The analysis of the dynamics of the electorate system by using q-distribution-a case study
Authors: Prenga Dode , Peqini Klaudio , Osmani Rudina 
Published: December 2021

Abstract
In this work we study the system of the votes, the mechanism of the electoral support formation, and also the elements of its dynamics, by analyzing the data from several election processes in Albania. Firstly, we evidence the specific features and the characteristics of the distributions of votes through a descriptive approach, and next we use those findings to identify the nature of the elementary processes of the agreement, the defects of the system and dynamical issues. The distributions of the votes for the majority or majority-like election as by polling stations reference results a two-parts function. The part of the distribution located in the small vote fraction fits to a power law or to a q-exponential function, therefore the foremost factor of the electoral support for the subjects populating this zone is based in the preferential attachment rule, with some modification. Consequently, the small subjects or independent candidates, realize their electoral attractiveness based on the individual performance. Also, their voters act rationally and usually gather sufficient information before deciding to support them. The bell-shaped part of the distribution which describes the votes of the candidates of the main parties, fits better to the q-gaussian functions. In this case, electoral support is affected strongly by the political activists (militants) which harvest local influences to convict people producing an extra support for the candidates of big parties, regardless of their performance and electoral values. This physiognomy is characteristic for all legislative and administrative majority voting or other majority-like elections as practically behave the closed-lists elections of 2009, 2013, 2017 and also the semi-opened list of the 2021. The distributions of the closed-list votes in the administrative elections are mostly of the exponential or q-exponential type. Also, the distributions based on the data from electoral constituencies which include many polling stations resulted q-exponentials for all types of elections. We connected the q-exponential form of the distribution with the electoral network failures, system deficiencies and heterogeneity effects. In 2021, the distributions of the votes for subjects is obtained similar to the typical recent majority voting distribution, a mix of the power law and q-gaussian functions. The distribution of the votes for the candidates on the semi-open list for those elections resulted a mix of two q-exponentials. We associated this last with the difficulties of the voters to understand new electoral rules and additional other causes of the non-electoral nature. Also, the electorate network might have suffered extra irregularity issues due to the inadequate sizes of elections units, etc. The distributions of the votes for the two main parties are found q-gaussians with q ∼ 1.32 and q ∼ 1.57 for the right and the left wing respectively. Based on the non-stationarity level measured by the q-value, significant redistribution events are expected for the left-wing network, whereas the right-wing network would experience fewer changes in ceteris paribus socio-electoral conditions. Interestingly, the mix of the votes for two main political parties has produced a q-gaussian with q=1.004, and subsequently, the joint system is found in a more relaxed state. Therefore, the compound network including two main parties is likely to not undergo significant redistribution of the votes in the near future. This means that the small subjects or the fresh-born ones are not likely to cause changes on the system. Based on the deductions for electoral agreement formation, we used our recently introduced q-opinion approach to model the electoral opinion formation. In this model, the q-opinion produces an additional term that multiplies the modified preferential attachment probability for the link establishment. Herein, the q-parameter is calculated by using an ad-hoc formula involving the performance of the candidate as utility function, which associates the agreement behavior as the response, with the candidate performance as the offer or the cause factor. The quantity q henceforth acts as activation-inhibition switch of the extra utility involved in the q-opinion model, and particularly it provides a nonzero voter’s support for the high-performance opponent candidates. The model has reproduced the distributions analyzed in this study. It resulted that many voters in this electorate system act rationally, despite their affiliations.