Generic Networks of Votings

In this paper, we analyse results of the 1 5 th International Henryk Wieniawski Violin Competition by comparing the properties of its results network to the properties of generic networks of votings . Suppose that a competition Comp is given. In this competition, k contestants are rated by n jurors. Suppose that the Borda count is used as a voting method, i.e. every juror gives k points for the best contestant, k −1 points for the second best contestant, and so on. In particular, the worts contestant gets 1 point. For such a competition, we create a weighted network N ( Comp ) in the following way. The node set of N ( Comp ) corresponds to jurors and the link set of N ( Comp ) consists of all links { J s , J t : s ≠ t } . For link l st connecting nodes J s and J t , we assign the weight w ( l st ) = w st , where w st = LF 2 ( α s α t − 1 ) . Here LF 2 is a Lehmer norm on the permutation group S k , whereas α s and α t denotes the votes of jurors J s and J t , respectively. In particular, for i = 1 , 2 , . . . , k , α s ( i ) is the number of points given to the i -th contestant by juror J s . The similar holds for juror J t . Note that α s and α t can be considered as elements of S k . Suppose now that the probability measure P is given on space V of all possible votings of a single juror, i.e. on space S k . Suppose that every juror votes independently according to P. We repeat such a voting process 100 times and for every j = 1 , 2 , . . . , 1 0 0 , we create a network N j in the way described above. In this paper, we compare some statistical properties of networks N j , for probability measures P being the convex combinations of two Dirac probability measures and a uniform probability measure, to the properties of network of jurors’ votings in the 2016 Wieniawski Competition.