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Motivation: Similarity-measure based clustering is a crucial problem appearing throughout scientific data analysis. Recently, a powerful new algorithm called Affinity Propagation (AP) based on message-passing techniques was proposed by Frey and Dueck \cite{Frey07}. In AP, each cluster is identified by a common exemplar all other data points of the same cluster refer to, and exemplars have to refer to themselves. Albeit its proved power, AP in its present form suffers from a number of drawbacks. The hard constraint of having exactly one exemplar per cluster restricts AP to classes of regularly shaped clusters, and leads to suboptimal performance, {\it e.g.}, in analyzing gene expression data. Results: 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 {\it 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., 11 pages, supplementary material: http://isiosf.isi.it/~weigt/scap_supplement.pdf

In this paper we extend replica bounds and free energy subadditivity arguments to diluted spin-glass 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., Latex file, 15 pages, 2 eps figures; Weak point revised and corrected; Misprints corrected

We study the problem of recovering a known cluster structure in a sparse network, also known as the planted partitioning problem, by means of statistical mechanics. We find a sharp transition from un-recoverable to recoverable structure as a function of the separation of the clusters. For multivariate data, such transitions have been observed frequently, but always as a function of the number of data points provided, i.e. given a large enough data set, two point clouds can always be recognized as different clusters, as long as their separation is non-zero. In contrast, for the sparse networks studied here, a cluster structure remains undetectable even in an infinitely large network if a critical separation is not exceeded. We give analytic formulas for this critical separation as a function of the degree distribution of the network and calculate the shape of the recoverability-transition. Our findings have implications for unsupervised learning and data-mining in relational data bases and provide bounds on the achievable performance of graph clustering algorithms., 4 Pages, 2 Figures

Combinatorial auctions are formulated as frustrated lattice gases on sparse random graphs, allowing the determination of the optimal revenue by methods of statistical physics. Transitions between computationally easy and hard regimes are found and interpreted in terms of the geometric structure of the space of solutions. We introduce an iterative algorithm to solve intermediate and large instances, and discuss competing states of optimal revenue and maximal number of satisfied bidders. The algorithm can be generalized to the hard phase and to more sophisticated auction protocols., 4 pages, 4 figures, minor changes, references added. To appear on PRL

Since their introduction, Boolean networks have been traditionally studied in view of their rich dynamical behavior under different update protocols and for their qualitative analogy with cell regulatory networks. More recently, tools borrowed from statistical physics of disordered systems and from computer science have provided a more complete characterization of their equilibrium behavior. However, the largest part of the results have been obtained in the thermodynamic limit, which is often far from being reached when dealing with realistic instances of the problem. The numerical analysis presented here aims at comparing - for a specific family of models - the outcomes given by the heuristic belief propagation algorithm with those given by exhaustive enumeration. In the second part of the paper some analytical considerations on the validity of the annealed approximation are discussed., Minor corrections

The determination and classification of fixed points of large Boolean networks is addressed in terms of constraint satisfaction problem. We develop a general simplification scheme that, removing all those variables and functions belonging to trivial logical cascades, returns the computational core of the network. The onset of an easy-to-complex regulatory phase is introduced as a function of the parameters of the model, identifying both theoretically and algorithmically the relevant regulatory variables., major revisions, extended results, version accepted for publication in PRL

Motivation: In the last few years a growing interest in biology has been shifting towards the problem of optimal information extraction from the huge amount of data generated via large scale and high-throughput techniques. One of the most relevant issues has recently become that of correctly and reliably predicting the functions of observed but still functionally undetermined proteins starting from information coming from the network of co-observed proteins of known functions. Method: The method proposed in this article is based on a message passing algorithm known as Belief Propagation, which takes as input the network of proteins physical interactions and a catalog of known proteins functions, and returns the probabilities for each unclassified protein of having one chosen function. The implementation of the algorithm allows for fast on-line analysis, and can be easily generalized to more complex graph topologies taking into account hyper-graphs, {\em i.e.} complexes of more than two interacting proteins., Comment: 12 pages, 9 eps figures, 1 additional html table

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., Comment: 40 pages, 29 eps figures

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 < \infty. 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., Comment: 9 pages, 1 figure

We study the computational complexity of a very basic problem, namely that of finding solutions to a very large set of random linear equations in a finite Galois Field modulo q. Using tools from statistical mechanics we are able to identify phase transitions in the structure of the solution space and to connect them to changes in performance of a global algorithm, namely Gaussian elimination. Crossing phase boundaries produces a dramatic increase in memory and CPU requirements necessary to the algorithms. In turn, this causes the saturation of the upper bounds for the running time. We illustrate the results on the specific problem of integer factorization, which is of central interest for deciphering messages encrypted with the RSA cryptosystem., Comment: 23 pages, 8 figures

We study an exactly solvable version of the famous random Boolean satisfiability problem, the so called random XOR-SAT problem. Rare events are shown to affect the combinatorial ``phase diagram'' leading to a coexistence of solvable and unsolvable instances of the combinatorial problem in a certain region of the parameters characterizing the model. Such instances differ by a non-extensive quantity in the ground state energy of the associated diluted spin-glass model. We also show that the critical exponent $\nu$, controlling the size of the critical window where the probability of having solutions vanishes, depends on the model parameters, shedding light on the link between random hyper-graph topology and universality classes. In the case of random satisfiability, a similar behavior was conjectured to be connected to the onset of computational intractability., Comment: 10 pages, 5 figures, to appear in J. Phys. A. v2: link to the XOR-SAT probelm added

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, \gamma_d=0.818 and \gamma_s=0.918 resp.), whereas for hard optimization problems like 3-SAT or Bicoloring it provides new upper bounds for their critical thresholds (\gamma_c^{var}=4.396 and \gamma_c^{var}=2.149 resp.)., Comment: 4 pages, 1 figure, accepted for publication in PRL