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 types 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 satisfiability-unsatisfiability (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 subdominant clusters even beyond the so-called rigidity transition where the thermodynamically relevant clusters become frozen. These nonequilibrium solutions belong to the most probable unfrozen clusters.

The minimum weight Steiner tree (MST) is an important combinatorial optimization problem over networks that has applications in a wide range of fields. Here we discuss a general technique to translate the imposed global connectivity constrain into many local ones that can be analyzed with cavity equation techniques. This approach leads to a new optimization algorithm for MST and allows us to analyze the statistical mechanics properties of MST on random graphs of various types.

The graph theoretic concept of maximal independent set arises in several practical problems in computer science as well as in game theory. A maximal independent set is defined by the set of occupied nodes that satisfy some packing and covering constraints. It is known that finding minimum-and maximum-density maximal independent sets are hard optimization problems. In this paper we use cavity method of statistical physics and Monte Carlo simulations to study the corresponding constraint satisfaction problem on random graphs. We obtain the entropy of maximal independent sets within the replica symmetric and one-step replica symmetry breaking frameworks, shedding light on the metric structure of the landscape of solutions and suggesting a class of possible algorithms. This is of particular relevance for the application to the study of strategic interactions in social and economic networks, where maximal independent sets correspond to pure Nash equilibria of a graphical game of public goods allocation.

The matching problem plays a basic role in combinatorial optimization and in statistical mechanics. In its stochastic variants, optimization decisions have to be taken given only some probabilistic information about the instance. While the deterministic case can be solved in polynomial time, stochastic variants are worst-case intractable. We propose an efficient method to solve stochastic matching problems which combines some features of the survey propagation equations and of the cavity method. We test it on random bipartite graphs, for which we analyze the phase diagram and compare the results with exact bounds. Our approach is shown numerically to be effective on the full range of parameters, and to outperform state-of-the-art methods. Finally we discuss how the method can be generalized to other problems of optimization under uncertainty.

The budding yeast Saccharomyces cerevisiae is the first eukaryote whose genome has been completely sequenced. It is also the first eukaryotic cell whose proteome (the set of all proteins) and interactome (the network of all mutual interactions between proteins) has been analyzed. In this paper we study the structure of the yeast protein complex network in which weighted edges between complexes represent the number of shared proteins. It is found that the network of protein complexes is a small world network with scale free behavior for many of its distributions. However we find that there are no strong correlations between the weights and degrees of neighboring complexes. To reveal non-random features of the network we also compare it with a null model in which the complexes randomly select their proteins. Finally we propose a simple evolutionary model based on duplication and divergence of proteins.

We discuss how inference can be performed when data are sampled from the nonergodic phase of systems with multiple attractors. We take as a model system the finite connectivity Hopfield model in the memory phase and suggest a cavity method approach to reconstruct the couplings when the data are separately sampled from few attractor states. We also show how the inference results can be converted into a learning protocol for neural networks in which patterns are presented through weak external fields. The protocol is simple and fully local, and is able to store patterns with a finite overlap with the input patterns without ever reaching a spin-glass phase where all memories are lost.

A fundamental problem in medicine and biology is to assign states, e.g., healthy or diseased, to cells, organs or individuals. State assignment or making a diagnosis is often a nontrivial and challenging process and, with the advent of omics technologies, the diagnostic challenge is becoming more and more serious. The challenge lies not only in the increasing number of measured properties and dynamics of the system (e.g., cell or human body) but also in the co-evolution of multiple states and overlapping properties, and degeneracy of states. We develop, from first principles, a generic rational framework for state assignment in cell biology and medicine, and demonstrate its applicability with a few simple theoretical case studies from medical diagnostics. We show how disease-related statistical information can be used to build a comprehensive model that includes the relevant dependencies between clinical and laboratory findings (signs) and diseases. In particular, we include disease-disease and sign-sign interactions and study how one can infer the probability of a disease in a patient with given signs. We perform comparative analysis with simple benchmark models to check the performances of our models. We find that including interactions can significantly change the statistical importance of the signs and diseases. This first principles approach, as we show, facilitates the early diagnosis of disease by taking interactions into accounts, and enables the construction of consensus diagnostic flow charts. Additionally, we envision that our approach will find applications in systems biology, and in particular, in characterizing the phenome via the metabolome, the proteome, the transcriptome, and the genome.

Circuit topology refers to the arrangement of interactions between objects belonging to a linearly ordered object set. Linearly ordered set of objects are common in nature and occur in a wide range of applications in economics, computer science, social science and chemical synthesis. Examples include linear bio-polymers, linear signaling pathways in cells as well as topological sorts appearing in project management. Using a statistical mechanical treatment, we study circuit topology landscapes of linear polymer chains with intra-chain contacts as a prototype of linearly sorted objects with interactions. We find generic features of the topological space and study the statistical properties of the space under the most basic constraints on the occupancy of arrangements and topological interactions. We observe that a set of correlated contact sites (a sector) could nontrivially influence the entropy of circuits as the number of involved sites increases. Finally, we discuss how constraints can be inferred from the information provided by local contact distributions in presence of a sector.