these ideas can be applied to the analysis of the dynamics of other glassy systems that can be either spin models without disorder or atomic and molecular glassy systems.Comment: 47 pages, 60 Fig

to real-world and random graphs

PACS. 05.20.-y – Classical statistical mechanics