# The emergence of informative higher scales in complex networks

## TL;DR

This paper introduces a measure called effective information to analyze the informational content of network paths, demonstrating how higher-scale structures can emerge and increase certainty in various real-world and simulated networks.

## Contribution

It derives a new measure for network information content and shows how macro-scale groupings can enhance effective information, revealing the emergence of informative higher scales.

## Key findings

- Higher effective information indicates more meaningful node relationships.
- Subgraph grouping into macro-nodes increases effective information.
- Informative higher scales are prevalent across diverse network types.

## Abstract

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.