# The emergence of informative higher scales in complex networks

arXiv: https://arxiv.org/abs/1907.03902 · HTML: https://mlsanity.com/paper/1907.03902/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jul 2019.

## Built on by (most cited first, 4 of 4)

- [Designing Ecosystems of Intelligence from First Principles](https://mlsanity.com/paper/2212.01354/) (2022)
- [Finding emergence in data by maximizing effective information](https://mlsanity.com/paper/2308.09952/) (2023)
- [Neural Information Squeezer for Causal Emergence](https://mlsanity.com/paper/2201.10154/) (2022)
- [M"obius transforms and Shapley values for vector-valued functions on weighted directed acyclic multigraphs](https://mlsanity.com/paper/2510.05786/) (2025)

## Abstract (arXiv)

The connectivity of a network contains information about the relationships between nodes, which can denote interactions, associations, or dependencies. We show that this information can be analyzed by measuring the uncertainty (and certainty) contained in paths along nodes and links in a network. Specifically, we derive from first principles a measure known as effective information and describe its behavior in common network models. Networks with higher effective information contain more information in the relationships between nodes. We show how subgraphs of nodes can be grouped into macro-nodes, reducing the size of a network while increasing its effective information (a phenomenon known as causal emergence). We find that informative higher scales are common in simulated and real networks across biological, social, informational, and technological domains. These results show that the emergence of higher scales in networks can be directly assessed and that these higher scales offer a way to create certainty out of uncertainty.