  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.
