Maximal modularity and the optimal size of parliaments

TL;DR

This paper investigates the relationship between parliament size and population using complex network models, finding that maximal modularity predicts a power-law scaling consistent with empirical data.

Contribution

It introduces a novel network-based model to analyze optimal parliament size and derives a non-monotonic modularity relation that predicts a power-law scaling law.

Findings

Maximal modularity peaks at a certain number of constituencies depending on population size.

The model predicts a power-law relation between parliament size and population.

Empirical data supports the predicted scaling law.

Abstract

An important question in representative democracies is how to determine the optimal parliament size of a given country. According to an old conjecture, known as the cubic root law, there is a fairly universal power-law relation, with an exponent close to 1/3, between the size of an elected parliament and the country's population. Empirical data in modern European countries support such universality but are consistent with a larger exponent. In this work, we analyze this intriguing regularity using tools from complex networks theory. We model the population of a democratic country as a random network, drawn from a growth model, where each node is assigned a constituency membership sampled from an available set of size $D$. We calculate analytically the modularity of the population and find that its functional relation with the number of constituencies is strongly non-monotonic, exhibiting a maximum that depends on the population size. The criterion of maximal modularity allows us to predict that the number of representatives should scale as a power-law in the size of the population, a finding that is qualitatively confirmed by the empirical analysis of real-world data.