Finite size corrections to random Boolean networks

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.Comment: Minor correction

Exhaustive enumeration unveils clustering and freezing in random 3-SAT

We study geometrical properties of the complete set of solutions of the random 3-satisfiability problem. We show that even for moderate system sizes the number of clusters corresponds surprisingly well with the theoretic asymptotic prediction. We locate the freezing transition in the space of solutions which has been conjectured to be relevant in explaining the onset of computational hardness in random constraint satisfaction problems.Comment: 4 pages, 3 figure

Mining association rules for label ranking

Lecture Notes in Computer Science Volume 6635, 2011.Recently, a number of learning algorithms have been adapted for label ranking, including instance-based and tree-based methods. In this paper, we continue this line of work by proposing an adaptation of association rules for label ranking based on the APRIORI algorithm. Given that the original APRIORI algorithm does not aim to obtain predictive models, two changes were needed for this achievement. The adaptation essentially consists of using variations of the support and confidence measures based on ranking similarity functions that are suitable for label ranking. Additionally we propose a simple greedy method to select the parameters of the algorithm. We also adapt the method to make a prediction from the possibly con icting consequents of the rules that apply to an example. Despite having made our adaptation from a very simple variant of association rules for classification, partial results clearly show that the method is making valid predictions. Additionally, they show that it competes well with state-of-the-art label ranking algorithms.This work was partially supported by project Rank! (PTDC/EIA/81178/2006) from FCT and Palco AdI project Palco3.0 financed by QREN and Fundo Europeu de Desenvolvimento Regional (FEDER). We thank the anonymous referees for useful comments

On the complexity of strongly connected components in directed hypergraphs

We study the complexity of some algorithmic problems on directed hypergraphs and their strongly connected components (SCCs). The main contribution is an almost linear time algorithm computing the terminal strongly connected components (i.e. SCCs which do not reach any components but themselves). "Almost linear" here means that the complexity of the algorithm is linear in the size of the hypergraph up to a factor alpha(n), where alpha is the inverse of Ackermann function, and n is the number of vertices. Our motivation to study this problem arises from a recent application of directed hypergraphs to computational tropical geometry. We also discuss the problem of computing all SCCs. We establish a superlinear lower bound on the size of the transitive reduction of the reachability relation in directed hypergraphs, showing that it is combinatorially more complex than in directed graphs. Besides, we prove a linear time reduction from the well-studied problem of finding all minimal sets among a given family to the problem of computing the SCCs. Only subquadratic time algorithms are known for the former problem. These results strongly suggest that the problem of computing the SCCs is harder in directed hypergraphs than in directed graphs.Comment: v1: 32 pages, 7 figures; v2: revised version, 34 pages, 7 figure

High-Order SNP Combinations Associated with Complex Diseases: Efficient Discovery, Statistical Power and Functional Interactions

There has been increased interest in discovering combinations of single-nucleotide polymorphisms (SNPs) that are strongly associated with a phenotype even if each SNP has little individual effect. Efficient approaches have been proposed for searching two-locus combinations from genome-wide datasets. However, for high-order combinations, existing methods either adopt a brute-force search which only handles a small number of SNPs (up to few hundreds), or use heuristic search that may miss informative combinations. In addition, existing approaches lack statistical power because of the use of statistics with high degrees-of-freedom and the huge number of hypotheses tested during combinatorial search. Due to these challenges, functional interactions in high-order combinations have not been systematically explored. We leverage discriminative-pattern-mining algorithms from the data-mining community to search for high-order combinations in case-control datasets. The substantially improved efficiency and scalability demonstrated on synthetic and real datasets with several thousands of SNPs allows the study of several important mathematical and statistical properties of SNP combinations with order as high as eleven. We further explore functional interactions in high-order combinations and reveal a general connection between the increase in discriminative power of a combination over its subsets and the functional coherence among the genes comprising the combination, supported by multiple datasets. Finally, we study several significant high-order combinations discovered from a lung-cancer dataset and a kidney-transplant-rejection dataset in detail to provide novel insights on the complex diseases. Interestingly, many of these associations involve combinations of common variations that occur in small fractions of population. Thus, our approach is an alternative methodology for exploring the genetics of rare diseases for which the current focus is on individually rare variations

A Polly Cracker system based on Satisfiability

This paper presents a public-key cryptosystem based on a subclass of the well-known satisfiability problem from propositional logic, namely the doubly-balanced 3-sat problem. We first describe the construction of an instance of our system starting from such a 3-sat formula. Then we discuss security issues: this is achieved on the one hand by exploring best methods to date for solving this particular problem, and on the other hand by studying (systems of multivariate) polynomial equation solving algorithms in this particular setting. The result of our investigations is that both types of method fail to break our instances. We end the paper with some complexity considerations and implementation results