We present a detailed study of the phase diagram of the Ising model in random graphs with arbitrary degree distribution. By using the replica method we compute exactly the value of the critical temperature and the associated critical exponents as a function of the minimum and maximum degree, and the degree distribution characterizing the graph. As expected, there is a ferromagnetic transition provided k ≤ 2 k 2 < ∞. However, if the fourth moment of the degree distribution is not finite then non-trivial scaling exponents are obtained. These results are analyzed for the particular case of power-law distributed random graphs. I. INTRODUCTIONThe increasing evidence that many physical, biological and social networks exhibit a high degree of wiring entanglement has led to the investigation of graph models with complex topological properties [1]. In particular, the possibility that some special nodes of the cluster (hubs) posses a larger probability to develop connections pointing to other nodes has been recently identified in scale-free networks [2,3]. These networks exhibit a power law degree distribution p k ∼ k −γ , where the exponent γ is usually larger than 2. This kind of degree distribution implies that each node has a statistically significant probability of having a large number of connections compared to the average degree of the network. Examples of such properties can be found in communication and social webs, along with many biological networks, and have led to the developing of several dynamical models aimed to the description and characterization of scale-free networks [2][3][4].Power law degree distributions are the signature of degree fluctuations that may alter the phase diagram of physical processes as in the case of random percolation [6,7] and spreading processes [8] that do not exhibit a phase transition if the degree exponent is γ ≤ 3. In this perspective, it is interesting to study the ordering dynamics of the Ising model in scale-free networks. The Ising model is, indeed, the prototypical model for the study of phase transitions and complex phenomena and it is often the starting point for the developing of models aimed at the characterization of ordering phenomena. For this reason, the Ising model and its variations are used to mimic a wide range of phenomena not pertaining to physics, such as the forming and spreading of opinions in societies and companies or the evolution and competition of species. Since social and biological networks are often characterized by scale-free properties, the study of the ferromagnetic phase transition in graphs with arbitrary degree distribution can find useful application in the study of several complex interacting systems and it has been recently pursued in Ref. [9]. The numerical simulations reported in Ref [9] show that in the case of a degree distribution with γ = 3 the Ising model has a critical temperature T c , characterizing the transition to an ordered phase, which scales logarithmically with the network size. Therefore, there is no ferromagnetic tran...

The electronic version of this article is the complete one and can be found online at http://genomebiology.com/2008/9/S1/S2Genome Biology 2008, 9:S2 http://genomebiology.com/2008/9/S1/S2 Genome Biology 2008, Volume 9, Suppl 1, Article S2 Peña-Castillo et al. S2.2 AbstractBackground: Several years after sequencing the human genome and the mouse genome, much remains to be discovered about the functions of most human and mouse genes. Computational prediction of gene function promises to help focus limited experimental resources on the most likely hypotheses. Several algorithms using diverse genomic data have been applied to this task in model organisms; however, the performance of such approaches in mammals has not yet been evaluated.

A major problem in evaluating stochastic local search algorithms for NP-complete problems is the need for a systematic generation of hard test instances having previously known properties of the optimal solutions. On the basis of statistical mechanics results, we propose random generators of hard and satisfiable instances for the 3-satisfiability problem. The design of the hardest problem instances is based on the existence of a first order ferromagnetic phase transition and the glassy nature of excited states. The analytical predictions are corroborated by numerical results obtained from complete as well as stochastic local algorithms.

Can a cluster structure in a sparse relational data set, i.e., a network, be detected at all by unsupervised clustering techniques? We answer this question by means of statistical mechanics making our analysis independent of any particular algorithm used for clustering. We find a sharp transition from a phase in which the cluster structure is not detectable at all to a phase in which it can be detected with high accuracy. We calculate the transition point and the shape of the transition, i.e., the theoretically achievable accuracy, analytically. This illuminates theoretical limitations of data mining in networks and allows for an understanding and evaluation of the performance of a variety of algorithms.

In this paper we extend replica bounds and free energy subadditivity arguments to diluted spinglass models on graphs with arbitrary, non-Poissonian degree distribution. The new difficulties specific of this case are overcome introducing an interpolation procedure that stresses the relation between interpolation methods and the cavity method. As a byproduct we obtain self-averaging identities that generalize the Ghirlanda-Guerra ones to the multi-overlap case.

We study the low temperature properties of p-spin glass models with finite connectivity and of some optimization problems. Using a one-step functional replica symmetry breaking Ansatz we can solve exactly the saddle-point equations for graphs with uniform connectivity. The resulting ground state energy is in perfect agreement with numerical simulations. For fluctuating connectivity graphs, the same Ansatz can be used in a variational way: For p-spin models (known as p-XOR-SAT in computer science) it provides the exact configurational entropy together with the dynamical and static critical connectivities (for p = 3, γ d = 0.818 and γs = 0.918 resp.), whereas for hard optimization problems like 3-SAT or Bicoloring it provides new upper bounds for their critical thresholds (γ The nature of the glassy phase and of the out-ofequilibrium dynamics of physical systems are two intertwined aspects of the behavior of many complex systems found in different fields, ranging from physics or biology to computer science and game theory. The existence and the characterization of long time states, and the question of relaxation times are important open issues of modern statistical mechanics and probability theory.Fully connected spin glasses have served as prototype models able to provide a highly non trivial static and off-equilibrium phenomenology already at the mean-field level [1,2]. However, some important features of complex real-world systems heavily rely on the connectivity pattern which is particularly simple in these systems.For instance, super-cooled liquids and structural glasses are characterized by a finite number of short range interactions for each particle (bounded by the so called kissing number of the particles), which leads to a complex structure of energy and entropy barriers in phase space. Heterogeneities in both the static and the off-equilibrium regimes witness such underlying constraints [3].In computer science, non-trivial ensembles of hard combinatorial optimization problems, the so called NPcomplete problems [4], typically map onto spin glasses with finite average connectivity (or degree) at zero temperature [5]. In the last years methods from statistical physics have been very useful in order to study phase transitions in such problems [6]. The hardest among these share characteristics that are largely independent on the specific algorithms adopted for their solution. Important features such as solution time or memory requirements are conjectured to be strictly related to the geometrical structure of their low temperature phase space [7]. For a survey on the state of the art we address the reader to two recent special issues [8].The main technical obstacle for the development of a statistical physics theory over finite degree graphs has been the extension of the Parisi replica symmetry breaking (RSB) scheme to the functional level [9,10]. The replica symmetric (RS) phases are described by a single probability distribution function of the effective fields, capturing the site-to-site fluctuations of th...

This limitation can be overcome by relaxing the AP hard constraints. A new parameter controls the importance of the constraints compared to the aim of maximizing the overall similarity, and allows to interpolate between the simple case where each data point selects its closest neighbor as an exemplar and the original AP. The resulting soft-constraint affinity propagation (SCAP) becomes more informative, accurate and leads to more stable clustering. Even though a new a priori free parameter is introduced, the overall dependence of the algorithm on external tuning is reduced, as robustness is increased and an optimal strategy for parameter selection emerges more naturally. SCAP is tested on biological benchmark data, including in particular microarray data related to various cancer types. We show that the algorithm efficiently unveils the hierarchical cluster structure present in the data sets. Further on, it allows to extract sparse gene expression signatures for each cluster.

The state-of-the-art error correcting codes are based on large random constructions (random graphs, random permutations, . . . ) and are decoded by linear-time iterative algorithms. Because of these features, they are remarkable examples of diluted mean-field spin glasses, both from the static and from the dynamic points of view. We analyze the behavior of decoding algorithms using the mapping onto statistical-physics models. This allows to understand the intrinsic (i.e. algorithm independent) features of this behavior.