Axiomatic aggregation of incomplete rankings

Aggregating multiple rankings in a database is an important task well studied by the database community. High-stakes application domains include hiring, lending, and education where multiple decision makers rank candidates and their input is then combined into an overall consensus ranking. While state-of-art aggregation methods succeed in giving equal say to each decision maker, to date no methods ensure fair representation for groups of candidates being ranked, who risk being impacted by discriminatory bias. We present the first solution to this open problem of guaranteeing fairness for disadvantaged groups of candidates, while still producing a good consensus of the base rankings. We design a family of exact fair algorithms achieving optimality for fair rank aggregation. We also develop approximate methods achieving fairness with guaranteed minimal approximation error scaling to millions of candidates in the rankings. A comparative study evaluates our proposed methods, revealing trade-offs between aggregation accuracy and different degrees of unfair bias in a rich variety of rank aggregation scenarios. Our real-world case study demonstrates that our solutions mitigate unfair bias using real-world data.

In this paper we explore the possibility of deriving consensus rankings by solving consensus optimization problems, characterizing consensus rankings as suitable complete order relations minimizing the average Kemeny-Snell distance to the individual rankings. This optimization problem can be expressed as a binary programming (BP) problem which can typically be solved reasonably efficiently. The underlying theory is discussed in Sect. 1. Applications of the proposed method given in Sect. 2 include a comparison to other mathematical programming (MP) approaches using the data set of Tse [9] and establishing a consensus ranking of marketing journals identified by domain experts from a subset of the Harzing journal quality list [2]. In Sect. 3 we discuss computational details and present the results of a benchmark experiment comparing the performance of the commercial solver CPLEX to three open source mixed integer linear programming (MILP) solvers

The problem of aggregating individual rankings to create an overall consensus ranking representative of the group is of longstanding interest in group decision making. The problem arises in situations where a group of k Decision Makers (DMs) are asked to rank order n alternatives. The question is how to combine the DMs' rankings into one consensus ranking. Several different approaches have been suggested to aggregate DM responses into a compromise or consensus ranking, however, none is generally recognised as being the best and the similarity of consensus rankings generated by these algorithms is largely unknown. In this paper, we propose a new Weighted-sum ordinal Consensus ranking Method (WCM) with the weights derived from a Sigmoid function. We run Monte Carlo simulation across a range of k and n to compare the similarity of the consensus rankings generated by our method with the best-known method of Borda-Kendall (BAK; Kendall, M. (1962) Rank correlation methods. New York, NY: Hafner) and two other commonly used techniques proposed by Beck, M.P. and Lin, B.W. (1983) 'Some heuristics for the consensus ranking problem', Computers and Operations Research, Vol. 10,

High-centrality nodes have disproportionate influence on the behavior of a network; therefore controlling such nodes can efficiently steer the system to a desired state. Existing multiplex centrality measures typically rank nodes assuming the layers are qualitatively similar. Many real systems, however, are comprised of networks heterogeneous in nature, for example, social networks may have both agnostic and affiliative layers. Here, we use rank aggregation methods to identify intervention targets in multiplex networks when the structure, the dynamics, and our intervention goals are qualitatively different for each layer. Our approach is to rank the nodes separately in each layer considering their different function and desired outcome, and then we use Borda count or Kemeny aggregation to identify a consensus ranking—top nodes in the consensus ranking are expected to effectively balance the competing goals simultaneously among all layers. To demonstrate the effectiveness of consensus ranking, we apply our method to a degree-based node removal procedure such that we aim to destroy the largest component in some layers, while maintaining large-scale connectivity in others. For any multi-objective intervention, optimal targets only exist in the Pareto-sense; we, therefore, use a weighted generalization of consensus ranking to investigate the trade-off between the competing objectives. We use a collection of model and real networks to systematically investigate how this trade-off is affected by multiplex network structure. We use the copula representation of the multiplex centrality distributions to generate model multiplex networks with given rank correlations. This allows us to separately manipulate the marginal centrality distribution of each layer and the interdependence between the layers, and to investigate the role of the two using both analytical and numerical methods.

Every economic system is certainly based on the ideology that provides the foundation and goal on the one hand and the axioms and principles on the other. Every economic system create a framework in which a socio-economic communities can take advantage of the natural resources and human for the sake of production and distribute them for consumption. In this case, Syed NawabHaider Naqvi transform the ethical values of Islam into a set of axioms, that is tauhid (Unity), equilibrium, free will, and responsibility. A set of axioms can be used as a reference in formulating economic behavior consistent. Axiomatic approach in Islamic economics is an analytical tool used to draw up the basic elements of Islamic economics. With sourced from the Al-Qur’an and the Hadith, the Islamic ethical axiom set has been built to meet the five characteristics logically fundamental axioms of the system.

Learning the true rank ordering among objects by aggregating a set of expert opinion rank order lists is an important and ubiquitous problem in many applications ranging from social choice theory to recommendation systems and search aggregation. We study the problem of unsupervised rank aggregation where no ground truth ordering information in available, neither about the true preference ordering among any set of objects nor about the quality of individual rank lists. Aggregating the often inconsistent and poor quality rank lists in such an unsupervised manner is a highly challenging problem, and standard consensus-based methods fall short in terms of both quality and scalability. In this manuscript we propose a novel framework to bypass these issues by using object attributes to augment the standard rank aggregation framework. We design algorithms that learn joint models on both rank lists and object features to obtain an aggregated rank ordering that is more accurate and robust, and also helps weed out rank lists of dubious validity. We validate our techniques on synthetic datasets where our algorithm is able to estimate the true rank ordering even when the rank lists are corrupted. Experiments on three real datasets, MQ2008, MQ2007 and OHSUMED, show that using object features can result in significant improvement in performance over existing rank aggregation methods that do not use object information. Furthermore, when at least some of the rank lists are of high quality, our methods are able to effectively exploit such information to output an aggregated rank ordering of high accuracy.

Context. The complexity of understanding the concept of multivalued dependencies (when setting out their theoretical foundations, an axiomatic approach is used) in relational (table) databases leads to problems of their modeling, which in turn can cause a violation of data integrity. A partial solution to these issues is facilitated by the search for equivalent axiomatic systems of multivalued dependencies that are minimal in terms of the number and power of their components. This, in turn, requires establishing the equivalence of these axiomatic systems and proving the independence of their elements. The object of the study is the axiomatisations of multivalued dependencies.The purpose of the work is to establish the equivalence of two axiomatic systems of multivalued dependencies, one of which is minimal in terms of the number and power of components, and to prove the independence of specified axiomatics. Method. Set-theoretic and logical-algebraic methods are used in the construction of proofs. The equivalence of two axiomatic systems of multivalued dependencies is established classically: the relation of syntactic inference is considered and sequences of multivalued dependencies are constructed that are proofs of the rules of each axiomatics from the components of the other. To build proof of independence of the axiomatic system, models of relational schemes are proposed, such that for an axiom or rule whose independence is being proved, it is impossible to construct proof from set of other axioms or rules.Results. Equivalence of the axiomatisation of multivalued dependencies proposed by a group of scientists Beeri, Fagin and Howard and the Biskup’s axiomatisation of multivalued dependencies has been established. The independence of the Biskup’s axiomatic system is proved in the sense that without loss of completeness it is impossible to remove neither the only axiom nor none of the inference rules.Conclusions. The independence of the Biskup’s axiomatisation of multivalued dependencies, which has no more in number and power of components than other axiomatic systems of multivalued dependencies, is proved. Using such axiomatic system has advantages in the development of CASE-tools (Computer-Aided Software Engineering Tools), which include the implementation of reducing the relational database schema to the fourth normal form, and also, when manually designing a logical model of a relational database.

Comparing and ranking information is an important topic in social and information sciences, and in particular on the web. Its objective is to measure the difference of the preferences of voters on a set of candidates and to compute a consensus ranking. Commonly, each voter provides a total order of all candidates. Recently, this approach was generalized to bucket orders, which allow ties. In this work we further generalize and consider total, bucket, interval and partial orders. The disagreement between two orders is measured by the nearest neighbor Spearman footrule distance, which has not been studied so far. For two bucket orders and for a total and an interval order the nearest neighbor Spearman footrule distance is shown to be computable in linear time, whereas for a total and a partial order the computation is NP-hard, 4-approximable and fixed-parameter tractable. Moreover, in contrast to the well-known efficient solution of the rank aggregation problem for total orders, we prove the NP-completeness for bucket orders and establish a 4-approximation.

One of the largest constituents of big data is the crowdsourced or user-generated data which contain a wide range of valuable information. However, they are inherently biased and possibly spammed, making trustworthy information extraction an imperative task. As a special case, we study reviewer-posted ratings for products. The biased ratings can lead to disappointed customers due to overrated products, and reduced revenues of business owners caused by undeserved negative ratings. To distill objective product quality measurements, most existing methods try to infer unbiased ratings from the raw ratings alone, and may not overcome the inherent bias to recover the underlying true ratings. Though improved bias corrections have been achieved with domain expert helps, the overhead of expert efforts can be rather expensive in practice. We exploit the variety of big data and adopt a multiple source mining approach, which finds trustworthy measurements without domain expert, but with knowledge crowdsourced and transferred from external domains. We address the challenges that the multiple data sources are 1) inherently heterogeneous, 2) at most only partially overlapping and 3) biased by themselves. We explore and analyze the strengths and weaknesses of various knowledge transfer strategies. We then propose Consensus Ranking Dual Transfer (CRDT) to handle the above challenges by identifying "anchor reviewers" as a bridge for robust "dual transfer", and removing bias in individual sources via consensus ranking aggregation. Experiments on real-world rating datasets demonstrate that the proposed approach can deliver more robust bias correcting effects than the baselines and can identify abnormal reviewers.

Grapevine is one of the most cultivated fruit crop worldwide with Vitis vinifera being the species with the highest economical importance. Being highly susceptible to fungal pathogens and increasingly affected by environmental factors, it has become an important agricultural research area, where gene expression analysis plays a fundamental role. Quantitative reverse transcription polymerase chain reaction (qRT-PCR) is currently amongst the most powerful techniques to perform gene expression studies. Nevertheless, accurate gene expression quantification strongly relies on appropriate reference gene selection for sample normalization. Concerning V. vinifera, limited information still exists as for which genes are the most suitable to be used as reference under particular experimental conditions. In this work, seven candidate genes were investigated for their stability in grapevine samples referring to four distinct stresses (Erysiphe necator, wounding and UV-C irradiation in leaves and Phaeomoniella chlamydospora colonization in wood). The expression stability was evaluated using geNorm, NormFinder and BestKeeper. In all cases, full agreement was not observed for the three methods. To provide comprehensive rankings integrating the three different programs, for each treatment, a consensus ranking was created using a non-weighted unsupervised rank aggregation method. According to the last, the three most suitable reference genes to be used in grapevine leaves, regardless of the stress, are UBC, VAG and PEP. For the P. chlamydospora treatment, EF1, CYP and UBC were the best scoring genes. Acquaintance of the most suitable reference genes to be used in grapevine samples can contribute for accurate gene expression quantification in forthcoming studies.

Aggregation of Rankings in Figure Skating

Consensus temporal order of amino acids and evolution of the triplet code

The need to meaningfully combine sets of rankings often comes up when one deals with ranked data. Although a number of heuristic and supervised learning approaches to rank aggregation exist, they require domain knowledge or supervised ranked data, both of which are expensive to acquire. In order to address these limitations, we propose a mathematical and algorithmic framework for learning to aggregate (partial) rankings without supervision. We instantiate the framework for the cases of combining permutations and combining top-k lists, and propose a novel metric for the latter. Experiments in both scenarios demonstrate the effectiveness of the proposed formalism.

Deriving weights from pairwise comparison ratio matrices: An axiomatic approach

Rough sets, proposed by Pawlak and rough fuzzy sets proposed by Dubois and Prade were expressed with the different computing formulas that were more complex and not conducive to computer operations. In this paper, we use the composition of a fuzzy matrix and fuzzy vectors in a given non-empty finite universal, constitute an algebraic system composed of finite dimensional fuzzy vectors and discuss some properties of the algebraic system about a basis and operations. We give an effective calculation representation of rough fuzzy sets by the inner and outer products that unify computing of rough sets and rough fuzzy sets with a formula. The basis of the algebraic system play a key role in this paper. We give some essential properties of the lower and upper approximation operators generated by reflexive, symmetric, and transitive fuzzy relations. The reflexive, symmetric, and transitive fuzzy relations are characterized by the basis of the algebraic system. A set of axioms, as the axiomatic approach, has been constructed to characterize the upper approximation of fuzzy sets on the basis of the algebraic system.