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.
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BibTeX
@misc{vazquez2026number,
title = {{The number of communities in real networks grows as a power law of their size}},
author = {Alexei Vazquez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18962},
eprint = {2609.18962},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18962},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - The number of communities in real networks grows as a power law of their size
AU - Alexei Vazquez
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.18962
UR - https://arxiv.org/abs/2609.18962
AB - 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.
ER -