# Mapping higher-order network flows in memory and multilayer networks with Infomap

arXiv: https://arxiv.org/abs/1706.04792 · HTML: https://mlsanity.com/paper/1706.04792/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jun 2017.

## Built on by (most cited first, 10 of 10)

- [The Map Equation Goes Neural: Mapping Network Flows with Graph Neural Networks](https://mlsanity.com/paper/2310.01144/) (2023)
- [Predicting Influential Higher-Order Patterns in Temporal Network Data](https://mlsanity.com/paper/2107.12100/) (2021)
- [Module-based regularization improves Gaussian graphical models when observing noisy data](https://mlsanity.com/paper/2303.16796/) (2023)
- [MDL-Pool: Adaptive Multilevel Graph Pooling Based on Minimum Description Length](https://mlsanity.com/paper/2409.10263/) (2024)
- [Deep Graph Learning will stall without Network Science](https://mlsanity.com/paper/2502.01177/) (2025)
- [An adaptation of InfoMap to absorbing random walks using absorption-scaled graphs](https://mlsanity.com/paper/2112.10953/) (2021)
- [Cognitive modelling with multilayer networks: Insights, advancements and future challenges](https://mlsanity.com/paper/2210.00500/) (2022)
- [Bayesian Detection of Mesoscale Structures in Pathway Data on Graphs](https://mlsanity.com/paper/2301.11120/) (2023)
- [FlowHON: Representing Flow Fields Using Higher-Order Networks](https://mlsanity.com/paper/2312.02243/) (2023)
- [Reliable data clustering with Bayesian community detection](https://mlsanity.com/paper/2510.15013/) (2025)

## Abstract (arXiv)

Comprehending complex systems by simplifying and highlighting important dynamical patterns requires modeling and mapping higher-order network flows. However, complex systems come in many forms and demand a range of representations, including memory and multilayer networks, which in turn call for versatile community-detection algorithms to reveal important modular regularities in the flows. Here we show that various forms of higher-order network flows can be represented in a unified way with networks that distinguish physical nodes for representing a~complex system's objects from state nodes for describing flows between the objects. Moreover, these so-called sparse memory networks allow the information-theoretic community detection method known as the map equation to identify overlapping and nested flow modules in data from a range of~different higher-order interactions such as multistep, multi-source, and temporal data. We derive the map equation applied to sparse memory networks and describe its search algorithm Infomap, which can exploit the flexibility of sparse memory networks. Together they provide a general solution to reveal overlapping modular patterns in higher-order flows through complex systems.