# Hierarchical Block Structures and High-resolution Model Selection in Large Networks

arXiv: https://arxiv.org/abs/1310.4377 · HTML: https://mlsanity.com/paper/1310.4377/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Oct 2013.

## Built on by (most cited first, 30 of 37)

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- [Disentangling homophily, community structure and triadic closure in networks](https://mlsanity.com/paper/2101.02510/) (2021)
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- [Graph Embedding with Hierarchical Attentive Membership](https://mlsanity.com/paper/2111.00604/) (2021)
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- [Implicit models, latent compression, intrinsic biases, and cheap lunches in community detection](https://mlsanity.com/paper/2210.09186/) (2022)
- [On the Fundamental Limits of Matrix Completion: Leveraging Hierarchical Similarity Graphs](https://mlsanity.com/paper/2109.05408/) (2021)
- [Analysing Multiscale Clusterings with Persistent Homology](https://mlsanity.com/paper/2305.04281/) (2023)
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- [Core-based Hierarchies for Efficient GraphRAG](https://mlsanity.com/paper/2603.05207/) (2026)
- [Subgraph nomination: Query by Example Subgraph Retrieval in Networks](https://mlsanity.com/paper/2101.12430/) (2021)
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- [Deep forecasting of translational impact in medical research](https://mlsanity.com/paper/2110.08904/) (2021)
- [Systematic assessment of the quality of fit of the stochastic block model for empirical networks](https://mlsanity.com/paper/2201.01658/) (2022)
- [Consistency between ordering and clustering methods for graphs](https://mlsanity.com/paper/2208.12933/) (2022)
- [Revealing Patient-Reported Experiences in Healthcare from Social Media using the DAPMAV Framework](https://mlsanity.com/paper/2210.04232/) (2022)
- [T-Stochastic Graphs](https://mlsanity.com/paper/2309.01301/) (2023)
- [Uncertainty quantification and posterior sampling for network reconstruction](https://mlsanity.com/paper/2503.07736/) (2025)
- [LLMs Between the Nodes: Community Discovery Beyond Vectors](https://mlsanity.com/paper/2507.22955/) (2025)
- [Cluster Attention for Graph Machine Learning](https://mlsanity.com/paper/2604.07492/) (2026)
- [SamBaS: Sampling-Based Stochastic Block Partitioning](https://mlsanity.com/paper/2108.06651/) (2021)

## Abstract (arXiv)

Discovering and characterizing the large-scale topological features in empirical networks are crucial steps in understanding how complex systems function. However, most existing methods used to obtain the modular structure of networks suffer from serious problems, such as being oblivious to the statistical evidence supporting the discovered patterns, which results in the inability to separate actual structure from noise. In addition to this, one also observes a resolution limit on the size of communities, where smaller but well-defined clusters are not detectable when the network becomes large. This phenomenon occurs not only for the very popular approach of modularity optimization, which lacks built-in statistical validation, but also for more principled methods based on statistical inference and model selection, which do incorporate statistical validation in a formally correct way. Here we construct a nested generative model that, through a complete description of the entire network hierarchy at multiple scales, is capable of avoiding this limitation, and enables the detection of modular structure at levels far beyond those possible with current approaches. Even with this increased resolution, the method is based on the principle of parsimony, and is capable of separating signal from noise, and thus will not lead to the identification of spurious modules even on sparse networks. Furthermore, it fully generalizes other approaches in that it is not restricted to purely assortative mixing patterns, directed or undirected graphs, and ad hoc hierarchical structures such as binary trees. Despite its general character, the approach is tractable, and can be combined with advanced techniques of community detection to yield an efficient algorithm that scales well for very large networks.