## Selecting representatives

### László Aszalós, Dávid Nagy

DOI: http://dx.doi.org/10.15439/2019F95

Citation: Communication Papers of the 2019 Federated Conference on Computer Science and Information Systems, M. Ganzha, L. Maciaszek, M. Paprzycki (eds). ACSIS, Vol. 20, pages 13–19 (2019)

Abstract. We use representatives to reduce complexity in many areas of life. Clusters are often replaced with their centre, and then these representatives are used to classify new objects. If the objects are described as a vector of real numbers, then the centre can be easily calculated. However, this method is unusable if only a similarity relation is given instead of coordinates of the object or the distances between the objects. Google can filter and rank relevant pages for a particular question; and here we follow a similar approach. The difference is that we have an undirected graph while the PageRank algorithm uses a directed one. In this article we show what conditions we set for our own ranking system. Following the description of the details of this method we demonstrate that it satisfies our criteria and how it selects the (mathematically proven) most typical elements of each cluster. Finally, we apply this method on several partitions of the natural numbers and on non-transitive tolerance relations to present the representatives of the numbers.

### References

- S Aeberhard, D Coomans, and O De Vel. Comparison of classifiers in high dimensional settings. Dept. Math. Statist., James Cook Univ., North Queensland, Australia, Tech. Rep, 92(02), 1992.
- Alon Altman and Moshe Tennenholtz. Ranking systems: the pagerank axioms. In Proceedings of the 6th ACM conference on Electronic commerce, pages 1–8. ACM, 2005.
- László Aszalós, Lajos Hajdu, and Attila Pethő. On a correlational clustering of integers. Indagationes Mathematicae, 27(1):173–191, 2016.
- László Aszalós and Tamás Mihálydeák. Rough classification in incomplete databases by correlation clustering. In 2015 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and Technology (IFSA-EUSFLAT-15). Atlantis Press, 2015.
- R. V. Mises and H. Pollaczek-Geiringer. Praktische verfahren der gleichungsauflösung . ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 9(2):152–164, 1929.
- Z. Néda, R. Sumi, M. Ercsey-Ravasz, M. Varga, B. Molnár, and Gy. Cseh. Correlation clustering on networks. Journal of Physics A: Mathematical and Theoretical, 42(34):345003, 2009.