Engagement in the electoral processes: Scaling laws and the role of political positions

We report on a statistical analysis of the engagement in the electoral processes of all Brazilian cities by considering the number of party memberships and the number of candidates for mayor and councillor. By investigating the relationships between the number of party members and the population of voters, we have found that the functional forms of these relationships are well described by sublinear power laws (allometric scaling) surrounded by a multiplicative log-normal noise. We have observed that this pattern is quite similar to those we previously reported for the relationships between the number of candidates (mayor and councillor) and population of voters [Europhys. Lett. 96, 48001 (2011)], suggesting that similar universal laws may be ruling the engagement in the electoral processes. We also note that the power-law exponents display a clear hierarchy, where the more influential is the political position the smaller is the value of the exponent. We have also investigated the probability distributions of the number of candidates (mayor and councillor), party memberships, and voters. The results indicate that the most influential positions are characterized by distributions with very short tails, while less influential positions display an intermediate power-law decay before showing an exponential-like cutoff. We discuss the possibility that, in addition to the political power of the position, limitations in the number of available seats can also be connected with this changing of behavior. We further believe that our empirical findings point out to an under-representation effect, where the larger the city is, the larger are the obstacles for more individuals to become directly engaged in the electoral process.

Figure 1

Scatter plot of the relationship between ${\log }_{10}N$ (population of voters) and ${\log }_{10}V$ (number of party memberships) when considering data from the year of 2008. We note that the relationship is approximated by a power-law tendency surrounded by fluctuations. Here, the green small dots are the values of ${\log }_{10}N$ and ${\log }_{10}V$ for each city and the circles are the average values of these points after binning the data in $w$ windows equally spaced in ${\log }_{10}V$. The dashed line is a linear fit to these average values and the power-law exponent is shown in the plot. The simulated results from Eq. (3) are represented by gray dots, which produce the window average values indicated by the gray squares.

Figure 2

Average power-law relationships between ${\log }_{10}N$ (population of voters) and ${\log }_{10}V$ (number of party memberships). Note that we have analyzed all the years in our database and the values of $\alpha$ (the slopes of these curves) are almost constant in time. The average value of $\alpha$ is shown in the plot and the dashed line represents a power law with this exponent. The gray squares represent the simulated results from Eq. (3).

Figure 3

Fluctuations around the power-law tendencies. (a) Variance ${\sigma }^2$ of the values of ${\log }_{10}N$ versus $\langle {\log }_{10}V\rangle$ evaluated after binning the data in $w$ windows equally spaced in ${\log }_{10}V$. We note the almost constant value of ${\sigma }^2$ over the population of voters. The horizontal dashed line is the average value of ${\sigma }^2$ over ${\log }_{10}V$ and over the years ($0.027\pm 0.002$). (b) Distribution of the normalized residuals ($\xi$) around the power-law tendencies. The dashed line is the Gaussian distribution with zero mean and unitary variance. In both plots, the gray squares represent the simulated results from Eq. (3).

Figure 4

The role of the political positions. Average values of the power-law exponents $\alpha$ for each political position. The exponents of the relationships between the number of councilor and mayoral candidates and the population of voters were obtained from Ref. [24]. We note that the average of $\alpha$ is smaller for the more influential political positions. The error bars are standard deviations from the mean value of $\alpha$.

Figure 5

Cumulative distributions of the number of mayoral candidates (circles), councillor candidates (diamonds), party memberships (triangles), and population of voters (squares) when considering data from the year 2008 (very similar distributions are obtained for the other years). We note the shift of the distributions with the increase in the number of available seats. We further observe that the distributions for mayor and councillor are characterized by exponential asymptotic behavior, while the distributions for party memberships and voters display an approximate power-law behavior before showing an exponential-like cutoff. The dashed lines show the regions where the distributions can be approximated by power laws (the exponents are shown in the plot).