scholarly journals An ABC-Problem for location and consensus functions on graphs

2016 ◽  
Vol 207 ◽  
pp. 15-28 ◽  
Author(s):  
F.R. McMorris ◽  
Henry Martyn Mulder ◽  
Beth Novick ◽  
Robert C. Powers
Keyword(s):  
Consensus Functions

On the Authoritativeness of Sunni Consensus

1986 ◽  
Vol 18 (4) ◽  
pp. 427-454 ◽  
Author(s):  
Weal B. Hallaq
Keyword(s):  
Legal Reasoning ◽  
Great Majority ◽  
The Body ◽  
Primary Sources ◽  
Legal Process ◽  
Mental Faculty ◽  
Original Case ◽  
Consensus Functions ◽  
Legal Judgments ◽  
Substantive Law

Sunni Islam recognizes four sources from and through which the laws governing its conduct are derived. These are the Qur'an, the Sunna of the Prophet, the consensus (ljmā') of the community and its scholars, and qiyās, the juridicological method of inference. The first two sources provide the jurist with the material from which he is to extract through qiyas and ijtihād (the disciplined exercise of mental faculty) the law which he believes to the best of his knowledge to be that decreed by God. Except for a relatively limited number of cases where the Qu'an and the Sunna offer already-formulated legal judgments, the great majority of furū' cases, which constitute the body of positive and substantive law, are derived by qiyas. Thus, qiyas may be used to “discover” the judgment of a new case provided that this case has not already been solved in the two primary sources. The process of legal reasoning which qiyas involves is charged with innumerable difficulties not the least of which is finding the circle of common similarity, the 'illa, between the original case in the texts and the new case which requires a legal judgment. Since finding the 'illa entails a certain amount of guesswork (zann) on the part of the jurist and since it is highly probable that the 'illa is extracted from a text which is not entirely reliable or a text capable of more than one interpretation, Sunni jurists deemed the results of qiyas to be probable (zannī). It is only at this point that consensus may enter into play in the legal process. Should Muslims, represented by their jurists, reach an agreement on the validity of a zanni legal judgment, such judgment is automatically transferred from the domain of juristic speculation to that of certainty (qat', yaqīn). Consensus then renders this judgment irrevocable, not to be challenged or reinterpreted by later generations. Furthermore, this judgment, being so irrevocable, acquires a validity tantamount to that of the Qur'an and the highly reliable traditions embodied in the Sunna of the Prophet. Thus, such a case with its established judgment becomes a precedent according to which another new legal question may be solved. It is only in this sense that consensus functions as a source of law, a source which is infallible.

scholarly journals Axioms for Consensus Functions on then-Cube

2017 ◽  
Vol 2017 ◽  
pp. 1-5
Author(s):  
C. Garcia-Martinez ◽  
F. R. McMorris ◽  
O. Ortega ◽  
R. C. Powers
Keyword(s):  
Metric Space ◽  
Minimax Criterion ◽  
Finite Metric Space ◽  
Finite Sequences ◽  
Consensus Functions

Apvalue of a sequenceπ=(x1,x2,…,xk)of elements of a finite metric space(X,d)is an elementxfor which∑i=1kdp(x,xi)is minimum. Thelp–function with domain the set of all finite sequences onXand defined bylp(π)={x:  xis apvalue ofπ}is called thelp–function on(X,d). Thel1andl2functions are the well-studied median and mean functions, respectively. In this note, simple characterizations of thelp–functions on then-cube are given. In addition, the center function (using the minimax criterion) is characterized as well as new results proved for the median and antimedian functions.

Social Network Optimization for Cluster Ensemble Selection

2020 ◽  
Vol 176 (1) ◽  
pp. 79-102
Author(s):  
Chenyue Zhao ◽  
Hosein Alizadeh ◽  
Behrouz Minaei ◽  
Majid Mohamadpoor ◽  
Hamid Parvin ◽  
...  
Keyword(s):  
Large Scale ◽  
Cluster Structure ◽  
Ensemble Methods ◽  
Quadratic Program ◽  
Maximization Problem ◽  
Similarity Matrix ◽  
Cluster Ensemble ◽  
Ensemble Selection ◽  
Consensus Functions ◽  
Consensus Partition

This paper studies the cluster ensemble selection problem for unsupervised learning. Given a large ensemble of clustering solutions, our goal is to select a subset of solutions to form a smaller yet better performing cluster ensemble than using all available solutions. The common way of aggregating the chosen solutions is accumulating the information of the selected results to a similarity matrix. This paper suggests transforming the similarity matrix to a modularity matrix and then applying a new consensus function which optimizes modularity measure in it. We represent the modularity maximization problem as a 0-1 quadratic program which can be exactly solved for small datasets. We also established a new greedy algorithm, namely sum linkage, to optimize the objective function specially for large scale datasets in a very short time. We show that the proposed consensus partition gets much closer to the actual cluster structure than the partitions obtained from the direct application of common cluster ensemble methods. The promising results compared with other most cited consensus functions show the excellent efficiency of the proposed method.

scholarly journals Removal independence and multi-consensus functions

1999 ◽  
Author(s):  
Mark Dwyer ◽  
Fred R. Mc Morris ◽  
Robert C. Powers
Keyword(s):  
Consensus Functions

An Ensemble Clusterer Framework based on Valid and Diverse Basic Small Clusters

2021 ◽  
pp. 1-31
Author(s):  
Tao Sun ◽  
Saeed Mashdour ◽  
Mohammad Reza Mahmoudi
Keyword(s):  
Clustering Algorithm ◽  
State Of The Art ◽  
Clustering Algorithms ◽  
Clustering Ensemble ◽  
High Quality ◽  
Consensus Function ◽  
Consensus Functions ◽  
Consensus Partition ◽  
Small Clusters ◽  
Minimum Quality

Clustering ensemble is a new problem where it is aimed to extract a clustering out of a pool of base clusterings. The pool of base clusterings is sometimes referred to as ensemble. An ensemble is to be considered to be a suitable one, if its members are diverse and any of them has a minimum quality. The method that maps an ensemble into an output partition (called also as consensus partition) is named consensus function. The consensus function should find a consensus partition that all of the ensemble members agree on it as much as possible. In this paper, a novel clustering ensemble framework that guarantees generation of a pool of the base clusterings with the both conditions (diversity among ensemble members and high-quality members) is introduced. According to its limitations, a novel consensus function is also introduced. We experimentally show that the proposed clustering ensemble framework is scalable, efficient and general. Using different base clustering algorithms, we show that our improved base clustering algorithm is better. Also, among different consensus functions, we show the effectiveness of our consensus function. Finally, comparing with the state of the art, we find that the clustering ensemble framework is comparable or even better in terms of scalability and efficacy.

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