# The Hyperspherical Geometry of Community Detection: Modularity as a Distance

arXiv: https://arxiv.org/abs/2107.02645 · HTML: https://mlsanity.com/paper/2107.02645/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 65 references that are in the map.
- First posted on arXiv 6 Jul 2021.

## Builds on (most cited first, 16 of 16)

- [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016)
- [From Louvain to Leiden: guaranteeing well-connected communities](https://mlsanity.com/paper/1810.08473/) (2018)
- [A Survey on Network Embedding](https://mlsanity.com/paper/1711.08752/) (2017)
- [Cooperative Game Theory Approaches for Network Partitioning](https://mlsanity.com/paper/1707.03587/) (2017)
- [Community detection in networks: A user guide](https://mlsanity.com/paper/1608.00163/) (2016)
- [Adjusting for Chance Clustering Comparison Measures](https://mlsanity.com/paper/1512.01286/) (2015)
- [Ground Truth Bias in External Cluster Validity Indices](https://mlsanity.com/paper/1606.05596/) (2016)
- [Multiscale dynamical embeddings of complex networks](https://mlsanity.com/paper/1804.03733/) (2018)
- [Scale-free Networks Well Done](https://mlsanity.com/paper/1811.02071/) (2018)
- [Different approaches to community detection](https://mlsanity.com/paper/1712.06468/) (2017)
- [Community Detection through Likelihood Optimization: In Search of a Sound Model](https://mlsanity.com/paper/1802.04472/) (2018)
- [Using synthetic networks for parameter tuning in community detection](https://mlsanity.com/paper/1906.04555/) (2019)
- [Multiway spectral community detection in networks](https://mlsanity.com/paper/1507.05108/) (2015)
- [Power-law relations in random networks with communities](https://mlsanity.com/paper/1603.09711/) (2015)
- [Geometric Multiscale Community Detection: Markov Stability and Vector Partitioning](https://mlsanity.com/paper/1708.03167/) (2017)
- [Positive semi-definite embedding for dimensionality reduction and out-of-sample extensions](https://mlsanity.com/paper/1711.07271/) (2017)

## Abstract (arXiv)

We introduce a metric space of clusterings, where clusterings are described by a binary vector indexed by the vertex-pairs. We extend this geometry to a hypersphere and prove that maximizing modularity is equivalent to minimizing the angular distance to some modularity vector over the set of clustering vectors. In that sense, modularity-based community detection methods can be seen as a subclass of a more general class of projection methods, which we define as the community detection methods that adhere to the following two-step procedure: first, mapping the network to a point on the hypersphere; second, projecting this point to the set of clustering vectors. We show that this class of projection methods contains many interesting community detection methods. Many of these new methods cannot be described in terms of null models and resolution parameters, as is customary for modularity-based methods. We provide a new characterization of such methods in terms of meridians and latitudes of the hypersphere. In addition, by relating the modularity resolution parameter to the latitude of the corresponding modularity vector, we obtain a new interpretation of the resolution limit that modularity maximization is known to suffer from.