In countries that employ proportional-representation elections with open lists, parties running for government give voters in each district a choice of candidates. On election day, citizens vote for both their preferred party and their preferred candidate from that party. Seats in each district are shared out to parties based on how well they did, and the parties then fill them with their most favoured candidates. The party with the most votes overall wins.

Santo Fortunato from the Institute for Scientific Interchange Foundation and Claudio Castellano from the University of Rome in Italy looked at election data sets from 1958, 1972 and 1987 in Italy, 2003 in Finland and 2005 in Poland, all of which used this election system. They began by plotting the distribution of the number of votes received by candidates, or v. They then assumed that, in addition to v, the distribution would be influenced by the number of candidates in each party and the total number of votes collected by each party. In actual fact, they found that the histogram was dependent on the ratio of the two quantities — in other words, the average number of votes received by a party’s candidates, or v0.

This in itself is not surprising because it simply means that, for example, candidates from a more popular party with a higher v0 would generally get more votes. After a closer inspection, however, they realized that the distribution was not dependant on both v and v0 separately, but only on v/v0 — a measure of the performance of a candidate compared with the other candidates in his or her party.

Remarkably, when they plotted the histogram of this single parameter for all five election data sets, they found that they all fell almost precisely onto the same curve. This suggests that it doesn’t matter, for example, how different polices are between parties, the distribution of votes will remain unchanged. “The elections considered span a period of 30 years, in which deep cultural, economic and social transformations have occurred,” say the researchers in their paper. “There is no hint of that in the data pattern.”

Talk the talk

In attempt to explain this universal distribution, Fortunato and Castellano supposed that a candidate gains popularity, and hence votes, by a word-of-mouth effect. They considered each candidate at the top of a tree-like structure, persuading a small number of close friends of to vote for them with a certain probability, who in turn would promote the candidate to more people, and so on. The number of contacts that each person would have to spread the word would have a distribution of its own, which the researchers assumed would follow a “power law” of a generic parameter, α.

Fortunato and Castellano simulated this structure for different values of the persuading probability, the minimum number of contacts a person could have and α. They discovered that their simulation could exactly replicate the curve underlying the election data sets. This, they claim, proves that different candidates in proportional elections gain votes by word of mouth.

The researchers are now looking to see if the word-of-mouth model fits other data, such as local elections for mayor. “Our preliminary results show that municipal elections in Rome, Sao Paolo and Rio de Janeiro give results fitting our curve,” Fortunato told physicsworld.com.