The number of communities in real networks grows as a power law of their size

Abstract

Most networks, from friendships to the internet, break up into communities: groups of nodes more densely connected to one another than to the rest. A basic question has remained open: how many communities should a network have, and how does that number grow with size? Simple models bracket the possibilities. In a small-world network built from dense modules -- a caveman graph of cliques -- each module is a community, so the number grows in proportion to size, as $n$. In self-similar, fractal networks it grows far more slowly, as the square root. Real networks could lie anywhere between. Here we measure the number of communities across close to a hundred real networks spanning four orders of magnitude, from a $34$-member club to millions of nodes, and within individual systems followed over decades as they grow. We find a power law with an exponent between one half and two thirds -- above the fractal value and below the linear limit. Simple models of network growth by local rules reproduce the same super-fractal exponent, a stable and measurable property of network organization. In short, network evolution herds a system's components into clusters, whose number grows as a power law of size.